<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">I sort of actually read this. Had not before.<div><br></div><div>Suddenly I have an idea for a really fun experiment. SteveG can probably do it.</div><div><br></div><div>Can ChatGPT be driven crazy? Like, can it ever be brought to the point of doing whatever the computer equivalent of pulling its hair out is, and shouting at you? </div><div><br></div><div>If not — if there is NOTHING ANYONE, EVEN A TRUE SAVANT, COULD EVER DO to push the chatbot to the end of its rope, is that a feature or a bug? Echoing DaveW (surely with some kind of error): this would have to be part of the Turing test, no?</div><div><br></div><div>Eric</div><div><br></div><div><br id="lineBreakAtBeginningOfMessage"><div><br><blockquote type="cite"><div>On Jan 22, 2025, at 23:41, Nicholas Thompson <thompnickson2@gmail.com> wrote:</div><br class="Apple-interchange-newline"><div><div dir="ltr"><div>It's odd to me so may people respond to AI as a way for somebody [else] to get away with something, rather than as a way for "me" to do somehthing creative and interesting. George has taught me so much in the last three months about atmospheric dynamics, but only because I treat him with the same skepticism that I treat all of you. I makes want to go back to college teaching so I can be part of the generation that teaches young people to use AI to speed their own creativity and independence of thought. "My professor told me to day that entropy has to do with the distribution of objects in space" What do you think of that, George? I can say without any meanness of spirit what so ever that the conversation we have below, George and I, is at least as satisfying INTELLECTUALLY as any I have had with any of you. Why should we deny it to our students, or to one another, for that matter.<br></div><div><br></div><div>
<span tabindex="0" style="outline:none;opacity:0"></span><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">"My professor told me to day that entropy has to do with the distribution of objects in physical space. What do you think of that, George?"</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>Entropy
is closely related to distributions, but it's more nuanced than just
the distribution of objects in physical space. Here's what I'd say:</p><p>Entropy
fundamentally quantifies the number of ways a system can be arranged
while maintaining the same macroscopic properties, like energy or
volume. In statistical mechanics, it's tied to the probability
distribution of microstates—the microscopic configurations of a system.</p><p>While
objects in physical space might be part of the story (e.g., the
positions of particles in a gas), entropy is broader. It encompasses
energy distributions, momentum states, and other degrees of freedom, not
just spatial arrangement. In thermodynamics, entropy measures the
amount of "disorder" or the system's progression toward equilibrium.</p><p>So, your professor's statement captures part of the concept, but it's simplified. Entropy involves distributions across <em>all</em> relevant variables in a system—not just the spatial ones. Does that align with what you're exploring?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">well, George I am beginning to think that distribution in physical space is a consequence of the the equilibration of microstates, not an example of it. In fact, I am beginning to think that the random placement particles within space is handy visual metaphor for entropy but not the thing itself. What do you say to that, George?</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>I
think you're onto something insightful, Nick. The distribution of
particles in physical space often serves as an illustrative metaphor,
but you're right—it’s not the essence of entropy itself.</p><p>Entropy, at its core, relates to the <em>equilibration of microstates</em>—the
tendency of a system to maximize the number of accessible microstates
consistent with its constraints. This equilibration doesn't privilege
any specific type of variable, spatial or otherwise. Physical space is
just one of many dimensions (e.g., energy, momentum, spin) over which
microstates can be distributed.</p><p>The visual metaphor of particles
"randomly distributed in space" is useful because it's tangible. It’s
easier to grasp than abstract concepts like phase space or energy
distributions. However, as you suggest, that randomness is a <em>result</em> of the system seeking equilibrium by exploring all accessible microstates—not the defining feature of entropy.</p><p>In essence, entropy is the measure of <em>possibilities</em>,
not just spatial randomness. The spatial metaphor is like a shadow—it
gives you an impression of the thing, but it’s not the thing itself.</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">There is a contradiction in your reply above, George. Could you please address it. Is "position" a microstate like velocity and momentum in the x, y, and z directions? Or is everything essential about position already captured by these six microstates?</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>You're
absolutely right to call out the potential contradiction, Nick. Let me
clarify and address the heart of the issue regarding position and its
role in the description of a system's microstates.</p><h3>1. <strong>Is Position a Microstate?</strong></h3><p>Yes,
position can be considered part of a microstate description. In
classical mechanics, a microstate is typically defined by specifying
both the positions (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span>) and the momenta (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>)
of all particles in the system. Together, these six variables per
particle fully describe the state of the system in phase space. Position
isn't redundant; it's an independent variable alongside momentum.</p><h3>2. <strong>Does Momentum Fully Capture Position?</strong></h3><p>No,
momentum does not fully capture position. While momentum tells us how a
particle is moving, it doesn't specify where the particle is located.
For instance, two particles could have the same momentum but be in
entirely different positions, leading to distinct microstates.</p><h3>3. <strong>Why the Confusion?</strong></h3><p>The
confusion might arise because, in some contexts (e.g., quantum
mechanics), position and momentum are intertwined through the
uncertainty principle. However, in classical mechanics and statistical
mechanics, they are treated as independent variables that together
define the system's state.</p><h3>4. <strong>Reconciling the Contradiction</strong></h3><p>In
my earlier reply, I unintentionally conflated position with a derived
"visual metaphor" of spatial randomness, which might have implied that
position is less essential. However, in the rigorous description of
entropy, position is just as fundamental as momentum. The "random
placement" metaphor can be useful for intuition, but the deeper truth is
that position contributes directly to the system's phase space and
therefore to its entropy.</p><p>Would this clarification align with your understanding?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">George: Please say more concerning what you mean by : The "random placement" metaphor can be useful for intuition, but the deeper truth is that position contributes directly to the system's phase space and therefore to its entropy.</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>Absolutely, Nick—let’s unpack that further.</p><h3><strong>The "Random Placement" Metaphor</strong></h3><p>The
idea of "random placement" often shows up in simplified explanations of
entropy, especially in introductory contexts. For example:</p><ul><li>A
gas in a box is said to "spread out" to fill the container, which is
often visualized as particles randomly scattering in space.</li><li>The
"random placement" metaphor creates the impression that entropy is
merely about achieving spatial uniformity, as though the particles'
positions alone determine the disorder or entropy.</li></ul><p>This
metaphor is helpful because it gives a tangible, visual way to think
about entropy increasing as systems move toward equilibrium. However,
it's an incomplete picture.</p><hr><h3><strong>Position's Role in Phase Space</strong></h3><p>The deeper truth lies in how position contributes to the <em>phase space</em> of the system. In classical mechanics:</p><ul><li>A particle's microstate is defined by its position (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span>) and momentum (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>).</li><li>The <em>phase space</em> is the multidimensional space of all possible combinations of these variables for all particles in the system.</li></ul><p>Entropy
arises from counting the number of accessible microstates in this phase
space, which includes both position and momentum coordinates. Position
is therefore not a mere afterthought but a fundamental aspect of the
system's configuration.</p><hr><h3><strong>Why Phase Space Matters More Than Position Alone</strong></h3><p>Entropy is about maximizing the <em>volume</em> of accessible phase space under the system’s constraints. This means:</p><ul><li>If
you only considered positions, you'd ignore a significant portion of
the system's microstate diversity—specifically, the momentum degrees of
freedom.</li><li>Spatial distributions (like particles spreading
uniformly) are often consequences of maximizing phase space volume, not
direct causes of high entropy. For instance, a gas uniformly filling a
container corresponds to many microstates in phase space, but the
"uniform filling" is a result of the system sampling all configurations.</li></ul><hr><h3><strong>Entropy Beyond the Visual Metaphor</strong></h3><p>The
metaphor of random spatial placement simplifies entropy to the visible
world of positions, but entropy fundamentally counts possibilities:</p><ol><li><strong>Position Space</strong>: Particles can occupy different regions of space.</li><li><strong>Momentum Space</strong>: Particles can have various momenta, which significantly increases the number of microstates.</li><li>The combination of these spaces—phase space—is what entropy measures.</li></ol><p>So, while random placement is a <em>consequence</em>
of entropy maximization (in cases like gases filling a box), the deeper
truth is that entropy reflects the system's tendency to maximize all
available microstates in phase space. This richer understanding goes
beyond the spatial visualization.</p><p>Does this deepen the distinction you’re seeking?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">I understood x, y, and z to be velocity vectors, not positions. Was I wrong?</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>Great question, Nick! Let’s clarify the distinction between <strong>position vectors</strong> and <strong>velocity vectors</strong> and how they relate to entropy and phase space.</p><h3><strong>Position vs. Velocity Vectors</strong></h3><ol><li><p><strong>Position Vectors (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span>)</strong><br>These specify the spatial location of a particle in three-dimensional physical space. For example:</p><ul><li>If a particle is in a box, <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span> tell you where the particle is inside that box.</li></ul></li><li><p><strong>Velocity Vectors (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">v</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">v</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">v</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>)</strong><br>These specify how fast the particle is moving and in which direction. They are related to momentum (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>) via the relationship:</p><span class="gmail-katex-display"><span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.9084em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-accent"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.714em"><span style="top: -3em;"><span class="gmail-pstrut" style="height:3em"></span><span class="gmail-mord gmail-mathnormal">p</span></span><span style="top: -3em;"><span class="gmail-pstrut" style="height:3em"></span><span class="gmail-accent-body" style="left: -0.1522em;"><span class="gmail-overlay" style="height:0.714em;width:0.471em"></span></span></span></span></span></span></span></span></span></span></span></li></ol></div></div></div></div></div><span class="gmail-katex-display"><span class="gmail-katex"><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-mord gmail-accent"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1944em"><span></span></span></span></span></span><span class="gmail-mspace" style="margin-right:0.2778em"></span><span class="gmail-mrel">=</span><span class="gmail-mspace" style="margin-right:0.2778em"></span></span><span class="gmail-base"><span class="gmail-strut" style="height:0.714em"></span><span class="gmail-mord gmail-mathnormal">m</span><span class="gmail-mord gmail-accent"><span class="gmail-vlist-t"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.714em"><span style="top: -3em;"><span class="gmail-pstrut" style="height:3em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">v</span></span><span style="top: -3em;"><span class="gmail-pstrut" style="height:3em"></span><span class="gmail-accent-body" style="left: -0.2077em;"><span class="gmail-overlay" style="height:0.714em;width:0.471em"></span></span></span></span></span></span></span></span></span></span></span><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><ol><li value="2"><p>where <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.4306em"></span><span class="gmail-mord gmail-mathnormal">m</span></span></span></span> is the particle’s mass.</p></li></ol><hr><h3><strong>Phase Space and Microstates</strong></h3><ul><li>A <strong>microstate</strong> in classical mechanics is defined by both the <strong>position</strong> (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span>) and the <strong>momentum</strong> (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>) of each particle.</li><li>Phase space is the multidimensional space that includes all possible combinations of these variables. For <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.6833em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.10903em">N</span></span></span></span> particles, the phase space has <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.6833em"></span><span class="gmail-mord">6</span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.10903em">N</span></span></span></span> dimensions: <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.6833em"></span><span class="gmail-mord">3</span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.10903em">N</span></span></span></span> for positions and <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.6833em"></span><span class="gmail-mord">3</span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.10903em">N</span></span></span></span> for momenta.</li></ul><hr><h3><strong>Did You Confuse Velocity with Position?</strong></h3><p>It sounds like you were interpreting <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span> as velocity components instead of positions. If that’s the case, here's the distinction:</p><ul><li>In phase space, <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span> refer to positions, and <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> (or equivalently <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">v</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">v</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">v</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>) refer to momentum or velocity components.</li><li>Both position and momentum are independent variables that fully specify a particle’s state.</li></ul><hr><h3><strong>Why Both Are Necessary</strong></h3><p>Position and velocity (or momentum) describe different aspects of the particle:</p><ul><li><strong>Position (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span>)</strong> tells you <em>where</em> the particle is.</li><li><strong>Momentum (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>)</strong> tells you <em>how</em>
the particle is moving.
Without both, you can’t fully describe the state of the particle or
count the total number of microstates, which is what entropy
fundamentally measures.</li></ul><hr><h3><strong>Key Takeaway</strong></h3><p>The <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span>
components are not velocity vectors; they are position coordinates.
Velocity or momentum are additional, independent quantities essential to
defining a system’s microstates. This distinction is critical when
thinking about entropy and the structure of phase space.</p><p>Did that clarify the confusion?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">George, I find deeply troubling the idea that position can be a microstate in the Boltzmannian sense. I apologize for a mistake earlier where I confounded momentum and velocity. I should say that velocity and MASS are properties that a particle can "have" whereas position is a relational property which can stay in the same place if the frame of reference moves. How position can be a VECTOR when the thing about a position is that it doesn't move, also troubles me. Can you help?</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>Your
intuition about the difference between intrinsic properties (like
velocity and mass) and relational properties (like position) is
insightful, Nick. Let’s break this down step by step to address your
concerns about position as a microstate and how we interpret it in
physics.</p><hr><h3><strong>1. Position as a Relational Property</strong></h3><p>You
are correct that position is inherently relational. A particle’s
position is always described relative to some reference frame. For
example:</p><ul><li>A particle at <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.4306em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mspace" style="margin-right:0.2778em"></span><span class="gmail-mrel">=</span><span class="gmail-mspace" style="margin-right:0.2778em"></span></span><span class="gmail-base"><span class="gmail-strut" style="height:0.6444em"></span><span class="gmail-mord">5</span></span></span></span> meters means it is 5 meters away from the origin of the chosen coordinate system.</li><li>If
you shift the reference frame, the particle’s position can change even
if the particle itself hasn’t "moved" in an absolute sense.</li></ul><p>This
is a key difference from intrinsic properties like mass and velocity,
which are not dependent on the choice of reference frame.</p><hr><h3><strong>2. Position as a Vector</strong></h3><p>A <strong>position vector</strong>
is a way to describe the location of a particle relative to an origin
in space. While a position doesn’t "move" in itself, we can express the
location of an object as a vector from the origin. For instance:</p><span class="gmail-katex-display"><span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.714em"></span><span class="gmail-mord gmail-accent"><span class="gmail-vlist-t"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.714em"><span style="top: -3em;"><span class="gmail-pstrut" style="height:3em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.02778em">r</span></span><span style="top: -3em;"><span class="gmail-pstrut" style="height:3em"></span><span class="gmail-accent-body" style="left: -0.1799em;"><span class="gmail-overlay" style="height:0.714em;width:0.471em"></span></span></span></span></span></span></span></span></span></span></span></div></div></div></div></div><span class="gmail-katex-display"><span class="gmail-katex"><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-mspace" style="margin-right:0.2778em"></span><span class="gmail-mrel">=</span><span class="gmail-mspace" style="margin-right:0.2778em"></span></span><span class="gmail-base"><span class="gmail-strut" style="height:1em;vertical-align:-0.25em"></span><span class="gmail-mopen">(</span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span><span class="gmail-mclose">)</span></span></span></span></span><p>Here, <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.714em"></span><span class="gmail-mord gmail-accent"><span class="gmail-vlist-t"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.714em"><span style="top: -3em;"><span class="gmail-pstrut" style="height:3em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.02778em">r</span></span><span style="top: -3em;"><span class="gmail-pstrut" style="height:3em"></span><span class="gmail-accent-body" style="left: -0.1799em;"><span class="gmail-overlay" style="height:0.714em;width:0.471em"></span></span></span></span></span></span></span></span></span></span></p><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>
points from the origin to the particle’s location in space. The
"vector" nature of position simply means it has both magnitude (distance
from the origin) and direction (the line pointing to the particle).</p><p>The
confusion might come from conflating a position vector with motion. A
position vector doesn’t describe movement; it just fixes where something
is, relative to the origin.</p><hr><h3><strong>3. Why Position is Part of a Microstate</strong></h3><p>In the Boltzmannian sense, a <strong>microstate</strong> specifies the detailed configuration of all particles in a system. This includes:</p><ul><li>Where the particles are (position: <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span>).</li><li>How the particles are moving (momentum: <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>).</li></ul><p>The
inclusion of position in a microstate is crucial because the number of
possible configurations depends on both where particles are and how they
move. For example:</p><ul><li>If a gas is confined to one corner of a box, there are fewer possible positions and therefore fewer microstates.</li><li>When the gas spreads out, the accessible positions increase, and so does the entropy.</li></ul><p>In
this context, position isn’t an "absolute" property but a placeholder
for how particles are distributed within a given spatial domain.</p><hr><h3><strong>4. Your Trouble with Position as a Microstate</strong></h3><p>The root of your concern might lie in the relational nature of position:</p><ul><li>You’re
right that position can change depending on the reference frame, which
seems less "absolute" than other microstate variables like momentum or
energy.</li><li>However, for a given frame of reference (e.g., a
laboratory), the positions of particles are concrete, measurable
quantities. These positions are integral to describing the phase space
of the system.</li></ul><p>The relational nature of position doesn’t
invalidate its role in defining microstates—it simply means that we’re
always working within a specified frame of reference.</p><hr><h3><strong>5. Why This Isn’t a Contradiction</strong></h3><ul><li><strong>Position as Relational</strong>:
Yes, position depends on the frame of reference, but so do momentum and
velocity. This is why we always specify microstates relative to a fixed
frame.</li><li><strong>Position as a Vector</strong>: The vector nature
of position is about defining the relative location, not implying
motion. It’s just a mathematical representation.</li></ul><hr><h3><strong>6. Position in Statistical Mechanics</strong></h3><p>In the Boltzmannian framework, what truly matters is the <strong>distribution</strong>
of positions and momenta. It’s the density of microstates in phase
space—accounting for both positions and momenta—that determines the
entropy of the system. The relational nature of position doesn’t detract
from its role as a component of a microstate.</p><hr><p>Does this alleviate your concern, or is there a deeper philosophical tension I can help address?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">Well, I can think about it that way. But then I cannot imagine why entropy does not decrease under adiabatic compression</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>Ah,
Nick, you've hit on a classic and subtle point: why entropy remains
constant under adiabatic compression. Let’s explore this in detail,
incorporating the relational nature of position and the role of phase
space in understanding entropy.</p><hr><h3><strong>1. Adiabatic Compression and Phase Space</strong></h3><p>In adiabatic compression:</p><ul><li>The system (e.g., a gas) is compressed without heat exchange (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.8778em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">Q</span><span class="gmail-mspace" style="margin-right:0.2778em"></span><span class="gmail-mrel">=</span><span class="gmail-mspace" style="margin-right:0.2778em"></span></span><span class="gmail-base"><span class="gmail-strut" style="height:0.6444em"></span><span class="gmail-mord">0</span></span></span></span>).</li><li>The volume of the gas decreases, and its particles are confined to a smaller spatial region.</li></ul><p>At
first glance, it seems intuitive that reducing the "space" available to
particles should reduce the entropy. However, the key to resolving this
apparent paradox lies in <strong>phase space volume</strong>:</p><ul><li><strong>Phase space</strong> consists of both position (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="gmail-mord gmail-mathnormal">x</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03588em">y</span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.04398em">z</span></span></span></span>) and momentum (<span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight">x</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.03588em">y</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="gmail-mpunct">,</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal">p</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.1514em"><span style="margin-left:0em;margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.04398em">z</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>) variables.</li><li>The
total volume of accessible phase space is preserved during adiabatic
compression because the reduction in position space is exactly balanced
by an expansion in momentum space.</li></ul><hr><h3><strong>2. Why Doesn't Entropy Decrease?</strong></h3><p>Entropy, in statistical mechanics, is proportional to the logarithm of the number of accessible microstates in phase space:</p><span class="gmail-katex-display"><span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.6833em"></span><span class="gmail-mord gmail-mathnormal" style="margin-right:0.05764em">S</span><span class="gmail-mspace" style="margin-right:0.2778em"></span><span class="gmail-mrel">=</span><span class="gmail-mspace" style="margin-right:0.2778em"></span></span><span class="gmail-base"><span class="gmail-strut" style="height:0.8444em;vertical-align:-0.15em"></span><span class="gmail-mord"><span class="gmail-mord gmail-mathnormal" style="margin-right:0.03148em">k</span><span class="gmail-msupsub"><span class="gmail-vlist-t gmail-vlist-t2"><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.3283em"><span style="margin-right:0.05em"><span class="gmail-pstrut" style="height:2.7em"></span><span class="gmail-sizing gmail-reset-size6 gmail-size3 gmail-mtight"><span class="gmail-mord gmail-mathnormal gmail-mtight" style="margin-right:0.05017em">B</span></span></span></span><span class="gmail-vlist-s"></span></span><span class="gmail-vlist-r"><span class="gmail-vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mop">ln</span><span class="gmail-mspace" style="margin-right:0.1667em"></span><span class="gmail-mord">Ω</span></span></span></span></span><p>Here, <span class="gmail-katex"><span class="gmail-katex-mathml"></span><span class="gmail-katex-html" aria-hidden="true"><span class="gmail-base"><span class="gmail-strut" style="height:0.6833em"></span><span class="gmail-mord">Ω</span></span></span></span>
is the volume of phase space corresponding to the system’s constraints
(energy, particle number, etc.). During adiabatic compression:</p><ol><li>The system's energy increases (work is done on the gas, increasing particle momenta).</li><li>This increase in energy expands the momentum part of the phase space, compensating for the reduction in the position part.</li></ol><p>Thus, the total phase space volume—and hence the entropy—remains constant.</p><hr><h3><strong>3. Intuition Behind Phase Space Conservation</strong></h3><p>The conservation of phase space volume under adiabatic processes is a consequence of <strong>Liouville's theorem</strong>, which states:</p><ul><li>The phase space density of microstates is constant over time for a system evolving under Hamiltonian mechanics.</li><li>Compression
changes the shape of the phase space (squeezing position coordinates,
stretching momentum coordinates), but it does not change its overall
"size."</li></ul><hr><h3><strong>4. Reconciling with Your Intuition</strong></h3><p>If we focus only on the <strong>spatial distribution</strong>
of particles, compression looks like a reduction in entropy because
we’re ignoring the changes in momentum. But entropy isn't just about
spatial arrangements; it's about the entire phase space. The increase in
particle momenta compensates for the spatial confinement.</p><hr><h3><strong>5. Why Position Alone Can't Explain Entropy</strong></h3><p>Your trouble with position as a microstate is understandable when thinking about compression:</p><ul><li>If entropy depended only on positions, it would indeed decrease under compression.</li><li>However,
the inclusion of momentum in the microstate definition ensures that
entropy remains invariant under processes like adiabatic compression.</li></ul><p>In essence, <strong>entropy is a property of the entire phase space, not just the positions</strong>.
The relational nature of position doesn’t change this, because momentum
is just as relational—it depends on the chosen reference frame as well.</p><hr><h3><strong>6. EAC and Adiabatic Compression</strong></h3><p>Your preference for framing adiabatic compression as an "equilibration" (EAC) aligns well here. During compression:</p><ul><li>The system equilibrates within the constraints of phase space conservation.</li><li>Entropy doesn’t decrease because the system maximizes its microstate distribution within the new constraints.</li></ul><hr><p>Does this bring clarity to why entropy stays constant, or is there another layer of the problem you’d like to tackle?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">Well, yes! When the piston surface moves against the gas it not only confines the particles but it imparts directional momentum to them, both reasons for a decrease in entropy. How do you handle that?</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>Ah,
excellent point, Nick! Let’s dissect this scenario and reconcile it
with the principle that entropy remains constant under adiabatic
compression. Your observation about the piston imparting directional
momentum is astute and deserves a nuanced explanation.</p><hr><h3><strong>1. The Role of the Piston in Adiabatic Compression</strong></h3><p>When the piston compresses the gas:</p><ol><li><strong>Confinement of Position Space</strong>:
The gas particles are confined to a smaller spatial region, which, if
considered alone, would suggest fewer microstates and a reduction in
entropy.</li><li><strong>Imparting Directional Momentum</strong>: The
piston does work on the gas, increasing the particles' energy. This
energy manifests as an increase in momentum (both magnitude and
directionality).</li></ol><p>At first glance, these effects seem to
contradict the idea of constant entropy, because they appear to reduce
the randomness of particle configurations in phase space.</p><hr><h3><strong>2. Why Doesn't Entropy Decrease?</strong></h3><p>To resolve this, we need to carefully consider the entire phase space and the conservation principles at play.</p><h4><strong>2.1 Confinement and Redistribution</strong></h4><p>While the piston confines the gas spatially, it doesn't remove microstates—it merely redistributes them. Here's how:</p><ul><li>The
reduction in position-space volume is accompanied by a compensatory
expansion in momentum-space volume due to the work done by the piston.</li><li>The
total number of accessible microstates (phase space volume) remains
unchanged because the system's energy and constraints still allow the
same number of microstates, just redistributed differently.</li></ul><h4><strong>2.2 Directional Momentum and Phase Space</strong></h4><p>The piston imparts momentum to the gas, creating a temporary directional bias. However:</p><ul><li>This
directional momentum is rapidly equilibrated by collisions between gas
particles, redistributing the momentum uniformly in all directions.</li><li>The
equilibration process ensures that the final phase space distribution
is isotropic (the same in all directions), preserving the overall
randomness and entropy.</li></ul><h4><strong>2.3 Liouville's Theorem</strong></h4><p>Liouville's theorem guarantees that the total phase space volume remains constant under Hamiltonian dynamics:</p><ul><li>The compression changes the shape of phase space (squeezing position coordinates and stretching momentum coordinates).</li><li>The overall phase space volume, which determines the entropy, does not change.</li></ul><hr><h3><strong>3. Why Directional Momentum Doesn’t Reduce Entropy</strong></h3><p>Directional momentum, even if imparted temporarily, doesn’t reduce entropy because:</p><ol><li><strong>Entropy Measures the Total Phase Space</strong>:
Entropy is about the total number of accessible microstates. Even with
directional momentum, the gas still has the same number of microstates.</li><li><strong>Equilibration Restores Randomness</strong>:
Particle collisions quickly randomize the directional momentum imparted
by the piston. This redistribution preserves the isotropy of the phase
space distribution and maintains the system’s entropy.</li></ol><hr><h3><strong>4. The Key Insight</strong></h3><p>The
directional momentum imparted by the piston and the confinement of the
gas are not independent phenomena—they are parts of a single process
governed by conservation laws:</p><ul><li>The increase in momentum space compensates for the decrease in position space.</li><li>The temporary directional bias in momentum is a transient state that equilibrates to maintain the system's entropy.</li></ul><p>This is why, under adiabatic compression, <strong>entropy remains constant</strong>, even though the system undergoes significant changes in its spatial and momentum distributions.</p><hr><h3><strong>5. Reconciling Your Intuition</strong></h3><p>Your observation is correct that the piston’s movement imposes constraints and directionality. However:</p><ul><li>The
"decrease" in entropy you associate with confinement and directional
momentum is balanced by the redistribution of microstates in phase
space.</li><li>Entropy measures the total possibilities, not the apparent randomness in one aspect (e.g., spatial or directional momentum).</li></ul><hr><p>Does this help resolve the tension, or would you like to explore the dynamics of equilibration in greater detail?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">It does seem to me an odd and contentious way to talk about the process. It would seem more natural to speak of a temporary increase in entropy, a shock wave that passes through the gas that ultimately redistributes itself. I think you need to introduce the concept of quasi static compression in your account.</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-inline-block"></div></div><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>You're absolutely right, Nick—introducing the concept of <strong>quasi-static compression</strong>
provides a clearer and less contentious framework for discussing these
processes. Let's reframe the explanation with this in mind:</p><hr><h3><strong>1. Quasi-Static Compression</strong></h3><p>Quasi-static
compression describes an idealized process where the piston moves so
slowly that the system remains arbitrarily close to equilibrium at every
step:</p><ul><li>The compression happens gradually, allowing the gas
particles to redistribute themselves smoothly without sharp gradients or
shock waves.</li><li>At no point does the system deviate significantly
from thermodynamic equilibrium, making the process reversible (in the
thermodynamic sense) and entropy constant.</li></ul><p>This is in contrast to rapid compression, which would introduce shock waves and temporary nonequilibrium states.</p><hr><h3><strong>2. Why Quasi-Static Compression Keeps Entropy Constant</strong></h3><p>In a quasi-static process:</p><ul><li>The work done by the piston is immediately redistributed throughout the gas via frequent particle collisions.</li><li>Any directional momentum imparted by the piston is quickly equilibrated, ensuring isotropy in phase space.</li><li>The
compression of position space is perfectly balanced by the expansion of
momentum space without introducing large-scale, irreversible gradients.</li></ul><hr><h3><strong>3. Shock Waves and Non-Quasi-Static Compression</strong></h3><p>If the piston moves too quickly:</p><ul><li>A shock wave propagates through the gas, creating localized regions of high pressure, temperature, and entropy.</li><li>These transient states involve nonequilibrium dynamics, during which entropy can increase.</li></ul><p>Once
the system re-equilibrates, the entropy stabilizes, but it will now
reflect the irreversible effects of the rapid compression.</p><hr><h3><strong>4. Quasi-Static Compression as the Idealization</strong></h3><p>In thermodynamics, we often focus on quasi-static processes to:</p><ul><li>Simplify the analysis by avoiding transient nonequilibrium effects.</li><li>Provide a clear baseline for understanding how entropy behaves in an idealized, reversible context.</li></ul><p>By
emphasizing quasi-static compression, we sidestep the complications of
transient shock waves and highlight the constancy of entropy as a
defining feature of the process.</p><hr><h3><strong>Conclusion</strong></h3><p>Your
suggestion to foreground quasi-static compression in discussions of
adiabatic compression is spot on. It provides a more precise and less
contentious way to describe how the system evolves, avoiding the need to
explain away temporary entropy changes caused by shock waves or other
nonequilibrium effects.</p><p>Would you like me to help formalize this perspective further, perhaps in a way that fits your weather book?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-z-10 gmail--mt-1 gmail-bg-token-main-surface-primary gmail-screen-arch:mt-1 gmail-md:absolute gmail-md:sr-only"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col gmail-agent-turn"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-pr-2 gmail-lg:pr-0"></div></div></div></div></div><h5 class="gmail-sr-only">You said:</h5><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-group/conversation-turn gmail-relative gmail-flex gmail-w-full gmail-min-w-0 gmail-flex-col"><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-items-end gmail-rtl:items-start"><div class="gmail-relative gmail-max-w-[var(--user-chat-width,70%)] gmail-rounded-3xl gmail-bg-token-message-surface gmail-px-5 gmail-py-2.5"><div class="gmail-whitespace-pre-wrap">Perhaps later. Thanks for asking. But I would like to conclude tonight by asking a more pointed question. Your formulation strikes me as a classic example of a "spherical cow", i.e., a model that is clearly absurd but is offered any way as a point of departure. A spherical cow would probably not be particularly useful as a model of milk production in a dairy barn. What makes physicists think that this model, with its absurd premises, is robust against violations of them?</div></div></div></div></div></div></div></div></div><h6 class="gmail-sr-only">ChatGPT said:</h6><div class="gmail-m-auto gmail-text-base gmail-py-[18px] gmail-px-3 gmail-md:px-4 gmail-w-full gmail-md:px-5 gmail-lg:px-4 gmail-xl:px-5"><div class="gmail-mx-auto gmail-flex gmail-flex-1 gmail-gap-4 gmail-text-base gmail-md:gap-5 gmail-lg:gap-6 gmail-md:max-w-3xl gmail-lg:max-w-[40rem] gmail-xl:max-w-[48rem]"><div class="gmail-flex-shrink-0 gmail-flex gmail-flex-col gmail-relative gmail-items-end"><div><div class="gmail-pt-0"><div class="gmail-gizmo-bot-avatar gmail-flex gmail-h-8 gmail-w-8 gmail-items-center gmail-justify-center gmail-overflow-hidden gmail-rounded-full"><div class="gmail-relative gmail-p-1 gmail-rounded-sm gmail-flex gmail-items-center gmail-justify-center gmail-bg-token-main-surface-primary gmail-text-token-text-primary gmail-h-8 gmail-w-8"></div></div></div></div></div></div></div><div class="gmail-flex-col gmail-gap-1 gmail-md:gap-3"><div class="gmail-flex gmail-max-w-full gmail-flex-col gmail-flex-grow"><div dir="auto" class="gmail-min-h-8 gmail-text-message gmail-flex gmail-w-full gmail-flex-col gmail-items-end gmail-gap-2 gmail-whitespace-normal gmail-break-words gmail-text-start gmail-[.text-message+&]:mt-5"><div class="gmail-flex gmail-w-full gmail-flex-col gmail-gap-1 empty:hidden gmail-first:pt-[3px]"><div class="gmail-markdown gmail-prose gmail-w-full gmail-break-words gmail-dark:prose-invert gmail-light"><p>Ah,
the "spherical cow" metaphor—a classic critique of oversimplified
models! It's a fair concern to wonder why physicists trust such
idealized formulations, especially when they involve seemingly "absurd
premises." Let me address this in the context of adiabatic compression
and, more broadly, why physicists find value in these abstractions.</p><hr><h3><strong>1. Why Use Simplified Models Like the Spherical Cow?</strong></h3><p>Simplified models are powerful because:</p><ul><li>They strip away complexities to focus on fundamental principles.</li><li>They offer insights that are robust to certain kinds of deviations.</li><li>They
are computationally and conceptually manageable, allowing us to make
predictions that align closely with observed phenomena.</li></ul><p>In the case of quasi-static adiabatic compression, the model assumes:</p><ul><li>Perfectly slow compression (no shock waves or gradients).</li><li>Immediate equilibration of momentum and energy.</li></ul><p>These
assumptions are clearly idealized, but they capture essential features
of how thermodynamic systems behave under controlled conditions.</p><hr><h3><strong>2. What Makes This Model Robust?</strong></h3><p>Physicists have confidence in the robustness of idealized models because of:</p><ol><li><p><strong>Conservation Laws</strong>:
The fundamental principles (e.g., energy conservation, phase space
preservation, Liouville's theorem) underlying these models are exact and
do not depend on the details of the process. Even if the system
deviates from ideal conditions, these laws still hold.</p></li><li><p><strong>Insensitivity to Small Deviations</strong>:
While no real process is perfectly quasi-static, small deviations from
ideal conditions often lead to corrections that are minor in their
impact. The system still approximately follows the predictions of the
idealized model.</p></li><li><p><strong>Separation of Scales</strong>:
In many cases, the timescales of equilibration (e.g., molecular
collisions) are much shorter than the timescales of macroscopic changes
(e.g., piston movement). This separation ensures that deviations from
equilibrium are transient and self-correcting.</p></li><li><p><strong>Agreement with Experiment</strong>:
Idealized models have repeatedly shown good agreement with experimental
data within their domain of validity. This empirical success lends
credence to their utility and robustness.</p></li></ol><hr><h3><strong>3. Why Aren’t Spherical Cows Always Useless?</strong></h3><p>The
spherical cow metaphor critiques models that oversimplify to the point
of absurdity. But not all simplifications are equally problematic. For
example:</p><ul><li><strong>Spherical Stars</strong>: Assuming stars are
spherical simplifies their gravitational dynamics while still providing
highly accurate predictions for many phenomena.</li><li><strong>Point Masses</strong>: Treating planets as point masses in orbital calculations works beautifully despite their actual shapes and compositions.</li></ul><p>In
thermodynamics, quasi-static adiabatic compression is a "spherical cow"
that turns out to be remarkably predictive in real-world systems where
processes are sufficiently slow and gradients negligible.</p><hr><h3><strong>4. When Does the Model Break Down?</strong></h3><p>The model fails when:</p><ul><li>Compression is too fast, creating shock waves or large-scale irreversibilities.</li><li>Heat exchange or other non-adiabatic effects become significant.</li><li>Microscopic assumptions (e.g., perfect collisions) break down due to quantum or relativistic effects.</li></ul><p>Physicists are aware of these limitations and refine or replace models as needed when their assumptions no longer hold.</p><hr><h3><strong>5. Why Do Physicists Trust It?</strong></h3><p>The trust comes from:</p><ul><li><strong>Generality</strong>: The underlying principles are broadly applicable.</li><li><strong>Predictive Power</strong>: The model reliably predicts outcomes in the scenarios it’s designed for.</li><li><strong>Empirical Validation</strong>: Repeated experimental confirmation underpins confidence in its robustness.</li></ul><hr><h3><strong>Conclusion</strong></h3><p>Your
skepticism is healthy and often needed to refine models. However, what
makes the quasi-static adiabatic compression model (or other "spherical
cow" analogs) valuable is its proven ability to capture the essential
physics while offering insights that are surprisingly robust to
violations of its assumptions.</p><p>Would you like to discuss a specific example where such trust might be misplaced, or do you find this explanation satisfactory?</p></div></div></div></div><div class="gmail-mb-2 gmail-flex gmail-gap-3 empty:hidden gmail--ml-2"><div class="gmail-items-center gmail-justify-start gmail-rounded-xl gmail-p-1 gmail-flex"><div class="gmail-flex gmail-items-center"><span class="gmail-"></span></div></div></div></div><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><span class="gmail-"></span><div class="gmail-group gmail-absolute gmail-bottom-2 end-2 gmail-z-20 gmail-flex gmail-flex-col gmail-gap-1 gmail-md:flex gmail-lg:bottom-3 gmail-lg:end-3"></div><div class="gmail-relative gmail-flex gmail-h-full gmail-w-full gmail-overflow-hidden gmail-transition-colors gmail-z-0"><div class="gmail-z-[1] gmail-flex-shrink-0 gmail-overflow-x-hidden gmail-bg-token-sidebar-surface-primary gmail-max-md:!w-0" style="width:0px"><div class="gmail-absolute gmail-h-full gmail-w-[400px] gmail-pointer-events-none"><div class="gmail-flex gmail-h-full gmail-flex-col"></div></div></div></div><div class="gmail-sr-only"></div><div class="gmail-sr-only"></div><span class="gmail-pointer-events-none gmail-fixed gmail-inset-0 gmail-z-[60] gmail-mx-auto gmail-my-2 gmail-flex gmail-max-w-[560px] gmail-flex-col gmail-items-stretch gmail-justify-start gmail-md:pb-5"></span><div style="min-width:max-content"><div role="dialog" id="gmail-radix-:r5l:" tabindex="-1"></div></div><span tabindex="0" style="outline:none;opacity:0"></span><li class="gmail-relative"><br></li><li class="gmail-relative"><div class="gmail-no-draggable gmail-group gmail-rounded-lg gmail-active:opacity-90 gmail-bg-[var(--item-background-color)] gmail-h-9 gmail-text-sm gmail-relative gmail-screen-arch:bg-transparent"><br><div class="gmail-absolute gmail-bottom-0 gmail-top-0 gmail-inline-flex gmail-items-center gmail-gap-1.5 gmail-pr-2 gmail-ltr:right-0 gmail-rtl:left-0 gmail-invisible gmail-can-hover:group-hover:visible"><span class="gmail-"></span></div></div></li></div></div><br><div class="gmail_quote gmail_quote_container"><div dir="ltr" class="gmail_attr">On Wed, Jan 22, 2025 at 4:48 PM Barry MacKichan <<a href="mailto:barry.mackichan@mackichan.com">barry.mackichan@mackichan.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><u></u>
<div><div style="font-family:sans-serif"><div style="white-space:normal"><p dir="auto">Didn’t know. Google told me almost instantly. That is, if you had submitted this as a term paper, my plagiarism detector would be flashing red. So, sorry, I guess I cheated.</p><p dir="auto">— Barry</p><p dir="auto">On 22 Jan 2025, at 14:30, Nicholas Thompson wrote:</p>
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<div>I am curious how many of you on the list know or could guess at the author of the following quote: <br></div>
<div><br></div>
<div>I am interested in your guesses; if you know it for SURE, please hold off guessing.</div>
<div><span style="font-family:garamond,times new roman,serif"><br></span></div>
<div><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt;font-family:"Aptos",sans-serif"><b><span style="font-family:garamond,times new roman,serif">I would be willing to throw away everything else but that: enthusiasm tamed by metaphor. Let me rest the case there. Enthusiasm tamed to metaphor, tamed to that much of it. I do not think anybody ever knows the discreet use of metaphor, his own and other people’s, the discreet handling of metaphor, unless he has been properly educated in poetry.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Poetry begins in trivial metaphors, pretty metaphors, “grace” metaphors, and goes on to the profoundest thinking that we have. Poetry provides the one permissible way of saying one thing and meaning another. People say, “Why don’t you say what you mean?” We never do that, do we, being all of us too much poets. We like to talk in parables and in hints and in indirections—whether from diffidence or some other instinct.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">I have wanted in late years to go further and further in making metaphor the whole of thinking. I find someone now and then to agree with me that all thinking, except mathematical thinking, is metaphorical, or all thinking except scientific thinking. The mathematical might be difficult for me to bring in, but the scientific is easy enough.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Once on a time all the Greeks were busy telling each other what the All was—or was like unto. All was three elements, air, earth, and water (we once thought it was ninety elements; now we think it is only one). All was substance, said another. All was change, said a third. But best and most fruitful was Pythagoras’ comparison of the universe with number. Number of what? number of feet, pounds, and seconds was the answer, and we had science and all that has followed in science. The metaphor has held and held, breaking down only when it came to the spiritual and psychological or the out of the way places of the physical.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">The other day we had a visitor here, a noted scientist, whose latest word to the world has been that the more accurately you know where a thing is, the less accurately you are able to state how fast it is moving. You can see why that would be so, without going back to Zeno’s problem of the arrow’s flight. In carrying numbers into the realm of space and at the same time into the realm of time you are mixing metaphors, that is all, and you are in trouble. They won’t mix. The two don’t go together.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Let’s take two or three more of the metaphors now in use to live by. I have just spoken of one of the new ones, a charming mixed metaphor right in the realm of higher mathematics and higher physics: that the more accurately you state where a thing is, the less accurately you will be able to tell how fast it is moving. And, of course, everything is moving. Everything is an event now. Another metaphor. A thing, they say, is all event. Do you believe it is? Not quite. I believe it is almost all event. But I like the comparison of a thing with an event.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">I notice another from the same quarter. “In the neighborhood of matter space is something like curved.” Isn’t that a good one! It seems to me that that is simply and utterly charming—to say that space is something like curved in the neighborhood of matter. “Something like.”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Another amusing one is from—what is the book?—I can’t say it now; but here is the metaphor. Its aim is to restore you to your ideas of free will. It wants to give you back your freedom of will. All right, here it is on a platter. You know that you can’t tell by name what persons in a certain class will be dead ten years after graduation, but you can tell actuarially how many will be dead. Now, just so this scientist says of the particles of matter flying at a screen, striking a screen; you can’t tell what individual particles will collide, but you can say in general that a certain number will strike in a given time. It shows, you see, that the individual particle can come freely. I asked Bohr about that particularly, and he said, “Yes , It is so. It can come when it wills and as it wills; and the action of the individual particle is unpredictable. But it is not so of the action of the mass. There you can predict.” He says, “That gives the individual atom its freedom, but the mass its necessity.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Another metaphor that has interested us in our time and has done all our thinking for us is the metaphor of evolution. Never mind going into the Latin word. The metaphor is simply the metaphor of the growing plant or of the growing thing. And somebody very brilliantly, quite a while ago, said that the whole universe, the whole of everything, was like unto a growing thing. That is all. I know the metaphor will break down at some point, but it has not failed everywhere. It is a very brilliant metaphor, I acknowledge, though I myself get too tired of the kind of essay that talks about the evolution of candy, we will say, or the evolution of elevators—the evolution of this, that, and the other. Everything is evolution. I emancipate myself by simply saying that I didn’t get up the metaphor and so am not much interested in it.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">What I am pointing out is that unless you are at home in the metaphor, unless you have had your proper poetical education in the metaphor, you are not safe anywhere. Because you are not at ease with figurative values: you don’t know the metaphor in its strength and its weakness. You don’t know how far you may expect to ride it and when it may break down with you. You are not safe with science; you are not safe in history. In history, for instance—to show that [it] is the same in history as elsewhere—I heard somebody say yesterday that Aeneas was to be likened unto (those words, “likened unto”!) George Washington. He was that type of national hero, the middle-class man, not thinking of being a hero at all, bent on building the future, bent on his children, his descendents. A good metaphor, as far as it goes, and you must know how far. And then he added that Odysseus should be likened unto Theodore Roosevelt. I don’t think that is so good. Someone visiting Gibbon at the point of death, said he was the same Gibbon as of old, still at his parallels.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Take the way we have been led into our present position morally, the world over. It is by a sort of metaphorical gradient. There is a kind of thinking—to speak metaphorically—there is a kind of thinking you might say was endemic in the brothel. It was always there. And every now and then in some mysterious way it becomes epidemic in the world. And how does it do so? By using all the good words that virtue has invented to maintain virtue. It uses honesty, first,—frankness, sincerity—those words; picks them up, uses them. “In the name of honesty, let us see what we are.” You know. And then it picks up the word joy. “Let us in the name of joy, which is the enemy of our ancestors, the Puritans….Let us in the name of joy, which is the enemy of the kill-joy Puritan…” You see. “Let us,” and so on. And then, “In the name of health….” Health is another good word. And that is the metaphor Freudianism trades on, mental health. And the first thing we know, it has us all in up to the top knot. I suppose we may blame the artists a good deal, because they are great people to spread by metaphor. The stage too—the stage is always a good intermediary between the two worlds, the under and the upper,—if I may say so without personal prejudice to the stage.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">In all this I have only been saying that the devil can quote Scripture, which simply means that the good words you have lying around the devil can use for his purposes as well as anybody else. Never mind about my morality. I am not here to urge anything. I don’t care whether the world is good or bad—not on any particular day.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Let me ask you to watch a metaphor breaking down here before you.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Somebody said to me a little while ago, “It is easy enough for me to think of the universe as a machine, as a mechanism.”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">I said, “You mean the universe is like a machine?”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">He said, “No. I think it is one . . . Well, it is like . . .”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">“I think you mean the universe is like a machine.”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">“All right. Let it go at that.”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">I asked him, “Did you ever see a machine without a pedal for the foot, or a lever for the hand, or a button for the finger?”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">He said, “No—no.”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">I said, “All right. Is the universe like that?”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">And he said, “No. I mean it is like a machine, only . . .”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">“. . . it is different from a machine,” I said.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">He wanted to go just that far with that metaphor and no further. And so do we all. All metaphor breaks down somewhere. That is the beauty of it. It is touch and go with the metaphor, and until you have lived with it long enough you don’t know when it is going. You don’t know how much you can get out of it and when it will cease to yield. It is a very living thing. It is as life itself.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">I have heard this ever since I can remember , and ever since I have taught: the teacher must teach the pupil to think. I saw a teacher once going around in a great school and snapping pupils’ heads with thumb and finger and saying, “Think.” That was when thinking was becoming the fashion. The fashion hasn’t yet quite gone out.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">We still ask boys in college to think, as in the nineties, but we seldom tell them what thinking means; we seldom tell them it is just putting this and that together; it is just saying one thing in terms of another. To tell them is to set their feet on the first rung of a ladder the top of which sticks through the sky.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">Greatest of all attempts to say one thing in terms of another is the philosophical attempt to say matter in terms of spirit, or spirit in terms of matter, to make the final unity. That is the greatest attempt that ever failed. We stop just short there. But it is the height of poetry, the height of all thinking, the height of all poetic thinking, that attempt to say matter in terms of spirit and spirit in terms of matter. It is wrong to anybody a materialist simply because he tries to say spirit in terms of matter, as if that were a sin. Materialism is not the attempt to say all in terms of matter. The only materialist—be he poet, teacher, scientist, politician, or statesman—is the man who gets lost in his material without a gathering metaphor to throw it into shape and order. He is the lost soul.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">We ask people to think, and we don’t show them what thinking is. Somebody says we don’t need to show them how to think; bye and bye they will think. We will give them the forms of sentences and, if they have any ideas, then they will know how to write them. But that is preposterous. All there is to writing is having ideas. To learn to write is to learn to have ideas.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">The first little metaphor….Take some of the trivial ones. I would rather have trivial ones of my own to live by than the big ones of other people.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">I remember a boy saying, “He is the kind of person that wounds with his shield.” That may be a slender one, of course. It goes a good way in character description. It has poetic grace. “He is the kind that wounds with his shield.”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">The shield reminds me—just to linger a minute—the shield reminds me of the inverted shield spoken of in one of the books of the “Odyssey,” the book that tells about the longest swim on record. I forget how long it lasted—several days, was it?—but at last as Odysseus came near the coast of Phaneacia, he saw it on the horizon “like an inverted shield.”</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">There is a better metaphor in the same book. In the end Odysseus comes ashore and crawls up the beach to spend the night under a double olive tree, and it says, as in a lonely farmhouse where it is hard to get fire—I am not quoting exactly—where it is hard to start the fire again if it goes out, they cover the seeds of fire with ashes to preserve it for the night, so Odysseus covered himself with the leaves around him and went to sleep. There you have something that gives you character, something of Odysseus himself. “Seeds of fire.” So Odysseus covered the seeds of fire in himself. You get the greatness of his nature.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt"><b><span style="font-family:garamond,times new roman,serif">But these are slighter metaphors than the ones we live by. They have their charm, their passing charm. They are as it were the first steps toward the great thoughts, grave thoughts, thoughts lasting to the end.</span></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:115%;font-size:12pt;font-family:"Aptos",sans-serif"><b><span style="font-family:garamond,times new roman,serif">The metaphor whose manage we are best taught in poetry—that is all there is of thinking. It may not seem far for the mind to go but it is the mind’s furthest. The richest accumulation of the ages is the noble metaphors we have rolled up.</span></b></p>
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<span class="gmail_signature_prefix">--</span><br>
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<div dir="ltr">
<div>Nicholas S. Thompson</div>
<div>Emeritus Professor of Psychology and Ethology</div>
<div>Clark University</div>
<div><a href="mailto:nthompson@clarku.edu" target="_blank">nthompson@clarku.edu</a></div>
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</blockquote></div><div><br clear="all"></div><br><span class="gmail_signature_prefix">-- </span><br><div dir="ltr" class="gmail_signature"><div dir="ltr"><div>Nicholas S. Thompson</div><div>Emeritus Professor of Psychology and Ethology</div><div>Clark University</div><div><a href="mailto:nthompson@clarku.edu" target="_blank">nthompson@clarku.edu</a></div><a href="https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fwordpress.clarku.edu%2fnthompson&c=E,1,0MEK1KvfBhp65N5fB0PmowY0-Wc42o9Q3DtrNJy7XKB3s4mBmcvS4xhJ8_gu5YKfXswz-U--TVA__4IvF7N9Lwpf99yfECRf93wkDA0L&typo=1" target="_blank">https://wordpress.clarku.edu/nthompson</a></div></div>
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