<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Jon, hi, <div class=""><br class=""></div><div class="">Sorry to be very late to reply. </div><div class=""><br class=""></div><div class="">I think the best entree into non-convexity is this very good review article by Hugo Touchette, which is what started me down the whole large-deviations road (on a pointer from Cosma):</div><div class=""><br class=""></div><div class=""><div class=""> author = "Touchette, Hugo",</div><div class=""> title = "The large deviation approach to statistical mechanics",</div><div class=""> journal = "Phys.~Rep.",</div><div class=""> volume = "478",</div><div class=""> pages = "1--69",</div><div class=""> year = "2009",</div><div class=""> note = "arxiv:0804.0327"</div><div class=""><br class=""></div><div class="">It’s not super-elaborated, but very well-organized. And unlike me, Touchette is a _really_ good scientific writer, so you don’t end up mystified what he is trying to say. Looking for the sections that discuss Legendre-Fenchel transforms, in place of the ordinary Legendre transform for the convex cases, will vector you in rapidly. </div><div class=""><br class=""></div><div class="">Because I have spent so many years painfully learning how and why to use ray-theoretic methods for dynamically evolving probability distributions (stuff that is widely understood by people like Nigel Goldenfeld or Dan Stein, but I was very very slow to get the point), I have a sense that there is tons to do and tons to relate. But I don’t know anywhere this stuff is all laid out didactically with good order and some fun, intuitive and non-messy examples. My sense of the field’s content and its potential is all just little anecdotes and this-or-that model that I worked out in my notes, but didn’t have anyplace to publish. Yet I am sure that all the other people who have worked on this, Like Bertini, di Sole, or Jona-Lasinio in Italy, all know and have worked through the same images.</div><div class=""><br class=""></div><div class="">I wish I had better than that to offer, </div><div class=""><br class=""></div><div class="">Eric</div><div class=""><br class=""></div><div class=""><br class=""></div><div><br class=""><blockquote type="cite" class=""><div class="">On Apr 21, 2022, at 6:33 AM, Jon Zingale <<a href="mailto:jonzingale@gmail.com" class="">jonzingale@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="ltr" class=""><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">ps.</div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">In: <a href="https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fredfish.com%2fpipermail%2ffriam_redfish.com%2f2022-March%2f092172.html&c=E,1,fwWPmfZWiAc7xhypuZwGFTvAr-AXk3qAHpFSTm33WXIRhNpxzLPcs0aFqLgPzIXnR2FvkQ_J41UNBmvjNudpyHPjJMbfv7VHAnJiddxhTrcSP23oilaX&typo=1" class="">https://redfish.com/pipermail/friam_redfish.com/2022-March/092172.html</a></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br class=""></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">EricS writes:</div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><pre style="white-space: pre-wrap;" class="">3. Then (actually, all along since the beginning of the construction) one needs to talk about what kind of aggregation operator we can apply to systems, and quantities that do accumulate under aggregation become the arguments of the state-function entropy, and the extensive state variables. (I say “accumulate” in favor of the more restrictive word “add”, because what we really require is that they are what are termed “scale factors” in large-deviation language, and we can admit a somewhat wider class of kinds of accumulation than just addition, though addition is the extremely common one.)</pre><pre style="white-space: pre-wrap;" class=""><br class=""></pre><pre style="white-space: pre-wrap;" class=""><span style="color:rgb(51,51,51);font-family:verdana,sans-serif;white-space:normal" class="">By somewhat "wider class" is it meant that one would require only the *multiplication by a scalar* condition of linearity without requiring additivity?</span><br class=""></pre></div></div>
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