[FRIAM] stygmergy, CA's, and [biological] development

[David Eric Smith]

There is a formalism for discrete-event dynamical systems known as “bond graphs”.  I haven’t read much about it, but Alan Perelson did some work on this when he was young and not famous.

Bond graphs seem to be a slightly more flexible construction than hypergraphs, and they contain a subset that are equivalent to hypergraphs.  

I bring them up here because bond graphs may admit a certain duality between states and events that interests me, as the right way to move the normal concept of Legendre/Lagrange-Hamilton duality for continuous dynamical systems into the limiting condition that all change is in discrete events, so the role of states and of the events that change states becomes maximally asymmetric, whereas in continuous Hamiltonian dynamics they are equivalent in the basic kinematics.

Speaking for the hypergraph, which I can describe without making the mistakes I would likely make for bond graphs: as I write the hypergraph, states are carried by one type of node (the “species” node) while events occur across one kind of link (a link connecting two “complexes”)   (Terminology from Horn and Jackson and Martin Feinberg for chemical reaction systems.)   The links between species and complexes, different in kind than the reaction-links connecting complexes, carry the system’s stoichiometry.

Anyway, If the usual duality between coordinates and their conjugate momenta in Hamiltonian dynamics were passed to this discrete-event limit, one would want to exchange the species nodes, which carry pure “coordinates” of state, with the links across which the events changing the state occur.  I forget now why I gave up trying to push this through sometime past, but I think it was that there is some non-equivalent role of nodes and links in the bipartite-graph way of representing the hypergraph that I could not see how to get past.  Bond graphs may have enough flexibility to find a proper involution exchanging states and events.

That old topic came to mind while I was watching the ant/pheromone thread, though not in any clean way.  From one view, ants have states, and pheromones mediate the dynamics by which they are updated.  But from another view, ants conduct the only events through which the state of the pheromone map is permitted to change.  Both of the foregoing characterizations are incomplete, as both ant and pheromone clearly have both state and dynamic properties.  But some more encompassing mapping that exchanged the part above was what I imagined.

I have seen a _very_ little bit of this category-theoretic mapping of large domains of representation to one another (through Barry Mazur), so I know that “objects” can be pretty heavy things, with a lot carried in the map.  But I was curious how either Legendre/Lagrange-Hamilton coordinate-momentum duality, or any state-event dualities in discrete-time systems might be classified, and whether there was some categorization in which the distinction between continuous-time dynamics and discrete-time dynamics (conceived as a limit, or as a primitive of its own) might be projected away.  

There is lots to do with entropy, quantum-classical correspondence principles, and the like, across this continuous-discrete divide.  Would a category representation that treated the continuous and discrete cases as the same also induce the map from entropy principles that I think is the right one to go from von Neumann entropies in the continuous, to the stochastic effective actions that are the correct entropies for the discrete case?  My above thoughts on it are surely not well-formed.

Eric




> On Oct 25, 2021, at 1:35 PM, Jon Zingale <jonzingale at gmail.com> wrote:
> 
> Thanks for understanding, Frank. Yes, the duality of vector spaces arises as a particular instance with perfect duality only in the case of finite-dimensional vector spaces. Here is a page outlining a broader discussion: https://ncatlab.org/nlab/show/duality <https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fncatlab.org%2fnlab%2fshow%2fduality&c=E,1,Y7Gt-tfBhsJqBcdlgik-MDtaA3N1XaKzxl7YQzYIaZkj8-mDlHauJpEwwk-M_VFDL4NbaZV8eKYD2vyxmHA5leI_7Z0I3Erc7ZWDcOA6sZ8,&typo=1>
> 
> Duality turns out to be a surprisingly general structural construction, and to the degree that categories aim to characterize phenomena (group -> symmetry, set -> size, Dyn -> behavior, etc...), I believe dualities in and between categories can guide quite a lot of rational exploration.
> 
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