However, I know of no relation between this circle of ideas and the funny analogy between probabilities and amplitudes that my talk was about.

*That* analogy really amounts to

probability theory : quantum theory :: L^{1} : L^{2}

and I like to joke that the next revolution in physics will involve L^{3} spaces.

(I don’t believe that: it’s just a joke, though you should look at Smolin’s paper.)

]]>(Sorry, I don’t have HTML-angles on this hilariously stupid “netbook” I’m using.)

]]>The symplectic structure is even more interesting and more useful when applied to the modern (post-classical) versions. Hamilton-Jacobi theory is amusing, but it’s less useful than out-and-out QM, and not significantly simpler.

2) Stat mech is basically the analytic continuation of QM, continued in the direction of imaginary time. This point and its ramifications are discussed in e.g. Feynman and Hibbs **Quantum Mechanics and Path Integrals** (1965). The classical limit of QM is obtained by the method of stationary phase, whereas the classical limit of stat mech is obtained by the method of steepest descent … so the two subjects are very nearly but not quite identical.

Again: If we’re going to make connections, it is even more interesting and more useful to connect the modern (post-classical) versions.

FWIW note that Planck invented QM as an outgrowth from stat mech … not directly from classical mechanics. So the connections are there, and have been since Day One.

3) Many (but not all) of the familiar formulas of thermodynamics can usefully be translated to the language of differential forms. In many cases all that is required is a re-interpretation of the symbols, leaving the form of the formula unchanged; for instance we interpret dE = T dS – P dV as a *vector* equation.

I say “not all” formulas because more than a few of the formulas you see in typical thermodynamics books are nonsense. This includes (almost) all expressions involving “dQ” or “dW”. Such things simply do not exist (except in trivial cases). Daniel Schroeder in **An Introduction to Thermal Physics** (1999) rightly calls them a crime against the laws of mathematics. With a modicum of self-discipline it is straightforward to do thermodynamics without committing such crimes.

Differential forms make thermodynamics simpler and more visually intuitive … and simultaneously more sophisticated, more powerful, and more correct.

Although there are fat books on the subject of differential topology, only the tiniest fraction of that is necessary for present purposes. An introductory discussion (including pictures) can be found at https://www.av8n.com/physics/thermo-forms.htm and the application to thermodynamics is worked out in some detail at https://www.av8n.com/physics/thermo/.

]]>which gives the Moran model of population genetics. It’s complex balanced, so we can understand it in detail. He uses the Kullback–Leibler divergence as a Lyapunov function for the rate equation. He uses the Anderson–Craciun–Kurtz theorem to get stationary distributions for the master equation.

His talk is reminding me that I should read this paper:

• David F. Anderson, Gheorghe Craciun, Manoj Gopalkrishnan and Carsten Wiuf, Lyapunov functions, stationary distributions, and non-equilibrium potential for chemical reaction networks.

]]>Also, with some small rate a *new* species tries to invade a randomly chosen site. When this happens we need to increase the size of our matrix, randomly choosing 0s and 1s in a new row and new column. The matrix entries are used to decide if the invasion succeeds, in the usual way.

When the mean field approximation obeys a standard Lotka–Volterra equation.

]]>Her program tries to explain how this works. It runs a simple lattice model that shows a transition between low-diversity and high-diversity states.

Consider a square lattice with states with each site carrying an individual of one species. Suppose we have a random matrix of 0s and 1s, saying which species can invade which other species. The entries of this matrix are chosen randomly and independently with probability of being 1.

]]>If you have the Java runtime environment enabled (which the powers that be are making ever more difficult, due to security concerns), you can see an applet that runs Mitarai’s program:

• Namiko Mitarai, Ecosystems with mutually exclusive interactions self-organize to a state of high diversity.

You can install the Java runtime environment here, and enable it by following the directions here.

]]>• Arjan van der Schaft, Shodhan Rao and Bayu Jayawardhana, On the mathematical structure of balanced chemical reaction networks governed by mass action kinetics.

An issue of terminology: what I call the Kolmogorov condition for detailed balance is called the Wegscheider condition when it’s applied to chemical reaction networks.

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