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Say one has a system of statistical physics whose entropy is given as a function of one or multiple variables; for example as $S(x)$. An example of such a system could be a osmosis system, or a polymer in a bath of water.

It seems to me that if $x$ represents a position, then solving $dS(x)/dx=0$ should give an equation of motion, as a system of statistical physics ought to evolve over time in the direction of maximal entropy. Is there a procedure, similar in generality as the Lagrangian or Hamiltonian formalism, that is able to produce equations of motions using entropy $S$, instead of $L$ or $H$ as the starting point?

Is solving $dS(x)/dx=0$ sufficient to get the equation of motion? A bit more is surely required as we haven't referenced time yet. How does the direction of maximal entropy relate to the 'motion' or evolution of a system over time?

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  • $\begingroup$ $dS(x)/dx=0$ at most determines stationary points. That is not a differential equation but just an equation. $\endgroup$
    – Valter Moretti
    Commented Dec 25, 2020 at 22:22
  • $\begingroup$ @ValterMoretti Looking into it further, I noticed the notion of an entropic force which is defined as the gradient of the entropy. $F(x)=\nabla S(x)$. Would this be an equation of motion, or am I still missing something? $\endgroup$
    – Anon21
    Commented Dec 26, 2020 at 0:06

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