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Why does entropy drive evolution equations?

Mark A. Peletier

Abstract

`Entropy' appears as driving force in many different evolution equations, both deterministic and stochastic, and in these equations this `entropy' also takes different forms. We show how all these examples can be understood as different instances of a common principle: Entropy drives evolutions because it characterizes the invariant measure of an underlying stochastic process. This interpretation explains the appearance of entropy, the different forms that entropy takes in these equations, and how entropy `drives' these evolution equations. We illustrate this common structure with examples from stochastic processes, gradient flows, and GENERIC systems.

Why does entropy drive evolution equations?

Abstract

`Entropy' appears as driving force in many different evolution equations, both deterministic and stochastic, and in these equations this `entropy' also takes different forms. We show how all these examples can be understood as different instances of a common principle: Entropy drives evolutions because it characterizes the invariant measure of an underlying stochastic process. This interpretation explains the appearance of entropy, the different forms that entropy takes in these equations, and how entropy `drives' these evolution equations. We illustrate this common structure with examples from stochastic processes, gradient flows, and GENERIC systems.
Paper Structure (53 sections, 9 theorems, 161 equations, 3 figures, 1 table)

This paper contains 53 sections, 9 theorems, 161 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The basic question
  3. Example: Damped oscillator
  4. Other examples
  5. The basic answer
  6. A core example
  7. Setup of the core example
  8. Interpretation of the core example
  9. ${\mathcal{S}}_n$ 'is equal to' the invariant measure after coarse-graining.
  10. The drift term ${\mathcal{S}}_n'(Y_t)/\beta$ in \ref{['eq:CG-example-SDE-Y']} is caused by the combination of coarse-graining and noise.
  11. Geometry gives another view on emergence of drift.
  12. Non-flat diffusion and connection to 'free energy'
  13. Further comments
  14. Entropy as 'remnant' of invariant measures
  15. Relative entropies and Sanov's theorem
  16. ...and 38 more sections

Key Result

Lemma 2.1

Define the process $X$ as in eq:CG-example-SDE-X and set $Y_t := \Phi(X_t) = |X_t|$ for all $t$, as above. The process $Y$ has the invariant measure $\nu_n := (\Phi_n)_\# \mu_n$ (see eqdef:push-forward), which can also be characterized as where $c_n'$ is a normalization constant. The function ${\mathcal{S}}_n$ satisfies

Figures (3)

  • Figure 1: Simulation of equation \ref{['eq:ODE-harmonic-oscillator']}. Note how $e$ is monotonically increasing, and the two other variables converge to zero. As $t\to\infty$, all energy is transferred from the mechanical components $P^2/2m+kQ^2/2$ to $e$, while keeping the sum ${\mathcal{E}}$ constant.
  • Figure 2: A particle $X$ released at time zero at $X_0$ (the large dot) visits neighbouring points as time increases (the small dots). These points are symmetrically spread around $X_0$ in $X$-space, but the curvature of the level sets of $\Phi_n$ causes their $Y$-values to be biased towards larger values of $\Phi_n$.
  • Figure 3: The setup of the Hamiltonian system consisting of two subsystems, 'System A' and a heat bath ('System B').

Theorems & Definitions (43)

  • Example A: The damped harmonic oscillator as a Generic system
  • Example B: Diffusion as gradient flow of entropy
  • Example C: Nonlinear diffusion as gradient flow of various entropies
  • Example D: The Boltzmann equation as Generic system
  • Definition 1.1
  • Remark 1.2: Entropy as volume of macrostates
  • Remark 1.3: Entropy and free energy
  • Remark 1.4: Novelty
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['l:CG-example-S_n']}
  • ...and 33 more