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Reversible birth-and-death dynamics in continuum: a de Bruijn-type identity for free-energy dissipation

Benedikt Jahnel, Jonas Köppl, Yannic Steenbeck, Alexander Zass

Abstract

We investigate free-energy dissipation in a continuous-time birth-and-death dynamics in $\mathbb{R}^d$. For these Markov processes, the class of reversible measures coincides with the infinite-volume Gibbs point processes for some sufficiently nice Hamiltonian. For a wide class of initial distributions, we derive a de~Bruijn-type identity that relates the time evolution of the specific relative entropy along trajectories to the Fisher information, in particular establishing the thermodynamic limit of the latter. Along the way, we analyze some fine properties of the considered dynamics, such as the existence and regularity of local densities, obtain a spatial ergodic theorem for the entropy production per unit volume, and derive a small-time exponential series expansion of the dynamics.

Reversible birth-and-death dynamics in continuum: a de Bruijn-type identity for free-energy dissipation

Abstract

We investigate free-energy dissipation in a continuous-time birth-and-death dynamics in . For these Markov processes, the class of reversible measures coincides with the infinite-volume Gibbs point processes for some sufficiently nice Hamiltonian. For a wide class of initial distributions, we derive a de~Bruijn-type identity that relates the time evolution of the specific relative entropy along trajectories to the Fisher information, in particular establishing the thermodynamic limit of the latter. Along the way, we analyze some fine properties of the considered dynamics, such as the existence and regularity of local densities, obtain a spatial ergodic theorem for the entropy production per unit volume, and derive a small-time exponential series expansion of the dynamics.
Paper Structure (35 sections, 39 theorems, 300 equations)

This paper contains 35 sections, 39 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Infinite-volume dynamics and convergence to equilibrium
  3. Contributions of this work
  4. Outline of the paper
  5. Setting and main results
  6. Point-process formalism
  7. Gibbs point processes
  8. Dynamics
  9. Main results
  10. Outlook
  11. Proof strategy
  12. The de Bruijn-type identity
  13. Regularity of log-densities
  14. Thermodynamic limit and de Brujin-type identity
  15. Impossibility of convergence in finite time
  16. ...and 20 more sections

Key Result

Proposition 2.2

Let $\mathcal{N}$ be a Poisson random measure on $\mathbb{R}^d\times[0,\infty)^3$ with intensity measure $\mathrm{d} x\otimes \mathrm{d} u \otimes\mathrm{e}^{-r}\mathrm{d} r\otimes\mathrm{d} s$, $\omega = \sum_{i\ge 1}\delta_{x_i} \in \Omega$, and $\underline\omega = \sum_{i\ge 1}\delta_{(x_i,\tau_i

Theorems & Definitions (82)

  • Remark 2.1
  • Proposition 2.2: Graphical representation
  • Remark 2.3
  • Proposition 2.4
  • Definition 2.5: Regular measures
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8: de Bruijn identity
  • Theorem 2.9: Attractor property
  • Proposition 2.10
  • ...and 72 more