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Entropy dissipation inequality for general binary collision models

Giada Basile, Dario Benedetto

TL;DR

This work develops a variational entropy-dissipation framework for general non-reversible binary collision models in the homogeneous Boltzmann setting by introducing a two-particle factorization condition that ensures the two-particle equilibrium measure factorizes. It provides a measure-flux formulation for both continuous-time Markov chains and the Boltzmann equation, proving an entropy balance that yields an $H$-theorem under the factorization assumption, i.e. $Ent(P_T|π) + E(Q|Υ_#V^P) ≤ Ent(P_0|π)$ (and its non-reversible counterpart). The authors construct explicit non-reversible collision kernels satisfying the factorization condition, including granular-gas-like kernels and socio-economic models such as opinion dynamics, with detailed expressions for the entropy dissipation in terms of forward and time-reversed fluxes. They illustrate the approach through two Kuramoto-type models and several opinion-dynamics variants, and provide a microscopic interpretation in terms of Kac-type walks, thereby extending entropy methods to a broad class of non-reversible kinetic processes with potential applications in physics and social sciences.

Abstract

We introduce a ``two-particle factorization'' condition which allows us to formulate the homogeneous Boltzmann equation for non-reversible collision kernels in terms of an entropy inequality. This formulation yields an H-Theorem. We provide some examples of non-reversible binary collision models with a concentration/dispersion mechanism, as in opinion dynamics, which satisfy this condition. As a preliminary step, we also provide an analogous variational formulation of non-reversible continuous time Markov chains, expressed in terms of an entropy dissipation inequality.

Entropy dissipation inequality for general binary collision models

TL;DR

This work develops a variational entropy-dissipation framework for general non-reversible binary collision models in the homogeneous Boltzmann setting by introducing a two-particle factorization condition that ensures the two-particle equilibrium measure factorizes. It provides a measure-flux formulation for both continuous-time Markov chains and the Boltzmann equation, proving an entropy balance that yields an -theorem under the factorization assumption, i.e. (and its non-reversible counterpart). The authors construct explicit non-reversible collision kernels satisfying the factorization condition, including granular-gas-like kernels and socio-economic models such as opinion dynamics, with detailed expressions for the entropy dissipation in terms of forward and time-reversed fluxes. They illustrate the approach through two Kuramoto-type models and several opinion-dynamics variants, and provide a microscopic interpretation in terms of Kac-type walks, thereby extending entropy methods to a broad class of non-reversible kinetic processes with potential applications in physics and social sciences.

Abstract

We introduce a ``two-particle factorization'' condition which allows us to formulate the homogeneous Boltzmann equation for non-reversible collision kernels in terms of an entropy inequality. This formulation yields an H-Theorem. We provide some examples of non-reversible binary collision models with a concentration/dispersion mechanism, as in opinion dynamics, which satisfy this condition. As a preliminary step, we also provide an analogous variational formulation of non-reversible continuous time Markov chains, expressed in terms of an entropy dissipation inequality.

Paper Structure

This paper contains 12 sections, 11 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-reversible jump processes
  3. Reversible Markov chains
  4. Non-reversible Markov chains
  5. Binary collision models
  6. Hard spheres
  7. Non-reversible collision kernels
  8. Microscopic interpretation
  9. How to construct collision kernels
  10. Some example
  11. Two Kuramoto type models
  12. Opinion dynamics models

Key Result

Proposition 2.1

Given $P_0\in {\mathcal{P}}({\mathcal{X}})$ with $\mathop{\rm Ent}\nolimits(P_0|\pi) < +\infty$, the pair $(P,{\mathcal{V}})\in {\mathcal{S}}$ is a measure-flux solution to the Kolmogorov equation with initial datum $P_0$ if and only if where $\Upsilon: [0,T] \times {\mathcal{X}} \times {\mathcal{X}} \to [0,T] \times {\mathcal{X}} \times {\mathcal{X}}$ is the map that exchanges the incoming and t

Figures (3)

  • Figure 1: On the left, the collision rate $\lambda$ for the first example as in Eq. \ref{['eq:hk1']} for $\delta = \pi/6$; on the right, the collision rate for the second example as in Eq. \ref{['eq:hk2']} for $\varepsilon = 0.1, 0.5, 0.9$.
  • Figure 2: The symmetric model. From the left to the right, the level sets of $g$, the equilibrium $\pi$ and the level sets of $\lambda$.
  • Figure 3: The asymmetric model. From the left to the right, the level sets of $g$, the equilibrium $\pi$ and the level sets of $\lambda$. The repulsive mechanism can be observed in the graph of $g$. In the graph of $\pi$ one can observe that neutral opinions are favored while opinions near $v=1$ are disfavored.

Theorems & Definitions (20)

  • Definition 2.1: Measure-flux solutions to the Kolmogorov equation
  • Proposition 2.1: First variational formulation
  • proof : Proof of Proposition \ref{['prop:markov']}
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.2
  • Proposition 2.3: Second variational formulation
  • proof
  • Proposition 2.4: Entropy balance for non-reversible discrete Markov chains
  • Proposition 2.5: Variational formulation for non-reversible Markov chain
  • ...and 10 more