Free energy dissipation and a decomposition of general jump diffusions on $\mathbb{R}^n$ without detailed balance

TL;DR

The paper extends stochastic thermodynamics to continuous-state jump diffusions driven by Brownian and Poisson noise. It derives a full free energy dissipation law $\frac{dF}{dt}=Q_{hk}-\beta^{-1}e_p$ and decomposes the generator as $\mathcal{L}=\mathcal{L}_s+\mathcal{L}_a$ in the weighted space $L^2(\rho_{ss})$, identifying a canonical conservative dynamics from $\mathcal{L}_a$ (with $F'_a(\rho)=0$) and a symmetric dissipative part from $\mathcal{L}_s$ that yields a nonlocal Fisher information $\mathcal{I}[\rho]$ governing dissipation. This framework clarifies how nonequilibrium stationary states are sustained by circulation currents without changing $\rho_{ss}$ and provides conditions under which free energy decays exponentially via MLSI-type inequalities for the symmetric sector. Numerical examples illustrate the distinct roles of symmetric and antisymmetric dynamics in nonequilibrium and equilibrium settings, highlighting the conceptual separation between housekeeping heat and genuine relaxation dissipation. Together, these results generalize the thermodynamic decomposition known for diffusions to the nonlocal, jump-driven regime.

Abstract

We analyze the thermodynamic structure of jump diffusions combining Brownian and Poisson noise, a class of stochastic dynamics relevant to nonequilibrium statistical physics. For such nonlocal dynamics, the free energy admits a full dissipation formula that decomposes into entropy production and housekeeping heat. A central result is a decomposition of the generator into symmetric and anti-symmetric parts with respect to the invariant measure $ρ_{ss}$. The symmetric sector corresponds to a reversible dynamics and yields a nonlocal Fisher information governing free-energy decay, whereas the anti-symmetric sector generates a canonical conservative flow that produces circulation but no dissipation. Several numerical examples demonstrate how this decomposition clarifies the structure of nonequilibrium stationary states in jump-driven systems.

Figures (2)

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Theorems & Definitions (17)

• Theorem 2.1
• Definition 2.1: Detailed balance
• Proposition 2.2
• Remark 3.1
• Theorem 3.1
• proof
• Remark 3.2
• Theorem 4.1
• proof
• Lemma 4.1