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

Technical Summary: Free Energy Dissipation and a Decomposition of General Jump Diffusions on $\mathbb{R}^n$ without Detailed Balance

Problem Statement
The paper addresses the thermodynamic structure of continuous-state jump-diffusion processes, a class of stochastic dynamics combining Brownian motion and Poisson-driven jumps. While the relationship between free energy, entropy production, and detailed balance is well-established for continuous diffusion processes (governed by local Fokker-Planck equations) and discrete-state Markov jump processes (Master equations), the non-equilibrium statistical mechanics of continuous-state jump diffusions remains underdeveloped. Specifically, there is a lack of explicit, computable formulas for free energy dissipation, entropy production, and housekeeping heat in the presence of non-local operators. The central challenge lies in extending the geometric and operator-theoretic decompositions (such as the Helmholtz-Hodge decomposition) to non-local settings, where the standard definition of probability current becomes non-local, and the characterization of "residual drift" and conservative dynamics is non-trivial.

Methodology
The authors analyze a general jump-diffusion process $(X_t)_{t \geq 0}$ on $\mathbb{R}^n$ defined by the stochastic differential equation:
$$dX_t = b(X_t)dt + \sqrt{2\beta^{-1}}a(X_{t-}) \circ dB_t + \int_{\mathbb{R}^n \setminus \{0\}} \sigma(X_{t-}, z) N(dt, dz)$$
where $B_t$ is a Brownian motion, $N$ is a Poisson random measure, and $\beta$ is the inverse temperature.

The methodology proceeds through three main steps:

1. Thermodynamic Formulation: The authors define the generalized free energy $F(\rho(t))$ as the relative entropy (scaled by $\beta^{-1}$) between the time-dependent density $\rho(t, x)$ and the invariant measure $\mu_{ss}$ with density $\rho_{ss}(x)$. By combining this definition with the non-local Fokker-Planck equation, they derive an explicit time-derivative formula for the free energy.
2. Decomposition of Dissipation: The derived dissipation rate is decomposed into two non-negative terms: the entropy production rate $e_p(t)$ and the housekeeping heat $Q_{hk}(t)$. This decomposition relies on defining both local probability currents (associated with diffusion) and non-local probability currents (associated with jumps).
3. Operator-Theoretic Splitting: The authors introduce a weighted Hilbert space $L^2(\mathbb{R}^n, \rho_{ss} dx)$ and analyze the adjoint structure of the generator $\mathcal{L}$ with respect to the invariant measure. They prove that $\mathcal{L}$ admits a canonical decomposition into a symmetric part ($\mathcal{L}_s$) and an anti-symmetric part ($\mathcal{L}_a$).

Key Contributions and Results

1. Free Energy Dissipation Identity:
   The paper establishes the identity:
   $$\frac{dF(\rho(t))}{dt} = Q_{hk}(t) - \beta^{-1} e_p(t)$$
   where $e_p(t) \geq 0$ represents the total entropy production rate (containing both local diffusive and non-local jump contributions) and $Q_{hk}(t) \geq 0$ represents the housekeeping heat required to maintain a non-equilibrium steady state (NESS). The authors prove a Clausius-type inequality $\theta \frac{dS}{dt} - \frac{dQ}{dt} = \theta e_p(t) \geq 0$.

2. Generator Decomposition ($\mathcal{L} = \mathcal{L}_s + \mathcal{L}_a$):
   A central result is the decomposition of the generator $\mathcal{L}$ into:

   • Symmetric Part ($\mathcal{L}_s$): This operator is self-adjoint with respect to $\mu_{ss}$. It generates a reversible dynamics that satisfies detailed balance. The free energy dissipation is driven entirely by this component and is quantified by a non-local Fisher information functional $I[\rho(t)]$, defined via the Dirichlet form of $\mathcal{L}_s$. Specifically, $\frac{dF}{dt} = -\beta^{-1} I[\rho(t)]$.
   • Anti-symmetric Part ($\mathcal{L}_a$): This operator is anti-symmetric with respect to $\mu_{ss}$. It generates a canonical conservative flow (divergence-free probability currents) that preserves the stationary measure. Crucially, the authors prove that the free energy dissipation associated with $\mathcal{L}_a$ vanishes identically ($F'_a(\rho) = 0$). This component accounts for the circulation and probability currents that break detailed balance without contributing to entropy decay.

3. Characterization of Detailed Balance:
   The paper refines the understanding of detailed balance by showing that in the absence of detailed balance, the stationary condition is satisfied globally by the formation of closed current loops generated by $\mathcal{L}_a$, rather than by the local vanishing of currents. The vanishing of the anti-symmetric part is shown to be equivalent to time-reversibility and detailed balance.

4. Numerical Illustrations:
   Two one-dimensional examples from intracellular active transport are presented:

   • Non-Equilibrium Case: A Lévy-driven model with a confining drift. The results show a non-zero steady-state housekeeping heat and entropy production, with the free energy decaying to zero while the system remains out of equilibrium. The decomposition clarifies that the persistent currents are driven by $\mathcal{L}_a$, while relaxation is driven by $\mathcal{L}_s$.
   • Equilibrium Case: A "reset-to-Gibbs" jump mechanism. In this scenario, the housekeeping heat vanishes, and the free energy dissipation is entirely equal to the entropy production rate, consistent with the absence of an anti-symmetric component.

Significance and Claims
The authors claim that this work provides a unified thermodynamic framework for jump-diffusion processes, extending classical results from continuous diffusions and discrete Markov chains to non-local dynamics. The primary significance lies in:

• Structural Clarity: Providing a clear geometric interpretation of non-equilibrium steady states by separating the dynamics into a dissipative, reversible sector ($\mathcal{L}_s$) and a non-dissipative, circulation-generating sector ($\mathcal{L}_a$).
• Quantification of Maintenance Costs: Explicitly identifying housekeeping heat as the energetic cost of maintaining non-equilibrium circulation, distinct from the dissipation associated with relaxation toward the steady state.
• Generalized Fisher Information: Introducing a non-local Fisher information functional that governs the rate of free energy decay, linking the spectral properties of the symmetric generator to the convergence rate of the system.

The paper concludes that this decomposition offers a systematic way to separate dissipative relaxation from driven circulation in complex systems, with potential applications in biophysics (e.g., intracellular transport) and machine learning (e.g., non-Gaussian generative modeling), though the paper focuses primarily on the theoretical derivation and illustrative examples rather than proposing new experimental protocols.