Consistent expansion of the Langevin propagator with application to entropy production

Stochastic thermodynamics is a developing theory for systems out of thermal equilibrium. It allows to formulate a wealth of nontrivial relations among thermodynamic quantities such as heat dissipation, excess work, and entropy production in generic nonequilibrium stochastic processes. A key quantity for the derivation of these relations is the propagator - the probability to observe a transition from one point in phase space to another after a given time. Here, applying stochastic Taylor expansions, we devise a formal expansion procedure for the propagator of overdamped Langevin dynamics. The three leading orders are obtained explicitly. The technique resolves the shortcomings of the current mathematical machinery for the calculation of the propagator. For the evaluation of the first two displacement cumulants, the leading order Gaussian propagator is sufficient. However, some functionals of the propagator, such as the entropy production, which we refer to as “first derivatives of the trajectory”, need to be evaluated to a previously-unrecognized higher order. The method presented here can be extended to arbitrarily higher orders in order to accurately compute any other functional of the propagator.

💡 Research Summary

The paper addresses a fundamental problem in stochastic thermodynamics: how to obtain an accurate short‑time propagator for overdamped Langevin dynamics, which is essential for computing trajectory‑dependent quantities such as entropy production. While the leading‑order Gaussian propagator (the “½‑order” term) is sufficient for evaluating the first two displacement cumulants (mean and variance), it fails to capture higher‑order corrections that become crucial when the observable involves a first derivative of the trajectory, e.g., the logarithmic ratio of forward and backward path probabilities that defines entropy production.

The authors start by formulating the overdamped Langevin equation and its associated Fokker‑Planck description, emphasizing the equivalence of Itô and Stratonovich interpretations at the level of the full path probability. They then develop a systematic stochastic Taylor expansion framework that can be applied to any numerical integration scheme. Explicitly, they treat the Euler‑Maruyama (first‑order) and Milstein (second‑order) schemes, and outline how higher‑order methods would be incorporated.

The central result is an expansion of the short‑time propagator up to order Δt^{3/2}. The propagator is written as the product of the leading Gaussian term P_{1/2} and a correction factor that contains two new functions, Φ and Ψ. Both Φ and Ψ are polynomial in the rank‑two tensor K = ΔxΔx/Δt and are of order unity. Their explicit forms are given in equations (53) and (65). The expansion reads schematically:

P(x+Δx, λ+Δλ | x, λ) = P_{1/2}(…) ×