[Paper Review] Continuous time-reversal and equality in the thermodynamic uncertainty relation

Andreas Dechant, Shin‐ichi Sasa|arXiv (Cornell University)|Oct 28, 2020

Advanced Thermodynamics and Statistical Mechanics47 references46 citations

TL;DR

This paper introduces a continuous time-reversal operation that interpolates between forward and reversed dynamics in nonequilibrium steady states of Markovian systems. By parameterizing the dynamics with a continuous θ ∈ [−1, 1], the authors derive tighter thermodynamic uncertainty relations (TURs), including a relative TUR (RTUR) that uses fluctuations relative to the local mean current. The key result is that equality in the TUR is achieved when the observable is the stochastic entropy production, and this occurs precisely at the equilibrium point (θ = 0) of the continuous time-reversal family, which is shown explicitly for a particle in a tilted periodic potential.

ABSTRACT

We introduce a continuous time-reversal operation which connects the time-forward and time-reversed trajectories in the steady state of an irreversible Markovian dynamics via a continuous family of stochastic dynamics. This continuous time-reversal allows us to derive a tighter version of the thermodynamic uncertainty relation (TUR) involving observables evaluated relative to their local mean value. Moreover, the family of dynamics realizing the continuous time-reversal contains an equilibrium dynamics halfway between the time-forward and time-reversed dynamics. We show that this equilibrium dynamics, together with an appropriate choice of the observable, turns the inequality in the TUR into an equality. We demonstrate our findings for the example of a particle diffusing in a tilted periodic potential.

Motivation & Objective

• To establish a continuous family of stochastic dynamics that interpolates between time-forward and time-reversed processes in nonequilibrium steady states.
• To derive tighter bounds on current fluctuations than the standard thermodynamic uncertainty relation (TUR).
• To identify conditions under which the TUR becomes an equality, particularly by introducing a relative TUR (RTUR) using local mean-subtracted current fluctuations.
• To demonstrate that the equilibrium dynamics at θ = 0, embedded in the continuous time-reversal family, enables exact equality in the TUR for the stochastic entropy production as the observable.

Proposed method

• Introduce a continuous time-reversal operation parameterized by θ ∈ [−1, 1], where θ = 1 corresponds to forward dynamics, θ = −1 to reversed dynamics, and θ = 0 to equilibrium.
• Define a modified drift vector aθ(x) = θνst(x) + B∇ln pst(x), which preserves the steady-state distribution pst(x) but scales the irreversible current by θ.
• Show that the stochastic entropy production Στθ scales linearly with θ, and that the path probabilities satisfy DKL(P−θ ∥ Pθ,†) = 0, implying symmetry under continuous time-reversal.
• Derive two tighter variants of the TUR: one using equilibrium fluctuations (Var0(Jτ)) and one using local mean-subtracted fluctuations (Var(δJτ)), both of which are tighter than the standard TUR.
• Use the Kullback-Leibler divergence and cumulant generating function techniques to derive a quadratic lower bound on the cumulant generating function, leading to a θ-independent variance bound.
• Apply the formalism to a particle in a tilted periodic potential, solving the Fokker-Planck equation to obtain explicit expressions for pst(x) and νst(x), and verify the equality condition at θ = 0.

Experimental results

Research questions

• RQ1 Can a continuous time-reversal operation be defined that connects time-forward and time-reversed dynamics in a smooth, parameterized way for nonequilibrium steady states?
• RQ2 Does this continuous time-reversal operation lead to tighter bounds on current fluctuations than the standard thermodynamic uncertainty relation?
• RQ3 Is there a specific observable and dynamics within the continuous time-reversal family for which the TUR becomes an equality?
• RQ4 Can the relative TUR (RTUR), based on fluctuations relative to the local mean current, be derived and shown to be tighter than the standard TUR?
• RQ5 What is the role of the equilibrium dynamics (θ = 0) in achieving equality in the TUR, and why does the stochastic entropy production serve as the optimal observable in this context?

Key findings

• The continuous time-reversal operation realizes a smooth interpolation between forward and reversed dynamics, with the equilibrium state (θ = 0) lying exactly halfway in the family.
• The relative TUR (RTUR) is derived as ⟨Jτ⟩² / Var(δJτ) ≤ 1/(2∆Sirr,τ), which is tighter than the standard TUR because Var(δJτ) < Var(Jτ) in general.
• The equilibrium fluctuation-based TUR, ⟨Jτ⟩² / Var0(Jτ) ≤ 1/(2∆Sirr,τ), is also tighter than the standard TUR, with Var0(Jτ) < Var(Jτ).
• For the stochastic entropy production Jτ = Στ, both the RTUR and the equilibrium-based TUR reduce to equality, demonstrating that the bound is saturated at the equilibrium point.
• The variance of the entropy production is exactly twice the average entropy production: Varθ(Στ) = 2∆Sirr,τ, and the mean and fluctuation parts are statistically independent.
• In the tilted periodic potential model, the steady-state distribution and current are explicitly computed, and the equality condition of the TUR is verified analytically at θ = 0 for the entropy production observable.