Time arrow in open-boundary one-dimensional stochastic dynamics

TL;DR

We address time irreversibility in a one-dimensional open-boundary Brownian system with a discontinuous temperature profile under finite timestep kicks. By deriving discrete-time forward/backward transition probabilities, performing detailed Brownian simulations, and comparing to a Fokker-Planck description, we reveal a violation of detailed balance near the temperature interface and identify a local gyration mechanism that yields zero net current in the NESS. To reconcile the simulations with continuous-time theory, we introduce an effective temperature profile that smooths the interface and recovers the NESS distribution predicted by the FP equation, with a width $w=\sqrt{2 (T_H-T_C)\Delta t}$. The results show that irreversibility emerges from the memory associated with discrete thermal kicks and suggests broad relevance to driven, nonuniform-temperature systems, including granular media, seismic activity, and microscopic engines.

Abstract

We consider the finite-timestep Brownian dynamics of a single particle confined in one dimension, with a nonuniform temperature profile. In such an open-boundary scenario, one cannot observe any net probability current in the nonequilibrium steady state (NESS). On the other hand, the nonequilibrium nature of this system is exhibited through the asymmetry in forward and backward transition probabilities, as is reported in this work through the stochastic simulation analysis and theoretical arguments. The irreversibility becomes prominent nearby the temperature interface. We propose that the observed irreversibility can be accounted for via a virtual-gyration scenario, while the collapse of virtual gyrations upon the one-dimensional coordinate leads to the absence of probability current.

Figures (4)

• Figure 1: Illustrations about the two scenarios. Arrows represent probability currents (width signifying magnitude). Points $C$, $B$, $A$ represent successive discrete positions from left to right, while point $A$ is at a higher temperature. (a) The chain-connecting scenario. Note that the net steady-state current between nodes has to be zero everywhere (detailed balance). (b) The loop scenario. In finite-timestep stochastic dynamics, thanks to the large diffusion parameter at $A$, point $A$ can reach $C$ directly over one single timestep $\Delta t$. This feature gives rise to the loop-like scenario and contributes to a "hidden" gyrating current at the steady state.
• Figure 2: Ratio of joint probabilities between forward and backward transitions. The counting region width $\delta$ is determined by the global root-mean-square (RMS) displacement of the Brownian particle. Blue dots: result where joint probabilities were extracted directly from simulation data; green dots: result evaluated from Eq. \ref{['eqn_ratio_predicted']}, via the NESS probability distribution $P(x)$ from simulation.
• Figure 3: Comparison of NESS probability distributions $P(x)$ for a Brownian particle in a harmonic potential with inhomogeneous temperature profile. Blue dots represent the simulation results; the orange solid line shows the theoretical prediction from Eq. \ref{['eqn_prob']} using the step-function temperature profile; the green dashed line represents the modified theoretical prediction using the effective temperature profile from Eq. \ref{['eqn_eff_temp']} (see inset).
• Figure 4: Probability current $j(x)$, evaluated via $j = P(x)v(x)$, where $v(x)$ is obtained through finite difference approximation of particle velocities. Note that for one-dimensional open-boundary systems, the ideal probability current has to be zero at all positions.