Stochastic entropy production arising from nonstationary thermal transport

We compute statistical properties of the stochastic entropy production associated with the nonstationary transport of heat through a system coupled to a time dependent nonisothermal heat bath. We study the one-dimensional stochastic evolution of a bound particle in such an environment by solving the appropriate Langevin equation numerically, and by using an approximate analytic solution to the Kramers equation to determine the behavior of an ensemble of systems. We express the total stochastic entropy production in terms of a relaxational or nonadiabatic part together with two components of housekeeping entropy production and determine the distributions for each, demonstrating the importance of all three contributions for this system. We compare the results with an approximate analytic model of the mean behavior and we further demonstrate that the total entropy production and the relaxational component approximately satisfy detailed fluctuation relations for certain time intervals. Finally, we comment on the resemblance between the procedure for solving the Kramers equation and a constrained extremization, with respect to the probability density function, of the spatial density of the mean rate of production of stochastic entropy.

Figure 1

Spatial trajectory of a particle according to stochastic dynamics in the presence of an environment where the temperature varies in time and space, illustrated by the text and background colors. The evolution of the stochastic entropy production $\mathrm{\Delta }{s}_{\mathrm{tot}}$ associated with the Brownian trajectory is sketched in the inset.

Figure 2

A comparison between the mean total stochastic entropy production obtained from the integral of the approximate Eq. (27) and the total stochastic entropy production averaged over $2\times {10}^7$ numerically generated particle trajectories, each with $40000$ time steps, shown with error bars. The time interval represents two cycles of temperature variation, preceded in the simulations by one cycle to establish an evolving pdf similar to Eq. (21).

Figure 3

Comparison of the average of $\mathrm{\Delta }{s}_1$ predicted analytically and calculated from the simulation of $2\times {10}^7$ particle trajectories over a time interval of two temperature cycles preceded by one cycle to establish the periodic stationary state.

Figure 4

As Fig. 3 but for the average of $\mathrm{\Delta }{s}_2$.

Figure 5

Numerical verification of an approximate detailed fluctuation relation for the pdf of total stochastic entropy production $\mathrm{\Delta }{s}_{\mathrm{tot}}$ in the time interval $7\pi /2\omega \le t\le 11\pi /2\omega$ (shown as a bar in the inset), driven by the change in temperature profile according to the evolving ${\kappa }_T\left(t\right)$. Values of $\mathrm{\Delta }{s}_{\mathrm{tot}}$ between $\pm 0.2$ from the simulation of $2\times {10}^7$ particle trajectories are collected into 400 bins, and the straight line represents the outcome expected in the absence of sampling errors.

Figure 6

As Fig. 5, but for $\mathrm{\Delta }{s}_1$.

Figure 7

Demonstration that a detailed fluctuation relation does not appear to hold for $\mathrm{\Delta }{s}_2$ in the interval $7\pi /2\omega \le t\le 11\pi /2\omega$.

Figure 8

Distributions of $\mathrm{\Delta }{s}_{\mathrm{tot}}, \mathrm{\Delta }{s}_1, \mathrm{\Delta }{s}_2$, and $\mathrm{\Delta }{s}_3$ generated in two cycles of temperature variation $2\pi /\omega \le t\le 6\pi /\omega$.