Generic Properties of Stochastic Entropy Production

We derive an Itô stochastic differential equation for entropy production in nonequilibrium Langevin processes. Introducing a random-time transformation, entropy production obeys a one-dimensional drift-diffusion equation, independent of the underlying physical model. This transformation allows us to identify generic properties of entropy production. It also leads to an exact uncertainty equality relating the Fano factor of entropy production and the Fano factor of the random time, which we also generalize to non-steady-state conditions.

Figure 1

Examples of nonequilibrium steady states. (a) Brownian particle driven by a constant nonconservative force in a periodic 1D sawtooth potential, $dX/dt=\mu [f-{\partial }_{x}U(X)]+\sqrt{2D}\xi$, with the potential $U(x)=(U_{0}x)/x^{*}$ for $x\in [0,x^{*}]$ and $U(x)=U_0(1-x)/(1-x^{*})$ for $x\in [x^{*},1]$. (b) 2D transport in a force field: $dX/dt=\mu f\cos (2\pi Y)+\sqrt{2D}{\xi }_x$ and $dY/dt=\sqrt{2D}{\xi }_y$. (c) Chiral active Brownian motion described by 3 degrees of freedom: position coordinates $dX/dt=\mu f\cos (\varphi )+\sqrt{2D}{\xi }_x$, $dY/dt=\mu f\sin (\varphi )+\sqrt{2D}{\xi }_y$, and orientation angle $d\varphi /dt={\mu }_{\varphi }\omega +\sqrt{2D_{\omega }}{\xi }_{\omega }$. In (b) and (c) $U=0$ and $f$ is an external nonconservative force.

Figure 2

Illustration of the decomposition of stochastic entropy production. In nonequilibrium steady states, the stochastic entropy production $S_{\mathrm{tot}}(t)$ (green) is given by the Boltzmann constant $k_B$ times the sum of the monotonically increasing entropic time $\tau (t)$ (orange), and the martingale process $M(t)$ (blue), see Eq. (11).

Figure 3

Generic properties of stochastic entropy production. Distributions of (a) entropy production at fixed $\tau =1$, (b) infimum of entropy production, (c) supremum of entropy production before the infimum, (d) number of crossings of entropy production, with $\mathrm{\Delta }=0.2k_B$. The symbols are obtained from numerical simulations of the three models sketched in Fig. 1 (blue squares, model A; red circles, model B; green diamonds, model C). The inset of (a) shows numerically estimated distributions of $S_{\mathrm{tot}}$ for the three models at fixed $t=1$ for comparison. The solid orange curves are the theoretical expressions (a) a Gaussian distribution with average $k_B\tau$ and variance $2k_B^2\tau$ (b) an exponential distribution with average $-k_B$ (c) Eq. (13); (d) Eq. (14). The dashed line in (c) is the theoretical distribution of minus the infimum for comparison. In all simulations, parameters are $\mathit{\mu }=\mathbb{I}$, $\mathit{D}=k_{B}T\mathbb{I}$, where $\mathbb{I}$ is the identity matrix, and $f=1$. In model A we chose $U_0=k_{B}T$ and $x^{*}=0.3$. In model C we chose $\omega =2$. Here, and in the following figures, each point represents an average over $10^6$ simulations.

Figure 4

Thermodynamic Fano factor equality. Long-time Fano factor of entropy production ${\sigma }_{S_{\mathrm{tot}}}^2/(k_B⟨S_{\mathrm{tot}}⟩)$ as a function of the external force $f$. The symbols are obtained from numerical simulations of the models shown in Fig. 1 (blue), Fig. 1 (red), and Fig. 1 (green). The solid lines are the prediction of Eq. (15) and have been calculated by means of Eq. (16) (see [34]). All the parameters of the numerical simulations except of the external force $f$ are the same as in Fig. 3.

Figure 5

Fano factor of stochastic entropy production out of steady state. The position of a Brownian particle is governed by the equation $dX/dt=-\mu {\kappa }_{f}X+\sqrt{2D}\xi$ with $\mu =1$ and $D=\mu k_{B}T$. The particle is initially at equilibrium with stiffness ${\kappa }_i=1$ (see inset). (a) Comparison between the exact value (orange line) of the long-time Fano factor of entropy production and the value obtained from numerical simulations (brown triangles) as a function of ${\kappa }_f$. The exact value is given by ${\sigma }_{S_{\mathrm{tot}}}^2/k_B⟨S_{\mathrm{tot}}⟩=({\kappa }_f/{\kappa }_i-1{)}^2/[({\kappa }_f/{\kappa }_i-1)-\log ({\kappa }_f/{\kappa }_i)]$; see Supplemental Material for details [34]. In simulations we measure $⟨\tau ⟩$, ${\sigma }_{\tau }^2$, and $\mathrm{\Omega }$ and use Eq. (17). The horizontal purple line is set to 2 for comparison. (b) Behavior of ${\sigma }_{\tau }^2/⟨\tau ⟩$ (solid line) and $2\mathrm{\Omega }/⟨\tau ⟩$ (dashed line) as a function of ${\kappa }_f$.