Local Fluctuation Theorem for Large Systems

The fluctuation theorem characterizes the distribution of the dissipation in nonequilibrium systems and proves that the average dissipation will be positive. For a large system with no external source of fluctuation, fluctuations in properties will become unobservable and details of the fluctuation theorem are unable to be explored. In this Letter, we consider such a situation and show how a fluctuation theorem can be obtained for a small open subsystem within the large system. We find that a correction term has to be added to the large system fluctuation theorem due to correlation of the subsystem with the surroundings. Its analytic expression can be derived provided some general assumptions are fulfilled, and its relevance is checked using numerical simulations.

Figure 1

A schematic diagram of the system studied. Wall particles are thermostated and represented in black ($c_i=0$), whereas fluid particles can be light gray ($c_i=1$, blue online) or dark gray ($c_i=-1$, red online).

Figure 2

Probability distribution functions for the local dissipation function in a subvolume 14.4 times smaller than the total volume and its Gaussian fit. Its mean value is $⟨{\Omega }_{\ell ,t}⟩\simeq 2.5$ and the asymmetry function, ${\rho }_{\ell }(A)=\ln [p({\Omega }_{\ell ,t}=A)/p({\Omega }_{\ell ,t}=-A)]$ is found to be a straight line of slope $3.147\pm 0.001$.

Figure 3

Evolution of the correlation length ${\ell }_0$ determined out of equilibrium using a fit of $C(x)=e^{-x/{\ell }_0}$ to the simulated data at various times. The dashed line is a fit to a function of $\sqrt{t}$.

Figure 4

The decay of spatial correlations in the integrated flux calculated with (9) (crosses) and a fit to the exponential model $e^{-x/{\ell }_0}$, where ${\ell }_0$ is the correlation length (solid line).

Figure 5

Slope of the asymmetry function [left hand side of Eq. (11)] versus the length of the subsystem. The dashed line is the theoretical result [right hand side of Eq. (11)], with the coefficient ${\ell }_0$ estimated using the data from Fig. 3.