Asymptotics of work distributions in nonequilibrium systems

The asymptotic behavior of the work distribution in driven nonequilibrium systems is determined using the method of optimal fluctuations. For systems described by Langevin dynamics the corresponding Euler-Lagrange equation together with the appropriate boundary conditions and an equation for the leading pre-exponential factor are derived. The method is applied to three representative examples and the results are used to improve the accuracy of free-energy estimates based on the application of the Jarzynski equation.

Figure 1

Histogram of work values as obtained from ${10}^7$ simulations of the Langevin Eq. (2) with $\beta =2$, $t_1=100$, and $V\left(x,t\right)$ given by Eqs. (26, 29). The full line is the asymptotic form of the work distribution as derived in Eq. (37). The inset shows two estimates for the free-energy difference $\Delta F$ together with their standard deviation as function of the sample size $n$. Circles are the standard estimate Eq. (38), squares give the improved one Eq. (39) incorporating the asymptotic behavior of $P\left(W\right)$ with $W^{\ast }=-4/3$. The dashed line is the exact result.

Figure 2

Comparison of the simulation results (circles) for $P\left(W\right)$ shown in Fig. 1 and the asymptotic behavior Eq. (37) (full line) on a logarithmic scale.

Figure 3

Histogram of 16200 work values obtained experimentally in [7] together with the asymptotic form (full line) derived from Eqs. (41, 42, 43). The inset shows the standard (circles) and the improved (squares) estimate for the free-energy difference. The exact result is $\Delta F=0$ (dashed line).