Variational formula for experimental determination of high-order correlations of current fluctuations in driven systems

For Brownian motion of a single-particle subject to a tilted periodic potential on a ring, we propose a formula for experimentally determining the cumulant generating function of time-averaged current without measurements of current fluctuations. We first derive this formula phenomenologically on the basis of two key relations: a fluctuation relation associated with Onsager’s principle of the least-energy dissipation in a sufficiently local region and an additivity relation by which spatially inhomogeneous fluctuations can be properly considered. We then derive the formula without any phenomenological assumptions. We also demonstrate its practical advantage by numerical experiments.

Figure 1

$G(h)$ for the Langevin equation (8) with $U(x)=U_0\cos (2\pi x/L)$. Quantities are converted to dimensionless forms by setting $\gamma =T=L=1$. We fix $f=1$. (Left) Numerical calculation of $K=2\mathit{TG}(h)$ in (14) for $U_0=0$ (red solid line), $3$ (green dashed line), and $5$ (blue dotted line). (Right) Experimental measurement for the system with $U_0=3$. We assume to know $T=\gamma =1$ and determined $G(h)$ experimentally from trajectories $(x(t)){}_{t=0}^{\tau }$ according to the method described in the text. $N=5$ and $\tau =250$ (triangle symbols), $1000$ (square symbols), and $4000$ (circle symbols). The error bars are within the symbols. The dashed line is the same as that on the left-hand side.