Thermodynamic formula for the cumulant generating function of time-averaged current

The cumulant generating function of time-averaged current is studied from an operational viewpoint. Specifically, for interacting Brownian particles under nonequilibrium conditions, we show that the first derivative of the cumulant generating function is equal to the expectation value of the current in a modified system with an extra force added, where the modified system is characterized by a variational principle. The formula reminds us of Einstein's fluctuation theory in equilibrium statistical mechanics. Furthermore, since the formula leads to the fluctuation-dissipation relation when the linear response regime is focused on, it is regarded as an extension of the linear response theory to that valid beyond the linear response regime. The formula is also related to previously known theories such as the Donsker-Varadhan theory, the additivity principle, and the least dissipation principle, but it is not derived from them. Examples of its application are presented for a driven Brownian particle on a ring subject to a periodic potential.

Figure 1

Left: $n$ resistances ($R_1,R_2,...,R_n$) are connected in series. We impose an electric potential $V$ on the circuit. Right: By using $n$ batteries, we impose an electric potential $V_i$ on each resistance $R_i$.

Figure 2

Numerical experiment for measurement of $G^F\left(h\right)$ (up) and $w_h^{F,\mathrm{opt}}\left(x\right)$ with $h=3$ (down). Quantities are converted to dimensionless forms by setting $\gamma =T=L=1$. We fix $f=1$ and $U_0=3$. We assumed to know $T=\gamma =1$ and experimentally determined $G^F\left(h\right)$ and $w_h^{F,\mathrm{opt}}\left(x\right)$ from trajectories ${\left[x\left(t\right)\right]}_{t=0}^{\tau }$ following the method described in the text. We set $m=10$, $C=10$, and $\tau =400000$. By taking 10 samples, we estimated $G^F\left(h\right)$ and $w_h^{F,\mathrm{opt}}\left(x\right)$. The obtained results are displayed with green dashed lines. Error bars are within the lines. The red lines were obtained from the evaluation of the largest eigenvalue of ${\mathcal{L}}_h^{\left(x\right)†}$.