Fluctuation-response relation of many Brownian particles under nonequilibrium conditions

We study many interacting Brownian particles under a tilted periodic potential. We numerically measure the linear response coefficient of the density field by applying a slowly varying potential transversal to the tilted direction. In equilibrium cases, the linear response coefficient is related to the intensity of density fluctuations in a universal manner, which is called a fluctuation-response relation. We then report numerical evidence that this relation holds even in nonequilibrium cases. This result suggests that Einstein’s formula on density fluctuations can be extended to driven diffusive systems when the slowly varying potential is applied in a direction transversal to the driving force.

Figure 1

Current $J$ in the $x_1$ direction as a function of $f$. The average values of $J$ for 10 samples are plotted for several values of $f$. The statistical error bars are smaller than the symbols. Inset: Close-up of the linear response regime. Since the two symbols with the same value of $U_0$ are close to each other, they are displayed in almost the same position.

Figure 2

Average values of ${⟨\mathrm{Re}\left[\delta {\widehat{\rho }}_{(0,1)}\right]⟩}_{\epsilon }$ over 100 samples are plotted as a function of $\epsilon$. $U_0=15$ and $K=200$. The statistical error bars are smaller than the symbols.

Figure 3

$C$ vs $TR$ for several values of $U_0,K$, and $f$. The conditions corresponding to the symbols are explained in the figure.

Figure 4

$(C-TR)∕TR$ as a function of the time step $\Delta t$. $U_0=15$ and $K=200$.

Figure 5

Schematic figure of a driven diffusive system under periodic boundary conditions.