Effective temperature in nonequilibrium steady states of Langevin systems with a tilted periodic potential

We theoretically study Langevin systems with a tilted periodic potential. It is known that the ratio $\Theta$ of the diffusion constant $D$ to the differential mobility ${\mu }_{\mathrm{d}}$ is not equal to the temperature of the environment (multiplied by the Boltzmann constant), except in the linear response regime, where the fluctuation dissipation theorem holds. In order to elucidate the physical meaning of $\Theta$ far from equilibrium, we analyze a modulated system with a slowly varying potential. We derive a large scale description of the probability density for the modulated system by use of a perturbation method. The expressions we obtain show that $\Theta$ plays the role of the temperature in the large scale description of the system and that $\Theta$ can be determined directly in experiments, without measurements of the diffusion constant and the differential mobility. Hence the relation $D={\mu }_{\mathrm{d}}\Theta$ among the independent measurable quantities $D$, ${\mu }_{\mathrm{d}}$, and $\Theta$ can be interpreted as an extension of the Einstein relation.

Figure 1

$\Theta ∕T$ as a function of $f\ell ∕T$ in the case that $U\left(x\right)=U_0\sin 2\pi x∕\ell$ with $U_0∕T=3.0$. $\Theta ∕T=1$ at $f=0$ and $\Theta ∕T>1$ for $f>0$. The inset displays $D\gamma ∕T$ as a function of $f\ell ∕T$ calculated using Eq. (7).

Figure 2

The left axis represents $\Gamma ∕\gamma$, which is displayed as a function of $f\ell ∕T$ (solid curve), and the right axis represents $F\ell ∕T$, also displayed as a function of $f\ell ∕T$ (dash-dotted curve), in the case that $U\left(x\right)=U_0\sin 2\pi x∕\ell$, with $U_0∕T=3.0$. The dotted line corresponds to $F=f$.

Figure 3

$⟨V⟩∕T$ versus $\Delta ∕T$ obtained in the numerical experiment. The dotted line represents a fitting curve of the form (27) in which $\Theta ∕T=1.67\left(4\right)$.