Einstein Relation Generalized to Nonequilibrium

The Einstein relation connecting the diffusion constant and the mobility is violated beyond the linear response regime. For a colloidal particle driven along a periodic potential imposed by laser traps, we test the recent theoretical generalization of the Einstein relation to the nonequilibrium regime which involves an integral over measurable velocity correlation functions.

Figure 1

(a) Experimental setup. (b) Typical trajectory of the angular particle position for a mean particle revolution time $\simeq 5.8\mathrm{s}$.

Figure 2

(a) Reconstructed potential $V(x)$. (b) Tilted potential. The colloidal particle is subjected to a constant driving force $f\simeq 0.06\mathrm{pN}$ and the periodic potential $V(x)$.

Figure 3

(a) Experimentally measured violation function $I(\tau )$ (solid line). (b) Comparison of the velocities involved in the violation function $I(\tau )$. For an ideal cosine potential, we sketch the probability distribution $p_s(x)$ (solid gray line) and the local mean velocity $v_s(x)$, together with the drift velocity and their mean $⟨\dot{x}⟩$ versus the angular particle position. The drift velocity is the deterministic part $F/\gamma$ of the actual velocity $\dot{x}$. The sign change in $I(\tau )$ at (2), (3), and (4) can be understood as follows. In a steady state, a single particle trajectory will start with highest probability in the shaded region, and, for an illustration, we choose its maximum as starting point (1) determining the value $v_s[x(t)]$ in Eq. (6). Neglecting thermal fluctuations, the particle would follow the dashed line, and during a small time step $\tau$ the product $F[x(t+\tau )]v_s[x(t)]$ is positive. If the particle passes (2), the product would become negative. The sign changes again if the particle passes (3) and then (4) and so on due to the periodic nature of the potential. Thermal noise and averaging over all trajectories does not change this behavior responsible for the oscillations of $I(\tau )$.

Figure 4

Experimental test of Eq. (5) for different driving forces $f$. The open bars show the measured diffusion coefficients $D$. The stacked bars are mobility (gray bar) and integrated violation (hatched bar), respectively.