Entropy Production and Time Asymmetry in Nonequilibrium Fluctuations

The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems: namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann’s constant.

Figure 1

Time series of typical paths $z(t)$ for the Brownian particle in the optical trap moving at the velocity $u$ for the forward process (upper curve) and $-u$ for the reversed process (lower curve) with $u=4.24\times {10}^{-6}\mathrm{m}/\mathrm{s}$.

Figure 2

(a) Entropy production of the Brownian particle versus the driving speed $u$. The solid line is given by Eq. (5). (b) Entropy production of the $RC$ electric circuit versus the injected current $I$. The solid line is the Joule law $d_{i}S/dt=R{I}^2/T$. The dots are the results of Eq. (9).

Figure 3

Measure of the heat dissipated by the Brownian particle along the forward and reversed paths of Fig. 1. The trap velocities are $\pm u$, with $u=4.24\times {10}^{-6}\mathrm{m}/\mathrm{s}$. We are searching for recurrences between the two processes. (a) Inset: A randomly selected trajectory in the time series. The probabilities of the corresponding forward (solid circles) and the backward (open circles) paths for $ϵ=8.4\mathrm{nm}$. These probabilities present an exponential decrease modulated by the fluctuations. (b) The dissipated heat given by the logarithm of the ratio of the forward and backward probabilities according to Eq. (6) for different values of $ϵ=k\times 0.558\mathrm{nm}$, with $k=11,\dots ,20$ in the range 6.1–11.2 nm. They are compared with the value (squares) directly calculated from Eq. (4). For small values of $ϵ$, the agreement is quite good for short time and within experimental errors for larger time.