Entropy production and time asymmetry in the presence of strong interactions

It is known that the equilibrium properties of open classical systems that are strongly coupled to a heat bath are described by a set of thermodynamic potentials related to the system's Hamiltonian of mean force. By adapting this framework to a more general class of nonequilibrium states, we show that the equilibrium properties of the bath can be well defined, even when the system is arbitrarily far from equilibrium and correlated with the bath. These states, which retain a notion of temperature, take the form of conditional equilibrium distributions. For out-of-equilibrium processes we show that the average entropy production quantifies the extent to which the system and bath state is driven away from the conditional equilibrium distribution. In addition, we show that the stochastic entropy production satisfies a generalized Crooks relation and can be used to quantify time asymmetry of correlated nonequilibrium processes. These results naturally extend the familiar properties of entropy production in weakly coupled systems to the strong coupling regime. Experimental measurements of the entropy production at strong coupling could be pursued using optomechanics or trapped-ion systems, which allow strong coupling to be engineered.

Figure 1

Schematic representation of the equality (15). The solid line represents the actual process given by the evolving distribution $\rho \left(t\right)=\rho (z_t;t)$ whilst the dashed line represents a hypothetical quasistatic process in which the system and bath distribution stays in the conditional equilibrium state $\sigma \left(t\right)=\sigma (z_t;t)\in {\mathcal{D}}_{\beta }$. The non-negative entropy production then quantifies the extent to which the system and bath are driven away from $\sigma \left(t\right)$, represented here as the distance of the blue line.