Frenetic Bounds on the Entropy Production

We give a systematic derivation of positive lower bounds for the expected entropy production (EP) rate in classical statistical mechanical systems obeying a dynamical large deviation principle. The logic is the same for the return to thermodynamic equilibrium as it is for steady nonequilibria working under the condition of local detailed balance. We recover there recently studied “uncertainty” relations for the EP, appearing in studies about the effectiveness of mesoscopic machines. In general our refinement of the positivity of the expected EP rate is obtained in terms of a positive and even function of the expected current(s) which measures the dynamical activity in the system, a time-symmetric estimate of the changes in the system’s configuration. Also underdamped diffusions can be included in the analysis.

Figure 1

Entropy production is bounded from below by two types of activity. For moving into a larger phase volume from condition $\rho$, thereby producing entropy, the dynamics must be sufficiently active inside $\rho$, represented by $L(\rho ,0)$, and must readily find an escape route, represented by $\psi (\rho ,j)$.