Dissipation, lag, and drift in driven fluctuating systems

This work deals with thermostated fluctuating systems subjected to driven transformations of the internal energetics. The main focus is on generally multidimensional systems with continuous configurational degrees of freedom over which overdamped Markovian fluctuations take place (diffusive regime of the motion). Mutual bounds are established between the average energy dissipation, the deviation between nonequilibrium probability density and underlying equilibrium distribution due to the system’s lag, and the statistical properties of the components of the directed flow induced by the transformation itself. The directed flow is here expressed in terms of time-dependent “drift velocity” associated with the probability current in a advection-like formulation of the nonstationary Fokker-Planck equation. Consideration of the drift makes that the bounds achieved here extend the inequality derived by Vaikuntanathan and Jarzynski [Europhys. Lett. 87, 60005 (2009)] involving only dissipation and lag. The key relations are then specified for the so-called stochastic pumps, i.e., systems that reach a periodic steady state in response of cyclic transformations and that are prototypes of nonautonomous dissipative converters of input energy into directed motion; a one-dimensional case model is adopted to illustrate the main features. Complementary results concerning bounds between the evolution rates of dissipation and lag, valid for both overdamped and underdamped dynamics, are also presented.