Equations of motion in nonequilibrium statistical mechanics for nonextensive systems

Recent investigations of nanoscale quantum device systems and small clusters of atoms and molecules have shown new features of both nonequilibrium dynamics and nonextensivity. The challenge is to understand theoretically the fast dynamical processes on time scales of femtoseconds in these systems. An overview of the approaches to time-dependent nonequilibrium statistical mechanics, including a critical review of the entropic methods due to Jaynes, Robertson, and Zubarev for the extensive nonequilibrium systems, reveals difficulties in setting up the associated dynamics. We therefore propose the use of the Lindblad equation for the density matrix in place of the usual unitary Liouville–von Neumann equation of motion because it meets all the required criteria (positive, trace-class, and includes the possibility of mixed to pure state evolution) for describing dissipative dynamics and the Tsallis prescription for handling the nonextensivity. We will explore briefly (i) the notion of decoherence, (ii) near equilibrium linear response, (iii) evolution of entropy, and (iv) action principle, based on these considerations. We also exhibit a time-dependent “mixed state” marker for describing the nonequilibrium state of a dissipative quantum oscillator, which is a prototype of an electronic device. This investigation, it is hoped, gives glimpses of insight into understanding the short time-scale dynamics of systems between the times determined by the uncertainty principle and the decoherence due to the heat bath.