Irreversibility and entropy production in transport phenomena, II: Statistical–mechanical theory on steady states including thermal disturbance and energy supply

Some general aspects of nonlinear transport phenomena are discussed on the basis of two kinds of formulations obtained by extending Kubo’s perturbational scheme of the density matrix and Zubarev’s non-equilibrium statistical operator formulation. Both formulations are extended up to infinite order of an external force in compact forms and their relationship is clarified through a direct transformation. In order to make it possible to apply these formulations straightforwardly to thermal disturbance, its mechanical formulation is given (in a more convenient form than Luttinger’s formulation) by introducing the concept of a thermal field E T which corresponds to the temperature gradient and by defining its conjugate heat operator A H = ∑ j h j r j for a local internal energy h j of the thermal particle j . This yields a transparent derivation of the thermal conductivity κ of the Kubo form and the entropy production ( d S / d t ) irr = κ E T 2 / T . Mathematical aspects of the non-equilibrium density-matrix will also be discussed. In Paper I (M. Suzuki, Physica A 390 (2011)1904), the symmetry-separated von Neumann equation with relaxation terms extracting generated heat outside the system was introduced to describe the steady state of the system. In this formulation of the steady state, the internal energy 〈 H 0 〉 t is time-independent but the field energy 〈 H 1 〉 t ( = − 〈 A 〉 t ⋅ F ) decreases as time t increases. To overcome this problem, such a statistical–mechanical formulation is proposed here as includes energy supply to the system from outside by extending the symmetry-separated von Neumann equation given in Paper I. This yields a general theory based on the density-matrix formulation on a steady state with energy supply inside and heat extraction outside and consequently with both 〈 H 0 〉 t and 〈 H 1 〉 t constant. Furthermore, this steady state gives a positive entropy production. esent general formulation of the current yields a compact expression of the time derivative of entropy production, which yields the plausible justification of the principle of minimum entropy production in the steady state even for nonlinear responses.