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

Abstract

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 straight-forwardly to thermal disturbance, a mechanical formulation of it 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 (dS/dt) irr = κE 2 T /T . Mathematical aspects on the nonequilibrium density-matrix will also be discussed. In Paper I (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. The present 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.

Theory of entropy production in quantum many-body systems

“…(67)), we derived the leading contribution to the electronic entropy production which is quadratic in the electric field. This is in contrast to past approaches 43,96 for the calculation of the Joule heating, which requires going to the second order in the electric field contribution to the density matrixρ 2;t for the calculation of the rate of change of the energy of the electrons. A field-theoretic approach 98,99 for the calculation of higher order terms in the entropy production, beyond the Born-Markov approximation will be treated elsewhere.…”

“…This exponential representation of the density matrix is not new; it is a generalized form 43,70 of the nonequilibrium statistical operator introduced by Zubarev, 32,33 and obtained for the case of steady states by Hershfield 71 . As discussed in the previous section, our new thermodynamic entropy operator,Ŝ t = −D lnρ t , is obtained fromŜ t by projecting to the space of operators diagonal in the basis {|α } of eigenstates ofĤ.…”

Order formation processes of complex systems including different parity order parameters

“…Then the total entropy S(t) ≡ S (m) (t)+S (η) (t) increases for certain parameters a, b and c in Eqs. (13) and (14). This is a typical behavior of our system.…”

“…This difference is based on the types of noises. Contrary to the traditional scaling theory [1], the present Langevin equations (13) and (14) include a "multiplicative" noise. Additionally, we have assumed conditions τ m ≫ τ η and m 2 (t ∼ t o ) = constant.…”

“…(56), the spin-glass model is different from the models described by Eqs. (13) and (14) in that the squared term of the order parameter q, namely −2z 2 K 4 q 2 /3, exists in the former model. This is because the spin-glass model belongs to a universality class different from that of the four-body interaction model, as shown in the previous studies [9].…”