First-Principle Derivation of Entropy Production in Transport Phenomena

The linear response framework was established by Kubo a half century ago, but no clear explanation of irreversibility namely entropy production has been given in this scheme. This has been now solved. The serious puzzle up to now is the following. Even using the linear response density matrix ρlr ρ0 + ρ1(t), it has been difficult to derive the entropy production. Surprisingly, the correct entropy production is given by the second-order term ρ2(t) as . It is shown to agree with the ordinary expression J·E/T σE2/T in the case of electric conduction for a static electric field E, where σ denotes the electric conductivity expressed by the famous canonical current-current time correlation functions in equilibrium. The present article gives a review of the derivation of entropy production (M.S., Physica A 390(2011)1904-1916) based on the first-principle of using the projected density matrix ρ2(t) or more generally ρeven(t), while the previous standard argument is due to the thermodynamic energy balance. This new derivation clarifies conceptually the physics of irreversibility in transport phenomena. In general, the transport phenomena are described by the odd part ρodd(t) of the density matrix and the entropy production (namely irreversibility) is described by the even part ρeven(t). These are related to each other through the coupled equations as , where "q" denotes "even" (symmetric) or "odd" (antisymmetric), Pq is the projection operator to the "q" part of ρ(t), and 1(t) denotes the partial Hamiltonian due to the external force such as in electric conduction. The concept of a stationary temperature Tst in steady states with current (say electric current) is also proposed by using the projected and symmetry-separated von Newmann equation introduced by the present author. The entropy production of the relevant steady state depends on this stationary temperature. A mechanical formulation of thermal conduction is given by introducing a thermal field ET and its conjugate "heat" operator for a local internal energy hj of the thermal particle j.