The linear response framework was established a half-century ago, but no persuasive direct derivation of entropy production has been given in this scheme. This long-term puzzle has now been solved in the present paper. The irreversible part of the entropy production in the present theory is given by (dS/dt) irr = (dU/dt)/T with the internal energy U(t) of the relevant system. Here, U(t) = β¨H 0 β© t = Tr H 0 Ο(t) for the Hamiltonian H 0 in the absence of an external force and for the density matrix Ο(t). As is well known, we have (dS/dt) irr = 0 if we use the linear-order density matrix Ο lr (t) = Ο 0 + Ο 1 (t).
Surprisingly, the correct entropy production is given by the second-order symmetric term
. This is shown to agree with the ordinary expression J β’ E/T = Ο E 2 /T in the case of electric conduction for a static electric field E, using the relations Tr
Here H 1 (t) denotes the partial Hamiltonian due to the external force such as H 1 (t) = -e β j r i β’E β‘ -Aβ’E in electric conduction. Thus, the linear response scheme is not closed within the first order of an external force, in order to manifest the irreversibility of transport phenomena. New schemes of steady states are also presented by introducing relaxation-type (symmetry-separated) von Neumann equations. The concept of stationary temperature T st is introduced, which is a function of the relaxation time Ο r characterizing the rate of extracting heat outside from the system. The entropy production in this steady state depends on the relaxation time. A dynamical-derivative representation method to reveal the irreversibility of steady states is also proposed. The present derivation of entropy production is directly based on the first principles of using the projected density matrix Ο 2 (t) or more generally symmetric density matrix Ο sym (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, using the symmetry of non-equilibrium states, and this manifests the duality of current and entropy production.