Positive Nonlinear Dynamical Group Uniting Quantum Mechanics and Thermodynamics

We discuss and motivate the form of the generator of a nonlinear quantum dynamical group “designed” so as to accomplish a unification of quantum mechanics (QM) and thermodynamics. We call this nonrelativistic theory Quantum Thermodynamics (QT). Its conceptual foundations differ from those of (von Neumann) quantum statistical mechanics (QSM) and (Jaynes) quantum information theory (QIT), but for thermodynamic equilibrium (TE) states it reduces to the same mathematics, and for zero entropy states it reduces to standard unitary QM. By restricting the discussion to a strictly isolated system (non-interacting, disentangled and uncorrelated) we show how the theory departs from the conventional QSM/QIT rationalization of the second law of thermodynamics. The nonlinear dynamical group of QT is construed so that the second law emerges as a theorem of existence and uniqueness of a stable equilibrium state for each set of mean values of the energy and the number of constituents. To achieve this QT assumes −kBTrρ ln ρ for the physical entropy and is designed to implement two fundamental ansatzs. The first is that in addition to the standard QM states described by idempotent density operators (zero entropy), a strictly isolated system admits also states that must be described by non-idempotent density operators (nonzero entropy). The second is that for such additional states the law of causal evolution is determined by the simultaneous action of a Schrödinger-von Neumanntype Hamiltonian generator and a nonlinear dissipative generator which conserves the mean values of the energy and the number of constituents, and (in forward time) drives the density operator in the ’direction’ of steepest entropy ascent (maximal entropy increase). The resulting positive nonlinear dynamical group (not just a semi-group) is well-defined for all nonequilibrium states, no matter how far from TE. Existence and uniqueness of solutions of the (Cauchy) initial state problem for all density operators, implies that the equation of motion can be solved not only in forward time, to describe relaxation towards TE, but also backwards in time, to reconstruct the ’ancestral’ or primordial lowest entropy state or limit cycle from which the system originates.