Mechanics and Thermodynamics can be fundamentally united by density operators with an ontic status obeying a locally maximum entropy production dynamics. But at what price?

Abstract

Perhaps the earliest explicit ansatz of a truly ontic status for the density operator has been proposed in [G.N. Hatsopoulos and E.P. Gyftopoulos, Found. Phys., Vol.6, 15, 127, 439, 561 (1976)]. Their self-consistent, unified quantum theory of Mechanics and Thermodynamics hinges on: (1) modifyng the ‘state postulate’ so that the full set of ontic individual states of a (strictly isolated and uncorrelated) quantum system is one-to-one with the full set of density operators (pure and mixed); and (2) complementing the remaining usual postulates of quantum theory with an ‘additional postulate’ which effectively seeks to incorporate the Second Law into the fundamental level of description. In contrast with the epistemic framework, where the linearity of the dynamical law is a requirement, the assumed ontic status of the density operator emancipates its dynamical law from the restrictive requirement of linearity. Indeed, when the ‘additional postulate’ is replaced by the dynamical ansatz of a (locally) steepest entropy ascent, nonlinear evolution equation for the density operator proposed in [G.P. Beretta, Sc.D. thesis, M.I.T., 1981, e-print quant-ph/0509116; and follow-up papers], the (Hatsopoulos-Keenan statement of the) Second Law emerges as a general theorem of the dynamics (about the Lyapunov stability of the equilibrium states). As a result, the ontic status is acquired not only by the density operator, but also by the entropy (which emerges as a microscopic property of matter, at the same level as energy), and by irreversibility (which emerges as a microscopic dynamical effect). This “adventurous scheme ... may end arguments about the arrow of time -- but only if it works” [J. Maddox, Nature, Vol.316, 11 (1985)]. Indeed, the scheme resolves both the Loschmidt paradox and the Schroedinger-Park paradox about the concept of ‘individual quantum state’. However, nonlinearity imposes a high price: the maximum entropy production (MEP) dynamical law does not have a universal structure like that of the Liouville-von Neumann equation obeyed by the density operator within the epistemic (statistical mechanics) view. Instead, much in the same way as the implications of the Second Law depend on the assumed model of a given physical reality, the MEP dynamical law for a composite system is model dependent: its structure depends on which constituent particles or subsystems are assumed as elementary and separable, i.e., incapable of no-signaling violations. See www.quantumthermodynamics.org for references.