[Paper Review] Maximal-entropy-production-rate nonlinear quantum dynamics compatible with second law, reciprocity, fluctuation-dissipation, and time-energy uncertainty relations

Gian-Paolo Beretta|arXiv (Cornell University)|Dec 7, 2001

Advanced Thermodynamics and Statistical Mechanics3 references3 citations

TL;DR

This paper proposes a nonlinear quantum dynamics framework based on maximal entropy production rate, compatible with the second law, reciprocity, fluctuation-dissipation, and time-energy uncertainty. It extends the steepest-entropy-ascent formalism via a closure ansatz that fixes the rate of entropy production using the time-energy uncertainty principle, yielding a consistent, separable, and well-behaved dynamics for isolated quantum systems, including composite systems.

ABSTRACT

In view of the recent quest for well-behaved nonlinear extensions of the traditional Schroedinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, in this paper, together with a review of the general features of the nonlinear quantum (thermo)dynamics I proposed in a series of papers [see references in G.P. Beretta, Found.Phys. 17, 365 (1987)], I show its exact equivalence with the maximal-entropy-production variational-principle formulation recently derived in S. Gheorghiu-Svirschevski, Phys.Rev. A 63, 022105 (2001). In addition, based on the formalism of general interest I developed for the analysis of composite systems, I show how the variational derivation can be extended to the case of a composite system to obtain the general form of my equation of motion, that turns out to be consistent with the demanding requirements of strong separability. Moreover, I propose a new intriguing fundamental ansatz: that the time evolution along the direction of steepest entropy ascent unfolds at the fastest rate compatible with the time-energy Heisenberg uncertainty relation. This ansatz provides a possible well-behaved general closure of the nonlinear dynamics, compatible with the nontrivial requirements of strong separability, and with no need of new physical constants. In any case, the time-energy uncertainty relation provides lower bounds to the internal-relaxation-time functionals and, therefore, upper bounds to the rate of entropy production.

Motivation & Objective

• To develop a well-behaved nonlinear extension of Schrödinger–von Neumann dynamics that explains loss of quantum coherence at the microscopic level.
• To ensure compatibility with the second law of thermodynamics, Onsager's reciprocity, and the fluctuation-dissipation theorem in all nonequilibrium states.
• To derive a closure for the entropy production rate using the time–energy uncertainty principle, ensuring physical consistency and separability.
• To generalize the steepest-entropy-ascent dynamics to composite systems while preserving strong separability and conservation laws.

Proposed method

• Proposes a nonlinear equation of motion for the density operator ρ, combining unitary evolution with a dissipative term based on steepest-entropy-ascent in the space of density operators.
• Uses the steepest-entropy-ascent ansatz to define the direction of the dissipative evolution, ensuring maximal entropy production along the steepest path in the state space.
• Introduces a variational principle formulation equivalent to the geometric steepest-ascent construction, providing dual mathematical formulations of the dynamics.
• Derives a lower bound on internal relaxation time functionals τ(ρ) from the time–energy uncertainty relation, constraining the rate of entropy production.
• Proposes a maximal-entropy-production-rate closure ansatz: the system evolves along the steepest entropy ascent direction at the highest rate allowed by the time–energy uncertainty principle.
• Extends the dynamics to composite systems using a tensor product structure, ensuring strong separability and conservation of local observables when the system is uncorrelated.

Experimental results

Research questions

• RQ1 Can a nonlinear quantum dynamics be constructed that consistently explains loss of quantum coherence while preserving fundamental thermodynamic principles?
• RQ2 How can the rate of entropy production in nonlinear quantum dynamics be consistently closed without introducing new physical constants?
• RQ3 Is it possible to unify Onsager's reciprocity and Callen's fluctuation–dissipation relations under a single nonlinear dynamics valid for all nonequilibrium states?
• RQ4 How can the steepest-entropy-ascent dynamics be extended to composite systems while maintaining strong separability and conservation of local energy and other observables?
• RQ5 Can the time–energy uncertainty principle be used to derive a fundamental upper bound on the entropy production rate, leading to a physically consistent closure of the dynamics?

Key findings

• The proposed dynamics satisfies the second law of thermodynamics by construction, as the entropy production rate is non-negative and maximal along the steepest-ascent direction.
• The dynamics provides a microscopic derivation and extension of Onsager's reciprocity and Callen's fluctuation–dissipation relations to all nonequilibrium states, both near and far from equilibrium.
• The time–energy uncertainty relation imposes a lower bound on the internal relaxation time functionals τ(ρ), which in turn implies an upper bound on the entropy production rate.
• The maximal-entropy-production-rate ansatz closes the dynamics by selecting the highest possible rate of entropy production compatible with the time–energy uncertainty principle, ensuring physical consistency.
• The dynamics is strongly separable: when the system is composed of uncorrelated subsystems, the evolution preserves the local conservation of energy and other observables, even for composite systems.
• The geometric steepest-entropy-ascent formulation is mathematically equivalent to the variational principle formulation proposed by Gheorghiu-Svirschevski, providing dual and consistent derivations of the same dynamics.