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arXiv:0910.1948

Towards a new approach of quantum dissipation in simple chemical systems

J. P. Badiali

Published Oct 12, 2009• 0 citations
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Abstract

We suggest a new approach for describing quantum dissipation in a small systems for which the system-plus-reservoir approach is not relevant. We first analyze the fact that equilibrium thermodynamics may reveal the existence of an underlying dynamics. This is true in the algebraic approach of quantum mechanics via the Tomita-Takesaki theorem. A similar result is obtained if we start from the path integral expression of the partition function, an equation of motion is introduced. In both cases a parameter has to be identified with the physical time. Several arguments are presented showing that it is so. By investigating the dynamics for short times we introduce an equilibrium condition from which a natural unit of time appears. The equation of motion is extended to non equilibrium situations for which a H-theorem can be derived. The equation of motion can be transformed into a quantum Smoluchovski equation, its solution is a probability having a clear physical meaning. The relaxation of this probability towards its equilibrium value requires a time during which a thermodynamic description is possible or not. A standard bistable model is investigated and the chemical rate $k(t)$ is calculated as a function of time, it appears to be a non monotonic function of time. In some conditions the stationary value of $k(t)$ can be identified with the Kramers result. Finally we compare our approach which is naturally irreversible with an analysis based on the Schrödinger equation which is reversible. From all these results it appears that our equation of motion or its Smoluchovski version appear as a realistic starting point for describing quantum dissipation in small systems.

DOI
10.48550/arxiv.0910.1948

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Chemical reactions are generally connected with energy transfer between reactands and their environment. There are different ways of taking this environment into account. The traditional system-plus-reservoir (S+R) approach couples the relevant system to a large number of environmental degrees of freedom, represented, e.g., by harmonic oscillators. However, this has the disadvantage—from a computational point of view—that a system of many coupled differential equations has to be solved due to the large number of degrees of freedom of the reservoir. The number of degrees of freedom can be drastically reduced if an effective description of the interaction between system and reservoir is applied. On the quantum mechanical level, the effect of the environment can be included, e.g., by the use of explicitly time-dependent Hamiltonians or nonlinear additions to the Schrödinger equation (SE). An advantage of the former method is the preservation of the canonical formalism and the linearity of the theory as well as the fact that its operators can be derived directly from the S+R approach. Therefore, it seems to be even more of a paradox that this method apparently violates the uncertainty principle. This violation does not occur in the nonlinear approaches but the nonlinearity seems, at first sight, to be problematic. However, it can be shown that a particular logarithmic nonlinear SE is equivalent to the approach using the time-dependent Hamiltonian and can be transformed into this. The fact that this transformation is nonunitary provides the key for the solution of the paradox and also allows one to show the equivalence of all three of the above-mentioned approaches. In investigating the classical counterpart, a noncanonical transformation is used between the dissipative physical level and the canonical formal level, which makes it even possible to transform the physical interacting system into a formal isolated system that can be treated in the conventional way. As a result, the effect of the interaction with the environment can be taken into account classically by noncanonical transformations and quantum mechanically by nonunitary transformations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 537–547, 1999
Starting from an algebraic approach of quantum physics it has been shown via the Tomita-Takesaki theorem and the KMS condition that the canonical density matrix contains the dynamics of the system provided we use a rescaling of time. In this paper we show that the path integral form of the partition function reveals a dynamics which is complementary of the one given by the Tomita-Takesaki theorem. To do that we work in the spirit of a Feynman'conjecture. We define the entropy as a measure of the disorder in space time. By using an equilibrium condition we introduce a natural time scale that it is precisely the one appearing in the Tomita-Takesaki theorem. For this time scale depending on the temperature but not on the system properties our definition of entropy is identical to the thermodynamic one. The underlying dynamics associated with the partition function allows us to derive a H\bf{H}H-theoremtheoremtheorem. It is obtained in the thermodynamic limit and provided we are in a regime in which the thermal fluctuations are larger than the quantum ones.
This work proposes a conceptual and ontological reformulation of classical dynamics, thermodynamics, and quantum mechanics based on a single postulate: dissipation is a fundamental property of all physical processes. Traditional dynamics relies on the idealization of isolated, reversible systems, yet no real physical system satisfies such conditions. By introducing an intrinsic dissipative energy term into the equations of motion, we suggest that dissipation entails an irreversible loss of physical information, from which a classical uncertainty principle naturally follows. Stationary states are reinterpreted as zero‑dissipation configurations, lacking internal temporal evolution, thereby offering an ontological basis for quantization, non‑locality, and the Heisenberg uncertainty principle. Quantum mechanics can thus be viewed as the mathematical description of these stationary states and of the dissipative transitions connecting them. The framework is conceptually consistent with fluctuation‑dissipation theory, open‑system quantum dynamics, decoherence theory, and relaxation phenomena in NMR spectroscopy. Several experimental proposals are outlined to discriminate this ontological model from both classical mechanics and standard quantum mechanics.