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arXiv:1502.03947 · Journal of Physics Conference Series

Irreversibility and decoherence in an ideal gas

János Polonyi

Published Jul 3, 2015• 2 citations
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Abstract

Different models are described where non-interacting particles generate dissipative effective forces by the mixing of infinitely many soft normal modes. The effective action is calculated for these models within the Closed Time Path formalism. This is a well known scheme for quantum systems but its application in classical mechanics presents a new, more unified derivation and treatment of dissipative forces within classical and quantum physics.

DOI
10.1088/1742-6596/626/1/012021

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Different models are described where non-interacting particles generate dissipative effective forces by the mixing of infinitely many soft normal modes. The effective action is calculated for these models within the Closed Time Path formalism. This is a well known scheme for quantum systems but its application in classical mechanics presents a new, more unified derivation and treatment of dissipative forces within classical and quantum physics.
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Phenomenological Equations of Motion for Dissipative Systems Lagrangian Hamiltonian and Hamilton-Jacobi Formulation of the Classical Dissipative Systems Noether's Theorem and Non-Noether Conservation Laws Dissipative Forces Derived from Many-Body Problems A Particle Coupled to a Field and the Damped Motion of a Central Particle Coupled to a Heat Bath Quantization of Dissipative Systems in General and of Explicitly Time-Dependent Hamiltonians in Particular Density Matrix and the Wigner Distribution Function for Damped Systems Path Integral Formulation of a Damped Harmonic Oscillator Quantization of the Motion of an Infinite Chain Heisenberg's Equations of Motion for a Particle Coupled to a Heat Bath Quantum Mechanical Models of Dissipative Systems and the Concept of Optical Potential.
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.
To study the phenomenon of decoherence the effective Lagrangian of a test particle interacting with an ideal gas is derived in the leading order of the perturbation expansion and in order $\ord{\partial^2_t}$. Two concepts of decoherence are distinguished and investigated in details: The general dynamical decoherence and the stationary decoherence, an approximation of the dynamical phenomenon corresponding to the assumption of a static test particle trajectory. Comparison of the stationary decoherence described by the present effective theory with the traditional description of decoherence based on collision theory reveals a qualitative similarity within their domain of applicability. However, the effective theory predicts comparable time scales for decoherence and dissipation in contrast to the deviation of this scales by many orders of magnitudes, estimated by referring to the collision approach. In particular the extremely fast decoherence predicted by the collisional approach can only be reproduced by the effective theory for parameter values which violate the applicability of both approaches. On the other hand, in harmonic systems, the true dynamical decoherence is indeed an extremely fast process involving a double exponential time dependence. It is argued that, in realistic systems, the dissipative effective dynamics is non-perturbative in nature and its asymptotic, relaxed state incorporates strong quantum fluctuations.