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arXiv:1109.2228 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Dynamical breakdown of time reversal invariance and causality

János Polonyi

Published Nov 14, 2011• 9 citations
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Abstract

Irreversibility and acausality of a subsystem are established in exactly soluble harmonic models with reversible and causal dynamics. It is shown that initial conditions, imposed on some dynamical degrees of freedom may break time reversal invariance for other degrees of freedom. This happens if observations carried out in any large but finite amount of time can not resolve the spectrum of the eliminated degrees of freedom, namely, when the spectrum has a condensation point at the ground state. Acausality follows due to the dominance of the dynamics by almost time-independent modes.

DOI
10.1103/physrevd.84.105021

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Irreversibility and acausality of a subsystem are established in exactly soluble harmonic models with reversible and causal dynamics. It is shown that initial conditions, imposed on some dynamical degrees of freedom may break time reversal invariance for other degrees of freedom. This happens if observations carried out in any large but finite amount of time can not resolve the spectrum of the eliminated degrees of freedom, namely, when the spectrum has a condensation point at the ground state. Acausality follows due to the dominance of the dynamics by almost time-independent modes.
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The role of the environment initial conditions in the breaking of the time reversal symmetry of effective theories and in generating the soft irreversibility is studied by the help of Closed Time Path formalism. The initial conditions break the time reversal symmetry of the solution of the equation of motion in a trivial manner. When open systems are considered then the initial conditions of the environment must be included in the effective dynamics. This is achieved by means of a generalized ϵ-prescription where the non-uniform convergence of the limit ϵ → 0 leaves behind a spontaneous breakdown of the time reversal symmetry.
Time-irreversibility of apparently irreversible phenomena observed in quantum systems with a very small number of degrees of freedom is studied quantitatively by a time-reversal test: the system evolved in the forward-time direction is time-reversed after applying a small perturbation at the reversal time and the separation between the time-reversed perturbed and the time-reversed unperturbed states is measured as a function of perturbation strength. If the system exhibits a normal diffusion, the time-reversal characteristics has a universal threshold of perturbation strength above which a complete loss of memory is realized. A remarkable fact is that, for almost all the model systems we examined, including discrete-time/continuous-time systems and deterministic/stochastic systems, the time reversal characteristics as well as the time-reversal dynamics itself converge asymptotically to universal behaviors independent of the details of the systems, which implies the prototype of quantum irreversibility. Only the diffusive behavior of the critical states in Harper equation deviates significantly from the universal behavior.
ABSTRACT A general approach to modeling irreversibility starting from microscopic reversibility is presented. The time t s up to which relevant degrees of freedom of a system are tracked is extremely much shorter than the spectral resolution time t e that would be necessary to resolve the spectrum of all degrees of freedom involved. A relaxator that breaks reversibility condenses in the Liouville operator of the relevant degrees of freedom. The irrelevant degrees of freedom act as an environment for the system. The irreversible relaxator Liouville equation contains memory effects and initial correlations of all degrees of freedom. Stationary states turn out to be generically unique and independent of the initial conditions and exceptions are due to degeneracies. Equilibrium states lie in the relaxator's kernel yielding a stationary Pauli master equation. Kinetic equations for one‐particle densities are constructed as special cases of relaxator Liouville dynamics. Kubo's linear response theory is generalized to relaxator Liouville dynamics and related to irreversibility within the system. In a weak coupling approximation between system and environment the relaxator can be reduced to environmental correlations and bilinear system operators. Markov approximation turns the relaxator Liouville dynamics into a semi‐group dynamics.
The spread of the time arrows from the environment to an observed subsystem is followed within a harmonic model. A similarity is pointed out between irreversibility and a phase with spontaneously broken symmetry. The causal structure of interaction might be lost in the irreversible case, as well. The Closed Time Path formalism is developed for classical systems and shown to handle the time arrow problem in a clear and flexible manner. The quantum case is considered, as well, and the common origin of irreversibility and decoherence is pointed out.
This paper develops a constraint-based account of dynamical models in physics, showing that reversible dynamics arise as highly restricted boundary cases of more general irreversible frameworks. Using open quantum systems (Lindblad master equation) and classical statistical mechanics (Boltzmann equation), we prove that reversible models form a proper subclass of irreversible models (R⊂I\mathcal{R} \subset \mathcal{I}R⊂I), obtained by imposing bijectivity, time-reversal invariance, and strict isolation. This structural inclusion clarifies how unitary Schrödinger evolution and Liouvillian dynamics emerge from dissipative, many-to-one processes, and reframes debates on the arrow of time, coarse-graining, and open-system foundations in terms of constraint hierarchies rather than fundamental reversibility.