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arXiv:quant-ph/0305084 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Decoherence of histories and hydrodynamic equations for a linear oscillator chain

J. J. Halliwell

Published Jul 25, 2003• 31 citations
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Abstract

We investigate the decoherence of histories of local densities for linear oscillators models. It is shown that histories of local number, momentum and energy density are approximately decoherent, when coarse grained over sufficiently large volumes. Decoherence arises directly from the proximity of these variables to exactly conserved quantities (which are exactly decoherent), and not from environmentally induced decoherence. We discuss the approach to local equilibrium and the subsequent emergence of hydrodynamic equations for the local densities.

DOI
10.1103/physrevd.68.025018

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We investigate the decoherence of histories of local densities for linear oscillators models. It is shown that histories of local number, momentum and energy density are approximately decoherent, when coarse grained over sufficiently large volumes. Decoherence arises directly from the proximity of these variables to exactly conserved quantities (which are exactly decoherent), and not from environmentally induced decoherence. We discuss the approach to local equilibrium and the subsequent emergence of hydrodynamic equations for the local densities.
Abstract Macroscopic systems are described most completely by local densities (particle number, momentum, and energy) yet the superposition states of such physical variables, indicated by the Everett interpretation, are not observed. In order to explain this, it is argued that histories of local number, momentum, and energy density are approximately decoherent when coarse-grained over sufficiently large volumes. Decoherence arises directly from the proximity of these variables to exactly conserved quantities (which are exactly decoherent), and not from environmentally induced decoherence. This chapter discusses the approach to local equilibrium and the subsequent emergence of hydrodynamic equations for the local densities. The results are general but the chapter focuses on a chain of oscillators as a specific example in which explicit calculations may be carried out. It discusses the relationships between environmentally-induced and conservation-induced decoherence, and presents a unified view of these two mechanisms.
In the context of the decoherent histories approach to quantum theory, it is shown that a class of macroscopic configurations consisting of histories of local densities (number, momentum, energy) exhibits negligible interference. This follows from the close connection of the local densities with the corresponding exactly conserved (and so exactly decoherent) quantities, and also from the observation that the eigenstates of local densities (averaged over a sufficiently large volume) remain approximate eigenstates under time evolution. The result is relevant to the derivation of hydrodynamic equations using the decoherent histories approach.
In the context of the decoherent histories approach to the quantum mechanics of closed systems, Gell-Mann and Hartle have argued that the variables typically characterizing the quasiclassical domain of a large complex system are the integrals over small volumes of locally conserved densities---hydrodynamic variables. The aim of this paper is to exhibit some simple models in which approximate decoherence arises as a result of local conservation. We derive a formula which shows the explicit connection between local conservation and approximate decoherence. We then consider a class of models consisting of a large number of weakly interacting components, in which the projections onto local densities may be decomposed into projections onto one of two alternatives of the individual components. The main example we consider is a one-dimensional chain of locally coupled spins, and the projections are onto the total spin in a subsection of the chain. We compute the decoherence functional for histories of local densities, in the limit when the number of components is very large. We find that decoherence requires two things: the smearing volumes must be sufficiently large to ensure approximate conservation, and the local densities must be partitioned into sufficiently large ranges to ensure protection against quantum fluctuations.
For a system consisting of a large collection of particles, variables that will generally become effectively classical are the local densities (number, momentum, energy). That is, in the context of the decoherent histories approach to quantum theory, it is expected that histories of these variables will be approximately decoherent and that their probabilities will be strongly peaked about hydrodynamic equations. This possibility is explored for the case of the diffusion of the number density of a dilute concentration of foreign particles in a fluid. This system has the appealing feature that the microscopic dynamics of each individual foreign particle is readily obtained and the approach to local equilibrium may be seen explicitly. It is shown that, for certain physically reasonable initial states, the probabilities for the histories of the number density are strongly peaked about evolution according to the diffusion equation. Decoherence of these histories is also shown for a class of initial states which includes nontrivial superpositions of number density. Histories of phase space densities are also discussed. The case of the histories of number, momentum, and energy density for more general systems, such as a dilute gas, is also discussed in outline. When the initial state is a local equilibrium state, it is shown that the histories are trivally decoherent and that the probabilities for histories are peaked about hydrodynamic equations. An argument for the decoherence of more general initial states is given.
The origin of classical predictability is investigated for the one dimensional harmonic chain considered as a closed quantum mechanical system. By comparing the properties of a family of coarse-grained descriptions of the chain, we conclude that local coarse grainings in this family are more useful for prediction than nonlocal ones. A quantum mechanical system exhibits classical behavior when the probability is high for histories having the correlations in time implied by classical deterministic laws. But approximate classical determinism holds only for certain coarse grainings and then only if the initial state of the system is suitably restricted. Coarse grainings by the values of the hydrodynamic variables (integrals over suitable volumes of densities of approximately conserved quantities) define the histories usually used in classical physics. But what distinguishes this coarse graining from others? This paper approaches this question by analyzing a family of coarse grainings for the linear harmonic chain. At one extreme in the family the chain is divided into local groups of N atoms. At the other extreme the N atoms are distributed nonlocally over the whole chain. Each coarse graining follows the average (center of mass) positions of the groups and ignores the ``internal'' coordinates within each group, these constituting a different environment for each coarse graining. For an initial condition where long wavelength modes are excited and short wavelength modes are distributed thermally we find that the coarse-grained positions obey deterministic equations of motion accompanied by noise. The noise is greater the more nonlocal the coarse graining. Further, the deterministic equations require more time steps to evolve over a given time interval for the nonlocal coarse grainings than for the local ones. A continuum limit is possible only for the near local coarse grainings. For parameters of the model characteristic of realistic situations these features strongly favor the local coarse grainings over the nonlocal ones for prediction. Each of these differences can be traced to the approximate conservation of the local center of mass momentum. We then consider the chain quantum mechanically and show that, for realistic parameters, all the coarse grainings decohere rapidly compared to dynamical time scales. We conclude that noise, decoherence, and computational complexity favor locality over nonlocality for deterministic predictability.
We study the emergence of decoherent histories in isolated systems based on exact numerical integration of the Schrödinger equation for a Heisenberg chain. We reveal that the nature of the system, which we switch from (i) chaotic to (ii) interacting integrable to (iii) noninteracting integrable, strongly impacts decoherence of coarse spin observables. From a finite size scaling law we infer a strong exponential suppression of coherences for (i), a weak exponential suppression for (ii), and no exponential suppression for (iii) on a relevant short (nonequilibrium) timescale. Moreover, for longer times we find stronger decoherence for (i) but the opposite for (ii), hinting even at a possible power-law decay for (ii) at equilibrium timescales. This behavior is encoded in the multitime properties of the quantum histories and it can not be explained by environmentally induced decoherence. Our results suggest that chaoticity plays a crucial role in the emergence of classicality in finite size systems.