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arXiv:1103.1589 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Time scales for dynamical relaxation to the Born rule

Mike D. Towler, N. J. Russell, Antony Valentini

Published Nov 30, 2011• 117 citations
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Abstract

We illustrate through explicit numerical calculations how the Born rule probability densities of non-relativistic quantum mechanics emerge naturally from the particle dynamics of de Broglie–Bohm pilot-wave theory. The time evolution of a particle distribution initially not equal to the absolute square of the wave function is calculated for a particle in a two-dimensional infinite potential square well. Under the de Broglie–Bohm ontology, the box contains an objectively existing ‘pilot wave’ which guides the electron trajectory, and this is represented mathematically by a Schrödinger wave function composed of a finite out-of-phase superposition of M energy eigenstates (with M ranging from 4 to 64). The electron density distributions are found to evolve naturally into the Born rule ones and stay there; in analogy with the classical case this represents a decay to ‘quantum equilibrium’. The proximity to equilibrium is characterized by the coarse-grained subquantum H -function which is found to decrease roughly exponentially towards zero over the course of time. The time scale τ for this relaxation is calculated for various values of M and the coarse-graining length ε . Its dependence on M is found to disagree with an earlier theoretical prediction. A power law, τ ∝ M −1 , is found to be fairly robust for all coarse-graining lengths and, although a weak dependence of τ on ε is observed, it does not appear to follow any straightforward scaling. A theoretical analysis is presented to explain these results. This improvement in our understanding of time scales for relaxation to quantum equilibrium is likely to be of use in the development of models of relaxation in the early Universe, with a view to constraining possible violations of the Born rule in inflationary cosmology.

DOI
10.1098/rspa.2011.0598

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We illustrate through explicit numerical calculations how the Born rule probability densities of non-relativistic quantum mechanics emerge naturally from the particle dynamics of de Broglie–Bohm pilot-wave theory. The time evolution of a particle distribution initially not equal to the absolute square of the wave function is calculated for a particle in a two-dimensional infinite potential square well. Under the de Broglie–Bohm ontology, the box contains an objectively existing ‘pilot wave’ which guides the electron trajectory, and this is represented mathematically by a Schrödinger wave function composed of a finite out-of-phase superposition of M energy eigenstates (with M ranging from 4 to 64). The electron density distributions are found to evolve naturally into the Born rule ones and stay there; in analogy with the classical case this represents a decay to ‘quantum equilibrium’. The proximity to equilibrium is characterized by the coarse-grained subquantum H -function which is found to decrease roughly exponentially towards zero over the course of time. The time scale τ for this relaxation is calculated for various values of M and the coarse-graining length ε . Its dependence on M is found to disagree with an earlier theoretical prediction. A power law, τ ∝ M −1 , is found to be fairly robust for all coarse-graining lengths and, although a weak dependence of τ on ε is observed, it does not appear to follow any straightforward scaling. A theoretical analysis is presented to explain these results. This improvement in our understanding of time scales for relaxation to quantum equilibrium is likely to be of use in the development of models of relaxation in the early Universe, with a view to constraining possible violations of the Born rule in inflationary cosmology.
Numerical simulations indicate that the Born rule does not need to be postulated in the de Broglie–Bohm pilot-wave theory, but tends to arise dynamically (relaxation to quantum equilibrium). These simulations were done for a particle in a two-dimensional box whose wave function obeys the non-relativistic Schrödinger equation and is therefore scalar. The chaotic nature of the de Broglie–Bohm trajectories, thanks to the nodes of the wave function which yield to vortices, is crucial for a fast relaxation to quantum equilibrium. For spinors, we typically do not expect any node. However, in the case of the Dirac equation, the de Broglie–Bohm velocity field has vorticity even in the absence of nodes. This observation raises the question of the origin of relaxation to quantum equilibrium for fermions. In this article, we provide numerical evidence to show that Dirac particles also undergo relaxation, by simulating the evolution of various non-equilibrium distributions for two-dimensional systems (the two-dimensional Dirac oscillator and the Dirac particle in a two-dimensional spherical step potential).
We study the origin of the Born probability rule ρ = | ψ | 2 in the de Broglie–Bohm pilot–wave formulation of quantum theory. It is argued that quantum probabilities arise dynamically, and have a status similar to thermal probabilities in ordinary statistical mechanics. This is illustrated by numerical simulations for a two–dimensional system. We show that a simple initial ensemble, with a non–standard distribution ρ ≠ | ψ | 2 of particle positions, evolves towards the quantum distribution to high accuracy. The relaxation process ρ →| ψ | 2 is quantified in terms of a coarse–grained H –function (equal to minus the relative entropy of ρ with respect to | ψ | 2 ), which is found to decrease approximately exponentially over time, with a time constant that accords with a simple theoretical estimate.
In the context of the de Broglie–Bohm pilot-wave theory, numerical simulations for simple systems have shown that states that are initially out of quantum equilibrium—thus violating the Born rule—usually relax over time to the expected | ψ | 2 distribution on a coarse-grained level. We analyse the relaxation of non-equilibrium initial distributions for a system of coupled one-dimensional harmonic oscillators in which the coupling depends explicitly on time through numerical simulations, focusing on the influence of different parameters such as the number of modes, the coarse-graining length and the coupling constant. We show that in general the system studied here tends to equilibrium, but the relaxation can be retarded depending on the values of the parameters, particularly to the one related to the strength of the interaction. Possible implications on the detection of relic non-equilibrium systems are discussed.
We consider a particle confined in a uniformly expanding two-dimensional square box from the point of the view of the de Broglie-Bohm pilot-wave theory. In particular we study quantum ensembles in which the Born Law is initially violated (quantum non-equilibrium). We show examples of such ensembles that start close to quantum equilibrium, as measured by the standard coarse-grained H-function, but diverge from it with time. We give an explanation of this result and discuss the possibilities that it opens.
The de Broglie–Bohm pilot-wave formulation of quantum theory allows the existence of physical states that violate the Born probability rule. Recent work has shown that in pilot-wave field theory on expanding space relaxation to the Born rule is suppressed for long-wavelength field modes, resulting in a large-scale power deficit [Formula: see text] which for a radiation-dominated expansion is found to have an approximate inverse-tangent dependence on [Formula: see text] (assuming that the width of the initial distribution is smaller than the width of the initial Born-rule distribution and that the initial quantum states are evenly-weighted superpositions of energy states). In this paper, we show that the functional form of [Formula: see text] is robust under changes in the initial nonequilibrium distribution — subject to the limitation of a subquantum width — as well as under the addition of an inflationary era at the end of the radiation-dominated phase. In both cases, the predicted deficit [Formula: see text] remains an inverse-tangent function of [Formula: see text]. Furthermore, with the inflationary phase the dependence of the fitting parameters on the number of superposed pre-inflationary energy states is comparable to that found previously. Our results indicate that, for the assumed broad class of initial conditions, an inverse-tangent power deficit is likely to be a fairly general and robust signature of quantum relaxation in the early universe.
We compare and contrast two distinct approaches to understanding the Born rule in de Broglie-Bohm pilot-wave theory, one based on dynamical relaxation over time (advocated by this author and collaborators) and the other based on typicality of initial conditions (advocated by the 'Bohmian mechanics' school). It is argued that the latter approach is inherently circular and physically misguided. The typicality approach has engendered a deep-seated confusion between contingent and law-like features, leading to misleading claims not only about the Born rule but also about the nature of the wave function. By artificially restricting the theory to equilibrium, the typicality approach has led to further misunderstandings concerning the status of the uncertainty principle, the role of quantum measurement theory, and the kinematics of the theory (including the status of Galilean and Lorentz invariance). The restriction to equilibrium has also made an erroneously-constructed stochastic model of particle creation appear more plausible than it actually is. To avoid needless controversy, we advocate a modest 'empirical approach' to the foundations of statistical mechanics. We argue that the existence or otherwise of quantum nonequilibrium in our world is an empirical question to be settled by experiment.