Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1306.1576 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Instability of quantum equilibrium in Bohm's dynamics

Samuel Colin, Antony Valentini

Published Sep 3, 2014• 29 citations
arXivHTMLPDF

Abstract

We consider Bohm's second-order dynamics for arbitrary initial conditions in phase space. In principle, Bohm's dynamics allows for 'extended' non-equilibrium, with initial momenta not equal to the gradient of phase of the wave function (as well as initial positions whose distribution departs from the Born rule). We show that extended non-equilibrium does not relax in general and is in fact unstable. This is in sharp contrast with de Broglie's first-order dynamics, for which non-standard momenta are not allowed and which shows an efficient relaxation to the Born rule for positions. On this basis, we argue that, while de Broglie's dynamics is a tenable physical theory, Bohm's dynamics is not. In a world governed by Bohm's dynamics, there would be no reason to expect to see an effective quantum theory today (even approximately), in contradiction with observation.

DOI
10.1098/rspa.2014.0288

Related papers

Modified Bohms Dynamics: A Stable Version

Kazemi, Mohammad Javad, Meysam Mashhadi, Mohammad Hosein Barati

Recently, it has been shown that the quantum equilibrium distribution in the original Bohm's model is unstable and so it isn't a tenable physical theory [Proc. R. Soc. A 470 20140288 (2014)]. In this paper we show that a natural modification of the Bohm's quantum force leads to a stable quantum equilibrium without any change to statistical predictions of this model in equilibrium state. Moreover, it is shown that an initial non-equilibrium phase space distribution relaxes to equilibrium distribution via a coarse grained H-theorem. But before end of the relaxation process, for example in the early universe, this modified quantum force can lead to the new predictions beyond standard quantum mechanics.

arXiv:1508.02658•Aug 11, 2015•0 citations

Relaxation to quantum equilibrium for Dirac fermions in the de Broglie–Bohm pilot-wave theory

Samuel Colin

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).

arXiv:1108.5496•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Dec 14, 2011•30 citations

Evolution of quantum non-equilibrium for coupled harmonic oscillators

Francisco Bento Lustosa, Nelson Pinto-Neto, Antony Valentini

In the context of de Broglie–Bohm pilot-wave theory, violations of the Born rule are allowed and can be considered as describing non-equilibrium distributions. We study the effects of interactions on quantum relaxation towards equilibrium for a system of one-dimensional coupled harmonic oscillators. We show by numerical simulations that interactions can delay or even prevent complete relaxation for some initial states. We also discuss how this effect might be relevant for cosmological scenarios and how non-equilibrium could be detected in some models.

arXiv:2205.13701•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Jan 1, 2023•8 citations

Quantum non-equilibrium and relaxation to equilibrium for a class of de Broglie–Bohm-type theories

Samuel Colin, Ward Struyve

The de Broglie–Bohm theory is about non-relativistic point-particles that move deterministically along trajectories. The theory reproduces the predictions of standard quantum theory, given that the distribution of particles over an ensemble of systems, all described by the same wavefunction ψ, equals the quantum equilibrium distribution |ψ| 2 . Numerical simulations done by Valentini and Westman (2005 Proc. R. Soc. A 461 253) have illustrated that non-equilibrium particle distributions may relax to quantum equilibrium after some time. Here we consider non-equilibrium distributions and their relaxation properties for a particular class of trajectory theories (first studied in detail by Deotto and Ghirardi (1998 Found. Phys. 28 1)) that are empirically equivalent to the de Broglie–Bohm theory in quantum equilibrium. In the examples we studied of such theories, we found a speed-up of the relaxation, compared to the ordinary de Broglie–Bohm theory. Hence non-equilibrium predictions that depend strongly on relaxation properties, such as those studied recently by Valentini, may vary across different trajectory theories. As such, these theories might be experimentally distinguishable.

arXiv:0911.2823•New Journal of Physics•Apr 1, 2010•31 citations

On quantum potential dynamics

Sheldon Goldstein, Ward Struyve

Non-relativistic de Broglie–Bohm theory describes particles moving under the guidance of the wave function. In de Broglieʼs original formulation, the particle dynamics is given by a first-order differential equation. In Bohmʼs reformulation, it is given by Newtonʼs law of motion with an extra potential that depends on the wave function—the quantum potential—together with a constraint on the possible velocities. It was recently argued, mainly by numerical simulations, that relaxing this velocity constraint leads to a physically untenable theory. We provide further evidence for this by showing that for various wave functions the particles tend to escape the wave packet. In particular, we show that for a central classical potential and bound energy eigenstates the particle motion is often unbounded. This work seems particularly relevant for ways of simulating wave function evolution based on Bohmʼs formulation of the de Broglie–Bohm theory. Namely, the simulations may become unstable due to deviations from the velocity constraint.

arXiv:1312.1990•Journal of Physics A Mathematical and Theoretical•Dec 11, 2014•18 citations

Quantum cosmology from the de Broglie–Bohm perspective

Nelson Pinto-Neto, J. C. Fabris

We review the main results that have been obtained in quantum cosmology from the perspective of the de Broglie-Bohm quantum theory. As it is a dynamical theory of assumed objectively real trajectories in the configuration space of the physical system under investigation, this quantum theory is not essentially probabilistic and dispenses the collapse postulate, turning it suitable to be applied to cosmology. In the framework of minisuperspace models, we show how quantum cosmological effects in the de-Broglie-Bohm's approach can avoid the initial singularity, and isotropize the Universe. We then extend minisuperspace in order to include linear cosmological perturbations. We present the main equations which govern the dynamics of quantum cosmological perturbations evolving in non-singular quantum cosmological backgrounds, and calculate some of their observational consequences. These results are not known how to be obtained in other approaches to quantum theory. In the general case of full superspace, we enumerate the possible structures of quantum space and time that emerge from the de Broglie-Bohm picture. Finally, we compare some of the results coming from the de Broglie-Bohm theory with other approaches, and discuss the physical reasons for some discrepancies that occur.

arXiv:1306.0820•Classical and Quantum Gravity•Jun 12, 2013•126 citations
We consider Bohm's second-order dynamics for arbitrary initial conditions in phase space. In principle, Bohm's dynamics allows for 'extended' non-equilibrium, with initial momenta not equal to the gradient of phase of the wave function (as well as initial positions whose distribution departs from the Born rule). We show that extended non-equilibrium does not relax in general and is in fact unstable. This is in sharp contrast with de Broglie's first-order dynamics, for which non-standard momenta are not allowed and which shows an efficient relaxation to the Born rule for positions. On this basis, we argue that, while de Broglie's dynamics is a tenable physical theory, Bohm's dynamics is not. In a world governed by Bohm's dynamics, there would be no reason to expect to see an effective quantum theory today (even approximately), in contradiction with observation.
Recently, it has been shown that the quantum equilibrium distribution in the original Bohm's model is unstable and so it isn't a tenable physical theory [Proc. R. Soc. A 470 20140288 (2014)]. In this paper we show that a natural modification of the Bohm's quantum force leads to a stable quantum equilibrium without any change to statistical predictions of this model in equilibrium state. Moreover, it is shown that an initial non-equilibrium phase space distribution relaxes to equilibrium distribution via a coarse grained H-theorem. But before end of the relaxation process, for example in the early universe, this modified quantum force can lead to the new predictions beyond standard quantum mechanics.
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).
In the context of de Broglie–Bohm pilot-wave theory, violations of the Born rule are allowed and can be considered as describing non-equilibrium distributions. We study the effects of interactions on quantum relaxation towards equilibrium for a system of one-dimensional coupled harmonic oscillators. We show by numerical simulations that interactions can delay or even prevent complete relaxation for some initial states. We also discuss how this effect might be relevant for cosmological scenarios and how non-equilibrium could be detected in some models.
The de Broglie–Bohm theory is about non-relativistic point-particles that move deterministically along trajectories. The theory reproduces the predictions of standard quantum theory, given that the distribution of particles over an ensemble of systems, all described by the same wavefunction ψ, equals the quantum equilibrium distribution |ψ| 2 . Numerical simulations done by Valentini and Westman (2005 Proc. R. Soc. A 461 253) have illustrated that non-equilibrium particle distributions may relax to quantum equilibrium after some time. Here we consider non-equilibrium distributions and their relaxation properties for a particular class of trajectory theories (first studied in detail by Deotto and Ghirardi (1998 Found. Phys. 28 1)) that are empirically equivalent to the de Broglie–Bohm theory in quantum equilibrium. In the examples we studied of such theories, we found a speed-up of the relaxation, compared to the ordinary de Broglie–Bohm theory. Hence non-equilibrium predictions that depend strongly on relaxation properties, such as those studied recently by Valentini, may vary across different trajectory theories. As such, these theories might be experimentally distinguishable.
Non-relativistic de Broglie–Bohm theory describes particles moving under the guidance of the wave function. In de Broglieʼs original formulation, the particle dynamics is given by a first-order differential equation. In Bohmʼs reformulation, it is given by Newtonʼs law of motion with an extra potential that depends on the wave function—the quantum potential—together with a constraint on the possible velocities. It was recently argued, mainly by numerical simulations, that relaxing this velocity constraint leads to a physically untenable theory. We provide further evidence for this by showing that for various wave functions the particles tend to escape the wave packet. In particular, we show that for a central classical potential and bound energy eigenstates the particle motion is often unbounded. This work seems particularly relevant for ways of simulating wave function evolution based on Bohmʼs formulation of the de Broglie–Bohm theory. Namely, the simulations may become unstable due to deviations from the velocity constraint.
We review the main results that have been obtained in quantum cosmology from the perspective of the de Broglie-Bohm quantum theory. As it is a dynamical theory of assumed objectively real trajectories in the configuration space of the physical system under investigation, this quantum theory is not essentially probabilistic and dispenses the collapse postulate, turning it suitable to be applied to cosmology. In the framework of minisuperspace models, we show how quantum cosmological effects in the de-Broglie-Bohm's approach can avoid the initial singularity, and isotropize the Universe. We then extend minisuperspace in order to include linear cosmological perturbations. We present the main equations which govern the dynamics of quantum cosmological perturbations evolving in non-singular quantum cosmological backgrounds, and calculate some of their observational consequences. These results are not known how to be obtained in other approaches to quantum theory. In the general case of full superspace, we enumerate the possible structures of quantum space and time that emerge from the de Broglie-Bohm picture. Finally, we compare some of the results coming from the de Broglie-Bohm theory with other approaches, and discuss the physical reasons for some discrepancies that occur.