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Dynamical origin of quantum probabilities
Open accessAbstract
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.
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Metadata
- Authors
- Valentini, A., & Westman, H.
- Affiliations
- Augustus College, 14 Augustus Road, London SW19 6LN, UK
- Chalmers University of Technology (SE)
- Corresponding author
- Valentini, Antony
- ORCID
- Valentini, Antony
- Venue
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Publisher
- The Royal Society
- Published
- 2005-01-08
- Volume
- 461
- Issue
- 2053
- Pages
- 253-272
- ISSN
- 1364-5021 (ISSN-L)
- 1471-2946
- OpenAlex
- W2130195092
- Cited by
- 226
- References
- 62
- Citations by year
- 2026: 3 · 2025: 8 · 2024: 4 · 2023: 14 · 2022: 10 · 2021: 11 · 2020: 12 · 2019: 36 · 2018: 9 · 2017: 7 · 2016: 8 · 2015: 12 · 2014: 10 · 2013: 4 · 2012: 18
- FWCI
- 3.48
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Statistical Mechanics and Entropy
- Keywords
- Born rule, Bohmian mechanics, quantum probability, H-function, Kullback-Leibler divergence, coarse-graining, two-dimensional quantum systems
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