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Brownian motion from a deterministic system of particles

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Abstract

Can Brownian motion arise from a deterministic system of particles? This paper addresses this question by analysing the derivation of Brownian motion as the limit of a deterministic hard-spheres gas with Lanford’s theorem. In particular, we examine the role of the Boltzmann-Grad limit in the loss of memory of the deterministic system and compare this derivation and the derivation of Brownian motion with the Langevin equation.

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Notes

  1. See also, e.g., Dürr et al. (1981) and Spohn (1980).

  2. For a historical approach to Brownian motion, which is not addressed in this paper, see e.g., Duplantier (2006), Maiocchi (1990), Genthon (2020).

  3. The BBGKY hierarchy (for Bogoliubov, Born, Green, Kirkwood, and Yvon) is a set of N integro-differential equations that describes the dynamics of the hard-spheres system (see e.g., Lanford, 1975, p. 85).

  4. Gallagher stresses that Bodineau et al.’s derivation ‘‘is an extension of the works (in particular, van Beijeren et al., 1980) where the linear Boltzmann equation was derived for long times’’(2019, p. 79).

  5. Lanford’s derivation of the Boltzmann equation stems from the Gibbs formulation of statistical mechanics. On the difference between the Boltzmann and Gibbs approaches to statistical mechanics, see Frigg (2008).

  6. We could reply that this restriction is only due to mathematical convergence issues, which might be removed in the future (Valente, 2014, p. 332). Moreover, we might stress that Lanford’s theorem is still informative from an in-principle point of view (Valente, 2014, p. 319). This paper does not aim at addressing this debate.

  7. In the same vein as the previous footnote, Valente replies that: “in spite of the above mentioned limitations, the striking point about Lanford’s theorem remains, namely that, for extremely diluted gases contained in a box, under suitable initial conditions one can derive the irreversible Boltzmann equation […]” (2014, p. 321). Again, it is outside the scope of this paper to discuss this point further.

  8. Norton discusses the Boltzmann-Grad limit in the context on his analysis of idealizations vs. approximations. For him, this example shows that the Boltzmann-Grad limit does not support idealization: it “has a limit system too impoverished to supply an inexact description of the finite systems” (2012, p. 16).

  9. The original notations in Golse’s paper (2014) are a for the diameter d, nkl for ωij, n for ω, and C(f) for C(ω).

  10. Golse and Degond argue that this property contributes to the appearance of irreversibility. We do not endorse this conclusion, and it is not within the scope of this paper to discuss the problem of irreversibility. However, we claim that this property is fully relevant to analyse the loss of memory of the deterministic hard-spheres gas.

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Acknowledgements

I am grateful to Anouk Barberousse, Laurent Desvillettes, Nicolas Fillion, Sébastien Rivat, and Laure Saint-Raymond for their helpful feedback on the manuscript. I also thank the two anonymous reviewers for their constructive comments. A previous version of the paper has been presented at the annual meeting of the British Society for the Philosophy of Science 2017, at a workshop on indeterminism organized by Augustin Baas, Michael Esfeld and Christian Wüthrich (Geneva, 2017), and at a workshop on philosophy of science organized by Sorin Bangu (Bucharest, 2018), and benefited from discussions with their audience.

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  1. IHPST — CNRS/University Paris 1 Panthéon-Sorbonne, 13 rue du Four, 75006, Paris, France

    Vincent Ardourel

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Ardourel, V. Brownian motion from a deterministic system of particles. Synthese 200, 29 (2022). https://doi.org/10.1007/s11229-022-03577-2

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