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Boltzmannian Non-Equilibrium and Local Variables

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Current Debates in Philosophy of Science

Part of the book series: Synthese Library ((SYLI,volume 477))


Boltzmannian statistical mechanics (BSM) partitions a system’s space of micro-states into cells and refers to these cells as ‘macro-states’. One of these cells is singled out as the equilibrium macro-state while the others are non-equilibrium macro-states. It remains unclear, however, how these states are characterised at the macro-level as long as only real-valued macro-variables are available. We argue that physical quantities like pressure and temperature should be treated as field-variables and show how field variables fit into the framework of our own version of BSM, the long-run residence time account of BSM. The introduction of field variables into the theory makes it possible to give a full macroscopic characterisation of the approach to equilibrium.

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  1. 1.

    This paper discusses BSM. For discussion of Gibbsian statistical mechanics see Frigg and Werndl (2021), and for a discussion of the relation between BSM and Gibbsian statistical mechanics see Werndl and Frigg (2017b) and Frigg and Werndl (2019).

  2. 2.

    Contemporary discussions of this argument can be found in Albert (2000, Ch. 3), Frigg (2008, Sec. 2), and Uffink (2007, Sec. 4).

  3. 3.

    For a discussion of these see Uffink’s (2007, Sec. 4) and Werndl and Frigg’s (2015a; 2015b).

  4. 4.

    At least if one thinks that local pressure is defined at every point in space.

  5. 5.

    See, for instance, Goldstein (2001).

  6. 6.

    The presentation of LBSM follows Werndl and Frigg (2015b). This paper focuses on deterministic systems. The generalisation to stochastic classical systems is spelled out in Werndl and Frigg (2017a), where statements of the relevant definitions and results can be found.

  7. 7.

    Being in thermodynamic equilibrium is an intrinsic property of the system, which offers a notion of ‘internal equilibrium’ (Guggenheim, 1967, 7). It contrasts with ‘mutual equilibrium’ (ibid., 8), which is the relational property of being in equilibrium with each other that two systems eventually reach after being put into thermal contact with each other. When defining equilibrium in BSM it is the internal equilibrium that we are interested in.

  8. 8.

    We state the definitions for continuous time. The corresponding definitions for discrete time are obtained by replacing the integrals by sums.

  9. 9.

    We assume that \(\varepsilon \) is small enough so that \(\alpha (1-\varepsilon )> \frac {1}{2}\).

  10. 10.

    We assume that \(\varepsilon <\gamma \).

  11. 11.

    For statements and discussions of LEH see Giberti et al. (2019), Jou et al. (2010, 14–15), Spohn (1991, 14), and Öttinger (2005, Ch. 2).

  12. 12.

    Nothing depends on this being a cube. The same construction can be made with a sphere.


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We would like to thank Cristián Soto for inviting us to participate in this project. We also would like to thank Sean Gryb, David Lavis, and Lamberto Rondoni for helpful discussions on the subject matter of the paper.

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Correspondence to Roman Frigg .

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Frigg, R., Werndl, C. (2023). Boltzmannian Non-Equilibrium and Local Variables. In: Soto, C. (eds) Current Debates in Philosophy of Science. Synthese Library, vol 477. Springer, Cham.

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