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Had We But World Enough, and Time... But We Don’t!: Justifying the Thermodynamic and Infinite-Time Limits in Statistical Mechanics

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Abstract

In this paper, I compare the use of the thermodynamic limit in the theory of phase transitions with the infinite-time limit in the explanation of equilibrium statistical mechanics. In the case of phase transitions, I will argue that the thermodynamic limit can be justified pragmatically since the limit behavior (i) also arises before we get to the limit and (ii) for values of N that are physically significant. However, I will contend that the justification of the infinite-time limit is less straightforward. In fact, I will point out that even in cases where one can recover the limit behavior for finite t, i.e. before we get to the limit, one cannot recover this behavior for realistic time scales. I will claim that this leads us to reconsider the role that the rate of convergence plays in the justification of infinite limits and calls for a revision of the so-called Butterfield’s principle.

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Notes

  1. An exception is Norton [15] who discusses the infinite-time limit in the explanation of reversible processes.

  2. Since the goal here is to relate the problem of the thermodynamic limit in the theory of phase transitions with the infinite-time limit in the explanation of equilibrium, I will be deliberately brief in my exposition of the problem of phase transitions. A more detailed treatment of these topics can be found, for instance, in Kadanoff [16], Butterfield [6], Batterman [17], and Butterfield and Buoatta [7].

  3. One needs to recognize that this only solves the first of the problems pointed out above and does not allow us to conclude that the same argument applies to other cases of phase transitions (problem (iii)), or to explain the role of the thermodynamic limit in renormalization group techniques (problem (ii)). These other problems have been studied extensively, for example, by Batterman [4], Morrison [5], Norton [8] and Butterfield himself [6, 7]. Since I do not have space to discuss these other issues here, I will restrict my analysis to the cases in which one can actually show that the values of the quantities that describe phase transitions in the thermodynamic limit are approximately the same as the values of the quantities before we get to the limit. The paradigmatic examples are the paramagnetic-ferromagnetic transition described above and the liquid-vapor transition at the critical point in which the compressibility behaves analogously to the magnetic susceptibility.

  4. I thank an anonymous referee for pointing this out.

  5. See Frigg [20], Uffink [21] and van Lith [22] for a more detailed description of the problems associated with the Gibbs formalism.

  6. Strictly speaking, this theorem was formulated in terms of metric transitivity instead of ergodicity. Metric transitivity is a property of dynamical systems that captures the same idea as ergodicity but in measure theoretic sense. For more details see Uffink [21, Sect. 6] and van Lith [22, Chap. 7].

  7. One can find arguments in this direction, for example, in Gallavotti [14] and Emch and Liu [11].

  8. For a quantitative estimation, see Gallavotti [14].

  9. A review of this attempts can be found in [24].

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Acknowledgements

I am grateful to Neil Dewar, Jos Uffink, Giovanni Valente and Charlotte Werndl for detailed feedback on a previous draft of the paper. Previous versions of this work have been presented at the at the workshop “Tatjana Afanassjewa and Her Legacy” hosted by the University of Salzburg and the workshop “The Second Law” hosted by the Munich Center for Mathematical Philosophy; I am grateful to the audiences and organizers for helpful feedback.

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    Patricia Palacios

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Palacios, P. Had We But World Enough, and Time... But We Don’t!: Justifying the Thermodynamic and Infinite-Time Limits in Statistical Mechanics. Found Phys 48, 526–541 (2018). https://doi.org/10.1007/s10701-018-0165-0

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