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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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.
KeywordsThermodynamic limit Phase transitions Infinite-time limit Rate of convergence Approximation Butterfield principle
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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