Abstract
Statistical physicists assume a probability distribution over micro-states to explain thermodynamic behavior. The question of this paper is whether these probabilities are part of a best system and can thus be interpreted as Humean chances. I consider two Boltzmannian accounts of the Second Law, viz. a globalist and a localist one. In both cases, the probabilities fail to be chances because they have rivals that are roughly equally good. I conclude with the diagnosis that well-defined micro-probabilities under-estimate the robust character of explanations in statistical physics.
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Notes
- 1.
- 2.
Probabilities are assumed to be attach to propositions, but, for convenience, I will sometimes also speak of the probabilities of events.
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- 4.
Things are in fact more complicated. Even if \(\Sigma (CM,TD)\) is an optimal improvement over \(\Sigma (CM)\), as far as entropy is concerned, we may be able to move to an even better system that covers other aspects of the pattern and entails a different probability function for the increase of entropy. But this possibility will not matter for the purposes of my argument.
- 5.
Note that the word “system” is ambiguous in this paper, it either refers to systems of representations (e.g. Lewis’s best system) or to physical systems. I assume that readers will always be able to find out which sort of systems I refer to.
- 6.
It is here assumed that the dynamics is homogeneous or stationary in time, i.e., if x(t) is a possible trajectory in phase space, so is x(t + T) with a constant T.
- 7.
Alternatively, we can shrink the support of the probability density over the initial condition by taking into account macro-information about later times, e.g. by conditioning on the macro-state 1 s after the initial time (cf. Frigg 2008, p. 679). Once again, we gain fit, but pay in simplicity.
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- 9.
Note that our natural dynamics of chances does not really mesh with PD. To explain the increase in entropy, a modification of PD would have to be combined with the natural dynamics.
- 10.
In this section, I focus on the localist strategy based upon the work of Albert. It closely parallels the globalist strategy as outlined above. Frigg and Hoefer (2013) take the localist strategy in a different way. But they run into problems parallel to those outlined in this section. See footnote 14 below for details.
- 11.
The same point applies to SL probabilities.
- 12.
To be fair to Frigg and Hoefer (2013), we should note that they explicitly bracket the question as to whether their strategy yields unique best probabilities (Sect. 4).
- 13.
Bohmians, of course, deny this, but they nevertheless grant quantum-mechanical probabilities.
- 14.
For an alternative strategy, Frigg and Hoefer (2013) do not just partition a suitable region of each phase space in two regions; rather, they assume a flat probability distribution over the initial macro-state in each phase-space. This then is supposed to imply a very high probability for trajectories that manifest thermo-dynamic behavior. But this suggestion runs into the same types of problems we have seen before. First, as Frigg and Hoefer (2013), Sect. 4 themselves acknowledge, it is not known whether a flat distribution over micro-states yields good fit. Second, there is an alternative probability model that is roughly equally good. The model distinguishes two sub-regions within the initial macro-state in each phase-space with the help of a physical characteristics. We assume a flat probability model for each of these sub-regions. In this way we improve fit, but have to pay in simplicity. The advantages of this model do not clearly balance the disadvantages and vice versa.
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Acknowledgements
I’m grateful to my commentators Luke Glynn, Radin Dardashti, Karim P. Y. Thebault and Mathias Frisch, to Georg Brun and to the participants at the Lausanne workshop for discussion. Thanks also to Michael Esfeld for the invitation and to Roman Frigg for sharing a yet unpublished manuscript with me.
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Beisbart, C. (2014). Good Just Isn’t Good Enough: Humean Chances and Boltzmannian Statistical Physics. In: Galavotti, M., Dieks, D., Gonzalez, W., Hartmann, S., Uebel, T., Weber, M. (eds) New Directions in the Philosophy of Science. The Philosophy of Science in a European Perspective, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-04382-1_36
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