## Abstract

Classical statistical mechanics posits probabilities for various events to occur, and these probabilities seem to be objective chances. This does not seem to sit well with the fact that the theory’s time evolution is deterministic. We argue that the tension between the two is only apparent. We present a theory of Humean objective chance and show that chances thus understood are compatible with underlying determinism and provide an interpretation of the probabilities we find in Boltzmannian statistical mechanics.

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## Notes

As is well known, there are different axiomatisations of probability. Nothing in what follows depends on which axiomatisation we chose.

We use this term in restrictive way: only statements having exactly that form are chance rules. Existence claims, statements about upper and lower bounds, or specifications of probability intervals are not chance rules in our sense.

This point has been made by Hájek (2007) for propensities. His arguments readily carry over to any notion of chance.

In this we disagree with Lewis, who thought it a major problem to prove that chances satisfy the axioms of probability. THOC

*defines*chance, and a function that does not satisfy the axioms of probability cannot be a chance function.Justifying PP is a thorny issue, and, unsurprisingly, one fraught with controversy. We refer the reader to Frigg and Hoefer (2010, Section 3.4) and references therein for a discussion.

Note that even if you believe that the world does contain necessary connections, powers or propensities, it still

*also*has a HM. The HM is just the panoply of actual events understood as purely occurrent, setting aside any modal aspects those events may possess. The Humean about chance then simply maintains that chance facts supervene on this HM.For a discussion of infinite sequences see (Elga 2004).

The assumption that macrostates can be indexed by an integer

*k*is a common idealisation in this context and we follow this convention here; see (Frigg 2008b) and references therein.This definition of TD-likeness is adapted from Lavis (2005). A different way of reformulating the Second Law emerges from (Albert 2000). We prefer an approach based on TD-likeness for the reasons outlined in (Frigg and Werndl 2011) and use it here because it is simpler than Albert’s. Noting we say about chance depends on this choice, though, and

*mutatis mutandis*our account of chance can also be applied to Albert’s transition probabilities.We base our discussion on the standard possible worlds definition of determinism; see (Earman 1986, Ch. 2).

Note that the term ‘Past Hypothesis’ is usually reserved for approaches in which the system under consideration is the entire universe; it then says that the universe came into being in a low entropy macrostate provided to us by modern Big Bang cosmology. We return to the issue of the nature of systems studied in SM Sect. 5.5.

Note that determining what will happen is not the same as entailing that the objective chance of something happening is equal to one (and

*mutatis mutandis*for not happening and chance equal to zero).Note that on our understanding of admissibility, it is not closed under logical conjunction, as Lewis supposed it to be.

That said, we are sympathetic with Lewis’ line on this point (1994, 479): we may, not unreasonably in light of actual science, hope that there is one

*robustly*Best System for our HM, or a small family of closely resembling cousins, that come out as Best under any reasonable ways of cashing out and weighing up the qualities of simplicity, strength and fit.Roughly, the Kolmogorov complexity is the length of the shortest computer programme that derives a certain result. With respect to a given language, the Kolmogorov complexity is an objective quantity.

In a phenomenon known as ‘gene surfing’, genetic drift becomes a much stronger evolutionary force in populations at the edge of a territorial expansion wave, because genes from the individuals at/near the edge of the wave will be disproportionately represented in the gene pool of the newly colonized regions in subsequent generations. Lehe et al. (2012) propose (in our terms) chance rules for the fixation of a favorable mutation as a function of distance of the individual in which the mutation occurs from the edge of the colonization wave.

For a discussion of this point see (Frisch 2011, forthcoming).

This is so even if the output of the rule is merely a chance rather than a yes/no determination. When it comes to the chance \( p(TS) \), which is almost always near-1, the information conveyed to the agent is nearly as strong as what is entailed by (but impossible to derive from) the deterministic laws plus IC’s.

In this section our discussion idealises by pretending that the histories of all sorts of different SM systems could be treated as representable via paths in a single phase space. This is an idealisation because systems with a different particle number

*N*have different phase spaces. We think that this is no threat to our approach. SM systems such as expanding gases and cooling solids are ubiquitous in HM and there will be enough of them for most*N*to ground a HBS supervenience claim. Those for which this is not the case (probably ones with very large*N*) can be treated along the lines of rare gambling devices such as dodecahedra: they will be seen as falling into the same class as more common systems and a flat distribution over possible initial conditions will be the best distribution in much the same way in which the 1/*n*rule is the best for all gambling devices.In Sect. 4 we argued that

*if allowed to compete*, some higher-level rules or laws may well deserve to make it into the Best System. Here the question is the prior one, which Lewis answered negatively: should such rules even be allowed to compete?In Schaffer’s notation,

*ch*<*p*_{ e′ },*w*,*t*> is the chance in world*w*, assessed at time*t*, that the proposition*p*_{ e }(asserting that event*e*happens) is true.

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## Acknowledgments

This paper was presented at the IHPST workshop “Probability in Biology and Physics” in Paris, February 2009. We would like to thank the organisers for the opportunity and the audience for stimulating comments. Furthermore, We would like to thank Nancy Cartwright, José Díez, Jossi Berkovitz, Mathias Frisch, Barry Loewer, Alan Hájek, Aidan Lyon, Kristina Musholt, Huw Price, Josefa Toribio, and Eric Winsberg for helpful discussions. Thanks are also due to two anonymous referees for helpful comments. RF acknowledges financial support from Grant FFI2012-37354 of the Spanish Ministry of Science and Innovation (MICINN). CH acknowledges the generous support of Spanish MICINN grants FFI2008-06418-C03-03 and FFI2011-29834-C03-03, AGAUR grant SGR2009-01528, and MICINN Consolider-Ingenio grant CSD2009-00056.

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Frigg, R., Hoefer, C. The Best Humean System for Statistical Mechanics.
*Erkenn* **80**
(Suppl 3), 551–574 (2015). https://doi.org/10.1007/s10670-013-9541-5

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DOI: https://doi.org/10.1007/s10670-013-9541-5