Abstract
In this paper we analyze the relation between the notion of typicality and the notion of probability and the related question of how the choice of measure in deterministic theories in physics may be justified. Recently it has been argued that although the notion of typicality is not a probabilistic notion, it plays a crucial role in underwriting probabilistic statements in classical statistical mechanics and in Bohm’s theory. We argue that even in theories with deterministic dynamics, like classical statistical mechanics and Bohm’s theory, the notion of probability can be understood as fundamentally objective, and that it is the notion of probability rather than typicality that may (sometimes) have an explanatory value.
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Notes
Determinism is compatible with probability that does not involve ignorance provided that certain conditions hold. For example, the set of predicted events should be infinite (the frequency interpretation of probability), or the events should be independent (satisfying the conditions for the Law of Large Numbers to hold). These conditions do not hold in the case discussed in this paper. Here we refer to predictions for finite sets of events in sequences where the events are dependent.
For the notion of macrostates in statistical mechanics, see Hemmo and Shenker (2012, Ch. 5).
See Hemmo and Shenker (2012, Ch. 6–7) for more details about the notions of probability and entropy in statistical mechanics.
Of course one can translate the transition probabilities to statements about a probability distribution over initial conditions also in cases where the measure is not invariant under the dynamics. But this is more complicated and perhaps less natural; see also Sect. 4.
Other issues that come up in the context of ergodic dynamics are that: (1) Since the sequences of states of a mechanical system are not strictly independent, the ergodic theorem does not satisfy the weak low of large numbers so that arbitrarily long but finite relative frequencies of macrostates need not approach their probabilities; and (2) real life thermodynamic systems are often non-ergodic, although some such systems may be empirically indistinguishable from ergodic systems; e.g. in the case of KAM dynamics. For further considerations, see Frigg et al. (2011).
We show in Hemmo and Shenker (2012, Ch. 7) that the measure of entropy need not in general be the Lebesgue measure.
Moreover, the notion of entropy, in the thermodynamic sense of the degree in which the energy in the system is exploitable to produce work, may be interpreted in statistical mechanics by the measure of the macrostate only if the Second Law of thermodynamics (in its probabilistic version) is true (see Hemmo and Shenker 2012, Ch. 1). But as we argue elsewhere (see our 2010, 2012, Ch. 13 and Albert 2000, Ch. 5) the Second Law of thermodynamics is not universally true in statistical mechanics.
The term ‘quantum equilibrium’ means that the evolution of the configuration q (as given by the ‘guiding’ equation) is probablistically equivariant with the evolution of the wavefunction (as given by the Schroedinger equation) in the following sense: if the probability distribution over the configuration q is given by the the absolute square of the wavefunction at some time t, then it is equivariant under the dynamics in the sense that the distribution is given by the absolute square of the wavefunction at all other times.
The quantum mechanical absolute squared measure is essentially the only measure that is invariant under the dynamics in Bohm’s theory (see Goldstein and Struyve 2007). This theorem in our view is not relevant to the issue of how typicality considerations may justify probabilistic statements in the same way that Liouville’s theorem is not relevant in the case of classical statistical mechanics.
Indeed, in his approach to statistical mechanics based on the GRW theory, Albert (2000, Ch. 7) attempts to dispense altogether with typicality considerations by conjecturing that each and every quantum state of a thermodynamic system will give rise to normal thermodynamic behavior. See Hemmo and Shenker (2012, Appendix B1).
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Acknowledgments
This research is supported by the Israel Academy of Science, Grant Number 713/10 and by the German-Israel Foundation, Grant Number 1054/09.
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Hemmo, M., Shenker, O. Probability and Typicality in Deterministic Physics. Erkenn 80 (Suppl 3), 575–586 (2015). https://doi.org/10.1007/s10670-014-9683-0
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DOI: https://doi.org/10.1007/s10670-014-9683-0