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Guttmann also considers but finds wanting the ergodic approach to statistical mechanics. The ergodic approach treats probabilities as relative waiting times during which a particle stays in a given portion of phase space (that is, the hypothetical space that describes all the potential states of a system). Thus for a particle that spends half its time in a given portion of phase space, the corresponding probability of that portion of phase space is 1/2. Although the ergodic approach offers a powerful way of understanding statistical mechanics and is a fertile area of mathematical research, it falls short as a justification for applying probabilities to particle systems where the individual particles are controlled by deterministic physical laws. Many assumptions with no physical justification need to be incorporated (e.g., assumptions about the limiting behavior of particle trajectories and about the type of flows in phase space).
Which leaves the pragmatist approach. As I indicated, I'm sympathetic to this approach. Indeed, I think it the best of the alternatives to the objectivist approaches that Guttmann considers. But Guttmann treats haphazardly what to my mind is the key question raised by the pragmatist approach. The pragmatist approach, when applied to statistical mechanics, enjoins us to use probabilities because they work—they enable us to construct a predictively ac curate and scientifically fecund theory. But why should they do so? As we have seen, physicists are not inclined to pursue this question, "be cause discussing it is not likely to improve our ability to make better predictions, to discover new effects, or to explain phenomena that we do not yet under stand." And that is why, Guttmann says, "it is the task of a philosopher" to answer this question.
Objectivist approaches fail because the underlying dynamics of individual particles, at least as far as statistical mechanics is concerned, is deterministic (bringing quantum mechanics into the mix doesn't help here since we still have to deal with ensembles of particles and statistics gets applied to the ensembles). And as we've seen, subjective and ergodic approaches are also unsuccessful, either failing to connect with physical reality or adding what amounts to a frequentist superstructure with all the problems that raises. Why then do probabilities work? More precisely, why is the mechanics for ensembles of particles statistical?
It's precisely at this point, just when he is homing in on the central question, that Guttmann seems repeatedly to lapse into the sorts of loose justifications that the founders and shapers of statistical mechanics employed to motivate their probabilities in the first place. Why, for instance, in an ensemble of particles does the future behavior of a particle become stochastically independent of its past behavior as the time separating past and future increases? As the time increases, the particle, being part of an ensemble, will undergo a lot of collisions with other particles. All those collisions will tend to wash out any influence the past behavior of the particle has on its (distant) future behavior. Justifications like this are intuitive and vague. But they can be given precise mathematical form, and such mathematical forms can in turn be used to construct a powerful physical theory, to wit, statistical mechanics.
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