The Loschmidt Reversibility Objection

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The Loschmidt Reversibility Objection

Some context

To summarize Boltzmann’s account from the last couple of lectures: he introduces a probabilistic reduction of the second law of thermodynamics, by claiming that an increase in entropy is extremely likely, but cannot be certain, as the micro-dynamics are time-reversible. His law is as follows, if entropy is far from the maximum possible for the system:

For any t2 > t1, Sʙ(t2) > Sʙ(t1)

So what does it mean for entropy to be probabilistic? One way to introduce probability is by appeal to “macro-probabilities” using the Proportionality Postulate, where the probability of a macro-state is proportional to the restricted Lebesgue measure, so that the approach to equilibrium is the evolution from an unlikely to a likely macro-state. Another way to introduce probability is by appeal to “micro-probabilities,” using the Statistical Postulate, where the probability that a micro-state lies in a sub-region A of a macro-region is a ratio of the measure of itself to that macro-region. In this second case the truth depends on the dynamics. 

The trouble with the first case (macro-probabilities): a system does not have to move from an improbable to a probable state. The number of improbable states might be higher. The trouble with the second case (micro-probabilities): time symmetry implies that if a system is in low entropy it must be so in virtue of some antecedent high entropy.

Loschmidt

Lodschmidt elaborates on the second problem, and claims further that if transition from low to high entropy is possible, it must be possible that transition can occur from high to low entropy. This argument can be formalized as follows:

P1. If a transition from state xᵢ to state xᶠ in time span ∆ is possible, then the transition from state Rxᶠ to state Rxᵢ in time span ∆ is possible as well

P2. The Boltzmann entropy is invariant under R

C. If a transition from entropy Sᵢ to higher entropy Sᶠ is possible, so is a transition from higher entropy Sᶠ to lower entropy Sᵢ

So Loschmidt’s objection reveals a contradiction with the 2^nd law, resulting in what is called “Loschmidt’s paradox.” The objection can be summarized quite simply as follows: it should not be possible to deduce an irreversible process from time-symmetric dynamics.

Frigg (2012)

According to Frigg it might be possible to respond to Loschmidt by a reminder that Boltzmann’s law is probabilistic, not universal, hence it can allow for some unwanted transitions even if the transitions are unlikely. However, this response would be inadequate: in virtue of time-invariance, if a system’s evolution from low to high entropy is very likely, the system is also very likely to have evolved from an antecedent higher entropy. This result is contrary to intuition, let alone Boltzmann’s law: it implies that my cold cup of tea, which has transitioned from its previous high temperature, is only in its current state in virtue of the antecedent lower temperature it must have had at the outset. 

Frigg also mentions Zermelo’s Recurrence Objection: Poincare had pointed out that almost every point in a system’s phase space lies on a trajectory that will return arbitrarily close to the same point after some finite time. For Zermelo, this means that entropy cannot continually increase; rather, there will be some period of time during which entropy decreases. So my cold cup of tea will eventually heat up again of its own accord, and this result also seems considerably counterintuitive.

Solving the problem

To solve Loschmidt’s objection our best option is to assume that an entire system has had some initial state of low entropy prior to any transitions. So, in short, we have to conditionalise the system’s initial state. This entails a replacement of the statistical postulate (which makes no reference to a system’s past.) To create an appropriate replacement, we restrict our consideration of states to those which have the right past – those which started off in the system’s initial state. Having restricted our consideration in this way, we can then ask what portion of micro-states has a higher entropy future, appealing to some image of the initial state under the dynamics of the system since the process started. Such an image can be formally conveyed in the following replacement of the statistical postulate, SP*: p(A) = m(AÇRt) / m(Rt). So, Frigg concludes, “if we choose A to be the set of those states that have a higher entropy future, then the probability given by SP* for a high entropy future has to come out high.” 

Results

What do you think about the solution Frigg describes? I’m not sure about it – does it indicate that the only way to maintain a probabilistic Boltzmannian account of entropy requires some convention from the outset? It reminds me of the work we did last term on the conventionality of geometry, where the stipulating of forces or rigid bodies had significant sway over the physics. Suppose we choose the big bang as our initial state of the system: this will require the assumption that laws of nature are universal in the sense of being valid all the time and everywhere, so that the system is sufficiently governed by the deterministic laws of classical mechanics throughout. That’s not directly problematic in itself, but it requires commitment. Furthermore, I wonder about ontological parsimony: how much of a transition or experiment is relevant to, -or warrants being included as- a part of the entire system?