The Gibbs Approach

I will give an overview of the Gibbs Approach as described by Frigg and the foundational issues that arise in this approach. I will end with some questions for discussion.

 

This approach differs quite fundamentally both from the Bolzmanian approach and everyday physics in that it does not describe and analyse the single system of interest. Instead SM now deals with an uncountably infinite collection of systems, an "ensemble". For all these systems the Hamiltonian is the same but they are distributed over different states (I guess, all possible states in phase space).
This ensemble as a whole is now the object of study. Its state is described by combining the states of all systems (i.e. the point in phase space of each) in a function ρ over the phase space. This function is treated as the probability density reflecting the probability of finding some random system from the ensemble to be in a state in a certain region of phase space.
This approach allows for a treatment of physical magnitudes observed in experiments as expectation values: such a magnitude is expressed by a function, and together with the density function the expected value can be calculated, which is the prediction of the formalism for the outcome of the experiment, the “phase average".
Now for equilibrium a necessary condition is the stationarity of the distribution, that is, if the macro-system is in equilibrium, the density function of the ensemble that describes that macro-state as an average does not change over time. An important kind of stationary distribution is the distribution that maximises Gibbsian entropy ${\mathrm{S<span\; style="font-size:\; xx-small;">G</span>}}_{}\left(\rho \right)$, given some assumption constraining energy and the number of particles. Depending on those assumptions three important distributions result, the micro-canonical, the canonical and the grand-canonical distribution.

This whole procedure is empirically very successful but conceptually puzzling. There are essentially three problems with it.

The first is the analysis of single macro-states in terms of ensembles. How can a theory make true statements and successful predictions about something that it doesn't describe? The typical textbook solution that combines ergodicity and time averages fails, because the idea of infinite time averages is untenable. One solution by Malament and Zabell makes use only of ergodicity and drops the time averages. Not all relevant systems are ergodic, but perhaps they are $ϵ$-ergodic. Another approach restricts the theory to systems with large degrees of freedom and the functions to so-called sum functions. These two guarantee that a system behaves as if it was ergodic. Both of these approaches are problematic and unresolved, according to Frigg.

The second problem is the problem of interpreting probability that we know from quantum mechanics. The three options on the market are frequentism, time averages and an epistemic interpretation. All three are difficult and controversial (also a familiar insight from quantum mechanics).
The third problem is that the Gibbs approach does not work for non-equilibrium states. For the formalism implies a constant Gibbs entropy, which conflicts with the idea of thermodynamical entropy. Furthermore there can't be a change from stationary distributions to non-stationary ones or vice versa. Frigg lists a number of approaches that try to adress this problem.

A couple of questions to discuss are:

• When describing a problem of frequentism Frigg claims that it is problematic to see an ensemble as an urn from which one system can be drawn. I don't see why that is. It does not seem absurd to think of this abstractly as choosing randomly one of all the systems in the ensemble, just like taking a ball from an urn.
• I wonder how different interpretations of probability could lead to different formalisms, as Frigg claims in (2008).
• Also I am not quite sure how exactly the approach solves the main problems of recurrence and reversal that the Bolzman approach faces. Does the formalism in terms of ensembles not have these problems? 

The Loschmidt Reversibility Objection

Get ready for more of my mind-blowing science...!

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?