Dispensing with the "a priori equal probability" postulate

Question

I find the "a priori equal probability postulate" in statistical mechanics terribly frustrating.

I look at two different states of a system, and they look very different. The particles are moving in different ways! I like to imagine that if you zoom in on the first system you might spot a dozen particles flying off, dancing along to their own chaotic 12-body problem, while in the second system 2 particles might be shooting off, trapped in a romantic orbit. I see absolutely no reason for assuming that two completely different micro-states with the same total energy will occur with equal frequency.

As far as I can see, Laplace's Principle of Insufficient Reason works great for decks of cards, because cards are all the same, except for their labels. But there's more than labelling going on with a physical system of atoms - different states are actually different!

So I want to ask about an alternative justification for this principle, which doesn't assume it a priori. I found the argument in Landau and Lifchitz, but they don't discuss how it relates to the aforementioned postulate.

We imagine that the microstate of our system at some time $t$ is governed by some probability distribution, $\rho(x;t)$ in phase space. Then the probability density function will evolve over time, but will remain constant at a point moving along with the phase-flow - this follows from Liouville's theorem and probabilistic common sense.

Now the only sensible definition for an equilibrium distribution I can see is one in which the pdf stops evolving, $\rho(x;t) = \rho(x;t+\Delta t)$ for all $\Delta t$. But this means that the pdf (which we can just call $\rho(x)$ now) must be constant along trajectories, and so is a constant of the motion.

But this means $\log \rho(x)$ is a constant of the motion. But this is additive (because probabilities multiply), and mechanical systems only have 7 additive integrals of the motion: energy, momentum and angular momentum. And if we assume Galilean relativity and say our system isn't spinning, then if the probability distribution is an equilibrium distribution, it must be a function of the energy alone (actually, to retain additivity, we must just have $\log \rho \propto \text{Energy}+\text{Const}$). And crucially, if we fix the energy, then $\rho$ must be uniform.

It seems to me this argument gives the "a priori equal probability postulate" with no major assumptions. We simply said:

1. Our system's state is governed by some probability distribution (seems reasonable).
2. This distribution, whatever it is, should evolve in time so it is constant with the phase flow (again, given Liouville's theorem this just seems obvious.)
3. This distribution shouldn't change over time (our macroscopic system at equilibrium doesn't seem to be changing, so whatever probability distribution we pick, it shouldn't change either.)

And from this we derive that our probability distribution is a function of the energy alone. If we add a fourth requirement:

4. The total energy should be constant.

Then we see the only way of satisfying all 4 of these reasonable-looking requirements is, remarkably, the microcanonical distribution. The equal probability postulate is our first beautiful theorem rather than a doubtful, unjustified assumption.

My question is, is this argument legitimate? If not, where is the flaw? And can this flaw be patched up? This postulate is so ubiquitous in statistical mechanics texts that I can't believe it can be dispensed with so simply.

Answer 1

I can see different subtleties in Landau's argument. First of all, it isn't entirely clear what is meant by "there are only seven additive constants of motion". To give an example, consider a single particle hamiltonian: $$H=\frac{\mathbf p^2}{2m}+m\omega ^2\frac{\mathbf q ^2}{2}.$$ For this hamiltonian there are several conserved quantities: $$e(\mathbf p,\mathbf q)=H(\mathbf p,\mathbf q),$$ $$e_1 (\mathbf p ,\mathbf q)=\frac{p_1^2}{2m}+m\omega ^2 \frac{q_1 ^2}{2},$$ etc.

Note that for a system of $N$ of these particles, the quantity: $$E(\{\mathbf p _i\},\{\mathbf q _i\})=\sum _{i=1}^N e_1(\mathbf p _i ,\mathbf q _i)$$ is conserved and perfectly qualifies as an additive constant of motion. As another example, in his treatise 1, Gibbs considers the possibility that for a system composed of $n$ oscillators with $k\leq n$ different frequencies $\omega _i$, a (canonical) distribution of the form:$$\log \rho = \alpha -\beta _1 E_1 -\beta _2 E_2 -\dots -\beta _k E_k$$ applies (here $E_i$ are the energies associated to the frequency $\omega _i$, that are separately conserved).

So it seems that the whole discussion is really making the assumption that the pdf depends only on $\mathbf P , \mathbf L , E$, which may be seen as the only constants of motion which aren't specific of any particular hamiltonian.

To better clarify this point, suppose that we have two completely separated systems "1" and "2", for which is separately true that the equilibrium pdf $\rho _i$ depends only on $E_i,P_i,L_i$. Since the systems don't interact with each other, statistical independence allows us to conclude that the composite system's pdf satisfies: $$\log \rho = \log \rho _1 +\log \rho _2,$$ however the left hand side is a function of $E_1,E_2,P_1,P_2,L_1,L_2$ and so it must be $\rho$. Landau's argument, "the pdf is additive and there are only 7 additive constants of motion", if taken literally, would allow us to conclude that the pdf of this composite system "1"+"2" depends only on $E,P,L$, which is ofcourse false in this situation. Indeed, it is precisely the (additional) assumption that also $\rho$ is a function only of $E=E_1+E_2$ etc., that allows us to conclude that $\rho$ is actually a linear function of those quantities. Clearly this assumption makes sense if the systems are actually in thermal contact and if they are large enough so that the surface energy beetween the two can be neglected.

In the particular case of the microcanonical ensemble, one evident problem in the assumption that the distribution is a function only of $E$, in particular a $\delta$ function, is that there could be subregions of this surface, having a finite measure, that are never reached during the system motion. In the example of the oscillators given above, the points of the surface with $E_1(p,q)\neq E_1(p(0),q(0))$ are never reached during the system's time evolution. I honestly don't know much about the subject, but the key word here is ergodicity. The ergodic theorem states that given a measure space $(X,\mu)$ with a measure preserving time evolution map $U^t$, the necessary and sufficient condition for the time averages to exist and agree with the ensemble average (with respect to $\mu$) is that the space is metrically indecomposable. This means that if $A\subset X$ and $U^t A = A$ for all $t$, then $\mu (A)=\mu (X)$ or $\mu (A)=0$. I think that Landau is alluding to this in a footnote of §4.

To summarize, if you want to obtain the microcanonical ensemble, at some point you have to introduce some additional assumptions which are external to classical mechanics. The "a-priori equal probability postulate" is a way to do this (probably not the only), which must be taken as a phenomenological assertion regarding systems, justified by its agreement with experiment.

1 J.W. Gibbs, Elementary principles in statistical mechanics

Answer 2

Without analyzing your (or Landau's) argument in depth, I'd like to point you to other alternatives to Laplacian Insufficient Reason. The idea of getting rid of Insufficient Reason is not new and was one of the drivers that motivated the great probability theorist and physicist Edwin T Jaynes, who was unsatisfied with Insufficient Reason unless clear physical or other symmetries of a system motivated it. A beautifully written summary (one of the best passages of technical writing of all time IMO) of his fundamental ideas is to be found in Section 2 of this paper:

E. T. Jaynes, "Information Theory and Statistical Mechanics", Phys. Rev. 106, number 4, pp 620-630, 1965

In short, he looks to the Bayesian notion of finding the most "unbiased" distribution of energies: what probability distribution leaves the maximum possible residual uncertainty about the problem, i.e. what probability distribution constrains the observed data (total energy E and mean energy per particle $\langle E\rangle$) the least so that a sequence of samples of individual particle energies from the probability distribution contains the most information, i.e. is least predictable. The answer is the microcanonical distribution. One is thus inferring least from the data.

Now Jayne's Maxent ideas are not universally accepted and there has been a great deal of work since. But my point is that you should not be shocked that there are challenges to Insufficient Reason, particularly by someone with as broad and deep outlook as Landau. Many physicists have refused to accept Insufficient Reason and sought either to find deeper justification of it, or supplant it.

Answer 3

Landau--Lifschitz is not reliable. I never recommend it to someone who does not already know the basics. It can often be inspiring....to an advanced physicist.
This "proof" if a famous fallacy. It relies on his argument that the distribution function of the combination of two independent subsystems is the product of their individual distribution functions. Now, when he argues in favour of this much earlier in the book, he explicitly points out that this is approximately true provided the interaction between all the subsystems of the total system is weak, each subsystem is small compared to the total, and their combination is also still small.
But he "forgets" all these conditions by the time he gets to the part of the book you cite. So you cannot assume that the logarithm is additive. Within certain limits and for certain purposes it is approximately additive...but not really.

There is a vast literature in the philosophy of science on how to justify the foundations of Statistical Mechanics, and Landau's line of reasoning is not accepted. I recommend Janneke van Lith (2001). Ergodic Theory, Interpretations of Probability and the Foundations of Statistical Mechanics. for a good survey of recent attempts. None are successfull (except mine ;-P)

Somewhere or other one must let the number of degrees of freedom go to infinity, or the statements of Stat Mech and Thermodynamics are not exactly true, only approximately true. His "proof" doesn't do this explicitly. So it is obviously wrong.

Dr. van Lith's doctoral thesis is open access http://dspace.library.uu.nl/bitstream/handle/1874/657/full.pdf?sequence=1

as is her review of Guttmann's misch-masch http://www.projects.science.uu.nl/igg/dis/guttmann.html