Concentration of measure in statistical mechanics

Question

I have just finished a course in advanced probability theory (martingales, Brownian motion, Ito calculus, concentration inequalities, Stein's method) and an undergraduate course in stat mech. I am interested in the connection between these two courses. As I am reading Wikipedia's page on concentration of measure, I note there is a section on this phenomenon in physics, where it is remarked that "All classical statistical physics is based on the concentration of measure phenomena"! This seemed an astounding fact to me, and since the page had no actual maths on it besides links to academic papers that were too opaque to parse, I wanted to ask about this here. In particular, this paragraph I would like to understand (and currently don't).

The fundamental idea (‘theorem’) about equivalence of ensembles in thermodynamic limit is exactly the thin shell concentration theorem. For each mechanical system consider the phase space equipped by the invariant Liouville measure (the phase volume) and conserving energy E. The microcanonical ensemble is just an invariant distribution over the surface of constant energy E obtained by Gibbs as the limit of distributions in phase space with constant density in thin layers between the surfaces of states with energy E and with energy E+ΔE. The canonical ensemble is given by the probability density in the phase space (with respect to the phase volume) $\rho =e^{\frac{F-E}{kT}}$ where quantities constants $F, T$ are defined by the conditions of probability normalisation and the given expectation of energy E.

I understand that the thin shell concentration theorem just refers to the fact that high-dimensional (say in $\mathbb{R}^n$) convex measures (eg high-dim Gaussian vectors) concentrate their norm around $\sqrt{n}$. But I don't know what the Louiville measure is (I haven't taken any physics besides stat mech) or what it has to do with this stat mech setting, or what all the talk about "surfaces of constant X" (what space are these surfaces embedded in?) is about.

Answer 1

It's an exaggeration to say that all of statistical mechanics is based on concentration of measure, but hopefully the sketch below is helpful.

The space in which all this is taking place is the topic of study in Lagrangian and Hamiltonian mechanics. A good reference is the book Mathematical Methods of Classical Mechanics by Vladimir Arnol'd, but for now it can just be thought of as the combination of the position $(x_1, x_2, x_3)$ and velocity $(v_1, v_2, v_3)$ for all of the particles of the gas. So the space is $\mathbb{R}^{6N}$ for an $N$-particle gas. In physics, a space like this one that contains both the position and the velocity data is called a phase space (the other common possibility is the configuration space, which just has the position data).

The Liouville measure is the way to integrate a volume on this space. For now you can just think of it as $dx_idv_i$, the product over all the $6N$ coordinates.

The microcanonical distribution is the distribution where all states have the same energy $E$, and the assumption of a priori equal probabilities holds. The set of states with energy $E$ forms a surface, $S_E$. (Generally, the level set of an equation forms a surface, the level set.) We can write the distribution function as $\frac{1}{Z_E} \cdot \frac{1}{\nabla E} dS_{E}$, where $Z_E$ is the microcanonical partition function with energy $E$, $\nabla E$ is the gradient of the energy function $E$ with respect to the coordinates $x_i, v_i$, and $dS_{E}$ is the surface volume element. (The gradient comes from changing variables.)

The canonical distribution doesn't fix the energy, but instead uses a Boltzmann weight with energy dependence; it has distribution function $\frac{1}{Z} e^{-\beta E}$, where $Z$ is the canonical partition function and $\beta$ is a thermodynamic parameter (inverse temperature).

It's a general result that in the thermodynamic limit $N \to \infty$, many physically meaningful calculations give the same result no matter which distribution is used (assuming we choose compatible values of $E$ in the microcanonical distribution and $\beta$ in the canonical distribution). It's not a mathematical theorem, because it depends on the exact nature of $E$ as a function of the $x_i$ and $v_i$, but it turns out to be true for many of the energy functions that show in practice, so it's considered a general result of statistical mechanics.

As an example, the Wikipedia article mentions that Khinchin proved this result in certain cases. The major steps in his proof were:

• Restrict to the case where the energy function can be written as the sum of functions, each depending on different variables. This often happens in practice; for example, in a monatomic ideal gas the energy function is just the kinetic energy $E = \sum \frac{1}{2}v_i^2$.
• Write the energy function as the sum of functions $E = E_S + E_L$, where $E_S$ and $E_L$ depend on different variables, and $E_S$ is small compared to $E_L$, so that $E_S = o(E_L)$.
• Show that in the canonical distribution, $E_S$ and $E_L$ are independent, and therefore the distribution of $E_S$ is just $\frac{1}{Z_S} e^{-\beta E_S}$.
• In the microcanonical distribution, $E_S$ and $E_L$ are not independent, because of the constraint $E = E_S + E_L$, where $E$ is fixed. Since $E_S = E - E_L$, we can compute the distribution of $E_S$ from the distribution of $E_L$. If $E_L = E_{L_1} + \ldots + E_{L_d}$ is itself the sum of small components, we can use concentration of measure or the central limit theorem to derive an asymptotic formula for the distribution of $E_L$. The distribution function of $E_S$ will turn out to be $\frac{1}{Z_S} e^{-\beta^{*}E_S} \cdot (1 + o(1))$, for some constant $\beta^{*}$.
• If we consider the canonical distribution with $\beta = \beta^{*}$, then we can show that if $E = E_S + E_L$ as above, then $E_S$ and $E_L$ are actually independent, and we just get the distribution of $E_S$ to be $\frac{1}{Z_S} e^{-\beta^{*} E_S}$.
• Since the errors in the distributions of $E_S$ are $o(1)$, the total error over the sums of all the small systems is $o(N)$. Thermodynamically relevant quantities are averages over the whole system (for instance, average kinetic energy of a gas, or the specific heat capacity). When we divide by $N$, the $\frac{o(N)}{N}$ term will vanish, so we get the same answer from both ensembles.