More on statistical mechanics

Referencing BOTH the Laplace transform AND the Laplace (saddle-point) approximation

The last post on the contact geometry of statistical (many-particle or many-agent) systems took me down a little rabbit hole.

Statistical origins of contact geometry

Looking up some of the concepts I mentioned there I ran into this fascinating question: why are the canonical and microcanonical partition functions Laplace transforms of each other (StackExchange)? This post attempts to reproduce the fascinating answer.

The context is a statistical system as before, with lots of possible (unobservable) microstates in some space Ω. The microcanonical partition function Ω(E) counts the microstates that have the same energy¹ E (say), so ρ(E)=|Ω(E)|/|Ω| is a probability distribution over E. Considering its Laplace transform, against a parameter β, corresponds to the canonical partition function (as mentioned on Wikipedia)

\(Z(\beta) := \mathcal{L}[\Omega](\beta)=\int_0^\infty \Omega(E)e^{-\beta E}dE =: e^{-\beta F(\beta)}\)

which implicitly defines the the free energy F(β):=-β⁻¹ log Z. Similarly, formally²,

\(\Omega(E) = \mathcal{L}^{-1}[Z](E) = \int_0^\infty Z(\beta) e^{\beta E} d\beta\)

But now because E and F(β) are extensive quantities—they scale with the number of particles N—we can apply the Laplace approximation here to get:

\(\Omega(E) = \int_0^\infty e^{-\beta F(\beta)} e^{\beta E}d\beta = \int_0^\infty e^{\beta N(e-f(\beta))}d\beta \sim e^{\beta^* N(e-f(\beta^*))}\)

where β* maximizes the expression in the exponent. The correspondence E↔β* establishes ensemble equivalence (for large N).

Why is this interesting?

We assumed nothing about the microstates, and so assume a uniform distribution over them. Yet when we consider a macroscopic parameter E, in the large-N limit, we see an exponential distribution arise, the Boltzmann distribution or Gibbs measure. That extends to several parameters and motivates the choice in the previous post³:

\(\rho(y;p) = e^{-w-p_iF^i(y)} %\quad\text{ or }\quad \rho(y;p) = e^{-s-p_i(F^i(y)-q^i)}\)

The pretty properties for observables that we saw previously

\(q^i := \langle F^i\rangle = \int \rho(y;p) F^i(y) d\Gamma = -\frac{\partial \log Z(p)}{\partial p_i} =-\frac{\partial w}{\partial p_i} \)

(yes use Ω instead of Γ, and β instead of p) follow from properties of the Laplace transform (Wiki) and how it interchanges multiplication with differentiation. It also shows that interchanging s:=w+pᵢqⁱ involves a (total) Laplace transformation. Notice as well that in thermodynamic limit ∂s/∂pᵢ=∂w/∂pᵢ+qⁱ=0 (for all i) so s is maximal there.

(Interestingly, the Laplace transformation apparently gives a useful reference even in cases where N is not large.)

1. Here I’ll use the terminology familiar from statistical physics.

2. Here we make the heroic assumption that the inverse is calculable and finite, which turns out to be only mildly heroic if you are familiar with terms like analytic continuation (to the complex plane) and contour integral (in complex analysis) which I won’t go into here. Here is the Wikipedia lemma if you’re so inclined.

3. Notice that this generalizes the form in which the canonical ensemble is usually presented:

\(\rho(y) = \frac{1}{Z(\beta)}e^{-\beta H(y)} = e^{-w-\beta H(y)} \quad \text{with} \quad w := \log Z(\beta) \text{ and } (p_1, F^1(y))=(\beta,H(y))\)