I would greatly appreciate any reference that solves the following problem (or a variant of it). If you know it is open, please say so.
Motivation: Suppose you have a solid (say a block of steel) surrounded by an ideal gas. Both are inside an insulated container. Thermodynamics tells us that after some time, they will both reach the same temperature. I want to justify this via statistical mechanics. To simplify the discussion, I will assume that the solid and the gas have the same number of atoms.
Model for the solid (simplified): Consider the lattice $$\mathcal{L}=\left\{(a,b,c):\ a,b,c\in\{0,1,\dots,n-1\} \right\}$$ These will be the equilibrium positions of the atoms of the solid. Thus, we have $N=n^3$ solid atoms. Let $x_i\in\mathbb{R}^3$ be the displacement of the $i^\text{th}$ solid atom from its position on the lattice, and let $v_i$ its velocity. To prevent the solid from roaming around, we impose the constraint that the atoms corresponding to the lattice positions $(0,0,0), (n-1,0,0)$ and $(0,n-1,0)$ are fixed, i.e., their displacements and velocities are always zero. About the interactions, I assume that two atoms interact only if they are nearest neighbours (NN) on the lattice. The interaction is done via a potential $V(\|x_i-x_j\|)$. The total potential energy of the solid is $$\mathcal{V}(x_1,\dots,x_N)=\sum_{i,j \text{ are NN}} V(\|x_i-x_j\|)$$
Hamiltonian and the energy surface: Let $u_1,\dots,u_N$ the velocities of the atoms of the ideal gas. The total Hamiltonian is
$$H(x_1,\dots,x_N,v_1,\dots,v_N,u_1,\dots,u_N)=\mathcal{V}(x_1,\dots,x_N)+\sum_i \|v_i\|^2+\sum_i\|u_i\|^2$$ Let $E$ be the total energy of the system. The energy (hyper)-surface is
$$H(x_1,\dots,x_N,v_1,\dots,v_N,u_1,\dots,u_N)=E$$
and is a $\ 2\cdot 3 (N-3)+3N-1=9N-19$ dimensional surface (since we have three points fixed).
The Goal: The goal is to prove (by adding perhaps some additional mild assumptions) that there exists an energy $E_{solid}^*$ such that if we sample a phase point from the energy surface according to the measure $\frac{d\Sigma}{\|\nabla H\|}$, then with overwhelming probability, the average energy of the solid:
$$\frac{\mathcal{V}(x_1,\dots,x_n)+\sum_i \|v_i\|^2}{N}$$
will be very close to $E_{solid}^*$.
Comments
- Quadratic is easy: If $V$ is quadratic, then with the usual change of variables (as in the equipartition theorem), we get $E_{solid}^*=\frac{2}{3}\cdot \frac{E}{N}$.
- Why non-quadratic (aka anharmonic) is difficult: Consider the energy surface
$$\mathcal{V}(x_1,\dots,x_n)=E_1$$
This is an $3(N-3)$ dimensional surface. Let $\Omega(E_1)$ be the "area" of that surface, according to the measure $\frac{d\Sigma}{\|\nabla \mathcal{V}\|}$. By standard statistical mechanics, we know that the distribution of the total potential energy of the solid has density
$$f(E_1)\propto \Omega(E_1)\cdot \left(\sqrt{E-E_1}\right) ^{6N-11}$$
where the second term comes from the quadratic part (velocities). To argue that $f(E_1)$ is very peaked at some energy, we need to analyze $\Omega(E_1)$. If $\mathcal{V}(x_1,\dots,x_n)$ was a "sum-function" i.e., $\mathcal{V}(x_1,\dots,x_n)=\sum_i F_i(x_i)$ then we could do it via the Central Limit Theorem (as in [1]). In some special cases, we can do some change of variables and reduce to the sum-function case. Here, new ideas are needed.
[1]: "Mathematical Foundations Of Statistical Mechanics", A. I. Khinchin
