Liyao Wang

In classical statistical mechanics, the heuristic that in the canonical ensemble the energy distribution is sharply peaked at the mean energy is crucial in justifying the equivalence between the canonical and microcanonical ensembles. It turns out that this is closely related with the fundamental notion of a typical set and the Shannon-McMillan-Breiman theorem in information theory. In this dissertation, we explore the connections between the two and establish some new rigorous results that are of interest to both fields. In the first part of this dissertation, we show that if the heat capacity $C_{v,n,\beta}$ is bounded by $nc$, with $n$ being the number of degrees of freedom and $c$ being a constant independent of $n$ and $\beta$, then the energy distribution is “sharply peaked” around the mean energy. Our result is expressed mathematically as a concentration inequality. It is also demonstrated that if the average third moment of $\beta(H_n-\langle H_n\rangle)$ is of order $o(n^{\frac{3}{2}})$, then the energy distribution is approximately a Gaussian distribution centered at the mean energy, with variance being $C_{v,n,\beta}T^2$. In the second part, we first show that if the Hamiltonian $H_n$ is convex, then the heat capacity $C_{v,n,\beta}$ is bounded by $n$, the number of degrees of freedom. The bound $n$ is achieved by some convex Hamiltonian and hence is optimal in this sense. The universal and optimal nature of the bound is perhaps surprising from a physical point of view. We also show that assuming convexity of $H_n$, the absolute value of the average third moment of $\beta(H_n-\langle H_n\rangle)$ is bounded by a (complicated) function $g(n)$, depending only on $n$. $g$ is plotted numerically up to some point and we suspect $g(n)=o(n^{\frac{3}{2}})$, which would imply the closeness to Gaussian of the energy distribution.