Boltzmann distribution

Question

Let's say we have n points, on which the Boltzmann distribution $P = \{p_1,p_2,...,p_n\}$ is defined. Is it generally true that $\prod_{i=1}^n p_i < \prod_{i=1}^m q_i$ if $Q = \{q_1,...,q_m \}$ is another Boltzmann distribution defined on only $m$ points among the original $n$ points with $m < n$?

The definition of Boltzmann distribution can be found, for example, via When do people actually use the maximum entropy distribution?. More precisely, $$p_i = \frac{e^{-\beta E_i}}{\sum_j e^{-\beta E_j}}$$.

Answer 1

The answer is yes. Indeed, by rescaling the $E_i$'s, without loss of generality (wlog) $\beta=1$. Also, by obvious induction on $m$, wlog $Q=\{p_1,\dots,p_{n-1}\}$.

Then the inequality in question becomes $$ \frac {\exp\big\{-\sum_{j=1}^n E_j\big\}}{\big(\sum_{j=1}^n e^{-E_j}\big)^n}< \frac {\exp\big\{-\sum_{j=1}^{n-1}E_j\big\}}{\big(\sum_{j=1}^{n-1}e^{-E_j}\big)^{n-1}},$$ which can be rewritten as $$e^{-E_n}\Big(\sum_{j=1}^{n-1}e^{-E_j}\Big)^{n-1}<\Big(\sum_{j=1}^n e^{-E_j}\Big)^n,$$ which is obvious, for any nonnegative values $E_j$ of energy.