I don't see what the big deal is. Let's try the ideal gas. For starters temperature ∝ Kinetic energy We have two rooms. One were molecules all travel at spee

I don't see what the big deal is. Let's try the ideal gas. For starters temperature ∝ Kinetic energy We have two rooms. One were molecules all travel at speed A and other room molecules travel at speed B and we open a small window. Eventually, over a long period of time the temerature in both rooms will settle at a temperature between that of B and of A. Let's try to formule this mathematically. This is something like the mean value theorem in calculus that f'(C)(B-A) = f(B) - f(A) for some intermediate value C. And here are function f(C) is the equilibrium temperature. In statistical mechanics we imagine we could count the number of particles -- 10^23 or 10^25 -- something very large. And some fraction M travel at speed A and N-M of them travel at speed B. And we count the probabilities of various mixtures occurring. Feynman Lectures on Computation is a great book https://www.amazon.com/Feynman-Lectures-Computation-Richard-P/dp/0738202967 https://www.amazon.com/Feynman-Lectures-Computation-Richard-...