On the Equilibrium Fluctuations in a Microcanonical Ensemble

Firstly, we describe fluctuations in the occupation numbers of the energy levels in the microcanonical ensemble using identities that we have derived for the purpose, which allow us to calculate the moments of the occupation numbers. Then we compute analytically the probability distribution of these fluctuations. We show that, for every system of finite N , fluctuations about the equilibrium distribution do in fact depend on the temperature. Indeed, at higher temperatures the fluctuations can be so large that the system does not fully converge on the Maxwell-Boltzmann distribution but actually fluctuates around it. We term this state, where not one but a region of macrostates closely fit the underlying distribution, a “fluctuating equilibrium”. Finally, we speculate on how this finding is applicable to financial and other thermodynamic-like systems.