Equilibrium distribution functions: connection with microscopic dynamics

Standard textbook derivations of the equilibrium distribution function rely on assumptions that may not satisfy all readers. Here, we present a straightforward approach to derive the equilibrium distribution function from the microscopic dynamics, and review how it can be used to obtain the expected expressions. In molecular dynamics simulations the equations of motion are often modified to simulate different ensembles or phenomena. We show that in some cases these equations will sample an equilibrium ensemble whereas in other cases they will not. For example, we find that for charged particles driven by a field, an equilibrium distribution is only possible when the system is confined. Furthermore, the approach correctly predicts that neither SLLOD shear flow dynamics nor constant temperature dynamics with a Berendsen thermostat sample any time-independent phase space distributions.