Inequivalence of Ensembles in Statistical Mechanics

For studying the thermodynamic properties of systems using statistical mechanics we propose an ensemble that lies in between the familiar canonical and microcanonical ensembles. From a comparative study of these ensembles we conclude that all these ensembles may not yield the same results even in the thermodynamic limit except at high temperatures. An investigation of the coupling between systems suggests that the state of thermodynamic equilibrium is a special case of statistical equilibrium. As a byproduct of this analysis we have obtained a general form for probability density function in an interval. I. INTRODUCTION In studying the thermodynamic properties of systems using statistical mechanics, we restrict ourselves to a constant energy surface since we know that the energy of the system under consideration is constant. The basic assumption of ergodicity which helps us replace the time average of observables by the corresponding phase averages leads us naturally to a constant density on the energy surface given by