work Heat and work can be distinguished if we take the underlying quantum nature of the system into account. For a closed system (no transfer of matter), heat in statistical mechanics is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in occupation numbers of the quantum energy levels of the system, without change in the values of the energy levels themselves. Work is the energy transfer associated with an ordered, macroscopic action on the system. If this action acts very slowly, then the adiabatic theorem of quantum mechanics implies that this will not cause jumps between energy levels of the system. In this case, the internal energy of the system only changes due to a change of the system's energy levels. The microscopic, quantum definitions of heat and work are the following: so that The two above definitions of heat and work are among the few expressions of statistical mechanics where the thermodynamic quantities defined in the quantum case find no analogous definition in the classical limit. The reason is that classical microstates are not defined in relation to a precise associated quantum microstate, which means that when work changes the total energy available for distribution among the classical microstates of the system, the energy levels (so to speak) of the microstates do not follow this change. The microstate in phase space Classical phase space The description of a classical system of F degrees of freedom may be stated in terms of a 2F dimensional phase space, whose coordinate axes consist of the F generalized coordinates qi of the system, and its F generalized momenta pi. The microstate of such a system will be specified by a single point in the phase space. But for a system with a huge number of degrees of freedom its exact microstate usually is not important. So the phase space can be divided into cells of the size h0 = ΔqiΔpi, each treated as a microstate. Now the microstates are discrete and countable and the internal energy U has no longer an exact value but is between U and U+δU, with . The number of microstates Ω that a closed system can occupy is proportional to its phase space volume: where is an Indicator function. It is 1 if the Hamilton function H(x) at the point x = (q,p) in phase space is between U and U+