the case of systems of few degrees of freedom. He introduced two further definitions of microcanonical entropy that do not depend on - the volume and surface entropy described above. (Note that the surface entropy differs from the Boltzmann entropy only by an -dependent offset.) The volume entropy and associated form a close analogy to thermodynamic entropy and temperature. It is possible to show exactly that ( is the ensemble average pressure) as expected for the first law of thermodynamics. A similar equation can be found for the surface (Boltzmann) entropy and its associated , however the "pressure" in this equation is a complicated quantity unrelated to the average pressure. The microcanonical and are not entirely satisfactory in their analogy to temperature. Outside of the thermodynamic limit, a number of artefacts occur. Nontrivial result of combining two systems: Two systems, each described by an independent microcanonical ensemble, can be brought into thermal contact and be allowed to equilibriate into a combined system also described by a microcanonical ensemble. Unfortunately, the energy flow between the two systems cannot be predicted based on the initial 's. Even when the initial 's are equal, there may be energy transferred. Moreover, the of the combination is different from the initial values. This contradicts the intuition that temperature should be an intensive quantity, and that two equal-temperature systems should be unaffected by being brought into thermal contact. Strange behavior for few-particle systems: Many results such as the microcanonical Equipartition theorem acquire a one- or two-degree of freedom offset when written in terms of . For a small systems this offset is significant, and so if we make the analogue of entropy, several exceptions need to be made for systems with only one or two degrees of freedom. Spurious negative temperatures: A negative occurs whenever the density of states is decreasing with energy. In some systems the density of states is not monotonic in energy, and so can change sign multiple times as the energy is increased. The preferred solution to these problems is avoid use of the microcanonical ensemble. In many realistic cases a system is thermostatted to a heat bath so that the energy is not precisely known. Then, a more accurate description is the canonical ensemble or grand canonical ensemble, both of which have complete correspondence to thermodynamics. Precise expressions for the ensemble The precise mathematical expression for a statistical ensemble depends on