Kinetic energy and microcanonical nonanalyticities in finite and infinite systems

abstract: In contrast to the canonical case, microcanonical thermodynamic functions can show nonanalyticities also for finite systems. In this paper we contribute to the understanding of these nonanalyticities by working out the relation between nonanalyticities of the microcanonical entropy and its configurational counterpart. If the configurational microcanonical entropy omega_N_c(v) has a nonanalyticity at v = v_c, then the microcanonical entropy omega_N(e) has a nonanalyticity at the same value e = v_c of its argument for any finite value of the number of degrees of freedom N. The presence of the kinetic energy weakens the nonanalyticities such that, if the configurational entropy is p-times differentiable, the entropy is p+\lfloor N/2 \rfloor -times differentiable. In the thermodynamic limit, however, the behaviour is very different: the nonanalyticities no longer occur at the same values of the arguments, and the nonanalyticity of the microcanonical entropy is shifted to a larger energy. These results give a general explanation of the peculiar behaviour previously observed for the mean-field spherical model. With the hypercubic model we provide a further example illustrating our results.