On The Calculation Of The Heat Capacity In Path Integral Monte Carlo Simulations

In Path Integral Monte Carlo simulations the systems partition function is mapped to an equivalent classical one at the expense of a temperature-dependent Hamiltonian with an additional imaginary time dimension. As a consequence the standard relation linking the heat capacityCvto the energy fluctuations, − 2, which is useful in standard classical problems with temperature-independent Hamiltonian, becomes invalid. Instead, it gets replaced by the general relation$\left\langle {C_\upsilon } \right\rangle _P = k_B N\beta ^2 \left( {\left\langle {E^2 } \right\rangle _P - \left\langle E \right\rangle _P^2 } \right) - k_B \beta ^2 \left\langle {\partial _\beta E} \right\rangle _P $for the intensive heat capacity estimator; β being the inverse temperature and the subscript P indicates the P-fold discretization in the imaginary time direction. This heatcapacity estimator has the advantage of being based directly on the energy estimatorand thus requires no extra computational effort and is suited for extensive phase diagramstudies. As an example, numerical results are presented for a two-dimensional fluid withinternal magnetic quantum degrees of freedom. We discuss in detail origin and consequences of the excess term. Due to the subtraction of two relatively large contributions ofsimilar absolute magnitude a large statistical effort would be necessary for very accurateheat capacity estimates.