General relations to obtain the time-dependent heat capacity from isothermal simulations

It is well-known that time-dependent correlation functions related to temperature and energy can crucially depend on the thermostatting mechanism used in computer simulations of molecular systems. We argue, however, that linear response functions must be considered as universal properties of physical systems. This implies that the classical fluctuation equation for the transient heat capacity, c_v(t), is not applicable to the thermostatted molecular dynamics (apart from long enough times). To improve on this point, we derive a number of exact general expressions for the frequency-dependent heat capacity in terms of energy correlation functions, valid for the Nosé-Hoover and some other thermostats. We also establish a general relation between auto- and cross correlation functions of energy and temperature. Recommendations on how to use these relations to maximize the numerical precision are provided. It is demonstrated that our approach allows us to obtain c_v(t) for a supercooled liquid system with high precision and over many decades in time reflecting all pertinent relaxation processes.