On the minimum entropy of a large system at low temperatures

This paper examines the general question: How far can one set lower limits on the entropy density S(T) of a large system at low temperatures by using data of a different type, e.g., scattering data or static susceptibilities? It is shown for an arbitrary one-component system that if we make a single unproved but highly plausible assumption about the fluctuations of large subvolumes, we can obtain an inequality relating S(T) to certain integrals over the temperature-dependent correlation functions of the system; and with one further very weak assumption the zero-temperature correlation functions alone determine a lower limit on S(T). It is then shown that the fluctuations of any locally conserved quantity give rise to a contribution to S of at least a constant times T³; in particular, by considering the density fluctuations we obtain the inequality S(T)≥ 1 2S_D(T), where S_D(T) is the "Debye" entropy which would arise from longitudinal phonons propagating at the hydrodynamic sound velocity determined by the macroscopic compressibility. (This result does not assume the existence of such phonons as good elementary excitations of the system.) Some other general results are derived as a byproduct: for instance, it is shown that any system obeying a diffusion equation of a certain type must have an entropy at least proportional to T.