Heat capacity in bits

Abstract

Information theory this century has clarified the 19th century work of Gibbs, and has shown that natural units for temperature kT, defined via 1/T=dS/dE, are energy per nat of information uncertainty. This means that (for any system) the total thermal energy E over kT is the log-log derivative of multiplicity with respect to energy, and (for all b) the number of base-b units of information lost about the state of the system per b-fold increase in the amount of thermal energy therein. For ``un-inverted'' (T>0) systems, E/kT is also a temperature-averaged heat capacity, equaling ``degrees-freedom over two'' for the quadratic case. In similar units the work-free differential heat capacity C_v/k is a ``local version'' of this log-log derivative, equal to bits of uncertainty gained per 2-fold increase in temperature. This makes C_v/k (unlike E/kT) independent of the energy zero, explaining in statistical terms its usefulness for detecting both phase changes and quadratic modes.