Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 16.1-4

THE CANONICAL FORMALISM; STATISTICAL MECHANICS IN HELMHOLTZ REPRESENTATION

Physics — Statistical Mechanics · canonical-formalism-heat-capacity-partition-sum

Problem

Show that

c_v = -\beta(\partial s/\partial \beta)_v

and thereby express

c_v

in terms of the partition sum and its derivatives (with respect to

\beta

). Answer:

c_v = N^{-1}k_B\beta^2 \frac{\partial^2 \ln Z}{\partial \beta^2}