  The explicit thermodynamic functions, in particular, the specific heat of a
spin system interacting with a spin bath which exerts finite dissipation on the
system are determined. We show that the specific heat is a sum of the products
of a thermal equilibration factor that carries the temperature dependence and a
dynamical correction factor, characteristic of the dissipative energy flow
under steady state from the system. The variation of specific heat with
temperature is accompanied by an abrupt transition that depends on these
dynamical factors characteristic of the finite system size.
