Thermodynamics of the scalar radiation in the presence of a reflecting plane wall

Abstract

This paper investigates further how the presence of a single reflecting plane wall modifies the usual Planckian forms in the thermodynamics of the massless scalar radiation in $N$-dimensional Minkowski spacetime. This is done in a rather unconventional way by integrating the energy density over space to obtain the internal energy and from that the Helmholtz free energy. The reflecting wall is modelled by assuming the Dirichlet or the Neumann boundary conditions on the wall. It is found that when $N>2$ integration over space eliminates dependence on the curvature coupling parameter $\xi$. Unexpectedly though, when $N=2$, the internal energy and the corresponding thermodynamics turn out to be dependent on $\xi$. For instance, the correction to the two-dimensional Planckian heat capacity is $\mp \xi k_{B}$ (minus for Dirichlet, plus for Neumann). Other aspects of this dependence on $\xi$ are also discussed. Results are confronted with those in the literature concerning related setups of reflecting walls (such as slabs) where conventional (i.e., global) approaches to obtain thermodynamics have been used.