Finite Temperature Casimir Effect of Scalar Field

Abstract: We derive analytic expressions for the Helmholtz free energy, Casimir force, and Casimir entropy for both one-dimensional and three-dimensional scalar fields with Dirichlet boundary conditions at finite temperature. We investigate the negative Casimir entropy problem in these systems, as well as for a scalar field in the bulk of a three-dimensional sphere, and find that this issue arises under different regularization prescriptions with differing counterterms. We argue against introducing any counterterms for the thermal corrections to the Casimir effect and predict that the thermal-fluctuation-induced Casimir force becomes repulsive in the high-temperature regime -- for instance, when $aT/(\pi\hbar{v})>0.2419$ for three-dimensional scalar fields.