On quantum and thermodynamic fluctuations of fermion fields in a finite volume

The effective potential $(V_{\text{eff}})$ of fermion fields enclosed in a finite region of space is calculated at nonzero temperatures. In a one-dimensional model, an exact expression for $V_{\text{eff}}$ is obtained in terms of Jacobi theta functions as a function of the temperature $(T)$ and length $(L)$ of the space. In four-dimensional space, the low- and hightemperature behaviors, and also the limit $T\to\hbar c/2L$ are studied. It is shown that the thermal corrections $(TL\ll1)$ to the vacuum energy density and the dimensional corrections $(TL\gg1)$ to the fermion free energy density are exponentially small.