Geometrically confined thermal field theory: Finite size corrections and phase transitions

Motivated by evidence for quark-gluon plasma signatures in small systems, we study a simple model of a massless, noninteracting scalar field confined with Dirichlet boundary conditions. We use this system to investigate the finite size corrections to thermal field–theoretically derived quantities compared to the usual Stefan-Boltzmann limit of an ideal gas not confined in any direction. Two equivalent expressions with different numerical convergence properties are found for the free energy in D rectilinear spacetime dimensions with c ≤ D − 1 spatial dimensions of finite extent. We find that the first law of thermodynamics generalizes such that the pressure depends on direction. For systems with finite dimension(s) but infinite volumes, such as a field constrained between two parallel plates or a rectangular tube, the relative fluctuations in energy are zero, and hence the canonical and microcanonical ensembles are equivalent. We present precise numerical results for the free energy, total internal energy, pressure, entropy, and heat capacity of our field between parallel plates, in a tube, and in finite volume boxes of various sizes in four spacetime dimensions. For temperatures and system sizes relevant for heavy ion phenomenology, we find large deviations from the Stefan-Boltzmann limit for these quantities, especially for the pressure. Our main result is the discovery that an isolated system of fields constrained between parallel plates reveals a divergent isoenergetic compressibility at a critical length Lc ∼ 1/T. This divergence constitutes a novel phase transition, which, unlike the usual temperature-driven phase transition, is driven solely by the size of the system.