Abstract
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 $c\ensuremath{\le}D\ensuremath{-}1$ 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 ${L}_{c}\ensuremath{\sim}1/T$. This divergence constitutes a novel phase transition, which, unlike the usual temperature-driven phase transition, is driven solely by the size of the system.
Highlights
Multiple experimental signals from the Relativistic Heavy Ion Collider (RHIC) [1,2] at Brookhaven National Laboratory (BNL) in Upton, New York, and the Large Hadron Collider (LHC) [3,4,5] at CERN in Geneva confirm the creation of the quark-gluon plasma (QGP) [6,7,8,9,10]; remarkably, these colliders recreate for the first time conditions similar to those existing in the early universe
We refer to [84,85,86,87,88] for somewhat related investigations, as well as to [89] regarding the importance of accounting for finite size effects in the different context of proton-proton collisions. While such a decoupling might seem irrelevant from a perturbative point of view, since this change of degrees of freedom across the two regions must involve nonperturbative physics, our system appears to be analogous to the quantum electrodynamics (QED) Casimir effect [46,50], which can be reproduced by imposing Dirichlet boundary conditions (DBCs)
Systems, focusing on the finite size corrections to the usual Stefan-Boltzmann thermodynamic properties computed in thermal field theory and demonstrating the emergence of a new geometric phase transition
Summary
Multiple experimental signals from the Relativistic Heavy Ion Collider (RHIC) [1,2] at Brookhaven National Laboratory (BNL) in Upton, New York, and the Large Hadron Collider (LHC) [3,4,5] at CERN in Geneva confirm the creation of the quark-gluon plasma (QGP) [6,7,8,9,10]; remarkably, these colliders recreate for the first time conditions similar to those existing in the early universe. Despite the long history of investigation into the thermodynamics of the QGP, an important aspect appears to have been overlooked by the HIC community: the effect of finite-sized rather than infinitely sized systems. We concentrate on the finite size corrections to thermodynamic quantities such as the equation of state (EoS) computed in thermal field theory in the usual Stefan-Boltzmann limit of an ideal gas of infinite size in all directions. What we will show is that for an isolated ideal gas constrained within parallel plates, the system resists compression until the separation of the plates is on the order of the thermal de Broglie wavelength; at this critical length Lc ∼ 1=T the susceptibility diverges and the system collapses This divergence of the susceptibility indicates a second order phase transition driven by the size of the system, which is conjugate to the pressure on the system. V we develop and discuss our main result: a novel phase transition, driven by the system size
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