Exactly solvable models: The road towards a rigorous treatment of phase transitions in finite systems

Abstract

Abstract. We discuss exact analytical solutions of a variety of statistical models recently obtained for finite systems by a novel powerful mathematical method, the Laplace-Fourier transform. Among them are a constrained version of the statistical multifragmentation model, the Gas of Bags Model and the Hills and Dales Model of surface partition. Thus, the LaplaceFourier transform allows one to study the nuclear matter equation of state, the equation of state of hadronic and quark gluon matter and surface partitions on the same footing. A complete analysis of the isobaric partition singularities of these models is done for finite systems. The developed formalism allows us, for the first time, to exactly define the finite volume analogs of gaseous, liquid and mixed phases of these models from the first principles of statistical mechanics and demonstrate the pitfalls of earlier works. The found solutions may be used for building up a new theoretical apparatus to rigorously study phase transitions in finite systems. The strategic directions of future research opened by these exact results are also discussed. There is always a sufficient amount of facts. Imagination is what we lack.D. I. Blokhintsev I. THEORETICAL DESCRIPTION OF PHASE TRANSITIONS IN FINITE SYSTEMSA rigorous theory of critical phenomena in finite systems was not built up to now. However, the experimental studies of phase transitions (PTs) in some systems demand the formulation of such a theory. In particular, the investigations of the nuclear liquid-gas PT [1, 2, 3] require the development of theoretical approaches which would allow us to study the critical phenomena without going into the thermodynamic limit V → ∞ (V is the volume of the system) because such a limit does not exist due the long range Coulomb interaction. Therefore, there is a great need in the theoretical approaches which may shed light on the "internal mechanism" of how the PTs happen in finite systems. The general situation in the theory of critical phenomena for finite (small) systems is not very optimistic at the moment because theoretical progress in this field has been slow. It is well known that the mathematical theory of phase transitions was worked out by T. D. Lee and C. N. Yang [4]. Unfortunately, there is no direct generic relation between the physical observables and zeros of the grand canonical partition in a complex fugacity plane. Therefore, we know very well what are the gaseous phase and liquid at infinite volumes: mixture of fragments of all sizes and ocean, respectively. This is known both for pure phases and for their mixture, but, despite some limited success [5], this general approach is not useful for the specific problems of critical phenomena in finite systems (see Sect. VIII below).The tremendous complexity of critical phenomena in finite systems prevented their systematic and rigorous theoretical study. For instance, even the best formulation of the statistical mechanics and thermodynamics of finite systems by Hill [6] is not rigorous while discussing PTs...

Effects of Induced Surface Tension in Nuclear and Hadron Matter

“…The new EoS is essentially more effective compared to the traditional multicomponent Hadron Resonance Gas Model (MHRGM) [3][4][5][6] and traditional Statistical Multifragmentation Model (SMM) [7][8][9][10], since one can easily use in it an arbitrarily large number of independent hard-core radii.…”

“…where z 1 = 4 is the degeneracy factor of nucleons, while the degeneracy factor for other fragments g is, for simplicity, chosen to be 1 (see a discussion in [8]). Here µ k is the baryonic chemical potential of k-nucleon fragment, τ 1.9 is the Fisher topological exponent and σ(T ) is the T -dependent proper surface tension coefficient with the following parametrization…”

Quark gluon bags as reggeons

“…The problem, however, arises from the fact that in the laboratory experiments we are dealing with finite systems. From the finite volume exact analytical solutions of the constrained statistical multifragmentation model (SMM) [6,7] and the GBM [9,26], found in [28] and [29,30], respectively, it is known that in thermally equilibrated finite system there is a nonnegligible probability of finding the small and not too heavy QGP bags, say with the mass of 10-15 GeV, even in the hadronic phase. Therefore, for finite volume systems created in high energy nuclear or elementary particle collisions such QGP bags could appear as any other metastable states in statistical mechanics, since in this case the statistical suppression is just a few orders of magnitude and not of the order of the Avogadro number.…”

“…Moreover, the finite volume solution of the GBM [29], in which the mean mass of the QGP bag is proportional to its volume, predicts the decay time τ n ≈ V πnV0T for the collective state n (n = 1, 2, 3, . .…”

“…. , V 0 ≈ 1 fm 3 [21,29]) of the mixed phase having the finite volume V and temperature T . Therefore, if a single statistical state with n ≥ 1 had a pressure close to zero, its life-time would be determined by the temperature and the volume.…”