[Paper Review] Exact Solution of the Statistical Multifragmentation Model and Liquid-Gas Phase Transition in Nuclear Matter

K. A. Bugaev, M. I. Gorenstein+2 more|ArXiv.org|Jul 27, 2000

Statistical Mechanics and Entropy3 citations

TL;DR

This paper presents an exact analytical solution of the Statistical Multifragmentation Model (SMM) in the thermodynamic limit using the grand canonical ensemble, incorporating excluded volume effects via the isobaric partition function. It rigorously demonstrates a first-order liquid-gas phase transition with a finite discontinuity in specific heat at the mixed-phase to gaseous-phase boundary, resolving long-standing ambiguities in finite-size numerical studies.

ABSTRACT

An exact analytical solution of the statistical multifragmentation model is found in thermodynamic limit. The system of nuclear fragments exhibits a 1-st order liquid-gas phase transition. The peculiar thermodynamic properties of the model near the boundary between the mixed phase and the pure gaseous phase are studied.

Motivation & Objective

• To establish a rigorous analytical framework for the Statistical Multifragmentation Model (SMM) in the thermodynamic limit.
• To resolve the open question of whether a first-order liquid-gas phase transition exists in nuclear matter within the SMM.
• To investigate the thermodynamic properties of the mixed phase and its boundary with the pure gaseous phase.
• To clarify the behavior of specific heat and energy density near the phase transition boundary.

Proposed method

• The study employs the grand canonical ensemble and introduces the isobaric partition function (IPF) to handle the constraint of fixed nucleon number and excluded volume effects.
• The IPF is analytically continued to find singularities that determine the pressure and thermodynamic properties in the thermodynamic limit.
• The system's phase behavior is derived from the singularities of the IPF: a simple pole for the gas phase and a singularity in the function F for the liquid phase.
• Excluded volume effects are modeled via a finite volume per fragment (b = 1/ρ₀), with ρ₀ = 0.16 fm⁻³, modifying the free volume in the partition function.
• The pressure and baryonic density in the liquid and gas phases are derived from the singularities sₗ and s₉, respectively, with ρₗ = 1/b and ρ₉ = ρ_id / (1 + bρ_id).
• The mixed phase is described by a linear combination of liquid and gas contributions, with λ(T) representing the volume fraction of the liquid phase.

Experimental results

Research questions

• RQ1 Does the Statistical Multifragmentation Model exhibit a first-order liquid-gas phase transition in the thermodynamic limit?
• RQ2 What are the thermodynamic properties of the mixed phase, particularly near the boundary with the pure gaseous phase?
• RQ3 How does the specific heat per nucleon behave near the phase transition boundary, and does it exhibit a divergence or a finite discontinuity?
• RQ4 What is the role of excluded volume effects in shaping the phase diagram and the stability of the mixed phase?
• RQ5 How does the volume fraction λ(T) of the liquid phase evolve with temperature in the mixed phase, and what is its impact on thermodynamic observables?

Key findings

• The SMM exhibits a first-order liquid-gas phase transition in the thermodynamic limit, confirmed by the existence of two distinct singularities in the isobaric partition function.
• The phase boundary between the mixed phase and the pure gaseous phase is characterized by a finite discontinuity in the specific heat per nucleon, cρ(T), due to a sharp, non-zero temperature derivative of the liquid volume fraction λ(T).
• The specific heat peak at constant density remains finite in the thermodynamic limit, with a non-zero width, contradicting earlier numerical expectations of a delta-like divergence.
• The energy density is continuous across the mixed-to-gas phase boundary, but its strong temperature dependence near the boundary leads to the observed peak in specific heat.
• The baryonic density in the liquid phase is fixed at ρₗ = 1/b = 0.16 fm⁻³, corresponding to normal nuclear density, while the gas phase density depends on chemical potential and temperature.
• The volume fraction λ(T) of the liquid phase drops abruptly to zero near the phase boundary, explaining the sharp rise in energy density and the resulting finite discontinuity in cρ(T).