[Paper Review] Exact Analytical Solution of the Constrained Statistical Multifragmentation Model and Phase Transitions in Finite Systems

K. A. Bugaev|ArXiv.org|Jul 12, 2005

Engineering Diagnostics and Reliability1 references3 citations

TL;DR

This paper presents an exact analytical solution for the constrained statistical multifragmentation model (CSMM) in finite volumes, using Laplace-Fourier transforms to analyze isobaric partition singularities. It establishes finite-volume analogs of gaseous, liquid, and mixed phases from first principles, showing that phase transitions emerge from the complex poles of the partition function, with real parts defining free energies and imaginary parts determining decay/formation times, thus enabling rigorous extraction of nuclear liquid-gas phase diagrams from finite-system experiments.

ABSTRACT

We discuss an exact analytical solution of a simplified version of the statistical multifragmentation model with the restriction that the largest fragment size cannot exceed the finite volume of the system. A complete analysis of the isobaric partition singularities of this model is done for finite volumes. It is shown that the real part of any simple pole of the isobaric partition defines the free energy of the corresponding state, whereas its imaginary part, depending on the sign, defines the inverse decay/formation time of this state. The developed formalism allows us, for the first time, to exactly define the finite volume analogs of gaseous, liquid and mixed phases of this model from the first principles of statistical mechanics and demonstrate the pitfalls of earlier works. The finite size effects for large fragments and the role of metastable (unstable) states are discussed.

Motivation & Objective

• To develop a rigorous, finite-volume extension of the statistical multifragmentation model (SMM) that accounts for geometric constraints on fragment size.
• To define finite-volume analogs of gaseous, liquid, and mixed phases unambiguously from statistical mechanics, avoiding mean-field approximations.
• To resolve inconsistencies in earlier works by showing that the mixed phase is not a simple mixture of pure phases but a superposition of multiple collective states.
• To demonstrate how phase transitions emerge in the thermodynamic limit from the behavior of complex poles in the isobaric partition function.
• To enable more reliable extraction of the nuclear liquid-gas phase diagram from experimental data on finite nuclei by incorporating finite-size corrections.

Proposed method

• The Laplace-Fourier transform is applied to the grand canonical partition function (GCP), reducing the analysis to the singularities (poles) of the isobaric partition function.
• The model enforces a finite upper limit on fragment size via the system volume $V$, replacing the infinite-volume assumption of prior SMM solutions.
• Simple poles in the complex $λ$-plane are analyzed: their real parts correspond to free energy densities, and their imaginary parts determine inverse decay/formation times.
• The behavior of poles under varying volume $V$ and chemical potential $\mu$ is studied to identify phase analogs and their evolution toward the thermodynamic limit.
• The method uses the exact solution of the constrained SMM to derive finite-volume corrections to mechanical pressure, entropy, particle number, and energy density.
• The analysis distinguishes metastable states ($Re(\lambda_n) \geq 0$) from mechanically unstable states ($Re(\lambda_n) < 0$) via the sign of the imaginary part of poles.

Experimental results

Research questions

• RQ1 How can the statistical multifragmentation model be exactly solved for finite volumes with a physically motivated constraint on the maximum fragment size?
• RQ2 What are the finite-volume analogs of gaseous, liquid, and mixed phases in the context of nuclear multifragmentation?
• RQ3 How do the singularities of the isobaric partition function correspond to thermodynamic states and phase transitions in finite systems?
• RQ4 How does the system approach the thermodynamic limit, and how do the poles of the partition function evolve to reproduce the known phase transition singularity?
• RQ5 What are the finite-size corrections to thermodynamic quantities like pressure, entropy, and particle density, and how do they affect experimental phase diagram reconstruction?

Key findings

• The gaseous phase is always stable, with an infinite decay/formation time, and is characterized by a single real pole $\lambda_0$ whose value $T\lambda_0$ gives the chemical potential and pressure.
• Complex conjugate pairs of poles $\lambda_{n>0}$ describe metastable or unstable states, with $-TRe(\lambda_{n>0})$ representing the free energy density and $bTIm(\lambda_{n>0})$ the inverse decay/formation time.
• The mixed phase is not a simple mixture of gaseous and liquid phases but a superposition of three or more collective states, each with distinct $\lambda_n$, leading to volume-like rather than surface-like free energy differences.
• For finite volumes, the liquid phase is represented by an infinite number of poles at the highest possible particle density ($\mu \to \infty$), while the gaseous phase appears when only one real pole exists.
• In the thermodynamic limit ($V \to \infty$), the number of poles increases indefinitely, and the imaginary parts of the closest poles vanish, forming an essential singularity that corresponds to the known liquid-phase singularity.
• Finite-size corrections to pressure and other thermodynamic quantities arise from the $T$- and $\mu$-dependence of $\lambda_n$, and must be included to accurately extract phase diagrams from experimental fragment yields.