Equation of state and distribution of particle sizes in Gibbs system

Abstract: In the framework of Gibbs statistical theory, the issue of the distribution of particle sizes forming the statistical system and the moments of this distribution are considered. This task is relevant for a wide variety of applications. The distribution for particle sizes and moments of this quantity are determined from probabilistic considerations. Particle size depends on interactions in the system, on the compressibility factor, on the number of interacting particles, on the volume of the system. The expressions for the intrinsic volume of particles are substituted into the equations of state written using the theory of excluded volume for various expressions of the exclusion factor. The equations of state thus obtained can be considered as a refinement of the equation of state, a transition to a higher level of description.

• The paper rigorously derives particle volume distribution from first principles, linking intrinsic particle size to system thermodynamics.
• It employs a grand canonical ensemble and excluded volume theory to refine the conventional equations of state for interacting systems.
• The framework bridges empirical parameterization with precise modeling for dense fluids, colloids, and nuclear matter.

Equation of State and Particle Size Distribution in a Gibbs System

Overview

This paper presents a rigorous statistical mechanical approach to the determination of the distribution and moments of particle sizes in a Gibbs system, with direct implications for the formulation and refinement of equations of state (EoS). Unlike conventional treatments that impose the intrinsic particle volume as an empirical parameter, this work derives the volume distribution and its statistical moments from first principles within the grand canonical framework. The particle size, now a function of system interactions, compressibility, and configuration, is then incorporated into established EoS via the theory of excluded volume, yielding more granular and theoretically consistent expressions for thermophysical properties.

Statistical Foundations and Derivation of Particle Size Distribution

The analysis begins with the definition of the configurational probability density in the grand canonical ensemble for a system of $n$ interacting particles, characterized by arbitrary pairwise-additive potentials within a volume $V$. The baseline probabilistic framework is extended to introduce the distribution $P^{(n)}(A_1, \ldots, A_n)$ of particle volumes ($A_k$), conditioned on spatial coordinates and the corresponding activities.

By conceptualizing the presence of a particle as the intersection of independent events—its location and its exclusion volume—the joint distribution of particle sizes is factorized. The recursive properties of occupation probabilities for interacting clusters of arbitrary size $n$ are exploited, with normalization and integration conditions incorporating the excluded volume restrictions, ensuring no overcounting as $n$ increases.

An explicit construction of the probability density for the distribution in the cluster is derived. For $n=1,2,\ldots$, the resulting forms generalize the exponential (one-parameter) distribution, with normalization factors represented using incomplete gamma functions. The structure naturally encapsulates both the mean and variance of the particle-size distribution, expressed as functions of system pressure, temperature, and activity.

The emergent result is that the effective particle size is a stochastic variable, whose distribution and statistical moments (e.g., mean, variance) are unambiguously linked to thermodynamic variables and the specific form of the interaction potential. In particular, for the ideal gas (zero-potential), the mean size recovers expected thermodynamic limits.

Incorporation into Equations of State

The theoretically derived intrinsic volume is then consistently substituted as the reference volume in a class of EoS constructed via the excluded volume approach. The author draws on the generative formalism established in prior work (Rusanov, [9]) for the exclusion factor $f$—the ratio of the mean excluded volume to mean intrinsic volume—as a function of the system's total packing fraction and density.

A continuum of EoS is thereby obtained, interpolating between and generalizing the van der Waals, Carnahan-Starling, Percus-Yevick, Guggenheim, and other well-known representations. Notably, transcendental and algebraic equations for the compressibility factor and pressure emerge, now possessing a direct dependence on the statistical moments of the size distribution and, in refined versions, on the core volume $d$ of non-compressible particles.

A significant analytic consequence is the elimination of the ambiguous or ad hoc status of the hard-core parameter in classical EoS. It is replaced by a self-consistently determined function of $T$, $P$, and the activity, ultimately reducing to the classical value only for limiting cases such as the ideal or infinite-dilution gas.

Analytical Properties and Limiting Cases

Detailed asymptotic analysis recovers the correct thermodynamic limits for the mean particle size as the system volume or particle number tends to zero or infinity. For the ideal gas, the mean size simplifies to $V/N$. In interacting systems, the mean and variance of sizes reflect compressibility and interparticle interactions.

In the case of nonzero hard core $d$, the mean size is bounded below by $d$ and varies with both $n$ (cluster size) and interaction parameters, showing nontrivial behavior, including monotonic or non-monotonic dependence on compressibility factors. For large systems, corrections beyond the leading order become negligible; the formalism remains valid for finite clusters and high densities, consistent with physical expectations around close packing and phase transitions.

Implications and Theoretical Significance

The formal methodology presented allows for systematic refinement of the EoS for fluids, colloidal suspensions, or nuclear matter, especially where the effective size of the constituent particles is variable and subject to interactions rather than fixed by construction. This approach parallels recent trends in condensed matter theory and chemical physics, recognizing the necessity to account for the statistical nature of molecular shape, volume, and correlation effects when constructing quantitative models for real substances.

By relating the exclusion parameter and hence the EoS to the derived statistical distribution, this work provides a pathway to close the gap between empirical parameterization and rigorous statistical mechanics—a crucial step for precise modeling in high-density, multi-component, or strongly interacting systems.

Moreover, the author notes the potential for further generalization to non-Gibbsian statistics (e.g., Tsallis statistics), where the resulting size distribution may acquire a power-law rather than exponential character, signaling important connections to non-extensive systems and complex fluids.

Outlook and Future Directions

Several future directions are evident. Extending this formalism offers new opportunities for deriving cluster size distributions and EoS for systems with non-pairwise interactions, polydispersity, and inhomogeneity. Application to non-equilibrium systems, dynamic clustering, and the calculation of higher-order virial coefficients using explicit size statistics is also anticipated.

The refinement of EoS via the statistical mechanics of effective particle size enables greater consistency in the physical interpretation of thermodynamic parameters, improved predictive capabilities for systems where volume exclusion, local structure, and polydisperse effects are significant, and bridges the gap between atomistic and mesoscopic models.

Conclusion

This work establishes a comprehensive statistical mechanical framework for deriving the distribution and mean of particle volumes in a general Gibbs system, embedding these quantities into a unified treatment of equations of state. By expressing the intrinsic particle size as a function of system thermodynamics and incorporating it rigorously into excluded volume EoS, the author advances the precision and consistency of modeling dense, structured, or interacting fluids. The framework is extendable, analytically robust, and theoretically sound, providing a foundation for future advances in statistical thermodynamics and the development of quantitatively accurate EoS for complex systems.

Reference: "Equation of state and distribution of particle sizes in Gibbs system" (1910.05034)