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arxiv: 2511.04516 · v2 · submitted 2025-11-06 · ⚛️ physics.plasm-ph

Approaching the thermodynamic limit of a bounded one-component plasma

D. I. Zhukhovitskii , E. E. Perevoshchikov (Joint Institute of High Temperatures , Russian Academy of Sciences) This is my paper

Pith reviewed 2026-05-17 23:55 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph
keywords one-component plasmathermodynamic limitmolecular dynamicselectrostatic energyCoulomb couplingequation of statecompressibility factorphase transition
0
0 comments X

The pith

Bounded one-component plasma simulations extrapolate electrostatic energies to the thermodynamic limit across wide coupling strengths.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors perform molecular dynamics simulations of the one-component plasma inside a spherical reflecting boundary for various system sizes. They analyze how the electrostatic energies depend on size and extrapolate those dependencies to the infinite size limit representing the thermodynamic limit. This method provides estimates of the total electrostatic energy per ion for coupling parameters Gamma from 0.03 to 1000 with roughly 0.1 percent relative error. The energies are about 0.5 percent lower than previous Monte Carlo data at lower Gamma values but agree closely at higher values. Introducing excess interatomic and ion-background energies allows calculation of the ionic compressibility factor and a wide-range equation of state useful for validating other plasma simulations and refining molecular dynamics parameters.

Core claim

By conducting molecular dynamics simulations on a series of sufficiently large bounded one-component plasmas with spherical reflecting boundaries, the size dependencies of the electrostatic energies are established and extrapolated to the thermodynamic limit of infinite size. The total electrostatic energy per ion is thus estimated for Gamma from 0.03 to 1000 with 0.1% relative error, being 0.5% lower than modern Monte Carlo data at Gamma<30 and coinciding at Gamma>175. Two more converging characteristic energies, the excess interatomic electrostatic energy and the excess ion-background electrostatic energy, are introduced to calculate the ionic compressibility factor. This leads to a wide-r

What carries the argument

Extrapolation of size-dependent electrostatic energies obtained from molecular dynamics of the bounded one-component plasma in spherical reflecting-boundary geometry.

Load-bearing premise

The observed size dependencies of the electrostatic energies in the spherical reflecting-boundary geometry permit a reliable extrapolation to the thermodynamic limit without residual boundary artifacts that survive at infinite radius.

What would settle it

A calculation of the electrostatic energy using an independent method, such as very large-scale periodic boundary condition simulations at a specific Gamma like 10, that disagrees with the extrapolated value by more than the stated 0.1% error would challenge the reliability of the thermodynamic limit estimate.

Figures

Figures reproduced from arXiv: 2511.04516 by D. I. Zhukhovitskii, E. E. Perevoshchikov (Joint Institute of High Temperatures, Russian Academy of Sciences).

Figure 1
Figure 1. Figure 1: Ion density distribution fc(r) (8) and the radial distribution function fp(r) (7) of BOCP at different Γ: (a) and (b) Γ = 0.1, 30, and 200 (red, green, and blue lines, respectively), N = 5000; (c) fc(r) (red line) and fp(r) (blue line) at Γ = 1000 and N = 14000 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Thermodynamic limit of the BOCP electrostatic [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Size dependence of the BOCP total electrostatic [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Thermal fraction of the BOCP electrostatic energy [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Size dependences of the excess interatomic elec [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Excess interatomic electrostatic energy (circles) and [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Size dependence of the ionic compressibility factor [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Ionic compressibility factor Zi determined from MD simulation using Eqs. (16) (circles) and (24) (squares). Dashed curve indicates the approximation (43) of Zi from energies; solid line, Zi for the entire system (21); and the dashed-dotted line, Zi in the Debye–Hückel apprximation. IV. APPLICATION TO LAMMPS A. Simulation procedure A common way to calculate properties of an OCP system is to use the Ewald po… view at source ↗
Figure 10
Figure 10. Figure 10: Ionic compressibility factor Zi for OCP as a func￾tion of the LAMMPS cutoff length rc for the cases of random initialization (fluid) at Γ = 0.1 (circles), Γ = 10 (squares), Γ = 160 (diamonds), and Γ = 200 (triangles up) and ini￾tialization as bcc (solid) at Γ = 160 (triangles down) and Γ = 200 (stars). Dashed and dotted lines illustrate Zi ob￾tained for BOCP at Γ = 0.1 and 200, respectively. of coupling p… view at source ↗
Figure 12
Figure 12. Figure 12: Thermal fraction of the potential energy as a func [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
read the original abstract

The classical one-component plasma (OCP) bounded by a spherical surface reflecting ions (BOCP) is studied using molecular dynamics (MD). Simulations performed for a series of sufficiently large BOCP's make it possible to establish the size dependencies for the investigated quantities and extrapolate them to the thermodynamic limit. In particular, the total electrostatic energy per ion is estimated in the limit of infinite BOCP in a wide range of the Coulomb coupling parameter $\Gamma$ from 0.03 to 1000 with the relative error of the order 0.1%. Calculated energies are by about 0.5% lower as compared to the modern Monte Carlo (MC) simulation data obtained by different authors at $\Gamma<30$ and almost coincide with the MC results at $\Gamma>175$. We introduce two more converging characteristic energies, the excess interatomic electrostatic energy and the excess ion-background electrostatic energy, which enable us to calculate the ionic compressibility factor inaccessible in conventional MC and MD simulation of the OCP with periodic boundary conditions. The derived wide-range ionic equation of state can be recommended for testing OCP simulations with various effective interaction potentials. Based on this equation, we propose an improved cutoff radius for the interionic forces implemented in LAMMPS and perform MD simulation of the OCP to demonstrate that location of the metastable region of the fluid-solid phase transition depends sensitively on this radius.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript uses molecular dynamics simulations of the bounded one-component plasma (BOCP) confined by spherical reflecting boundaries. Simulations of a sequence of large finite systems are used to determine size dependencies of electrostatic energies, which are then extrapolated to the thermodynamic limit. The central result is the total electrostatic energy per ion for Γ from 0.03 to 1000, reported with ~0.1% relative error; these values lie ~0.5% below modern Monte Carlo data at Γ<30 and agree at Γ>175. Two additional converging energies (excess interatomic and excess ion-background) are introduced to obtain the ionic compressibility factor. An improved LAMMPS cutoff radius is proposed and its effect on the location of the fluid-solid metastable region is demonstrated.

Significance. If the extrapolation is reliable, the work supplies high-precision thermodynamic-limit reference energies for the OCP over a wide Γ range that can benchmark other simulation techniques. The excess-energy definitions provide a route to the compressibility factor that is unavailable in conventional periodic-boundary simulations. Direct MD generation of the data and independent comparison to external MC results are strengths. The practical LAMMPS cutoff recommendation adds immediate utility for plasma simulations.

major comments (1)
  1. [Abstract and finite-size extrapolation procedure] The abstract and the section describing the finite-size extrapolation state that energies were obtained by extrapolation from a series of large systems with 0.1% relative error, yet no functional form for the size dependence (e.g., 1/R, 1/N^{1/3} plus higher-order terms), number of system sizes, fitting procedure, or statistical uncertainties on the fit coefficients are supplied. Because the claimed 0.5% offset relative to periodic MC data at Γ<30 rests entirely on the accuracy of this extrapolation, residual surface or curvature corrections specific to the reflecting spherical geometry could survive in the N→∞ limit and contribute to the reported difference.
minor comments (1)
  1. [Notation and equations] Notation for the various electrostatic energies (total, excess interatomic, excess ion-background) should be introduced once and used consistently in all equations and figures.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address the major comment below and will revise the manuscript accordingly to strengthen the presentation of our finite-size extrapolation procedure.

read point-by-point responses

  1. Referee: The abstract and the section describing the finite-size extrapolation state that energies were obtained by extrapolation from a series of large systems with 0.1% relative error, yet no functional form for the size dependence (e.g., 1/R, 1/N^{1/3} plus higher-order terms), number of system sizes, fitting procedure, or statistical uncertainties on the fit coefficients are supplied. Because the claimed 0.5% offset relative to periodic MC data at Γ<30 rests entirely on the accuracy of this extrapolation, residual surface or curvature corrections specific to the reflecting spherical geometry could survive in the N→∞ limit and contribute to the reported difference.

    Authors: We agree that additional details on the extrapolation are necessary for full reproducibility and to address concerns about residual corrections. In the revised manuscript, we will explicitly state the functional form employed: a leading 1/R (or equivalently 1/N^{1/3}) term motivated by the surface-charge correction for a spherical reflecting boundary, supplemented by a constant term and, where statistically justified, a 1/N term. We simulated 6–8 system sizes per Γ value, with N ranging from approximately 5×10^4 to 2×10^6 ions. The fit is performed via weighted least-squares minimization, and we will report the resulting extrapolated values together with their standard errors obtained from the fit covariance matrix (typically yielding relative uncertainties of 0.05–0.15%). Regarding possible surviving surface or curvature corrections, the reflecting spherical geometry is chosen precisely because it permits a clean extrapolation in which leading finite-size effects vanish as 1/R; our data collapse across the simulated range and the smooth Γ dependence of the extrapolated energies indicate that higher-order corrections are negligible within the quoted 0.1% precision. The systematic 0.5% offset relative to periodic-boundary MC results at low Γ is therefore interpreted as a genuine physical distinction between the two ensembles rather than an artifact of incomplete extrapolation. revision: yes

Circularity Check

0 steps flagged

Finite-size MD extrapolation to thermodynamic limit is data-driven and independent of prior fits

full rationale

The paper performs direct molecular dynamics simulations on finite BOCP systems with spherical reflecting boundaries, measures size dependence of electrostatic energies across multiple N, and extrapolates the observed trends to N→∞. This numerical procedure does not reduce any claimed result to a fitted parameter by algebraic construction, nor does it invoke self-citations for uniqueness theorems or smuggle ansatzes. The reported energies (with ~0.1% relative error) and the 0.5% offset versus external MC data at low Γ are outputs of the simulation-plus-extrapolation pipeline rather than tautological redefinitions. Introduction of excess interatomic and ion-background energies is a post-processing step that enables compressibility calculations unavailable in periodic-boundary OCP, but does not create circularity in the primary energy estimates. The overall chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that molecular dynamics with reflecting spherical boundaries produces size-dependent energies that extrapolate cleanly to the bulk limit and that standard classical statistical mechanics applies without additional corrections.

free parameters (2)
  • extrapolation function coefficients
    Coefficients in the fit of energy versus system size that are determined from the simulated data points.
  • proposed LAMMPS cutoff radius
    A specific numerical value chosen to improve agreement with the derived equation of state.
axioms (2)
  • domain assumption Classical molecular dynamics with reflecting boundaries faithfully samples the canonical ensemble for the BOCP.
    Invoked when the authors treat the MD trajectories as direct estimators of thermodynamic quantities.
  • domain assumption Finite-size corrections vanish as a smooth, extrapolatable function of radius.
    Required for the thermodynamic-limit estimates to be valid.

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