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M. A. Shpot

Publications and source records attributed to M. A. Shpot.

At least 19 recordsLinked to original sources
arXiv2026

Phase diagram of a double-occupancy cell model of a fluid with Curie-Weiss interaction

A double-occupancy cell model of a fluid with Curie-Weiss interaction is studied. First, we show that the model is isomorphic to the Blume-Capel model on a complete graph through a simple transformation from spin to occupancy variables. We then investigate its phase behavior within the grand-canonical ensemble using a combination of analytical and numerical methods. Despite its simplicity, the model exhibits a remarkably rich thermodynamic behavior depending on the ratio between the local repulsive and global attractive interactions. We identify regimes characterized by a single critical point, two distinct critical points, tricritical behavior, and triple-point formation. For sufficiently strong repulsion, the system possesses three fluid phases of different densities, leading to both gas-liquid and liquid-liquid coexistence. The locations of the critical, tricritical, and triple points are determined, and the corresponding phase diagrams are constructed. These results demonstrate that the competition between double-occupancy repulsion and long-range attraction is sufficient to generate complex phase behavior in a minimal multiple-occupancy lattice-gas model.

cond-mat.stat-mech↗
arXiv2026

The cell fluid model with Curie-Weiss interactions: special cases and analytical results

Inspired by previous extensive numerical studies of a cell fluid model with Curie-Weiss interactions, we concentrate on some analytically tractable special cases in its description. The key ingredient of the model is a competition between global attraction and local repulsion interactions between particles with coupling constants $J_1$ and $J_2$, respectively. We provide analytical results in several limiting cases, including the ideal-gas limit $J_1=J_2=0$ and the strong-repulsion limit $J_2\gg J_1$. For $J_2\gg J_1$, a detailed analytical study is presented. We derive explicit expressions for the critical point parameters, the equation of state, and the binodal and spinodal curves in closed form. The equation of state is found to be in full agreement with that of the van der Waals lattice gas, and the order parameter satisfies the standard Curie-Weiss equation. In a neighborhood of the critical point, a Landau expansion is shown to have the same form and symmetry as that of the classical lattice gas within the mean-field approximation. Moreover, based on the explicit knowledge of a few leading terms in the asymptotic expansion of the deformed exponential function governing the physics of the cell model, we extend its validity range to include the marginal case of thermodynamic stability, $J_1=J_2$. In particular, this extension makes a consideration of the ideal-gas limit $J_1=J_2=0$ formally legitimate. For the generic marginal case $J_1=J_2\ne0$ systematically avoided in previous works, we present numerical data and phase diagrams that augment their findings for $J_2>J_1$.

cond-mat.stat-mech↗
arXiv2025

Thermodynamic response functions in a cell fluid model with Curie-Weiss interaction. I. Supercritical region

Thermodynamic response functions, including the isothermal compressibility, the thermal pressure coefficient, and the thermal expansion coefficient, isochoric and isobaric heat capacities are explicitly derived for a many-particle system interacting through a Curie-Weiss-type potential. These calculations are based on an exact equation of state previously obtained for a cell fluid model in the grand canonical ensemble. The resulting response functions are presented graphically as functions of temperature, density, and chemical potential within the supercritical region.

cond-mat.stat-mech↗
arXiv2025

Triple point in a cell fluid model with effective temperature-dependent attraction

We study a cell fluid model of a many-particle system with Curie-Weiss-type interaction potential. It is considered as an open system in a fixed volume partitioned into a large number of congruent cubic cells. The interaction potential comprises two competing components: a global uniform attraction acting between all particle pairs in the volume and a short-range repulsion between particles occupying the same cell. Previous studies have established that this model admits an exact solution, exhibits multiple critical points, and undergoes a sequence of first-order phase transitions. Despite variations in the interaction strengths, no triple point appears as long as these parameters remain fixed. We demonstrate that incorporating effective {temperature-dependent} attractive interactions fundamentally alters the phase behavior of the cell model. This modification preserves the model's exact solvability while resulting in the emergence of a triple point in the phase diagram.

cond-mat.stat-mech↗
arXiv2025

Entropy of the cell fluid model with Curie-Weiss interaction

Entropy of the cell fluid model with Curie-Weiss interaction is obtained in analytical form as a function of temperature and chemical potential. A parametric equation is derived representing the entropy as a function of density. Features of both the entropy per particle and the entropy per cell are investigated at the entropy-density and entropy-chemical potential planes. The considered cell model is a multiple-occupancy model and possesses an infinite sequence of first-order phase transitions at sufficiently low temperatures. We find that the entropy exhibits pronounced minima at around integer-valued particle densities, which may be a generic feature of multiple-occupancy models.

cond-mat.stat-mech↗
arXiv2025

The free propagator of strongly anisotropic systems with free surfaces

A brief overview of fluctuation-induced forces in statistical systems with film geometry at the critical point and the calculation of Casimir amplitudes, which characterize these forces quantitatively, is presented. Particular attention is paid to the special features of strongly anisotropic $m$-axis systems at the Lifshitz point, specifically, in the case of a "$perpendicular$" orientation of surfaces with free boundary conditions. Beyond the simplest one-loop approximation, calculations of Casimir amplitudes are impossible without knowledge of the Gaussian propagator, which corresponds to the lines of Feynman diagrams in the perturbation theory. We present an explicit expression for such a propagator in the case of an anisotropic system confined by parallel surfaces $perpendicular$ to one of the anisotropy axes. Using this propagator, we reproduce the one-loop result derived earlier in an essentially different way. The knowledge of the propagator provides the possibility of higher-order calculations in perturbation theory.

cond-mat.stat-mech↗
arXiv2025

A multiple occupancy cell fluid model with competing attraction and repulsion interactions

An analytically solvable cell fluid model with unrestricted cell occupancy, infinite-range Curie-Weiss-type attraction and short-range intra-cell repulsion is studied within the grand-canonical ensemble. Building on an exact single-integral representation of the grand partition function, we apply Laplace's method to obtain asymptotically exact expressions for the pressure, density and equation of state. The model exhibits a hierarchy of first-order transitions, each terminating at a critical point. We determine the coordinates of the first five such points. Recasting the formalism in dimensionless variables highlights the explicit temperature dependence of all thermodynamic functions. This enables us to derive a closed-form expression for the entropy. The results reveal pronounced entropy minima around integer cell occupancies and reproduce density-anomaly isotherm crossings analogous to those in core-softened models.

cond-mat.stat-mech↗
arXiv2025

Influence of attractive parts of interaction potentials on critical point parameters

We investigate how microscopic features of interparticle potentials influence macroscopic critical point parameters. Our analytical calculations are based on the cell model for continuous many-particle systems. We explore two types of pair interactions described by the Morse potential and a Curie-Weiss-type potential. For Morse fluids, we present numerical results obtained with microscopic parameters corresponding to the alkali metals sodium (Na) and potassium (K). The calculated dimensionless critical point parameters for liquid Na and K, expressed in dimensional units, allow for direct comparison with available experimental and simulation data. For the Curie-Weiss cell model with competing interactions, which exhibits a sequence of first-order phase transitions, we examine the critical parameters for the first three critical points. We analyze our results by varying the attractive part of the Morse potential and the Curie-Weiss attraction strength, providing insights into how these microscopic characteristics change critical point coordinates.

cond-mat.stat-mech↗
arXiv2025

A new special function related to a discrete Gauss-Poisson distribution and some physics of the cell model with Curie-Weiss interactions

Inspired by previous studies in statistical physics [see, in particular, Kozitsky at al., A phase transition in a Curie-Weiss system with binary interactions, Condens. Matter Phys. 23, 23502 (2020)] we introduce a discrete Gauss-Poisson probability distribution function \begin{equation}\label{GPD}\tag{A1} p_{GP}(n ;z,r)=\left[R(r;z)\right]^{-1}\frac{\mbox{e}^{zn}}{n!}\,\mbox{e}^{-\frac 12\,rn^2} \end{equation} with support on $\mathbb N_0$ and parameters $z\in\mathbb R$ and $r\in\mathbb R_+$. The probability mass function $p_{GP}(n ;z,r)$ is normalized by the special function $R(r;z)$, given by the infinite sum \begin{equation}\label{R}\tag{A2} R(r;z)=\sum_{n=0}^\infty\frac{\mbox{e}^{zn}}{n!}\,\mbox{e}^{-\frac 12\,rn^2}, \end{equation} possessing extremely intersting mathematical properties. We present an asymptotic estimate $R^{(\rm as)}(r;z\gg1)$ for the function $R(r;z)$ with large arguments $z$, along with similar formulas for its logarithm and logarithmic derivative. These functions exhibit very interesting oscillatory behavior around their asymptotics, for parameters $r$ above some threshold value $r^*$. Some implications of our findings are discussed in the context of the Curie-Weiss cell model of simple fluids.

cond-mat.stat-mech↗
arXiv2024

A Ramanujan's hypergeometric transformation formula, its validity range and implications

We extend the validity range of a Ramanujan's hypergeometric transformation formula proved by Berndt, Bhargava and Garvan, Trans. Amer. Math. Soc. 347, 4163 (1995) and study its implications. Relations to special values of complete elliptic integrals of the first kind in the singular value theory are established. Consequently, we derive several closed-form evaluations of hypergeometric functions $_2F_1$ with different sets of parameters and arguments. Connections with other hypergeometric transformations and some recent results are discussed.

math.CA↗
arXiv2021

Boundary conformal field theory at the extraordinary transition: The layer susceptibility to $O(\varepsilon)$

We present an analytic calculation of the layer (parallel) susceptibility at the extraordinary transition in a semi-infinite system with a flat boundary. Using the method of integral transforms put forward by McAvity and Osborn [Nucl. Phys. B 455 (1995) 522] in the boundary CFT we derive the coordinate-space representation of the mean-field propagator at the transition point. The simple algebraic structure of this function provides a practical possibility of higher-order calculations. Thus we calculate the explicit expression for the layer susceptibility at the extraordinary transition in the one-loop approximation. Our result is correct up to order $O(\varepsilon)$ of the $\varepsilon=4-d$ expansion and holds for arbitrary width of the layer and its position in the half-space. We discuss the general structure of our result and consider the limiting cases related to the boundary operator expansion and (bulk) operator product expansion. We compare our findings with previously known results and less complicated formulas in the case of the ordinary transition. We believe that analytic results for layer susceptibilities could be a good starting point for efficient calculations of two-point correlation functions. This possibility would be of great importance given the recent breakthrough in bulk and boundary conformal field theories in general dimensions.

hep-th↗
arXiv2020

Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions

We consider two integrals over $x\in [0,1]$ involving products of the function $ζ_1(a,x)\equiv ζ(a,x)-x^{-a}$, where $ζ(a,x)$ is the Hurwitz zeta function, given by $$\int_0^1ζ_1(a,x)ζ_1(b,x)\,dx\quad\mbox{and}\quad \int_0^1ζ_1(a,x)ζ_1(b,1-x)\,dx$$ when $\Re (a,b)>1$. These integrals have been investigated recently in \cite{SCP}; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function $ζ(x)$ and two free parameters $a$ and $b$. The limiting forms of these sums when $a+b$ takes on integer values are considered.

math.CA↗
arXiv2017

A Feynman integral in Lifshitz-point and Lorentz-violating theories in $R^D\oplus R^m$

We evaluate a one-loop, two-point, massless Feynman integral $I_{D,m}(p,q)$ relevant for perturbative field theoretic calculations in strongly anisotropic $d=D+m$ dimensional spaces given by the direct sum $\mathbb R^D\oplus\mathbb R^m$. Our results are valid in the whole convergence region of the integral for generic (non-integer) co-dimensions $D$ and $m$. We obtain series expansions of $I_{D,m}(p,q)$ in terms of powers of the variable $X:=4p^2/q^4$, where $p=|\bm p|$, $q=|\bm q|$, $\bm p\in\mathbb R^D$, $\bm q\in\mathbb R^m$, and in terms of generalised hypergeometric functions $_3F_2(-X)$, when $X<1$. These are subsequently analytically continued to the complementary region $X\ge 1$. The asymptotic expansion in inverse powers of $X^{1/2}$ is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

hep-th↗
arXiv2016

Integrals of products of Hurwitz zeta functions and the Casimir effect in $ϕ^4$ field theories

We evaluate two integrals over $x\in [0,1]$ involving products of the function $ζ_1(a,x)\equiv ζ(a,x)-x^{-a}$ for $\Re (a)>1$, where $ζ(a,x)$ is the Hurwitz zeta function. The evaluation of these integrals for the particular case of integer $a\geq 2$ is also presented. As an application we calculate the $O(g)$ weak-coupling expansion coefficient $c_{1}(\varepsilon)$ of the Casimir energy for a film with Dirichlet-Dirichlet boundary conditions, first stated by Symanzik [Schrödinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1-44] in the framework of $gϕ^4_{4-\varepsilon}$ theory.

math.CA↗
arXiv2015

The Clausenian hypergeometric function $_3F_2$ with unit argument and negative integral parameter differences

New explicit as well as manifestly symmetric three-term summationformulas are derived for the Clausenian hypergeometric series $_3F_2(1)$ with negative integral parameter differences. Our results generalize and naturally extend several similar relations published, in recent years, by many authors. An appropriate and useful connection is established with the quite underestimated 1974 paper by P. W. Karlsson.

math.CA↗
arXiv2012

Lifshitz-point correlation length exponents from the large-n expansion

The large-n expansion is applied to the calculation of thermal critical exponents describing the critical behavior of spatially anisotropic d-dimensional systems at m-axial Lifshitz points. We derive the leading nontrivial 1/n correction for the perpendicular correlation-length exponent nu_{L2} and hence several related thermal exponents to order O(1/n). The results are consistent with known large-n expansions for d-dimensional critical points and isotropic Lifshitz points, as well as with the second-order epsilon expansion about the upper critical dimension d^*=4+m/2 for generic m\in[0,d]. Analytical results are given for the special case d=4, m=1. For uniaxial Lifshitz points in three dimensions, 1/n coefficients are calculated numerically. The estimates of critical exponents at d=3, m=1 and n=3 are discussed.

hep-th↗
arXiv2010

On conjectured local generalizations of anisotropic scale invariance and their implications

The theory of generalized local scale invariance of strongly anisotropic scale invariant systems proposed some time ago by Henkel [Nucl. Phys. B \textbf{641}, 405 (2002)] is examined. The case of so-called type-I systems is considered. This was conjectured to be realized by systems at m-axial Lifshitz points; in support of this claim, scaling functions of two-point cumulants at the uniaxial Lifshitz point of the three-dimensional ANNNI model were predicted on the basis of this theory and found to be in excellent agreement with Monte Carlo results [Phys. Rev. Lett. \textbf{87}, 125702 (2001)]. The consequences of the conjectured invariance equations are investigated. It is shown that fewer solutions than anticipated by Henkel generally exist and contribute to the scaling functions if these equations are assumed to hold for all (positive and negative) values of the d-dimensional space (or space time) coordinates $(t,\bm{r})\in \mathbb{R}\times\mathbb{R}^{d-1}$. Specifically, a single rather than two independent solutions exists in the case relevant for the mentioned fit of Monte Carlo data for the ANNNI model. Renormalization-group improved perturbation theory in $4+m/2-ε$ dimensions is used to determine the scaling functions of the order-parameter and energy-density two-point cumulants in momentum space to two-loop order. The results are mathematically incompatible with Henkel's predictions except in free-field-theory cases. However, the scaling function of the energy-density cumulant we obtain for m=1 upon extrapolation of our two-loop RG results to d=3 differs numerically little from that of an effective free field theory.

cond-mat.stat-mech↗