arXiv subjects

Igor Travěnec

Publications and source records attributed to Igor Travěnec.

12 recordsLinked to original sources

arXiv2026

Disordered ground states in one-dimensional exactly solvable fluids

One-dimensional fluids of classical hard-core particles of diameter $a$, interacting in pairs via a soft repulsive (monotonically decreasing) potential of finite range $φ(x)=\varepsilon \left[ (a'-x)/(a'-a)\right]^{1/ν}$ $(a\le x\le a')$ with real positive parameters $\varepsilon$ and $ν$, are studied in an isothermal-isobaric ensemble. If $a'\le 2a$, the pairwise interactions are reduced to nearest-neighbour interactions, which allows for an exact solution of the thermal equilibrium. We focus on the $T\to 0$ ground states, specifically on the equation of state for the mean distance between nearest neighbours $l_0$ and the pair correlation function $g_0(x)$. If $φ(x)$ is concave $(ν\ge 1)$, there exists an ``incompressibility'' pressure $p_i=\varepsilon/(a'-a)$ such that the ground state is an equidistant chain of particles with spacing $l_0=a'$ for $0 p_i$. If $ν>1$ (strict concavity), the ground state at $p=p_i$ is disordered with $l_0=\left[ νa +(ν-1)a'\right]/(2ν-1)$ and $g_0(x)$ being a superposition of weighted Dirac delta functions over discrete positions. If $ν=1$ (linear ramp), the ground state at $p=p_i$ is disordered with $l_0=(a+a')/2$ and the continuous $g_0(x)$ is a superposition of Heaviside step functions multiplied by polynomials in $x$. The isothermal susceptibility at $T=0$ is nonzero for concave $φ(x)$ ($ν\ge 1$) at $p=p_i$ and, therefore, the corresponding disordered ground states are non-hyperuniform, i.e., they resemble disordered fluids at nonzero temperatures. It turns out that pair correlation functions of disordered ground states, which occur only at a single pressure $p_i$, extend their predictive power to thermodynamic states at nonzero temperatures over a wider range of pressures around $p_i$.

cond-mat.stat-mech↗

arXiv2025

Structural transitions in interacting lattice systems

We consider two-dimensional systems of point particles located on rectangular lattices and interacting via pairwise potentials. The goal of this paper is to investigate the phase transitions (and their nature) at fixed density for the minimal energy of such systems. The 2D rectangle lattices we consider have an elementary cell of sides $a$ and $b$, the aspect ratio is defined as $Δ=b/a$ and the inverse particle density $A = a b$; therefore, the ``symmetric'' state with $Δ=1$ corresponds to the square lattice and the ``non-symmetric'' state to the rectangular lattice with $Δ\ne 1$. For certain types of the interaction potential, by changing continuously the particle density, such lattice systems undertake at a specific value of the (inverse) particle density $A^*$ a structural transition from the symmetric to the non-symmetric state. The structural transition can be either of first order ($Δ$ unstick from its symmetric value $Δ=1$ discontinuously) or of second order ($Δ$ unstick from $Δ=1$ continuously); the first and second-order phase transitions are separated by the so-called tricritical point. We develop a general theory on how to determine the exact values of the transition densities and the location of the tricritical point. The general theory is applied to the double Yukawa and Yukawa-Coulomb potentials.

math-ph↗

arXiv2025

Equidistant versus bipartite ground states for 1D classical fluids at fixed particle density

We study the ground-state properties of one-dimensional fluids of classical (i.e., non-quantum) particles interacting pairwisely via a potential, at the fixed particle density $ρ$. Restricting ourselves to periodic configurations of particles, two possibilities are considered: an equidistant chain of particles with the uniform spacing $A=1/ρ$ and its simplest non-Bravais modulation, namely a bipartite lattice composed of two equidistant chains, shifted with respect to one another. Assuming the long range of the interaction potential, the equidistant chain dominates if $A$ is small enough, $0<A<A_c$. At a critical value of $A=A_c$, the system undergoes a continuous second-order phase transition from the equidistant chain to a bipartite lattice. The energy and the order parameter are singular functions of the deviation from the critical point $A-A_c$ with universal (i.e., independent of the model's parameters) mean-field values of critical exponents. The tricritical point at which the curve of continuous second-order transitions meets with the one of discontinuous first-order transitions is determined. The general theory is applied to the Lennard-Jones model with the $(n,m)$ Mie potential for which the phase diagram is constructed. The inclusion of a hard-core around each particle reveals a non-universal critical phenomenon with an $m$-dependent critical exponent.

math-ph↗

arXiv2025

Low-temperature anomalies in one-dimensional exactly solvable fluids

Previous experiments and numerical simulations have revealed that a limited number of two- and three-dimensional particle systems contract in volume upon heating isobarically. This anomalous phenomenon is known as negative thermal expansion (NTE). The present paper focuses on the possibility of NTE in exactly solvable one-dimensional fluids. Firstly, the quantization of classical pure hard rods (free of NTE) does not induce NTE which indicates an unimportant role of quantum mechanics in the topic. Secondly, the classical hard rods with various types of soft nearest-neighbor interactions that contain a basin of attraction with only one minimum are investigated. The ground-state analysis reveals that, for certain potentials, increasing the pressure can lead to a discontinuous jump in the mean spacing between particles. The low-temperature analysis of the exact equation of state indicates that the NTE anomaly is present if the curvature of the soft potential increases with the distance between particles or if the potential exhibits a singularity within the basin of attraction. Isotherms of the compressibility factor, which measures the deviation of the thermodynamic behavior of a real gas from that of an ideal gas, demonstrate typical plateau or double-plateau shapes in large intervals of particle density.

cond-mat.soft↗

arXiv2025

1D fluids with repulsive nearest-neighbour interactions: Low-temperature anomalies

A limited number of 2D and 3D materials under a constant pressure contract in volume upon heating isobarically; this anomalous phenomenon is known as the negative thermal expansion (NTE). In this paper, the NTE anomaly is observed in 1D fluids of classical particles interacting pairwisely with two competing length scales: the hard-core diameter $a$ and the finite range $a'>a$ of a soft repulsive potential component. If $a'\le 2a$, the pair interactions reduce themselves to nearest neighbours which permits a closed-form solution of thermodynamics in the isothermal-isobaric ensemble characterized by temperature $T$ and pressure $p$. We focus on the equation of state (EoS) which relates the average distance between particles (reciprocal density) $l$ to $T$ and $p\ge 0$. The EoS is expressible explicitly in terms of elementary or special functions for specific, already known and new, cases like the square shoulder, the linear and quadratic ramps as well as certain types of logarithmic interaction potentials. The emphasis is put on low-$T$ anomalies of the EoS. Firstly, the equidistant ground state as the function of the pressure can exhibit, at some ``compressibility'' pressures, a jump in chain spacing from $a'$ to $a$. Secondly, the analytical structure of the low-$T$ expansion of $l(T,p)$ depends on ranges of $p$-values. Thirdly, the presence of the NTE anomaly depends very much on the shape of the core-softened potential.

cond-mat.stat-mech↗

arXiv2023

On off-critical zeros of lattice energies in the neighborhood of the Riemann zeta function

The Riemann zeta function $ζ(s):= \sum_{n=1}^{\infty} 1/n^s$ can be interpreted as the energy per point of the lattice $\mathbb{Z}$, interacting pairwisely via the Riesz potential $1/r^s$. Given a parameter $Δ\in (0,1]$, this physical model is generalized by considering the energy per point $E(s,Δ)$ of a periodic one-dimensional lattice alternating the distances between the nearest-neighbour particles as $2/(1+Δ)$ and $2Δ/(1+Δ)$, keeping the lattice density equal to one independently of $Δ$. This energy trivially satisfies $E(s,1)=ζ(s)$ at $Δ=1$, it can be easily expressed as a combination of the Riemann and Hurwitz zeta functions, and extended analytically to the punctured $s$-plane $\mathbb{C} \setminus \{ 1\}$. In this paper, we perform numerical investigations of the zeros of the energy $\{ ρ=ρ_x+{\rm i}ρ_y\}$, which are defined by $E(ρ,Δ)=0$. The numerical results reveal that in the Riemann limit $Δ\to 1^-$ theses zeros include the anticipated critical zeros of the Riemann zeta function with $\Re(ρ_x)=\frac{1}{2}$ as well as an unexpected -- comparing to the Riemann Hypothesis -- infinite series of off-critical zeros. The analytic treatment of these off-critical zeros shows that their imaginary components are equidistant and their real components diverge logarithmically to $-\infty$ as $Δ\to 1^-$, i.e., they become invisible at the Riemann's $Δ=1$.

math.NT↗

arXiv2022

Interplay between critical and off-critical zeros of two-dimensional Epstein zeta functions

The two-dimensional Epstein zeta function formulated on a rectangular lattice with spacings $a_x=1$ and $a_y=Δ$, $ζ^{(2)}(s,Δ) = \frac{1}{2} \sum_{j,k} (j^2+Δ^2 k^2)^{-s}$ $(\Re(s)>1)$ where the sum goes over all integers except of the origin $(j,k)=(0,0)$, is studied. It can be analytically continued to the whole complex $s$-plane except for the point $s=1$. The nontrivial zeros $\{ ρ=ρ_x+{\rm i}ρ_y \}$ of the Epstein zeta function, defined by $ζ^{(2)} (ρ,Δ)=0$, split into ``critical'' zeros (on the critical line $ρ_x=\frac{1}{2}$) and ``off-critical'' zeros ($ρ_x\ne\frac{1}{2}$). According to the present numerical calculation, the critical zeros form open or closed curves $ρ_y(Δ)$ in the plane $(Δ,ρ_y)$. Two nearest critical zeros merge at special points, referred to as left/right edge zeros, which are defined by a divergent tangent ${\rm d}ρ_y/{\rm d}Δ\vert_{Δ^*}$. Each of these edge zeros gives rise to a continuous curve of off-critical zeros which can thus be generated systematically. As a rule, each curve of off-critical zeros joins a pair of left and right edge zeros. It is shown that in the regions of small/large values of the anisotropy parameter $Δ$ the Epstein zeta function can be approximated adequately by a function which reveals an equidistant distribution of critical zeros along the imaginary axis in the limits $Δ\to 0$ and $Δ\to\infty$. It is also found that for each $Δ\in (0,Δ_c^*]\cup [1/Δ_c^*,\infty)$ with $Δ_c^*\approx 0.141733$ there exists a pair of \emph{real} off-critical zeros, their $ρ_x$ components go to the borders $0$ and $1$ of the critical region in the limits $Δ\to 0,\infty$. As a rule, each curve of off-critical zeros joins a pair of left and right edge zeros.

math.NT↗

arXiv2022

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

The Riesz potential $f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard-Jones type potentials $f_{n,m}^{\rm{LJ}}(r):=a r^{-n}-b r^{-m}$, $n>m$ that are widely used in Molecular Simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices of Riesz and Lennard-Jones energies. We discuss the minimality of the Body-Centred-Cubic lattice (BCC), Face-Centred-Cubic lattice (FCC), Simple Hexagonal lattices (SH) and Hexagonal Close-Packing structure (HCP), globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for $s<0$ and the HCP lattice is computed to have higher energy than the FCC (for $s>3/2$) and BCC (for $s<3/2$) lattices. In the Lennard-Jones case with exponents $3<m<n$, the ground state among lattices is confirmed to be a FCC lattice whereas a HCP phase occurs once added to the investigated structures. Furthermore, phase transitions of type ``FCC-SH" and ``FCC-HCP-SH" (when the HCP lattice is added) as the inverse density $V$ increases are observed for a large spectrum of exponents $(n,m)$. In the SH phase, the variation of the ratio $Δ$ between the inter-layer distance $d$ and the lattice parameter $a$ is studied as $V$ increases. In the critical region of exponents $0<m<n<3$, the SH phase with an extreme value of the anisotropy parameter $Δ$ dominates. If one limits oneself to rigid lattices, the BCC-FCC-HCP phase diagram is found. For $-2<m<n<0$, the BCC lattice is the only energy minimizer. Choosing $-4<m<n<-2$, the FCC and SH latices become minimizers.

math-ph↗

arXiv2021

Generation of off-critical zeros for hypercubic Epstein zeta-functions

We study the Epstein zeta-function formulated on the $d$-dimensional hypercubic lattice, $ζ^{(d)}(s) = (1/2)\sideset{}{'}\sum_{n_1,\ldots,n_d} (n_1^2+\cdots+n_d^2)^{-s/2}$ where $\Re(s)>d$ and the summation runs over all integers excluding the origin. An analytical continuation of the Epstein function to the whole complex $s$-plane is constructed for spatial dimension $d$ being a continuous variable ranging from $0$ to $\infty$. We are interested in zeros $ρ=ρ_x+{\rm i}ρ_y$ defined by $ζ^{(d)}(ρ) = 0$. Besides the trivial zeros, there exist "critical" zeros (on the critical line) with $ρ_x=\frac{d}{2}$ and "off-critical" zeros (off the critical line) with $ρ_x \ne \frac{d}{2}$. Our numerical results reveal that critical zeros form closed or semi-open curves $ρ_y(d)$ which enclose disjunctive regions of the complex plane $(ρ_x= d/2,ρ_y)$. Each curve involves a number of left/right edge points $ρ^*$, defined by an infinite tangent ${\rm d}ρ_y/{\rm d}d\vert_{ρ^*}$, which give rise to two conjugate tails of off-critical zeros with continuously varying dimension $d$. The curves of critical and off-critical zeros exhibit a singular expansion around edge points whose derivation resembles to the one around a critical point of mean-field type (with exponent $1/2$ for the order parameter) in many-body statistical models. Further it turns out that for each $d>9.24555\ldots$ there exists a conjugate pair of {\em real} off-critical zeros which tend to the boundaries $0$ and $d$ of the critical strip in the limit $d\to\infty$. As a by-product of the formalism, we derive an exact result for $\lim_{d\to 0} ζ^{(d)}(s)/d$ and an equidistant distribution of critical zeros along the imaginary axis in the limit $d\to\infty$.

math.NT↗

arXiv2019

Two-dimensional Wigner crystals of classical Lennard-Jones particles

The ground-state of two-dimensional (2D) systems of classical particles interacting pairwisely by the generalized Lennard-Jones potential is studied. Taking the surface area per particle $A$ as a free parameter and restricting oneself to periodic Bravais lattices with one particle per unit cell, Bétermin L [2018 Nonlinearity {\bf 31} 3973] proved that the hexagonal, rhombic, square and rectangular structures minimize successively the interaction energy per particle as $A$ increases. We show here that the second-order transitions between the rhombic/square and square/rectangular phases are of mean-field type. The aim of this paper is to extend Bétermin's analysis to periodic 2D lattices with more than one particle per elementary cell. Being motivated by previous works dealing with other kinds of models, we propose as the ground-state the extensions of the 2D rectangular (1-chain) lattice, namely the "zig-zag" (2-chain), 3-chain, 4-chain, etc. structures possessing 2, 3, 4, etc. particles per unit cell, respectively. By using a recent technique of lattice summation we find for the standard Lennard-Jones potential that their ground-state energy per particle approaches systematically as the number of particles per unit cell increases to the one of a phase separated state (the optimal hexagonal lattice). We analyze analytically the low-density limit $A\to\infty$ and the limiting hard-core case of the generalized Lennard-Jones potential.

cond-mat.other↗

arXiv2017

Tricriticality for dimeric Coulomb molecular crystals in ground state

We study the ground-state properties of a system of dimers. Each dimer consists in a pair of equivalent charges at a fixed distance, immersed in a neutralizing homogeneous background. All charges interact pairwisely by Coulomb potential. The dimer centers form a two-dimensional rectangular lattice with the aspect ratio $α\in [0,1]$ and each dimer is allowed to rotate around its center. The previous numerical simulations, made for the more general Yukawa interaction, indicate that only two basic dimer configurations can appear: either all dimers are parallel or they have two different angle orientations within alternating (checkerboard) sublattices. As the dimer size increases, two second-order phase transitions, related to two kinds of the symmetry breaking in dimer's orientations, were reported. In this paper, we use a recent analytic method based on an expansion of the interaction energy in Misra functions which converges quickly and provides an analytic derivation of the critical behaviour. Our main result is that there exists a specific aspect ratio of the rectangular lattice $α^*=0.71410684000071\ldots$ which divides the space of model's phases onto two distinct regions. If the lattice aspect ratio $α>α^*$, we recover both types of the second-order phase transitions and find that they are of mean-field type with the critical exponent $β= 1/2$. If $α<α^*$, the phase transition associated with the discontinuity of dimer's angles on alternating sublattices becomes of first order. For $α=α^*$, the first- and second-order phase transitions meet at the tricritical point, characterized by the different critical index $β= 1/4$. Such phenomenon is known from literature about the Landau theory of one-component fields, but in our two-component version the scenario is more complicated.

cond-mat.stat-mech↗

arXiv2016

Separation of Wigner structures for 2D equimolar binary mixtures of Coulomb particles

We study the lowest energy configurations of an equimolar binary mixture of classical pointlike particles with charges $Q_1$ and $Q_2$, such that $q=Q_2/Q_1\in [0,1]$. The particles interact pairwisely via 3D Coulomb potential and are confined to a 2D plane with a homogeneous neutralizing background charge density. In a recent paper by M. Antlanger and G. Kahl [Cond. Mat. Phys. {\bf 16}, 43501 (2013)], using numerical computations based on evolutionary algorithm, six fully mixed structures were identified for $0\le q\lesssim 0.59$, while the separation of $Q_1$ and $Q_2$ pure hexagonal phases minimizes the energy for $0.59\lesssim q<1$. Here, we introduce a novel structure which consists in the separation of two phases, the pure hexagonal one formed by a fraction of particles with the larger charge $Q_1$ and the other fixed one containing different numbers of $Q_1$ and $Q_2$ charges. Using an analytic method based on an expansion of the interaction energy in Misra functions we show that this novel structure provides the lowest energy in two intervals of $q$ values, $0<q\lesssim 0.04707$ and $0.58895\lesssim q\lesssim 0.61367$. This fact might inspire numerical methods, for both Coulomb and Yukawa interactions, to test more general separations which go beyond the separation of two pure phases.

cond-mat.other↗