arXiv subjects

Ladislav Šamaj

Publications and source records attributed to Ladislav Šamaj.

At least 19 recordsLinked to original sources

arXiv2026

Counterions between walls with surface charge modulations: Exact Poisson-Boltzmann solutions

We study the thermal equilibrium of a classical system of identical point charges moving between two walls with parallel surfaces at a distance $d$, charged symmetrically with position-dependent surface charge densities. Specifically, we study the effect of surface charge modulation on the effective force (pressure) between the walls in the Poisson-Boltzmann limit. Based on Monte Carlo simulations at high temperatures, it is predicted that surface charge modulation reduces the pressure between the walls compared to uniformly charged wall surfaces with the same average surface charge density. We restrict ourselves to surface charge modulations in only one direction and, using a general solution of the two-dimensional Liouville equation, we construct exact solutions for the electrostatic potential. The surface charge density on the walls is generated inversely from this potential, which means that our exact results apply to a limited set of models with a specific variation of the surface charge density with distance $d$. Explicit analytical results show that surface charge modulation can both increase (small distances $d$) and decrease (large $d$) the pressure between the walls.

cond-mat.stat-mech↗

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↗

arXiv2026

A strong-coupling theory for polarizable symmetrically charged walls with counterions only

A pair of parallel polarizable planar walls at distance d is considered. The walls are symmetrically charged with a uniform surface charge density, neutralized by mobile point counterions moving between them. The case of repulsive particle images is studied in the strong-coupling (SC) regime. Of interest is the dependence of the effective inter-wall interaction (pressure), mediated by the mobile counterions, as a function of the distance d. It is shown that previous virial SC single-particle theories work well at small d when the dielectric jump is small; for intermediate and large dielectric jumps they are inadequate even in the SC region. Here, we propose a Wigner-type SC theory based on harmonic deviations of particles from their ground-state monolayer or bilayer Wigner structures formed inside the space between the dielectric walls. Our Monte-Carlo simulations are in very good agreement with the Wigner SC predictions, even down to moderate coupling constants ($Ξ> 10$).

cond-mat.soft↗

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↗

arXiv2025

Contact value theorem for electric double layers with modulated surface charge density

The contact value theorem was originally derived for Coulomb fluids of mobile charged particles in thermal equilibrium, in the presence of interfaces carrying a {\em uniform} surface charge density and in the absence of dielectric discontinuities. It relates the pressure (the effective force) between two parallel electric double layers to the particle number density and the surface charge density at the interface, separately for each of the two electric double layers. In this paper, we generalise the contact value theorem to electric double layers with interfaces carrying a {\em modulated} surface charge density. The derivation is based on balance of forces exerted on interfaces. The relevance of particular terms of the contact value theorem is tested on an exactly solvable two-dimensional Coulomb system with counterions only at the coupling constant $Γ=2$.

cond-mat.stat-mech↗

arXiv2024

Effective interaction between guest charges immersed in 2D jellium

The model under study is an infinite 2D jellium of pointlike particles with elementary charge $e$, interacting via the logarithmic potential and in thermal equilibrium at the inverse temperature $β$. Two cases of the coupling constant $Γ\equiv βe^2$ are considered: the Debye-Hückel limit $Γ\to 0$ and the free-fermion point $Γ=2$. In the most general formulation, two guest particles, the one with charge $q e$ (the valence $q$ being an arbitrary integer) and the hard core of radius $σ>0$ and the pointlike one with elementary charge $e$, are immersed in the bulk of the jellium at distance $d\ge σ$. Two problems are of interest: the asymptotic large-distance behavior of the excess charge density induced in the jellium and the effective interaction between the guest particles. Technically, the induced charge density and the effective interaction are expressed in terms of multi-particle correlations of the pure (translationally invariant) jellium system. It is shown that the separation form of the induced charge density onto its radial and angle parts, observed previously in the limit $Γ\to 0$, is not reproduced at the coupling $Γ=2$. Based on an exact expression for the effective interaction between guest particles at $Γ=2$, oppositely ($q=0,-1,-2,\ldots$) charged guest particles always attract one another while likely ($q=1,2,\ldots$) charged guest particles repeal one another up to a certain distance $d$ between them and then the mutual attraction takes place up to asymptotically large (finite) distances.

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

Ordered ground state configurations of the asymmetric Wigner bilayer system -- revisited: an unsupervised clustering algorithm analysis

We have re-analysed the rich plethora of ground state configurations of the asymmetric Wigner bilayer system that we had recently published in a related diagram of states [M. Antlanger \textit{et al.}, Phys. Rev. Lett. \textbf{117}, 118002 (2016)], comprising roughly 60~000 state points in the phase space spanned by the distance between the plates and the charge asymmetry parameter of the system. In contrast to this preceding contribution where the classification of the emerging structures was carried out ``by hand'', we have used this time machine learning concepts, notably based on a principal component analysis and a $k$-means clustering approach: using a 30-dimensional feature vector for each emerging structure (containing relevant information, such as the composition of the configuration as well as the most relevant order parameters) we were able to re-analyse these ground state configurations in a considerably more systematic and comprehensive manner than we could possibly do in the previously published classification scheme. Indeed we were now able to identify new structures in previously unclassified regions of the parameter space and could considerably refine the previous classification scheme, identifying thereby a rich wealth of new emerging ground state configurations. Thorough consistency checks confirm the validity of the newly defined diagram of states.

cond-mat.soft↗

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↗

arXiv2022

Electric double layers with modulated surface charge density: Exact 2D results

Electric double layers (EDL) with counterions only, say electrons with the elementary charge $-e$, in thermal equilibrium at the inverse temperature $β$ are considered. In particular, we study the effect of the surface charge modulation on the particle number density profile and the pressure. The mobile particles are constrained to the surface of a 2D cylinder and immersed in vacuum (no dielectric image charges). An EDL corresponds to the end-circle of the cylinder which carries a (fixed) position-dependent line charge density. The geometries of one single EDL and two EDLs at distance $d$ are considered; the particle density profile is studied for both geometries, the effective interaction of two EDLs is given by the particle pressure on either of the line walls. For any coupling constant $Γ\equiv βe^2 = 2\times {\rm integer}$, there exists a mapping of the 2D one-component Coulomb system onto the 1D interacting anticommuting-field theory defined on a chain of sites. Using specific transformation symmetries of anticommuting variables, the contact value theorem is generalized to the EDL with modulated line charge density. For the free-fermion coupling $Γ=2$ it is shown that, under certain conditions, the matrix of interaction strengths between anticommuting variables diagonalizes itself which permits one to obtain exact formulas for the particle density profile as well as the pressure. The obtained results confirm the previous indications of weak-coupling and Monte Carlo observations that the surface charge inhomogeneity implies an enhancement of the counterion density at the contact with the charged line and a diminution of the pressure between two parallel lines in comparison with the uniformly charged ones (with the same mean charge densities).

cond-mat.stat-mech↗

arXiv2022

Ising ferromagnets and antiferromagnets in an imaginary magnetic field

We study classical Ising spin-$\frac{1}{2}$ models on the 2D square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution which prevents from application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field we calculate with a high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent $β$ and the anomaly number $c$ are shown to be constant along the critical line, equal to their values $β=\frac{1}{8}$ and $c=\frac{1}{2}$ for the 2D Ising in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: it varies chaotically with the size of the system.

cond-mat.stat-mech↗

arXiv2022

Like-charge attraction at the nanoscale: ground-state correlations and water destructuring

Like-charge attraction, driven by ionic correlations, challenges our understanding of electrostatics both in soft and hard matter. For two charged planar surfaces confining counterions and water, we prove that even at relatively low correlation strength, the relevant physics is the ground-state one, oblivious of fluctuations. Based on this, we derive a simple and accurate interaction pressure, that fulfills known exact requirements and can be used as an effective potential. We test this equation against implicit-solvent Monte Carlo simulations and against explicit-solvent simulations of cement and several types of clays. We argue that water destructuring under nanometric confinement drastically reduces dielectric screening, enhancing ionic correlations. Our equation of state at reduced permittivity therefore explains the exotic attractive regime reported for these materials, even in absence of multivalent counterions.

cond-mat.soft↗

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↗

arXiv2020

Attraction of like-charged walls with counterions only: Exact results for the 2D cylinder geometry

We study a 2D system of identical mobile particles on the surface of a cylinder of finite length $d$ and circumference $W$, immersed in a medium of dielectric constant $\varepsilon$. The two end-circles of the cylinder are like-charged with the fixed uniform charge densities, the particles of opposite charge $-e$ ($e$ being the elementary charge) are coined as ``counterions''; the system as a whole is electroneutral. Such a geometry is well defined also for finite numbers of counterions $N$. Our task is to derive an effective interaction between the end-circles mediated by the counterions in thermal equilibrium at the inverse temperature $β$. The exact solution of the system at the free-fermion coupling $Γ\equiv βe^2/\varepsilon =2$ is used to test the convergence of the pressure as the (even) number of particles increases from $N=2$ to $\infty$. The pressure as a function of distance $d$ is always positive (effective repulsion between the like-charged circles), decaying monotonously; the numerical results for $N=8$ counterions are very close to those in the thermodynamic limit $N\to\infty$. For the couplings $Γ=2γ$ with $γ=1,2,\ldots$, there exists a mapping of the continuous two-dimensional (2D) Coulomb system with $N$ particles onto the one-dimensional (1D) lattice model of $N$ sites with interacting sets of anticommuting variables. This allows one to treat exactly the density profile, two-body density and the pressure for the couplings $Γ=4$ and $6$, up to $N=8$ particles. Our main finding is that the pressure becomes negative at large enough distances $d$ if and only if both like-charged walls carry a nonzero charge density. This indicates a like-attraction in the thermodynamic limit $N\to\infty$ as well, starting from a relatively weak coupling constant $Γ$ in between 2 and 4.

cond-mat.stat-mech↗