arXiv subjects

Yacov Kantor

Publications and source records attributed to Yacov Kantor.

At least 19 recordsLinked to original sources

arXiv2026

Large clusters in a correlated percolation model

We consider a correlated site percolation problem on a cubic lattice of size $L^3$, with $16\le L\le 512$. The sites of an initially full lattice are removed by a random walk of ${\cal N}=uL^3$ steps. When the parameter $u$ crosses a threshold $u_c=3.15$, a large system transitions between percolating and non-percolating states. We study the $L$-dependence of the mean mass (number of sites) $M_r$ of the $r$th largest cluster, as well as $r$-dependence of $M_r$ for various system sizes $L$ at $u_c$. We demonstrate that $M_r\sim L^{5/2}/r^{5/6}$ for moderate or large $L$ and $r\gg 1$, and also conclude that for {\em any} $r$ the fractal dimensions of the clusters are $5/2$.

cond-mat.stat-mech↗

arXiv2024

Critical exponents of correlated percolation of sites not visited by a random walk

We consider a $d$-dimensional correlated percolation problem of sites {\em not} visited by a random walk on a hypercubic lattice $L^d$ for $d=3$, 4 and 5. The length of the random walk is ${\cal N}=uL^d$. Close to the critical value $u=u_c$, many geometrical properties of the problem can be described as powers (critical exponents) of $u_c-u$, such as $β$, which controls the strength of the spanning cluster, and $γ$, which characterizes the behavior of the mean finite cluster size $S$. We show that at $u_c$ the ratio between the mean mass of the largest cluster $M_1$ and the mass of the second largest cluster $M_2$ is independent of $L$ and can be used to find $u_c$. We calculate $β$ from the $L$-dependence of $M_2$ and $γ$ from the finite size scaling of $S$. The resulting exponent $β$ remains close to 1 in all dimensions. The exponent $γ$ decreases from $\approx 3.9$ in $d=3$ to $\approx1.9$ in $d=4$ and $\approx 1.3$ in $d=5$ towards $γ=1$ expected in $d=6$, which is close to $γ=4/(d-2)$.

cond-mat.stat-mech↗

arXiv2022

Quantum particle in a spherical well confined by a cone

We consider the quantum problem of a particle in either a spherical box or a finite spherical well confined by a circular cone with an apex angle $2θ_0$ emanating from the center of the sphere, with $0<θ_0<π$. This non-central potential can be solved by an extension of techniques used in spherically-symmetric problems. The angular parts of the eigenstates depend on azimuthal angle $φ$ and polar angle $θ$ as $P_λ^m(\cosθ){\rm e}^{imφ}$ where $P_λ^m$ is the associated Legendre function of integer order $m$ and (usually noninteger) degree $λ$. There is an infinite discrete set of values $λ=λ_i^m$ ($i=0,1,3,\dots$) that depend on $m$ and $θ_0$. Each $λ_i^m$ has an infinite sequence of eigenenergies $E_n(λ_i^m)$, with corresponding radial parts of eigenfunctions. In a spherical box the discrete energy spectrum is determined by the zeros of the spherical Bessel functions. For several $θ_0$ we demonstrate the validity of Weyl's continuous estimate ${\cal N}_W$ for the exact number of states $\cal N$ up to energy $E$, and evaluate the fluctuations of $\cal N$ around ${\cal N}_W$. We examine the behavior of bound states in a well of finite depth $U_0$, and find the critical value $U_c(θ_0)$ when all bound states disappear. The radial part of the zero energy eigenstate outside the well is $1/r^{λ+1}$, which is not square-integrable for $λ\le 1/2$. ($0<λ\le 1/2$ can appear for $θ_0>θ_c\approx 0.726π$ and has no parallel in spherically-symmetric potentials.) Bound states have spatial extent $ξ$ which diverges as a (possibly $λ$-dependent) power law as $U_0$ approaches the value where the eigenenergy of that state vanishes.

quant-ph↗

arXiv2021

Percolation Perspective on Sites Not Visited by a Random Walk in Two Dimensions

We consider the percolation problem of sites on an $L\times L$ square lattice with periodic boundary conditions which were unvisited by a random walk of $N=uL^2$ steps, i.e. are vacant. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with $u$. The clusters of vacant sites are not fractal but have fractal boundaries of dimension 4/3. The lattice size $L$ is the only large length scale in this problem. The typical mass (number of sites $s$) in the largest cluster is proportional to $L^2$, and the mean mass of the remaining (smaller) clusters is also proportional to $L^2$. The normalized (per site) density $n_s$ of clusters of size (mass) $s$ is proportional to $s^{-τ}$, while the volume fraction $P_k$ occupied by the $k$th largest cluster scales as $k^{-q}$. We put forward a heuristic argument that $τ=2$ and $q=1$. However, the numerically measured values are $τ\approx1.83$ and $q\approx1.20$. We suggest that these are effective exponents that drift towards their asymptotic values with increasing $L$ as slowly as $1/\ln L$ approaches zero.

cond-mat.stat-mech↗

arXiv2019

Percolation of sites not removed by a random walker in $d$ dimensions

How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of $N=uL^d$ steps on a $d$-dimensional hypercubic lattice of size $L^d$ (with periodic boundaries). We systematically explore dependence of the probability $Π_d(L,u)$ of percolation (existence of a spanning cluster) of sites not removed by the RW on $L$ and $u$. The concentration of unvisited sites decays exponentially with increasing $u$, while the visited sites are highly correlated -- their correlations decaying with the distance $r$ as $1/r^{d-2}$ (in $d>2$). Upon increasing $L$, the percolation probability $Π_d(L,u)$ approaches a step function, jumping from 1 to 0 when $u$ crosses a percolation threshold $u_c$ that is close to 3 for all $3\le d\le6$. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with $ν=2/(d-2)$. There is no percolation threshold at the lower critical dimension of $d=2$, with the percolation probability approaching a smooth function $Π_2(\infty,u)>0$.

cond-mat.stat-mech↗

arXiv2018

Localization of random walks to competing manifolds of distinct dimensions

We consider localization of a random walk (RW) when attracted or repelled by multiple extended manifolds of different dimensionalities. In particular, we focus on $(d-1)$- and $(d-2)$-dimensional manifolds in $d$-dimensional space, where attractive interactions are (fully or marginally) relevant. The RW can then be in one of four phases where it is localized to neither, one, or both manifolds. The four phases merge at a special multi-critical point where (away from the manifolds) the RW spreads diffusively. Extensive numerical analyses on two dimensional RWs confined inside or outside a rectangular wedge confirm general features expected from a continuum theory, but also exhibit unexpected attributes, such as a reentrant localization to the corner while repelled by it.

cond-mat.stat-mech↗

arXiv2017

Pinning and unbinding of (ideal) polymers from a wedge corner

A polymer repelled by unfavorable interactions with a uniform flat surface may still be pinned to attractive edges and corners. This is demonstrated by considering adsorption of a two-dimensional ideal polymer to an attractive corner of a repulsive wedge. The well-known mapping between the statistical mechanics of an ideal polymer and the quantum problem of a particle in a potential is then used to analyze the singular behavior of the unbinding transition of the polymer. The divergence of the localization length is found to be governed by an exponent that varies continuously with the angle (when reflex). Numerical treatment of the discrete (lattice) version of such an adsorption problem confirms this behavior.

cond-mat.stat-mech↗

arXiv2017

Nonequilibrium interactions between ideal polymers and a repulsive surface

We use Newtonian and overdamped Langevin dynamics to study long flexible polymers dragged by an external force at a constant velocity $v$. The work $W$ by that force depends on the initial state of the polymer and the details of the process. Jarzynski equality can be used to relate the non-equilibrium work distribution $P(W)$ obtained from repeated experiments to equilibrium free energy difference $ΔF$ between the initial and final states. We use the power law dependence of the geometrical and dynamical characteristics of the polymer on the number of monomers $N$ to suggest the existence of a critical velocity $v_c(N)$, such that for $v<v_c$ the reconstruction of $ΔF$ is an easy task, while for $v$ significantly exceeding $v_c$ it becomes practically impossible. We demonstrate the existence of such $v_c$ analytically for ideal polymer in free space and numerically for a polymer being dragged away from a repulsive wall. Our results suggest that the distribution of the dissipated work $W_{\rm d}=W-ΔF$ in properly scaled variables approaches a limiting shape for large $N$.

cond-mat.stat-mech↗

arXiv2017

Attractive and repulsive polymer-mediated forces between scale-free surfaces

We consider forces acting on objects immersed in, or attached to, long fluctuating polymers. The confinement of the polymer by the obstacles results in polymer-mediated forces that can be repulsive (due to loss of entropy) or attractive (if some or all surfaces are covered by adsorbing layers). The strength and sign of the force in general depends on the detailed shape and adsorption properties of the obstacles, but assumes simple universal forms if characteristic length scales associated with the objects are large. This occurs for scale-free shapes (such as a flat plate, straight wire, or cone), when the polymer is repelled by the obstacles, or is marginally attracted to it (close to the depinning transition where the absorption length is infinite). In such cases, the separation $h$ between obstacles is the only relevant macroscopic length scale, and the polymer mediated force equals ${\cal A} \, k_{B}T/h$, where $T$ is temperature. The amplitude ${\cal A}$ is akin to a critical exponent, depending only on geometry and universality of the polymer system. The value of ${\cal A}$, which we compute for simple geometries and ideal polymers, can be positive or negative. Remarkably, we find ${\cal A}=0$ for ideal polymers at the adsorption transition point, irrespective of shapes of the obstacles, i.e. at this special point there is no polymer-mediated force between obstacles (scale-free or not).

cond-mat.stat-mech↗

arXiv2016

Winding angles of long lattice walks

We study the winding angles of random and self-avoiding walks on square and cubic lattices with number of steps $N$ ranging up to $10^7$. We show that the mean square winding angle $\langleθ^2\rangle$ of random walks converges to the theoretical form when $N\rightarrow\infty$. For self-avoiding walks on the square lattice, we show that the ratio $\langleθ^4\rangle/\langleθ^2\rangle^2$ converges slowly to the Gaussian value 3. For self avoiding walks on the cubic lattice we find that the ratio $\langleθ^4\rangle/\langleθ^2\rangle^2$ exhibits non-monotonic dependence on $N$ and reaches a maximum of 3.73(1) for $N\approx10^4$. We show that to a good approximation, the square winding angle of a self-avoiding walk on the cubic lattice can be obtained from the summation of the square change in the winding angles of $\ln N$ independent segments of the walk, where the $i$-th segment contains $2^i$ steps. We find that the square winding angle of the $i$-th segment increases approximately as $i^{0.5}$, which leads to an increase of the total square winding angle proportional to $(\ln N)^{1.5}$.

cond-mat.stat-mech↗

arXiv2015

Long Polymers Near Wedges and Cones

We perform a Monte Carlo study of $N$-step self-avoiding walks, attached to the corner of an impenetrable wedge in two dimensions ($d=2$), or the tip of an impenetrable cone in $d=3$, of sizes ranging up to $N=10^6$ steps. We find that the critical exponent $γ_α$ which determines the dependence of the number of available conformations on $N$ for a cone/wedge with opening angle $α$, is in good agreement with the theory for $d=2$. We study the end-point distribution of the walks in the allowed space and find similarities to the known behavior of random walks (ideal polymers) in the same geometry. For example the ratio between the mean square end-to-end distances of a polymer near the wedge and a polymer in free space depends linearly on $γ_α$ as is known for ideal polymers. We show that the end-point distribution of polymers attached to a wedge does not separate into a product of angular and radial functions, as it does for ideal polymers in the same geometry. The angular dependence of the end-position of polymers near the wedge differs from theoretical predictions.

cond-mat.stat-mech↗

arXiv2015

Diffusion in the Presence of Scale-Free Absorbing Boundaries

Scale-free surfaces, such as cones, remain unchanged under a simultaneous expansion of all coordinates by the same factor. Probability density of a particle diffusing near such absorbing surface at large time approaches a simple form that incorporates power-law dependencies on time and distance from a special point, such as apex of the cone, which are characterized by a single exponent $η$. The same exponent is used to describe the number of spatial conformations of long ideal polymer attached to the special point of a repulsive surface of the same geometry and can be used in calculation of entropic forces between such polymers and surfaces. We use the solution of diffusion equation near such surfaces to find the numerical values of $η$, as well as to provide some insight into the behavior of ideal polymers near such surfaces.

cond-mat.stat-mech↗

arXiv2014

Entropic pressure in lattice models for polymers

In lattice models local pressure on a surface is derived from the change in the free energy of the system due to the exclusion of a certain boundary site, while the total force on the surface can be obtained by a similar exclusion of all surface sites. In these definitions, while the total force on the surface of a lattice system matches the force measured in a continuous system, the local pressure does not. Moreover, in a lattice system, the sum of the local pressures is not equal to the total force as is required in a continuous system. The difference is caused by correlation between occupations of surface sites as well as finite displacement of surface elements used in the definition of the pressures and the force. This problem is particularly acute in the studies of entropic pressure of polymers represented by random or self-avoiding walks on a lattice. We propose a modified expression for the local pressure which satisfies the proper relation between the pressure and the total force, and show that for ideal polymers in the presence of scale-invariant boundaries it produces quantitatively correct values for continuous systems. The required correction to the pressure is non-local, i.e., it depends on long range correlations between contact points of the polymer and the surface.

cond-mat.stat-mech↗

arXiv2014

Ideal Polymers near Scale-Free Surfaces

The number of allowed configurations of a polymer is reduced by the presence of a repulsive surface resulting in an entropic force between them. We develop a method to calculate the entropic force, and detailed pressure distribution, for long ideal polymers near a scale-free repulsive surface. For infinite polymers the monomer density is related to the electrostatic potential near a conducting surface of a charge placed at the point where the polymer end is held. Pressure of the polymer on the surface is then related to the charge density distribution in the electrostatic problem. We derive explicit expressions for pressure distributions and monomer densities for ideal polymers near a two- or three-dimensional wedge, and for a circular cone in three dimensions. Pressure of the polymer diverges near sharp corners in a manner resembling (but not identical to) the electric field divergence near conducting surfaces. We provide formalism for calculation of all components of the total force in situations without axial symmetry.

cond-mat.stat-mech↗

arXiv2012

Polymer-mediated entropic forces between scale-free objects

The number of configurations of a polymer is reduced in the presence of a barrier or an obstacle. The resulting loss of entropy adds a repulsive component to other forces generated by interaction potentials. When the obstructions are scale invariant shapes (such as cones, wedges, lines or planes) the only relevant length scales are the polymer size R_0 and characteristic separations, severely constraining the functional form of entropic forces. Specifically, we consider a polymer (single strand or star) attached to the tip of a cone, at a separation h from a surface (or another cone). At close proximity, such that h<<R_0, separation is the only remaining relevant scale and the entropic force must take the form F=AkT/h. The amplitude A is universal, and can be related to exponents ηgoverning the anomalous scaling of polymer correlations in the presence of obstacles. We use analytical, numerical and epsilon-expansion techniques to compute the exponent ηfor a polymer attached to the tip of the cone (with or without an additional plate or cone) for ideal and self-avoiding polymers. The entropic force is of the order of 0.1 pN at 0.1 micron for a single polymer, and can be increased for a star polymer.

cond-mat.soft↗

arXiv2011

Entropic force of polymers on a cone tip

We consider polymers attached to the tip of a cone, and the resulting force due to entropy loss on approaching a plate (or another cone). At separations shorter than the polymer radius of gyration R_g, the only relevant length scale is the tip-plate (or tip-tip) separation h, and the entropic force is given by F=A kT/h. The universal amplitude A can be related to (geometry dependent) correlation exponents of long polymers. We compute A for phantom polymers, and for self-avoiding (including star) polymers by epsilon-expansion, as well as by numerical simulations in 3 dimensions.

cond-mat.stat-mech↗

arXiv2010

First Passage Distributions in a Collective Model of Anomalous Diffusion with Tunable Exponent

We consider a model system in which anomalous diffusion is generated by superposition of underlying linear modes with a broad range of relaxation times. In the language of Gaussian polymers, our model corresponds to Rouse (Fourier) modes whose friction coefficients scale as wavenumber to the power $2-z$. A single (tagged) monomer then executes subdiffusion over a broad range of time scales, and its mean square displacement increases as $t^α$ with $α=1/z$. To demonstrate non-trivial aspects of the model, we numerically study the absorption of the tagged particle in one dimension near an absorbing boundary or in the interval between two such boundaries. We obtain absorption probability densities as a function of time, as well as the position-dependent distribution for unabsorbed particles, at several values of $α$. Each of these properties has features characterized by exponents that depend on $α$. Characteristic distributions found for different values of $α$ have similar qualitative features, but are not simply related quantitatively. Comparison of the motion of translocation coordinate of a polymer moving through a pore in a membrane with the diffusing tagged monomer with identical $α$ also reveals quantitative differences.

cond-mat.stat-mech↗

arXiv2009

Configurations of polymers attached to probes

We study polymers attached to spherical (circular) or paraboloidal (parabolic) probes in three (two) dimensions. Both self-avoiding and random walks are examined numerically. The behavior of a polymer of size $R_0$ attached to the tip of a probe with radius of curvature $R$, differs qualitatively for large and small values of the ratio $s=R_0/R$. We demonstrate that the scaled compliance (inverse force constant) $S/R_0^2$, and scaled mean position of the polymer end-point $ /R$ can be expressed as a function of $s$. Scaled compliance is anisotropic, and quite large in the direction parallel to the surface when $R_0\sim R$. The exponent $γ$, characterizing the number of polymer configurations, crosses over from a value of $γ_1$ - characteristic of a planar boundary - at small $s$ to one reflecting the overall shape of the probe at large $s$. For a spherical probe the crossover is to an unencumbered polymer, while for a parabolic probe we cannot rule out a new exponent.

cond-mat.stat-mech↗