arXiv subjects

V. A. Avetisov

Publications and source records attributed to V. A. Avetisov.

12 recordsLinked to original sources

arXiv2021

Fractional Brownian motion meets topology: statistical and topological properties of globular macromolecules with volume interactions

In the paper we investigate statistical and topological properties of fractional Brownian polymer chains, equipped with the short-range volume interactions. The attention is paid to statistical properties of collapsed conformations with the fractal dimension $D_f\ge 2$ in the three-dimensional space, which are analyzed both numerically and \textit{via} the mean-field Flory approach. Our study is motivated by an attempt to mimic the conformational statistics of collapsed unknotted polymer rings, which are known to form compact hierarchical crumpled globules (CG) with $D_f=3$ at large scales. Replacing the topologically-stabilized CG state by a self-avoiding fractal path adjusted to the fractal dimension $D_f=3$ we tremendously simplify the problem of generating compact self-avoiding conformations since we wash out the topological constraints from the consideration. We make use of the Monte-Carlo simulations to prepare the equilibrium ensemble of swollen chains with various fractal dimensions. A combination of the Flory arguments with statistical analysis of the conformations from simulations allows one to infer the dependence of the critical exponent of the swollen chains on the fractal dimension of the seed chain. We show that with the increase of $D_f$, typical conformations become more territorial and less knotted. Distributions of the knot complexity, $P(χ)$ for various fractal dimensions of the swollen chains suggest a close relationship between statistical and topological properties of fractal paths with volume interactions.

cond-mat.soft↗

arXiv2014

Islands of stability in motif distributions of random networks

We consider random non-directed networks subject to dynamics conserving vertex degrees and study analytically and numerically equilibrium three-vertex motif distributions in the presence of an external field, $h$, coupled to one of the motifs. For small $h$ the numerics is well described by the "chemical kinetics" for the concentrations of motifs based on the law of mass action. For larger $h$ a transition into some trapped motif state occurs in Erdős-Rényi networks. We explain the existence of the transition by employing the notion of the entropy of the motif distribution and describe it in terms of a phenomenological Landau-type theory with a non-zero cubic term. A localization transition should always occur if the entropy function is non-convex. We conjecture that this phenomenon is the origin of the motifs' pattern formation in real evolutionary networks.

cond-mat.stat-mech↗

arXiv2013

Statistical model of intra-chromosome contact maps

The statistical properties of intra-chromosome maps obtained by a genome-wide chromosome conformation capture method (Hi-C) are described in the framework of the hierarchical crumpling model of heteropolymer chain with quenched disorder in the primary sequence. We conjecture that the observed Hi-C maps are statistical averages over many different ways of hierarchical genome folding, and show that the existence of quenched primary structure coupled with hierarchical folding can induce the observed fine structure of intra-chromosome contact maps.

q-bio.GN↗

arXiv2013

Fractal globule as an artificial molecular machine

The relaxation of an elastic network, constructed by a contact map of a fractal (crumpled) polymer globule is investigated. We found that: i) the slowest mode of the network is separated from the rest of the spectrum by a wide gap, and ii) the network quickly relaxes to a low--dimensional (one-dimensional, in our demonstration) manifold spanned by slowest degrees of freedom with a large basin of attraction, and then slowly approaches the equilibrium not escaping this manifold. By these dynamic properties, the fractal globule elastic network is similar to real biological molecular machines, like myosin. We have demonstrated that unfolding of a fractal globule can be described as a cascade of equilibrium phase transitions in a hierarchical system. Unfolding manifests itself in a sequential loss of stability of hierarchical levels with the temperature change.

cond-mat.stat-mech↗

arXiv2010

On the motifs distribution in random hierarchical networks

The distribution of motifs in random hierarchical networks defined by nonsymmetric random block--hierarchical adjacency matrices, is constructed for the first time. According to the classification of U. Alon et al of network superfamilies by their motifs distributions, our artificial directed random hierarchical networks falls into the superfamily of natural networks to which the class of neuron networks belongs. This is the first example of ``handmade'' networks with the motifs distribution as in a special class of natural networks of essential biological importance.

q-bio.QM↗

arXiv2010

Chaotic Hamiltonian systems revisited: Survival probability

We consider the dynamical system described by the area--preserving standard mapping. It is known for this system that $P(t)$, the normalized number of recurrences staying in some given domain of the phase space at time $t$ (so-clled "survival probability") has the power--law asymptotics, $P(t)\sim t^{-ν}$. We present new semi--phenomenological arguments which enable us to map the dynamical system near the chaos border onto the effective "ultrametric diffusion" on the boundary of a tree--like space with hierarchically organized transition rates. In the frameworks of our approach we have estimated the exponent $ν$ as $ν=\ln 2/\ln (1+r_g)\approx 1.44$, where $r_g=(\sqrt{5}-1)/2$ is the critical rotation number.

nlin.CD↗

arXiv2009

On scale-free and poly-scale behaviors of random hierarchical network

In this paper the question about statistical properties of block--hierarchical random matrices is raised for the first time in connection with structural characteristics of random hierarchical networks obtained by mipmapping procedure. In particular, we compute numerically the spectral density of large random adjacency matrices defined by a hierarchy of the Bernoulli distributions $\{q_1,q_2,...\}$ on matrix elements, where $q_γ$ depends on hierarchy level $γ$ as $q_γ=p^{-μγ}$ ($μ>0$). For the spectral density we clearly see the free--scale behavior. We show also that for the Gaussian distributions on matrix elements with zero mean and variances $σ_γ=p^{-νγ}$, the tail of the spectral density, $ρ_G(λ)$, behaves as $ρ_G(λ) \sim |λ|^{-(2-ν)/(1-ν)}$ for $|λ|\to\infty$ and $0<ν<1$, while for $ν\ge 1$ the power--law behavior is terminated. We also find that the vertex degree distribution of such hierarchical networks has a poly--scale fractal behavior extended to a very broad range of scales.

cond-mat.dis-nn↗

arXiv2008

Random Hierarchical Matrices: Spectral Properties and Relation to Polymers on Disordered Trees

We study the statistical and dynamic properties of the systems characterized by an ultrametric space of states and translationary non-invariant symmetric transition matrices of the Parisi type subjected to "locally constant" randomization. Using the explicit expression for eigenvalues of such matrices, we compute the spectral density for the Gaussian distribution of matrix elements. We also compute the averaged "survival probability" (SP) having sense of the probability to find a system in the initial state by time $t$. Using the similarity between the averaged SP for locally constant randomized Parisi matrices and the partition function of directed polymers on disordered trees, we show that for times $t>t_{\rm cr}$ (where $t_{\rm cr}$ is some critical time) a "lacunary" structure of the ultrametric space occurs with the probability $1-{\rm const}/t$. This means that the escape from some bounded areas of the ultrametric space of states is locked and the kinetics is confined in these areas for infinitely long time.

cond-mat.dis-nn↗

arXiv2008

First Passage Time Distribution and Number of Returns for Ultrametric Random Walk

In this paper, we consider a homogeneous Markov process ξ(t;ω) on an ultrametric space Q_p, with distribution density f(x,t), x in Q_p, t in R_+, satisfying the ultrametric diffusion equation df(x,t)/dt =-Df(x,t). We construct and examine a random variable τ(ω) that has the meaning the first passage times. Also, we obtain a formula for the mean number of returns on the interval (0,t] and give its asymptotic estimates for large t.

math-ph↗

arXiv2004

Non-Degenerate Ultrametric Diffusion

General non-degenerate p-adic operators of ultrametric diffusion are introduced. Bases of eigenvectors for the introduced operators are constructed and the corresponding eigenvalues are computed. Properties of the corresponding dynamics (i.e. of the ultrametric diffusion) are investigated.

cond-mat.other↗

arXiv2002

p-Adic description of characteristic relaxation in complex systems

This work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the p-adic analysis. We show that three types of relaxation fitted into the Kohlrausch-Williams-Watts law, the power decay law, or the logarithmic decay law, are similar random processes. Inherently, these processes are ultrametric and are described by the p-adic master equation. The physical meaning of this equation is explained in terms of a random walk constrained by a hierarchical energy landscape. We also discuss relations between the relaxation kinetics and the energy landscapes.

cond-mat.dis-nn↗

arXiv1999

Application of p-adic analysis to models of spontaneous breaking of the replica symmetry

Methods of p-adic analysis are applied to the investigation of the spontaneous symmetry breaking in the models of spin glasses. A p-adic expression for the replica matrix is given and moreover the replica matrix in the models of spontaneous breaking of the replica symmetry in the simplest case is expressed in the form of the Vladimirov operator of p-adic fractional differentiation. Also the model of hierarchical diffusion (that was proposed to describe relaxation of spin glasses) investigated using p-adic analysis.

cond-mat.dis-nn↗