arXiv subjects

B. Waclaw

Publications and source records attributed to B. Waclaw.

At least 19 recordsLinked to original sources

arXiv2015

Condensation in models with factorized and pair-factorized stationary states

Non-equilibrium real-space condensation is a phenomenon in which a finite fraction of some conserved quantity (mass, particles, etc.) becomes spatially localised. We review two popular stochastic models of hopping particles that lead to condensation and whose stationary states assume a factorized form: the zero-range process and the misanthrope process, and their various modifications. We also introduce a new model - a misanthrope process with parallel dynamics - that exhibits condensation and has a pair-factorized stationary state.

cond-mat.stat-mech↗

arXiv2013

Condensation in stochastic mass transport models: beyond the zero-range process

We consider an extension of the zero-range process to the case where the hop rate depends on the state of both departure and arrival sites. We recover the misanthrope and the target process as special cases for which the probability of the steady state factorizes over sites. We discuss conditions which lead to the condensation of particles and show that although two different hop rates can lead to the same steady state, they do so with sharply contrasting dynamics. The first case resembles the dynamics of the zero-range process, whereas the second case, in which the hop rate increases with the occupation number of both sites, is similar to instantaneous gelation models. This new "explosive" condensation reveals surprisingly rich behaviour, in which the process of condensate's formation goes through a series of collisions between clusters of particles moving through the system at increasing speed. We perform a detailed numerical and analytical study of the dynamics of condensation: we find the speed of the moving clusters, their scattering amplitude, and their growth time. We finally show that the time to reach steady state decreases with the size of the system.

cond-mat.stat-mech↗

arXiv2013

Mechanically driven growth of quasi-two dimensional microbial colonies

We study colonies of non-motile, rod-shaped bacteria growing on solid substrates. In our model, bacteria interact purely mechanically, by pushing each other away as they grow, and consume a diffusing nutrient. We show that mechanical interactions control the velocity and shape of the advancing front, which leads to features that cannot be captured by established Fisher-Kolmogorov models. In particular, we find that the velocity depends on the elastic modulus of bacteria or their stickiness to the surface. Interestingly, we predict that the radius of an incompressible, strictly two-dimensional colony cannot grow linearly in time. Importantly, mechanical interactions can also account for the nonequilibrium transition between circular and branching colonies, often observed in the lab.

physics.bio-ph↗

arXiv2012

Quantum widening of CDT universe

The physical phase of Causal Dynamical Triangulations (CDT) is known to be described by an effective, one-dimensional action in which three-volumes of the underlying foliation of the full CDT play a role of the sole degrees of freedom. Here we map this effective description onto a statistical-physics model of particles distributed on 1d lattice, with site occupation numbers corresponding to the three-volumes. We identify the emergence of the quantum de-Sitter universe observed in CDT with the condensation transition known from similar statistical models. Our model correctly reproduces the shape of the quantum universe and allows us to analytically determine quantum corrections to the size of the universe. We also investigate the phase structure of the model and show that it reproduces all three phases observed in computer simulations of CDT. In addition, we predict that two other phases may exists, depending on the exact form of the discretised effective action and boundary conditions. We calculate various quantities such as the distribution of three-volumes in our model and discuss how they can be compared with CDT.

hep-th↗

arXiv2010

The various facets of random walk entropy

We review various features of the statistics of random paths on graphs. The relationship between path statistics and Quantum Mechanics (QM) leads to two canonical ways of defining random walk on a graph, which have different statistics and hence different entropies. Generic random walk (GRW) is in correspondence with the field-theoretical formalism, whereas maximal entropy random walk (MERW), introduced by us in a recent work, is motivated by the Feynman path-integral formulation of QM. GRW maximizes entropy locally (neighbors are chosen with equal probabilities), in contrast to MERW which does so globally (all paths of given length and endpoints are equally probable). The stationary distribution for MERW is given by the ground state of a quantum-mechanical problem where nodes whose degree is smaller than average act as repulsive impurities. We investigate static and dynamical properties GRW and MERW in a variety of examples in one and two dimensions. The most spectacular difference arises in the case of weakly diluted lattices, where a particle performing MERW gets eventually trapped in the largest nearly spherical region which is free of impurities. We put forward a quantitative explanation of this localization effect in terms of a classical Lifshitz phenomenon.

cond-mat.stat-mech↗

arXiv2010

Spectrum of the Product of Independent Random Gaussian Matrices

We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the large-N limit is rotationally symmetric in the complex plane and is given by a simple expression rho(z,\bar{z}) = 1/(Mπσ^2} |z|^{-2+2/M} for |z|<σ, and is zero for |z|> σ. The parameter σcorresponds to the radius of the circular support and is related to the amplitude of the Gaussian fluctuations. This form of the eigenvalue density is highly universal. It is identical for products of Gaussian Hermitian, non-Hermitian, real or complex random matrices. It does not change even if the matrices in the product are taken from different Gaussian ensembles. We present a self-contained derivation of this result using a planar diagrammatic technique for Gaussian matrices. We also give a numerical evidence suggesting that this result applies also to matrices whose elements are independent, centered random variables with a finite variance.

cond-mat.stat-mech↗

arXiv2009

Mass condensation in one dimension with pair-factorized steady states

We consider stochastic rules of mass transport which lead to steady states that factorize over the links of a one-dimensional ring. Based on the knowledge of the steady states, we derive the onset of a phase transition from a liquid to a condensed phase that is characterized by the existence of a condensate. For various types of weight functions which enter the hopping rates, we determine the shape of the condensate, its scaling with the system size, and the single-site mass distribution as characteristic static properties. As it turns out, the condensate's shape and its scaling are not universal, but depend on the competition between local and ultralocal interactions. So we can tune the shape from a delta-like envelope to a parabolic-like or a rectangular one. While we treat the liquid phase in the grand-canonical formalism, we develop a different analytical approach for the condensed phase. Its predictions are well confirmed by numerical simulations. Possible extensions to higher dimensions are indicated.

cond-mat.stat-mech↗

arXiv2009

Tuning the shape of the condensate in spontaneous symmetry breaking

We investigate what determines the shape of a particle condensate in situations when it emerges as a result of spontaneous breaking of translational symmetry. We consider a model with particles hopping between sites of a one-dimensional grid and interacting if they are at the same or at neighboring nodes. We predict the envelope of the condensate and the scaling of its width with the system size for various interaction potentials and show how to tune the shape from a delta-peak to a rectangular or a parabolic-like form.

cond-mat.stat-mech↗

arXiv2009

Pair-factorized steady states on arbitrary graphs

Stochastic mass transport models are usually described by specifying hopping rates of particles between sites of a given lattice, and the goal is to predict the existence and properties of the steady state. Here we ask the reverse question: given a stationary state that factorizes over links (pairs of sites) of an arbitrary connected graph, what are possible hopping rates that converge to this state? We define a class of hopping functions which lead to the same steady state and guarantee current conservation but may differ by the induced current strength. For the special case of anisotropic hopping in two dimensions we discuss some aspects of the phase structure. We also show how this case can be traced back to an effective zero-range process in one dimension which is solvable for a large class of hopping functions.

cond-mat.stat-mech↗

arXiv2008

Approaching the thermodynamic limit in equilibrated scale-free networks

We discuss how various models of scale-free complex networks approach their limiting properties when the size N of the network grows. We focus mainly on equilibrated networks and their finite-size degree distributions. Our results show that subleading corrections to the scaling of the position of the cutoff are strong even for networks of order 10^9 nodes. We observe also a logarithmic correction to the scaling for degenerated graphs when the degree distribution follows a power law k^(-3). We study also the distribution of the maximal degree and show that it may have a different scaling than the cutoff and, moreover, it approaches the thermodynamic limit much faster. We present also some results on the cutoff function and the distribution of the maximal degree in equilibrated networks.

cond-mat.stat-mech↗

arXiv2008

Localization of maximal entropy random walk

We define a new class of random walk processes which maximize entropy. This maximal entropy random walk is equivalent to generic random walk if it takes place on a regular lattice, but it is not if the underlying lattice is irregular. In particular, we consider a lattice with weak dilution. We show that the stationary probability of finding a particle performing maximal entropy random walk localizes in the largest nearly spherical region of the lattice which is free of defects. This localization phenomenon, which is purely classical in nature, is explained in terms of the Lifshitz states of a certain random operator.

cond-mat.dis-nn↗

arXiv2008

Power laws in zero-range processes on random networks

We study statistical properties of a zero-range process (ZRP) on random networks. We derive an analytic expression for the distribution of particles (also called node occupation distribution) in the steady state of the ZRP in the ensemble of uncorrelated random graphs. We analyze the dependence of this distribution on the node-degree distribution. In particular, we show that when the degree distribution is tuned properly, one can obtain scale-free fluctuations in the distribution of particles. Such fluctuations lead to a power law in the distribution of particles, just like in the ZRP with the hopping rate u(m)=1+b/m on homogeneous graphs.

cond-mat.stat-mech↗

arXiv2008

Causal and homogeneous networks

Growing networks have a causal structure. We show that the causality strongly influences the scaling and geometrical properties of the network. In particular the average distance between nodes is smaller for causal networks than for corresponding homogeneous networks. We explain the origin of this effect and illustrate it using as an example a solvable model of random trees. We also discuss the issue of stability of the scale-free node degree distribution. We show that a surplus of links may lead to the emergence of a singular node with the degree proportional to the total number of links. This effect is closely related to the backgammon condensation known from the balls-in-boxes model.

cond-mat.dis-nn↗

arXiv2008

Counting metastable states of Ising spin glasses on arbitrary graphs

Using a field-theoretical representation of the Tanaka-Edwards integral we develop a method to systematically compute the number N_s of 1-spin-stable states (local energy minima) of a glassy Ising system with nearest-neighbor interactions and random Gaussian couplings on an arbitrary graph. In particular, we use this method to determine N_s for K-regular random graphs and d-dimensional regular lattices for d=2,3. The method works also for other graphs. Excellent accuracy of the results allows us to observe that the number of local energy minima depends mainly on local properties of the graph on which the spin glass is defined.

cond-mat.dis-nn↗

arXiv2007

Free zero-range processes on networks

A free zero-range process (FRZP) is a simple stochastic process describing the dynamics of a gas of particles hopping between neighboring nodes of a network. We discuss three different cases of increasing complexity: (a) FZRP on a rigid geometry where the network is fixed during the process, (b) FZRP on a random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical network whose topology continuously changes during the process in a way which depends on the current distribution of particles. The case (a) provides a very simple realization of the phenomenon of condensation which manifests as the appearance of a condensate of particles on the node with maximal degree. The case (b) is very interesting since the averaging over typical ensembles of graphs acts as a kind of homogenization of the system which makes all nodes identical from the point of view of the FZRP. In the case (c), the distribution of particles and the dynamics of network are coupled to each other. The strength of this coupling depends on the ratio of two time scales: for changes of the topology and of the FZRP. We will discuss a specific example of that type of interaction and show that it leads to an interesting phase diagram.

cond-mat.stat-mech↗

arXiv2007

Statistical mechanics of complex networks

The science of complex networks is a new interdisciplinary branch of science which has arisen recently on the interface of physics, biology, social and computer sciences, and others. Its main goal is to discover general laws governing the creation and growth as well as processes taking place on networks, like e.g. the Internet, transportation or neural networks. It turned out that most real-world networks cannot be simply reduced to a compound of some individual components. Fortunately, the statistical mechanics, being one of pillars of modern physics, provides us with a very powerful set of tools and methods for describing and understanding these systems. In this thesis, we would like to present a consistent approach to complex networks based on statistical mechanics, with the central role played by the concept of statistical ensemble of networks. We show how to construct such a theory and present some practical problems where it can be applied. Among them, we pay attention to the problem of finite-size corrections and the dynamics of a simple model of mass transport on networks.

cond-mat.stat-mech↗

arXiv2007

Condensation in zero-range processes on inhomogeneous networks

We investigate the role of inhomogeneities in zero-range processes in condensation dynamics.We consider the dynamics of balls hopping between nodes of a network, and find that the condensation is triggered by the ratio k_1/k of the highest degree k_1 to the average degree k. Although the condensate takes on the average an extensive number of balls, its occupation can oscillate in a wide range. We show that in systems with strong inhomogeneity, the typical melting time of the condensate grows exponentially with the number of balls.

cond-mat.stat-mech↗

arXiv2007

Finite size effects in Barabasi-Albert growing networks

We investigate the influence of the network's size on the degree distribution in Barabasi-Albert model of growing network with initial attractiveness. Our approach based on spectral moments allows to treat analytically several variants of the model and to calculate the cut-off function giving finite size corrections to the degree distribution. We study the effect of initial configuration as well as of addition more than one link per time step. The results indicate that asymptotic properties of the cut-off depend only on the exponent in the power law describing the tail of the degree distribution.

cond-mat.stat-mech↗