arXiv subjects

R. K. P. Zia

Publications and source records attributed to R. K. P. Zia.

At least 73 records · Page 4Linked to original sources

arXiv2006

Factorised Steady States in Mass Transport Models on an Arbitrary Graph

We study a general mass transport model on an arbitrary graph consisting of $L$ nodes each carrying a continuous mass. The graph also has a set of directed links between pairs of nodes through which a stochastic portion of mass, chosen from a site-dependent distribution, is transported between the nodes at each time step. The dynamics conserves the total mass and the system eventually reaches a steady state. This general model includes as special cases various previously studied models such as the Zero-range process and the Asymmetric random average process. We derive a general condition on the stochastic mass transport rules, valid for arbitrary graph and for both parallel and random sequential dynamics, that is sufficient to guarantee that the steady state is factorisable. We demonstrate how this condition can be achieved in several examples. We show that our generalized result contains as a special case the recent results derived by Greenblatt and Lebowitz for $d$-dimensional hypercubic lattices with random sequential dynamics.

cond-mat.stat-mech↗

arXiv2005

Canonical Analysis of Condensation in Factorised Steady State

We study the phenomenon of real space condensation in the steady state of a class of mass transport models where the steady state factorises. The grand canonical ensemble may be used to derive the criterion for the occurrence of a condensation transition but does not shed light on the nature of the condensate. Here, within the canonical ensemble, we analyse the condensation transition and the structure of the condensate, determining the precise shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as $L^{1/2}$ where $L$ is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. Our results are asymptotically exact and can also be interpreted within the framework of sums of random variables. We further analyse two additional cases: one where the condensation transition is somewhat different from the usual second order phase transition and one where there is no true condensation transition but instead a pseudocondensate appears at superextensive densities.

cond-mat.stat-mech↗

arXiv2005

Spurious phase in a model for traffic on a bridge

We present high-precision Monte Carlo data for the phase diagram of a two-species driven diffusive system, reminiscent of traffic across a narrow bridge. Earlier studies reported two phases with broken symmetry; the existence of one of these has been the subject of some debate. We show that the disputed phase disappears for sufficiently large systems and/or sufficiently low bulk mobility.

cond-mat.stat-mech↗

arXiv2005

Exact dynamics of a reaction-diffusion model with spatially alternating rates

We present the exact solution for the full dynamics of a nonequilibrium spin chain and its dual reaction-diffusion model, for arbitrary initial conditions. The spin chain is driven out of equilibrium by coupling alternating spins to two thermal baths at different temperatures. In the reaction-diffusion model, this translates into spatially alternating rates for particle creation and annihilation, and even negative ``temperatures'' have a perfectly natural interpretation. Observables of interest include the magnetization, the particle density, and all correlation functions for both models. Two generic types of time-dependence are found: if both temperatures are positive, the magnetization, density and correlation functions decay exponentially to their steady-state values. In contrast, if one of the temperatures is negative, damped oscillations are observed in all quantities. They can be traced to a subtle competition of pair creation and annihilation on the two sublattices. We comment on the limitations of mean-field theory and propose an experimental realization of our model in certain conjugated polymers and linear chain compounds.

cond-mat.stat-mech↗

arXiv2005

The Nature of the Condensate in Mass Transport Models

We study the phenomenon of real space condensation in the steady state of a class of one dimensional mass transport models. We derive the criterion for the occurrence of a condensation transition and analyse the precise nature of the shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as $L^{1/2}$ where $L$ is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. We interpret these results within the framework of sums of random variables.

cond-mat.stat-mech↗

arXiv2005

Will jams get worse when slow cars move over?

Motivated by an analogy with traffic, we simulate two species of particles (`vehicles'), moving stochastically in opposite directions on a two-lane ring road. Each species prefers one lane over the other, controlled by a parameter $0 \leq b \leq 1$ such that $b=0$ corresponds to random lane choice and $b=1$ to perfect `laning'. We find that the system displays one large cluster (`jam') whose size increases with $b$, contrary to intuition. Even more remarkably, the lane `charge' (a measure for the number of particles in their preferred lane) exhibits a region of negative response: even though vehicles experience a stronger preference for the `right' lane, more of them find themselves in the `wrong' one! For $b$ very close to 1, a sharp transition restores a homogeneous state. Various characteristics of the system are computed analytically, in good agreement with simulation data.

cond-mat.stat-mech↗

arXiv2005

Gas permeation through a polymer network

We study the diffusion of gas molecules through a two-dimensional network of polymers with the help of Monte Carlo simulations. The polymers are modeled as non-interacting random walks on the bonds of a two-dimensional square lattice, while the gas particles occupy the lattice cells. When a particle attempts to jump to a nearest-neighbor empty cell, it has to overcome an energy barrier which is determined by the number of polymer segments on the bond separating the two cells. We investigate the gas current $J$ as a function of the mean segment density $ρ$, the polymer length $\ell$ and the probability $q^{m}$ for hopping across $m$ segments. Whereas $J$ decreases monotonically with $ρ$ for fixed $\ell$, its behavior for fixed $ρ$ and increasing $\ell$ depends strongly on $q$. For small, non-zero $q$, $J$ appears to increase slowly with $\ell$. In contrast, for $q=0$, it is dominated by the underlying percolation problem and can be non-monotonic. We provide heuristic arguments to put these interesting phenomena into context.

cond-mat.soft↗

arXiv2004

Construction of the factorized steady state distribution in models of mass transport

For a class of one-dimensional mass transport models we present a simple and direct test on the chipping functions, which define the probabilities for mass to be transferred to neighbouring sites, to determine whether the stationary distribution is factorized. In cases where the answer is affirmative, we provide an explicit method for constructing the single-site weight function. As an illustration of the power of this approach, previously known results on the Zero-range process and Asymmetric random average process are recovered in a few lines. We also construct new models, namely a generalized Zero-range process and a binomial chipping model, which have factorized steady states.

cond-mat.stat-mech↗

arXiv2004

The Effects of Next-Nearest-Neighbor Interactions on the Orientation Dependence of Step Stiffness: Reconciling Theory with Experiment for Cu(001)

Within the solid-on-solid (SOS) approximation, we carry out a calculation of the orientational dependence of the step stiffness on a square lattice with nearest and next-nearest neighbor interactions. At low temperature our result reduces to a simple, transparent expression. The effect of the strongest trio (three-site, non pairwise) interaction can easily be incorporated by modifying the interpretation of the two pairwise energies. The work is motivated by a calculation based on nearest neighbors that underestimates the stiffness by a factor of 4 in directions away from close-packed directions, and a subsequent estimate of the stiffness in the two high-symmetry directions alone that suggested that inclusion of next-nearest-neighbor attractions could fully explain the discrepancy. As in these earlier papers, the discussion focuses on Cu(001).

cond-mat.mtrl-sci↗

arXiv2004

Bond percolation of polymers

We study bond percolation of $N$ non-interacting Gaussian polymers of $\ell$ segments on a 2D square lattice of size $L$ with reflecting boundaries. Through simulations, we find the fraction of configurations displaying {\em no} connected cluster which span from one edge to the opposite edge. From this fraction, we define a critical segment density $ρ_{c}^L(\ell)$ and the associated critical fraction of occupied bonds $p_{c}^L(\ell)$, so that they can be identified as the percolation threshold in the $L \to \infty$ limit. Whereas $p_{c}^L(\ell)$ is found to decrease monotonically with $\ell$ for a wide range of polymer lengths, $ρ_{c}^L(\ell)$ is non-monotonic. We give physical arguments for this intriguing behavior in terms of the competing effects of multiple bond occupancies and polymerization.

cond-mat.stat-mech↗

arXiv2004

Factorised Steady States in Mass Transport Models

We study a class of mass transport models where mass is transported in a preferred direction around a one-dimensional periodic lattice and is globally conserved. The model encompasses both discrete and continuous masses and parallel and random sequential dynamics and includes models such as the Zero-range process and Asymmetric random average process as special cases. We derive a necessary and sufficient condition for the steady state to factorise, which takes a rather simple form.

cond-mat.stat-mech↗

arXiv2004

Complete Solution of the Kinetics in a Far-from-equilibrium Ising Chain

The one-dimensional Ising model is easily generalized to a \textit{genuinely nonequilibrium} system by coupling alternating spins to two thermal baths at different temperatures. Here, we investigate the full time dependence of this system. In particular, we obtain the evolution of the magnetisation, starting with arbitrary initial conditions. For slightly less general initial conditions, we compute the time dependence of all correlation functions, and so, the probability distribution. Novel properties, such as oscillatory decays into the steady state, are presented. Finally, we comment on the relationship to a reaction-diffusion model with pair annihilation and creation.

cond-mat.stat-mech↗

arXiv2004

Fluctuations and correlations in an individual-based model of biological coevolution

We extend our study of a simple model of biological coevolution to its statistical properties. Staring with a complete description in terms of a master equation, we provide its relation to the deterministic evolution equations used in previous investigations. The stationary states of the mutationless model are generally well approximated by Gaussian distributions, so that the fluctuations and correlations of the populations can be computed analytically. Several specific cases are studied by Monte Carlo simulations, and there is excellent agreement between the data and the theoretical predictions.

q-bio.PE↗

arXiv2003

Relevance of space anisotropy in the critical behavior of m-axial Lifshitz points

The critical behavior of $d$-dimensional systems with $n$-component order parameter $\bmϕ$ is studied at an $m$-axial Lifshitz point where a wave-vector instability occurs in an $m$-dimensional subspace ${\mathbb R}^m$ ($m{>}1$). Field theoretic renormalization group techniques are exploited to examine the effects of terms in the Hamiltonian that break the rotational symmetry of the Euclidean group ${\mathbb E}(m)$. The framework for considering general operators of second order in $\bmϕ$ and fourth order in the derivatives $\partial_α$ with respect to the Cartesian coordinates $x_α$ of ${\mathbb R}^m$ is presented. For the specific case of systems with cubic anisotropy, the effects of having an additional term, $\sum_{α=1}^m(\partial_α^2\bmϕ)^2$, are investigated in an $ε$ expansion about the upper critical dimension $d^{*}(m)=4+m/2$. Its associated crossover exponent is computed to order $ε^2$ and found to be positive, so that it is a \emph{relevant} perturbation on a model isotropic in ${\mathbb R}^m$.

cond-mat.stat-mech↗

arXiv2003

Punctuated equilibria and 1/f noise in a biological coevolution model with individual-based dynamics

We present a study by linear stability analysis and large-scale Monte Carlo simulations of a simple model of biological coevolution. Selection is provided through a reproduction probability that contains quenched, random interspecies interactions, while genetic variation is provided through a low mutation rate. Both selection and mutation act on individual organisms. Consistent with some current theories of macroevolutionary dynamics, the model displays intermittent, statistically self-similar behavior with punctuated equilibria. The probability density for the lifetimes of ecological communities is well approximated by a power law with exponent near -2, and the corresponding power spectral densities show 1/f noise (flicker noise) over several decades. The long-lived communities (quasi-steady states) consist of a relatively small number of mutualistically interacting species, and they are surrounded by a ``protection zone'' of closely related genotypes that have a very low probability of invading the resident community. The extent of the protection zone affects the stability of the community in a way analogous to the height of the free-energy barrier surrounding a metastable state in a physical system. Measures of biological diversity are on average stationary with no discernible trends, even over our very long simulation runs of approximately 3.4x10^7 generations.

nlin.AO↗

arXiv2003

The totally asymmetric exclusion process with extended objects, a model for protein synthesis

The process of protein synthesis in biological systems resembles a one dimensional driven lattice gas in which the particles have spatial extent, covering more than one lattice site. We expand the well studied Totally Asymmetric Exclusion Process (TASEP), in which particles typically cover a single lattice site, to include cases with extended objects. Exact solutions can be determined for a uniform closed system. We analyze the uniform open system through two approaches. First, a continuum limit produces a modified diffusion equation for particle density profiles. Second, an extremal principle based on domain wall theory accurately predicts the phase diagram and currents in each phase. Finally, we briefly consider approximate approaches to a non-uniform open system with quenched disorder in the particle hopping rates and compare these approaches with Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2003

Watching a drunkard for ten nights: A study of distributions of variances

For any physical observable in statistical systems, the most frequently studied quantities are its average and standard deviation. Yet, its full distribution often carries extremely interesting information and can be invoked to put any surprising properties of the individual moments into perspective. As an example, we consider a problem concerning simple random walks which was posed in a recent text. When a drunk is observed over L nights, taking N steps per night, and the number of steps to the right is recorded for each night, an average and a variance based on these data can be computed. When the variance is used to estimate p, the probability for the drunk to step right, complex values for p are frequently found. To put such obviously nonsensical results into context, we study the full probability distribution for the variance of the data string. We discuss the connection of our results to the problem of data binning and provide two other brief examples to demonstrate the importance of full distributions.

cond-mat↗

arXiv2003

Flicker Noise in a Model of Coevolving Biological Populations

We present long Monte Carlo simulations of a simple model of biological macroevolution in which births, deaths, and mutational changes in the genome take place at the level of individual organisms. The model displays punctuated equilibria and flicker noise with a 1/f-like power spectrum, consistent with some current theories of evolutionary dynamics.

nlin.AO↗