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arXiv:cond-mat/0305691 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Persistence properties of a system of coagulating and annihilating random walkers

Supriya Krishnamurthy, R. Rajesh, Oleg V. Zaboronski

Published Oct 3, 2003• 29 citations
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Abstract

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability $1/(q\ensuremath{-}1)$ or coagulate with probability $(q\ensuremath{-}2)/(q\ensuremath{-}1).$ In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate $\overline{P}(m,t),$ the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly $(m\ensuremath{-}1)$ coagulations. Using perturbative renormalization group analysis for $d<2,$ we show that, if the number of coagulations m is much less than the typical number $M(t),$ then $\overline{P}(m,t)\ensuremath{\sim}{m}^{\ensuremath{\zeta}/d}{t}^{\ensuremath{-}\ensuremath{\theta}},$ with $\ensuremath{\theta}=dQ+Q(Q\ensuremath{-}1/2)\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2}),$ $\ensuremath{\zeta}=(2Q\ensuremath{-}1)\ensuremath{\epsilon}+(2Q\ensuremath{-}1)(Q\ensuremath{-}1)(1/2+AQ){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3}),$ where $Q=(q\ensuremath{-}1)/q,$ $\ensuremath{\epsilon}=2\ensuremath{-}d$ and $A=\ensuremath{-}0.006\dots{}.$ $M(t)$ is shown to scale as $M(t)\ensuremath{\sim}{t}^{d/2\ensuremath{-}\ensuremath{\delta}},$ where $\ensuremath{\delta}=d(1\ensuremath{-}Q)+(Q\ensuremath{-}1)(Q\ensuremath{-}1/2)\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2}).$ In two dimensions, we show that $\overline{P}(m,t)\ensuremath{\sim}\mathrm{ln}{(t)}^{Q(3\ensuremath{-}2Q)}\mathrm{ln}{(m)}^{(2Q\ensuremath{-}{1)}^{2}}{t}^{\ensuremath{-}2Q}$ for $m\ensuremath{\ll}{t}^{2Q\ensuremath{-}1}.$ We also derive an exact nonperturbative relation between the exponents: namely $\ensuremath{\delta}(Q)=\ensuremath{\theta}(1\ensuremath{-}Q).$ The one-dimensional results corresponding to $\ensuremath{\epsilon}=1$ are compared with results from Monte Carlo simulations.

DOI
10.1103/physreve.68.046103

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A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at $x=0$ at time $t=0$, the persistence probability $P(x,t)$ that a site at $x$ has not been visited till time $t$ has been calculated. $P(x,t)$ behaves as $(t/|x|-1)^{-α}$ with $α\sim 0.3$ while the global fraction ${\cal{P}}(t) = \sum_xP(x,t)/2t$ of sites remaining unvisited at time $t$ attains a constant value. $F(x,t)$, the probability that the site at $x$ is visited for the first time at $t$ behaves as $(t/|x|-1)^{-β}/|x|$ where $β= 1+ α$ for $t/|x|>> 1$,and ${\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/t$. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.

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We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q\ensuremath−1)1/(q\ensuremath{-}1)1/(q\ensuremath−1) or coagulate with probability (q\ensuremath−2)/(q\ensuremath−1).(q\ensuremath{-}2)/(q\ensuremath{-}1).(q\ensuremath−2)/(q\ensuremath−1). In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P‾(m,t),\overline{P}(m,t),P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m\ensuremath−1)(m\ensuremath{-}1)(m\ensuremath−1) coagulations. Using perturbative renormalization group analysis for d<2,d<2,d<2, we show that, if the number of coagulations m is much less than the typical number M(t),M(t),M(t), then P‾(m,t)\ensuremath∼m\ensuremathζ/dt\ensuremath−\ensuremathθ,\overline{P}(m,t)\ensuremath{\sim}{m}^{\ensuremath{\zeta}/d}{t}^{\ensuremath{-}\ensuremath{\theta}},P(m,t)\ensuremath∼m\ensuremathζ/dt\ensuremath−\ensuremathθ, with \ensuremathθ=dQ+Q(Q\ensuremath−1/2)\ensuremathϵ+O(\ensuremathϵ2),\ensuremath{\theta}=dQ+Q(Q\ensuremath{-}1/2)\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2}),\ensuremathθ=dQ+Q(Q\ensuremath−1/2)\ensuremathϵ+O(\ensuremathϵ2), \ensuremathζ=(2Q\ensuremath−1)\ensuremathϵ+(2Q\ensuremath−1)(Q\ensuremath−1)(1/2+AQ)\ensuremathϵ2+O(\ensuremathϵ3),\ensuremath{\zeta}=(2Q\ensuremath{-}1)\ensuremath{\epsilon}+(2Q\ensuremath{-}1)(Q\ensuremath{-}1)(1/2+AQ){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3}),\ensuremathζ=(2Q\ensuremath−1)\ensuremathϵ+(2Q\ensuremath−1)(Q\ensuremath−1)(1/2+AQ)\ensuremathϵ2+O(\ensuremathϵ3), where Q=(q\ensuremath−1)/q,Q=(q\ensuremath{-}1)/q,Q=(q\ensuremath−1)/q, \ensuremathϵ=2\ensuremath−d\ensuremath{\epsilon}=2\ensuremath{-}d\ensuremathϵ=2\ensuremath−d and A=\ensuremath−0.006….A=\ensuremath{-}0.006\dots{}.A=\ensuremath−0.006…. M(t)M(t)M(t) is shown to scale as M(t)\ensuremath∼td/2\ensuremath−\ensuremathδ,M(t)\ensuremath{\sim}{t}^{d/2\ensuremath{-}\ensuremath{\delta}},M(t)\ensuremath∼td/2\ensuremath−\ensuremathδ, where \ensuremathδ=d(1\ensuremath−Q)+(Q\ensuremath−1)(Q\ensuremath−1/2)\ensuremathϵ+O(\ensuremathϵ2).\ensuremath{\delta}=d(1\ensuremath{-}Q)+(Q\ensuremath{-}1)(Q\ensuremath{-}1/2)\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2}).\ensuremathδ=d(1\ensuremath−Q)+(Q\ensuremath−1)(Q\ensuremath−1/2)\ensuremathϵ+O(\ensuremathϵ2). In two dimensions, we show that P‾(m,t)\ensuremath∼ln(t)Q(3\ensuremath−2Q)ln(m)(2Q\ensuremath−1)2t\ensuremath−2Q\overline{P}(m,t)\ensuremath{\sim}\mathrm{ln}{(t)}^{Q(3\ensuremath{-}2Q)}\mathrm{ln}{(m)}^{(2Q\ensuremath{-}{1)}^{2}}{t}^{\ensuremath{-}2Q}P(m,t)\ensuremath∼ln(t)Q(3\ensuremath−2Q)ln(m)(2Q\ensuremath−1)2t\ensuremath−2Q for m\ensuremath≪t2Q\ensuremath−1.m\ensuremath{\ll}{t}^{2Q\ensuremath{-}1}.m\ensuremath≪t2Q\ensuremath−1. We also derive an exact nonperturbative relation between the exponents: namely \ensuremathδ(Q)=\ensuremathθ(1\ensuremath−Q).\ensuremath{\delta}(Q)=\ensuremath{\theta}(1\ensuremath{-}Q).\ensuremathδ(Q)=\ensuremathθ(1\ensuremath−Q). The one-dimensional results corresponding to \ensuremathϵ=1\ensuremath{\epsilon}=1\ensuremathϵ=1 are compared with results from Monte Carlo simulations.
Sinai's model of diffusion in one dimension with random local bias is studied by a real space renormalization group which yields asymptotically exact long time results. The distribution of the position of a particle and the probability of it not returning to the origin are obtained, as well as the two-time distribution which exhibits ``aging'' with $\mathrm{ln}t/\mathrm{ln}{t}^{\ensuremath{'}}$ scaling and a singularity at $x(t)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}x({t}^{\ensuremath{'}})$. The effects of a small uniform force are also studied. Extension to motion of many domain walls yields nonequilibrium time dependent correlations for the 1D random field Ising model with Glauber dynamics and ``persistence'' exponents of 1D reaction-diffusion models with random forces.
diffusion in disordered environments, random quantum spin chains Abstract. — We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P pr(t) that the expectation value 〈x(t) 〉 of the walker’s position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a logarithmic dependence P pr(t) ∼ ln(t) −θ with a new exponent θ = 0.191 ± 0.002. We note on a complete correspondence between the average persistence of random walks and the magnetization autocorrelation function of the transverse-field Ising chain, in the homogeneous and disordered case. First passage time or persistence problems have a long history in the physical literature [1]. Recently they gained a lot of interest, since the persistence exponents, describing the asymptotic behavior of first passage time probabilities, are shown to be independent dynamical critical exponents, which have been calculated for various models exactly[2, 3]. Not much is known about analogous quantities in systems with quenched disorder, for instance the random walk (or diffusion) in a disordered environment [4], which is in the one-dimensional case the Sinai-model[5]. For this model the first passage / persistence exponent for a single walker has been determined by us in a previous work [6]. In this letter we introduce and study the concept of average persistence of random walks both in homogeneous and random environments. We consider a random walk with nearest neighbor hopping in one dimension defined by the Master equation pi(t = 0) = δi,1,
In this work, we investigate the critical behavior of a model describing the parity-conserving branching and annihilating process of random walkers. The model is simulated on a one dimensional lattice on which the sites can be occupied by multiple particles with a finite annhilation probability. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and spacial correlation length critical exponents. Further, we follow the short-time critical relaxation to provide a set of relevant dynamical critical exponents. We check the validity of the hyperscaling relation for both sets of stationary and dynamical critical exponents. These are consistent with the BARW-PC universality class.
We study the persistence properties of the T=0T=0T=0 coarsening dynamics of one-dimensional q-state Potts model using a modified mean-field approximation (MMFA). In this approximation, the spatial correlations between the interfaces separating spins with different Potts states is ignored, but the correct time dependence of the mean density P(t)P(t)P(t) of persistent spins is imposed. For this model, it is known that P(t)P(t)P(t) follows a power-law decay with time, $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}(q)},$ where $\ensuremath{\theta}(q)$ is the q-dependent persistence exponent. We study the spatial structure of the persistent region within the MMFA. We show that the persistent site pair correlation function P2(r,t){P}_{2}(r,t)P2​(r,t) has the scaling form P2(r,t)=P(t)2f(r/t1/2){P}_{2}{(r,t)=P(t)}^{2}{f(r/t}^{1/2})P2​(r,t)=P(t)2f(r/t1/2) for all values of the persistence exponent $\ensuremath{\theta}(q).$ The scaling function has the limiting behavior $f(x)\ensuremath{\sim}{x}^{\ensuremath{-}2\ensuremath{\theta}}$ $(x\ensuremath{\ll}1)$ and $f(x)\ensuremath{\rightarrow}1$ $(x\ensuremath{\gg}1).$ We then show within the independent interval approximation (IIA) that the distribution n(k,t)n(k,t)n(k,t) of separation k between two consecutive persistent spins at time t has the asymptotic scaling form ${n(k,t)=t}^{\ensuremath{-}2\ensuremath{\varphi}}{g(t,k/t}^{\ensuremath{\varphi}}),$ where the dynamical exponent has the form $\ensuremath{\varphi}=\mathrm{max}(1/2,\ensuremath{\theta}).$ The behavior of the scaling function for large and small values of the arguments is found analytically. We find that for small separations $k\ensuremath{\ll}{t}^{\ensuremath{\varphi}},n(k,t)\ensuremath{\sim}{P(t)k}^{\ensuremath{-}\ensuremath{\tau}},$ where $\ensuremath{\tau}=\mathrm{max}[2(1\ensuremath{-}\ensuremath{\theta}),2\ensuremath{\theta}],$ while for large separations $k\ensuremath{\gg}{t}^{\ensuremath{\varphi}},$ g(t,x)g(t,x)g(t,x) decays exponentially with x. The unusual dynamical scaling form and the behavior of the scaling function is supported by numerical simulations.
The authors first show that the movement of a domain wall in a highly anisotropic Ising model of spins in two dimensions can be mapped to a random walk movement of labelled walkers in one dimension. This movement is correlated in the sense that the transition probability of a given walker, labelled i, depends upon the current positions of its labelled neighbours (i-1 and i+1). A Monte Carlo simulation of the walk for various numbers of walkers (up to a maximum of 50) and for quite long times (up to a maximum of 25*10 4 Monte Carlo steps) yields useful and interesting information about the dynamics. They verify that (i) the centre of mass of the walkers executes an unbiased random walk for all times starting from the lowest times; and (ii) the moment of inertia of the walkers asymptotically reaches a constant value which scales with the number of walkers. They also discuss the relation of this work to the dynamics in the 2D Ising model, identifying phi with the dynamical exponent z. The random walk equivalence also provides a geometrical interpretation of the known results for the solid-on-solid model.
A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at x=0x=0x=0 at time t=0t=0t=0, the persistence probability P(x,t)P(x,t)P(x,t) that a site at xxx has not been visited till time ttt has been calculated. P(x,t)P(x,t)P(x,t) behaves as (t/∣x∣−1)−α(t/|x|-1)^{-α}(t/∣x∣−1)−α with α∼0.3α\sim 0.3α∼0.3 while the global fraction P(t)=∑xP(x,t)/2t{\cal{P}}(t) = \sum_xP(x,t)/2tP(t)=∑x​P(x,t)/2t of sites remaining unvisited at time ttt attains a constant value. F(x,t)F(x,t)F(x,t), the probability that the site at xxx is visited for the first time at ttt behaves as (t/∣x∣−1)−β/∣x∣(t/|x|-1)^{-β}/|x|(t/∣x∣−1)−β/∣x∣ where β=1+αβ= 1+ αβ=1+α for t/∣x∣>>1t/|x|>> 1t/∣x∣>>1,and F(t)=∑xF(x,t)/2t∼1/t{\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/tF(t)=∑x​F(x,t)/2t∼1/t. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.