Novel Coarsening in the Zero Temperature Dynamics of an Ising Chain with Kinetically Induced Field
This paper has been withdrawn by the authors.
cond-mat.stat-mech↗
arXiv subjects
Satya N. Majumdar
Publications and source records attributed to Satya N. Majumdar.
Phase Transition in the Takayasu Model with Desorption
We study a lattice model where particles carrying different masses diffuse, coalesce upon contact, and also unit masses adsorb to a site with rate $q$ or desorb from a site with nonzero mass with rate $p$. In the limit $p=0$ (without desorption), our model reduces to the well studied Takayasu model where the steady-state single site mass distribution has a power law tail $P(m)\sim m^{-τ}$ for large mass. We show that varying the desorption rate $p$ induces a nonequilibrium phase transition in all dimensions. For fixed $q$, there is a critical $p_c(q)$ such that if $p p_c(q)$, $P(m)\sim \exp(-m/m^*)$ for large $m$. The model is studied analytically within a mean field theory and numerically in one dimension.
cond-mat.stat-mech↗
Analytical Studies of Strategies for Utilization of Cache Memory in Computers
We analyze quantitatively several strategies for better utilization of the {\em cache} or the {\em {fast access}} memory in computers. We define a performance factor $α$ that denotes the fraction of the cache area utilized when the main memory is accessed at random. We calculate $α$ exactly for different competing strategies, including the hash-rehash and the skewed-associative strategies which were earlier analyzed via simulations.
cond-mat.stat-mech↗
Nonequilibrium Phase Transitions in Models of Aggregation, Adsorption, and Dissociation
We study nonequilibrium phase transitions in a mass-aggregation model which allows for diffusion, aggregation on contact, dissociation, adsorption and desorption of unit masses. We analyse two limits explicitly. In the first case mass is locally conserved whereas in the second case local conservation is violated. In both cases the system undergoes a dynamical phase transition in all dimensions. In the first case, the steady state mass distribution decays exponentially for large mass in one phase, and develops an infinite aggregate in addition to a power-law mass decay in the other phase. In the second case, the transition is similar except that the infinite aggregate is missing.
cond-mat.stat-mech↗
Nonequilibrium Phase Transition in a Model of Diffusion, Aggregation and Fragmentation
We study the nonequilibrium phase transition in a model of aggregation of masses allowing for diffusion, aggregation on contact and fragmentation. The model undergoes a dynamical phase transition in all dimensions. The steady state mass distribution decays exponentially for large mass in one phase. On the contrary, in the other phase it has a power law tail and in addition an infinite aggregate. The model is solved exactly within a mean field approximation which keeps track of the distribution of masses. In one dimension, by mapping to an equivalent lattice gas model, exact steady states are obtained in two extreme limits of the parameter space. Critical exponents and the phase diagram are obtained numerically in one dimension. We also study the time dependent fluctuations in an equivalent interface model in (1+1) dimension and compute the roughness exponent $χ$ and the dynamical exponent z analytically in some limits and numerically otherwise. Two new fixed points of interface fluctuations in (1+1) dimension are identified. We also generalize our model to include arbitrary fragmentation kernels and solve the steady states exactly for some special choices of these kernels via mappings to other solvable models of statistical mechanics.
cond-mat.stat-mech↗
Persistence in Nonequilibrium Systems
This is a brief review of recent theoretical efforts to understand persistence in nonequilibrium systems. Some of the recent experimental results are also briefly mentioned. I also discuss recent generalizations of persistence in various directions and conclude with a summary of open questions.
cond-mat.stat-mech↗
Residence Time Distribution for a Class of Gaussian Markov Processes
We study the distribution of residence time or equivalently that of ``mean magnetization" for a family of Gaussian Markov processes indexed by a positive parameter $α$. The persistence exponent for these processes is simply given by $θ=α$ but the residence time distribution is nontrivial. The shape of this distribution undergoes a qualitative change as $θ$ increases, indicating a sharp change in the ergodic properties of the process. We develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary $α$. For some special values of $α$, we obtain closed form expressions of the distribution function.
cond-mat.stat-mech↗
Analytical results for random walk persistence
In this paper, we present the detailed calculation of the persistence exponent $θ$ for a nearly-Markovian Gaussian process $X(t)$, a problem initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. New resummed perturbative and non-perturbative expressions for $θ$ are obtained, which suggest a connection with the result of the alternative independent interval approximation (IIA). The perturbation theory is extended to the calculation of $θ$ for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and non-perturbative expressions for the persistence exponent $θ(X_0)$, describing the probability that the process remains bigger than $X_0\sqrt{ }$.
cond-mat.stat-mech↗
Persistence with Partial Survival
We introduce a parameter $p$, called partial survival, in the persistence of stochastic processes and show that for smooth processes the persistence exponent $θ(p)$ changes continuously with $p$, $θ(0)$ being the usual persistence exponent. We compute $θ(p)$ exactly for a one-dimensional deterministic coarsening model, and approximately for the diffusion equation. Finally we develop an exact, systematic series expansion for $θ(p)$, in powers of $ε=1-p$, for a general Gaussian process with finite density of zero crossings.
cond-mat.stat-mech↗
Phase Transition in a Conserved-Mass Model of Aggregation and Dissociation
We introduce a new model of aggregation of particles where in addition to diffusion and aggregation upon contact, a single unit of mass can dissociate from a conglomerate. This dissociation move conserves the total mass and leads to a striking behaviour in the steady state. As the parameters are varied, the system undergoes a dynamical phase transition in all dimensions. In one phase the mass distribution decays exponentially for large mass whereas in the other phase there is a power law distribution of masses which coexists with an infinite mass aggregate. The model is investigated analytically within mean field theory, and numerically in one dimension.
cond-mat.stat-mech↗
Survival Probability of a Mobile Particle in a Fluctuating Field
We study the the survival probability P(t) upto time t, of a test particle moving in a fluctuating external field. The particle moves according to some prescribed deterministic or stochastic rules and survives as long as the external field that it sees at its own location does not change sign. This is a natural generalization of the "static persistence" (when the particle is at rest) that has generated considerable recent interests. Two types of motions of the particle are considered. In one case, the particle adopts a strategy to live longer and in the other it just diffuses randomly. Three different external fields were considered: (i) the solution of diffusion equation, (ii) the "colour" profile of the q-state Potts model undergoing zero temperature coarsening dynamics and (iii) spatially uncorrelated Brownian signals. In most cases studied, $P(t)\sim t^{-θ_m}$ for large t. The exponent $θ_m$ is calculated via numerically, analytically by approximate methods and in some cases exactly. It is shown in some special cases that the survival probability of the mobile particle is related to the persistence of special "patterns" present in the initial configuration of a phase ordering system.
cond-mat.stat-mech↗
Far from equilibrium dynamics of the Bose gas
We study the approach to equilibrium of a Bose gas to a superfluid state. We point out that dynamic scaling, characteristic of far from equilibrium phase-ordering systems, should hold. We stress the importance of a non-dissipative Josephson precession term in driving the system to a new universality class. A model of coarsening in d=2, involving a quench between two temperatures below the equilibrium superfluid transition temperature (T_c), is exactly solved and demonstrates the relevance of the Josephson term. Numerical results on quenches from above T_c in d=2,3 provide evidence for the scaling picture postulated.
cond-mat↗
Survival Probability of a Gaussian Non-Markovian Process: Application to the T=0 Dynamics of the Ising Model
We study the decay of the probability for a non-Markovian stationary Gaussian walker not to cross the origin up to time $t$. This result is then used to evaluate the fraction of spins that do not flip up to time $t$ in the zero temperature Monte-Carlo spin flip dynamics of the Ising model. Our results are compared to extensive numerical simulations.
cond-mat↗
Nontrivial Exponent for Simple Diffusion
The diffusion equation \partial_tϕ= \nabla^2ϕis considered, with initial condition ϕ( _x_ ,0) a gaussian random variable with zero mean. Using a simple approximate theory we show that the probability p_n(t_1,t_2) that ϕ( _x_ ,t) [for a given space point _x_ ] changes sign n times between t_1 and t_2 has the asymptotic form p_n(t_1,t_2) \sim [\ln(t_2/t_1)]^n(t_1/t_2)^{-θ}. The exponent θhas predicted values 0.1203, 0.1862, 0.2358 in dimensions d=1,2,3, in remarkably good agreement with simulation results.
cond-mat↗
Phase transition of a Bose gas in a harmonic potential
We consider a dilute Bose gas confined by a harmonic potential. We define an appropriate thermodynamic limit and analyze the properties of the phases and phase transition in this limit. Critical properties in the presence of the potential are found to be different from, though simply related, to those in the usual translationally invariant case. We argue that the properties of magnetically trapped rubidium\cite{Wieman} and sodium\cite{ketterle} gases (in which Bose-Einstein condensation has been recently observed) are well approximated by our thermodynamic limit except in a narrow window of temperature around the critical temperature. We also consider the effect of the confining potential on the non-equilibrium dynamics following a rapid quench to the ordered side and give a scaling description of the late time universal dynamics.
cond-mat↗
Critical Properties of Random Quantum Potts and Clock Models
We study zero temperature phase transitions in two classes of random quantum systems -the $q$-state quantum Potts and clock models. For models with purely ferromagnetic interactions in one dimension, we show that for strong randomness there is a second order transition with critical properties that can be determined exactly by use of an RG procedure. Somewhat surprisingly, the critical behaviour is completely independent of $q$ (for $2 \leq q < \infty$). For the $q > 4$ clock model, we suggest the existence of a novel multicritical point at intermediate randomness. We also consider the $T = 0$ transition from a paramagnet to a spin glass in an infinite range model. Assuming that the transition is second order, we solve for the critical behaviour and find $q$ independent exponents.
cond-mat↗
Non-equilibrium Dynamics Following a Quench to the Critical Point in a Semi-infinite System
We study the non-equilibrium dynamics (purely dissipative and relaxational) in a semi-infinite system following a quench from the high temperature disordered phase to its critical temperature. We show that the local autocorrelation near the surface of a semi-infinite system decays algebraically in time with a new exponent which is different from the bulk. We calculate this new non-equilibrium surface exponent in several cases, both analytically and numerically.
cond-mat↗
Coarsening in the q-State Potts Model and the Ising Model with Globally Conserved Magnetization
We study the nonequilibrium dynamics of the $q$-state Potts model following a quench from the high temperature disordered phase to zero temperature. The time dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale $L(t)\sim t^{1/2}$. In particular, the autocorrelation function decays as $L(t)^{-λ(q)}$. We illustrate these properties by solving exactly the kinetic Potts model in $d=1$. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general $q$ and $d$. We establish a correspondence between the two-point correlations of the $q$-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization $(2/q-1)$. The dynamics of this Ising model is solved exactly in the large q limit, and in the limit of a large number of components $n$ for the order parameter. For general $q$ and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent $λ(q)$. These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.
cond-mat↗