Dynamics of the two-dimensional directed Ising model: zero-temperature coarsening

We investigate the laws of coarsening of a two-dimensional system of Ising spins evolving under single-spin-flip irreversible dynamics at low temperature from a disordered initial condition. The irreversibility of the dynamics comes from the directedness, or asymmetry, of the influence of the neighbours on the flipping spin. We show that the main characteristics of phase ordering at low temperature, such as self-similarity of the patterns formed by the growing domains, and the related scaling laws obeyed by the observables of interest, which hold for reversible dynamics, are still present when the dynamics is directed and irreversible, but with different scaling behaviour. In particular the growth of domains, instead of being diffusive as is the case when dynamics is reversible, becomes ballistic. Likewise, the autocorrelation function and the persistence probability (the probability that a given spin keeps its sign up to time $t$) have still power-law decays but with different exponents.Comment: 29 pages, 36 figure

Nonequilibrium dynamics of the zeta urn model

We consider a mean-field dynamical urn model, defined by rules which give the rate at which a ball is drawn from an urn and put in another one, chosen amongst an assembly. At equilibrium, this model possesses a fluid and a condensed phase, separated by a critical line. We present an analytical study of the nonequilibrium properties of the fluctuating number of balls in a given urn, considering successively the temporal evolution of its distribution, of its two-time correlation and response functions, and of the associated \fd ratio, both along the critical line and in the condensed phase. For well separated times the \fd ratio admits non-trivial limit values, both at criticality and in the condensed phase, which are universal quantities depending continuously on temperature.Comment: 30 pages, 1 figur

Single-spin-flip dynamics of the Ising chain

We consider the most general single-spin-flip dynamics for the ferromagnetic Ising chain with nearest-neighbour influence and spin reversal symmetry. This dynamics is a two-parameter extension of Glauber dynamics corresponding respectively to non-linearity and irreversibility. The associated stationary state measure is given by the usual Boltzmann-Gibbs distribution for the ferromagnetic Hamiltonian of the chain. We study the properties of this dynamics both at infinite and at finite temperature, all over its parameter space, with particular emphasis on special lines and points.Comment: 31 pages, 18 figure

Temporal Correlations and Persistence in the Kinetic Ising Model: the Role of Temperature

We study the statistical properties of the sum $S_t=\int_{0}^{t}dt' \sigma_{t'}$, that is the difference of time spent positive or negative by the spin $\sigma_{t}$, located at a given site of a $D$-dimensional Ising model evolving under Glauber dynamics from a random initial configuration. We investigate the distribution of $S_{t}$ and the first-passage statistics (persistence) of this quantity. We discuss successively the three regimes of high temperature ($T>T_{c}$), criticality ($T=T_c$), and low temperature ($T<T_{c}$). We discuss in particular the question of the temperature dependence of the persistence exponent $\theta$, as well as that of the spectrum of exponents $\theta(x)$, in the low temperature phase. The probability that the temporal mean $S_t/t$ was always larger than the equilibrium magnetization is found to decay as $t^{-\theta-\frac12}$. This yields a numerical determination of the persistence exponent $\theta$ in the whole low temperature phase, in two dimensions, and above the roughening transition, in the low-temperature phase of the three-dimensional Ising model.Comment: 21 pages, 11 PostScript figures included (1 color figure

Statistics of the occupation time for a class of Gaussian Markov processes

We revisit the work of Dhar and Majumdar [Phys. Rev. E 59, 6413 (1999)] on the limiting distribution of the temporal mean M_{t}=t^{-1}\int_{0}^{t}du \sign y_{u}, for a Gaussian Markovian process $y_{t}$ depending on a parameter $\alpha$, which can be interpreted as Brownian motion in the scale of time $t^{\prime}=t^{2\alpha}$. This quantity, for short the mean `magnetization', is simply related to the occupation time of the process, that is the length of time spent on one side of the origin up to time t. Using the fact that the intervals between sign changes of the process form a renewal process in the time scale t', we determine recursively the moments of the mean magnetization. We also find an integral equation for the distribution of $M_{t}$. This allows a local analysis of this distribution in the persistence region $(M_t\to\pm1)$, as well as its asymptotic analysis in the regime where $\alpha$ is large. We finally put the results thus found in perspective with those obtained by Dhar and Majumdar by another method, based on a formalism due to Kac.Comment: latex, 31 page