Nonequilibrium dynamics of random field Ising spin chains: exact results via real space renormalization group.

The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function and the two-time autocorrelations . We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.

Random walkers in one-dimensional random environments: exact renormalization group analysis.

Sinai's model of diffusion in one dimension with random local bias is studied by a real space renormalization group, which yields exact results at long times. The effects of an additional small uniform bias force are also studied. We obtain analytically the scaling form of the distribution of the position x(t) of a particle, the probability of it not returning to the origin, and the distributions of first passage times, in an infinite sample as well as in the presence of a boundary and in a finite but large sample. We compute the distribution of the meeting time of two particles in the same environment. We also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward. These quantities obey multifractal scaling, characterized by generalized persistence exponents theta(g) which we compute. In the presence of a small bias, the number of returns to the origin becomes finite, characterized by a universal scaling function which we obtain. The full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained, as well as detailed analytical information about correlations between directions and times of successive jumps. The two-time distribution of the positions of a particle, x(t) and x(t') with t>t', is also computed exactly. It is found to exhibit "aging" with several time regimes characterized by different behaviors. In the unbiased case, for t-t' approximately t'alpha with alpha>1, it exhibits a ln t/ln t' scaling, with a singularity at coinciding rescaled positions x(t)=x(t'). This singularity is a novel feature, and corresponds to particles that remain in a renormalized valley. For closer times alpha<1, the two-time diffusion front exhibits a quasiequilibrium regime with a ln(t-t')/ln t' behavior which we compute. The crossover to a t/t' aging form in the presence of a small bias is also obtained analytically. Rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are also characterized in detail. Connections with the Green function of a one-dimensional Schrödinger problem and quantum spin chains are discussed.

Reaction diffusion models in one dimension with disorder.

We study a large class of one-dimensional reaction diffusion models with quenched disorder using a real space renormalization group method (RSRG) which yields exact results at large time. Particles (e.g., of several species) undergo diffusion with random local bias (Sinai model) and may react upon meeting. We obtain a detailed description of the asymptotic states (i.e., attractive fixed points of the RSRG), such as the large time decay of the density of each specie, their associated universal amplitudes, and the spatial distribution of particles. We also derive the spectrum of nontrivial exponents which characterize the convergence towards the asymptotic states. For reactions which lead to several possible asymptotic states separated by unstable fixed points, we analyze the dynamical phase diagram and obtain the critical exponents characterizing the transitions. We also obtain a detailed characterization of the persistence properties for single particles as well as more complex patterns. We compute the decay exponents for the probability of no crossing of a given point by, respectively, the single particle trajectories (theta) or the thermally averaged packets (theta). The generalized persistence exponents associated to n crossings are also obtained. Specifying to the process A+A--> or A with probabilities (r,1-r), we compute exactly the exponents delta(r) and psi(r) characterizing the survival up to time t of a domain without any merging or with mergings, respectively, and the exponents deltaA(r) and psiA(r) characterizing the survival up to time t of a particle A without any coalescence or with coalescences, respectively. theta, psi, and delta obey hypergeometric equations and are numerically surprisingly close to pure system exponents (though associated to a completely different diffusion length). The effect of additional disorder in the reaction rates, as well as some open questions, are also discussed.