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.
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.
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 <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. 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 <S0(t)Sx(t)><S0(t('))Sx(t('))> 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.
In a recent letter [PRL 80 (1998) 3539] Fisher, Le Doussal and Monthus report new predictions for the persistence properties of Sinai's model, which they obtain by using an approximate real space renormalization group scheme. In this comment we show that these predictions can be derived in a rigorous and more transparent way by reducing the quantities of interest to particular statistical properties of a simple random walk in a homogeneous environment. We also point out that they are valid in a much more general context: our results are neither restricted to the random force model but are valid for any asymmetric hopping model, nor to the vicinity of the critical point.
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,
Using exact expressions for the persistence probability and for the leading eigenvalue of the Fokker-Planck operator of a random walk in a random environment, we establish a fundamental relation between the statistical properties of anomalous diffusion and the critical and off-critical behavior of random quantum spin chains. Many exact results are obtained from this correspondence, including the space and time correlations of surviving random walks and the distribution of the gaps of the corresponding Fokker-Planck operator. In turn we derive analytically the dynamical exponent of the random transverse-field Ising spin chain in the Griffiths-McCoy region.
This is a pedagogical account on reaction-diffusion systems and their relationship with integrable quantum spin chains.Reaction-diffusion systems are paradigmatic examples of non-equilibrium systems.Their long-time behaviour is strongly influenced through fluctuation effects in low dimensions which renders the habitual mean-field kinetic equations inapplicable.Starting from the master equation rewritten as a Schrödinger equation with imaginary time, the associated quantum Hamiltonian of certain one-dimensional reaction-diffusion models is closely related to integrable magnetic chains.The relationship with the Hecke algebra and its quotients allows to identify integrable reaction-diffusion models and, through the Baxterization procedure, relate them to the solutions of Yang-Baxter equations which can be solved via the Bethe ansatz.Methods such as spectral and partial integrability, free fermions, similarity transformations or diffusion algebras are reviewed, with several concrete examples treated explicitly.An outlook on how the recently-introduced concept of local scale invariance might become useful in the description of nonequilibrium ageing phenomena is presented, with particular emphasis on the kinetic Ising model with Glauber dynamics.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.