arXiv subjects
Publications and source records attributed to Haye Hinrichsen.
The conserved Manna model with a planar absorbing boundary is studied in various space dimensions. We present a heuristic argument that allows one to compute the surface critical exponent in one dimension analytically. Moreover, we discuss the mean field limit that is expected to be valid in d>4 space dimensions and demonstrate how the corresponding partial differential equations can be solved.
We study the dynamic properties of a model for wetting with two competing adsorbates on a planar substrate. The two species of particles have identical properties and repel each other. Starting with a flat interface one observes the formation of homogeneous droplets of the respective type separated by nonwet regions where the interface remains pinned. The wet phase is characterized by slow coarsening of competing droplets. Moreover, in 2+1 dimensions an additional line of continuous phase transition emerges in the bound phase, which separates an unordered phase from an ordered one. The symmetry under interchange of the particle types is spontaneously broken in this region and finite systems exhibit two metastable states, each dominated by one of the species. The critical properties of this transition are analyzed by numeric simulations.
We consider a nonequilibrium process on a timeline with discrete sites which evolves by a non-Markovian update rule in such a way that an active site at time t activates one or several sites in the future at time t+dt. The time intervals dt are distributed algebraically as dt^(-1-kappa), where 0<kappa<1 is a control paramter. Depending on the activation rate, the system display a nonequilibrium phase transition which may be interpreted as directed percolation transition driven by temporal Levy flights in the limit of zero space dimensions. The critical properties are investigated by extensive numerical simulations and compared with field-theoretic predictions.
Local Scale Invariance (LSI) is a theory for anisotropic critical phenomena designed in the spirit of conformal invariance. For a given representation of its generators it makes non-trivial predictions about the form of universal scaling functions. In the past decade several representations have been identified and the corresponding predictions were confirmed for various anisotropic critical systems. Such tests are usually based on a comparison of two-point quantities such as autocorrelation and response functions. The present work highlights a potential problem of the theory in the sense that it may predict any type of two-point function. More specifically, it is argued that for a given two-point correlator it is possible to construct a representation of the generators which exactly reproduces this particular correlator. This observation calls for a critical examination of the predictive content of the theory.
Estimating entropies from limited data series is known to be a non-trivial task. Naive estimations are plagued with both systematic (bias) and statistical errors. Here, we present a new 'balanced estimator' for entropy functionals Shannon, Rényi and Tsallis) specially devised to provide a compromise between low bias and small statistical errors, for short data series. This new estimator out-performs other currently available ones when the data sets are small and the probabilities of the possible outputs of the random variable are not close to zero. Otherwise, other well-known estimators remain a better choice. The potential range of applicability of this estimator is quite broad specially for biological and digital data series.
Recently Dantas, Oliveira and Stilck [J. Stat. Mech. (2007) P08009] studied how the one-dimensional diffusive contact process crosses over from the critical behavior of directed percolation to an effective mean field behaviour when the diffusion rate is sent to infinity. They showed that this crossover can be described in terms of a crossover exponent $ϕ$, finding the boundaries 3 <= $ϕ$ <= 4 in one spatial dimension. In the present work we refine and extend this result up to four spatial dimensions by a field-theoretic calculation and extensive numerical simulations.
We study the phase transition between survival and extinction in an epidemic process with long-range interactions and immunization. This model can be viewed as the well-known general epidemic process (GEP) in which nearest-neighbor interactions are replaced by Levy flights over distances r which are distributed as P(r) ~ r^(-d-sigma). By extensive numerical simulations we confirm previous field-theoretical results obtained by Janssen et al. [Eur. Phys. J. B7, 137 (1999)].
Recently Baumann et al. [arXiv:0709.3228v1] studied the phase-ordering kinetics of the two-dimensional Ising model with uniform spatially quenched disorder by Monte-Carlo simulations. They found that the two-time response and correlation functions are in agreement with the predictions of local scale invariance generalised to z!=2. The present paper shows why this is not true and suggests an alternative approach which leads to a much better agreement with the numerical results.
Ageing phenomena are observed in a large variety of dynamical systems exhibiting a slow relaxation from a non-equilibrium initial state. Ageing can be characterised in terms of the linear response R(t,s) at time t to a local perturbation at time s<t. Usually one distinguishes two dynamical regimes, namely, the quasi-stationary regime, where the response is translationally invariant in time, and the ageing regime, where this invariance is broken. In general these two regimes are separate in the sense that the two limits of (a) taking t,s to infinity while keeping t/s fixed, and (b) taking t,s to infinity with fixed t-s, give different results. In recent years, ageing was also investigated in the context of homogeneous critical systems such as the Glauber-Ising model and the contact process. Here we argue that, in contrast to a widespread believe, homogeneous critical systems do not have a separate quasi-stationary regime, hence the two limits do commute. Moreover, it is discussed under which conditions two particular exponents, denoted as a and a' in the literature, have to be identical.
We show that a time series $x_t$ evolving by a non-local update rule $x_t = f (x_{t-n},x_{t-k})$ with two different delays $k<n$ can be mapped onto a local process in two dimensions with special time-delayed boundary conditions provided that $n$ and $k$ are coprime. For certain stochastic update rules exhibiting a non-equilibrium phase transition this mapping implies that the critical behavior does not depend on the short delay $k$. In these cases, the autocorrelation function of the time series is related to the critical properties of directed percolation.
This review article gives an overview of recent progress in the field of non-equilibrium phase transitions into absorbing states with long-range interactions. It focuses on two possible types of long-range interactions. The first one is to replace nearest-neighbor couplings by unrestricted Levy flights with a power-law distribution P(r) ~ r^(-d-sigma) controlled by an exponent sigma. Similarly, the temporal evolution can be modified by introducing waiting times Dt between subsequent moves which are distributed algebraically as P(Dt)~ (Dt)^(-1-kappa). It turns out that such systems with Levy-distributed long-range interactions still exhibit a continuous phase transition with critical exponents varying continuously with sigma and/or kappa in certain ranges of the parameter space. In a field-theoretical framework such algebraically distributed long-range interactions can be accounted for by replacing the differential operators nabla^2 and d/dt with fractional derivatives nabla^sigma and (d/dt)^kappa. As another possibility, one may introduce algebraically decaying long-range interactions which cannot exceed the actual distance to the nearest particle. Such interactions are motivated by studies of non-equilibrium growth processes and may be interpreted as Levy flights cut off at the actual distance to the nearest particle. In the continuum limit such truncated Levy flights can be described to leading order by terms involving fractional powers of the density field while the differential operators remain short-ranged.
The stochastic differential equation $\dot{x}(t) = ax(t) + bx(t-τ) + c x(t) ξ(t)$ with a time-delayed feedback and a multiplicative Gaussian noise is shown to be related to Kardar-Parisi-Zhang universality class of growing surfaces.
In contrast to recent claims by Enss, Henkel, Picone, and Schollwoeck [J. Phys. A 37, 10479] it is shown that the critical autoresponse function of the 1+1-dimensional contact process is not in agreement with the predictions of local scale invariance.
A biologically motivated model for spatio-temporal coexistence of two competing species is studied by mean-field theory and numerical simulations. In d>1 dimensions the phase diagram displays an extended region where both species coexist, bounded by two second-order phase transition lines belonging to the directed percolation universality class. The two transition lines meet in a multicritical point, where a non-trivial critical behavior is observed.
As a canonical model for wetting far from thermal equilibrium we study a Kardar-Parisi-Zhang interface growing on top of a hard-core substrate. Depending on the average growth velocity the model exhibits a non-equilibrium wetting transition which is characterized by an additional surface critical exponent theta. Simulating the single-step model in one spatial dimension we provide accurate numerical estimates for theta and investigate the distribution of contact points between the substrate and the interface as a function of time. Moreover, we study the influence of finite-size effects, in particular the time needed until a finite substrate is completely covered by the wetting layer for the first time.
The non-equilibrium phase transition in models for epidemic spreading with long-range infections in combination with incubation times is investigated by field-theoretical and numerical methods. Here the spreading process is modelled by spatio-temporal Levy flights, i.e., it is assumed that both spreading distance and incubation time decay algebraically. Depending on the infection rate one observes a phase transition from a fluctuating active phase into an absorbing phase, where the infection becomes extinct. This transition between spreading and extinction is characterized by continuously varying critical exponents, extending from a mean-field regime to a phase described by the universality class of directed percolation. We compute the critical exponents in the vicinity of the upper critical dimension by a field-theoretic renormalization group calculation and verify the results in one spatial dimension by extensive numerical simulations.
We show that the static friction force which must be overcome to render a sticking contact sliding is reduced if an external torque is also exerted. As a test system we study a planar disk lying on horizontal flat surface. We perform experiments and compare with analytical results to find that the coupling between static friction force and torque is nontrivial: It is not determined by the Coulomb friction laws alone, instead it depends on the microscopic details of friction. Hence, we conclude that the macroscopic experiment presented here reveals details about the microscopic processes lying behind friction.
If a body is resting on a flat surface, the maximal static friction force before motion sets in is reduced if an external torque is also applied. The coupling between the static friction force and static friction torque is nontrivial as our studies for a tripod lying on horizontal flat surface show. In this article we report on a series of experiments we performed on a tripod and compare these with analytical and numerical solutions. It turns out that the coupling between force and torque reveals information about the microscopic properties at the onset to sliding.