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A class of non-local contact processes is introduced and studied using mean-field approximation and numerical simulations. In these processes particles are created at a rate which decays algebraically with the distance from the nearest particle. It is found that the transition into the absorbing state is continuous and is characterized by continuously varying critical exponents. This model differs from the previously studied non-local directed percolation model, where particles are created by unrestricted Levy flights. It is motivated by recent studies of non-equilibrium wetting indicating that this type of non-local processes play a role in the unbinding transition. Other non-local processes which have been suggested to exist within the context of wetting are considered as well.
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In this paper we give explicit examples of power-law correlated stationary Markovian processes y(t) where the stationary pdf shows tails which are gaussian or exponential. These processes are obtained by simply performing a coordinate transformation of a specific power-law correlated additive process x(t), already known in the literature, whose pdf shows power-law tails 1/x^a. We give analytical and numerical evidence that although the new processes (i) are Markovian and (ii) have gaussian or exponential tails their autocorrelation function still shows a power-law decay <y(t) y(t+T)>=1/T^b where b grows with a with a law which is compatible with b=a/2-c, where c is a numerical constant. When a<2(1+c) the process y(t), although Markovian, is long-range correlated. Our results help in clarifying that even in the context of Markovian processes long-range dependencies are not necessarily associated to the occurrence of extreme events. Moreover, our results can be relevant in the modeling of complex systems with long memory. In fact, we provide simple processes associated to Langevin equations thus showing that long-memory effects can be modeled in the context of continuous time stationary Markovian processes.
We study two dimensional stripe forming systems with competing repulsive interactions decaying as $r^{-alpha}$. We derive an effective Hamiltonian with a short range part and a generalized dipolar interaction which depends on the exponent $alpha$. An approximate map of this model to a known XY model with dipolar interactions allows us to conclude that, for $alpha <2$ long range orientational order of stripes can exist in two dimensions, and establish the universality class of the models. When $alpha geq 2$ no long-range order is possible, but a phase transition in the KT universality class is still present. These two different critical scenarios should be observed in experimentally relevant two dimensional systems like electronic liquids ($alpha=1$) and dipolar magnetic films ($alpha=3$). Results from Langevin simulations of Coulomb and dipolar systems give support to the theoretical results.
We present an effective evolution equation for a coarse-grained distribution function of a long-range-interacting system preserving the symplectic structure of the non-collisional Boltzmann, or Vlasov, equation. We first derive a general form of such an equation based on symmetry considerations only. Then, we explicitly derive the equation for one-dimensional systems, finding that it has the form predicted on general grounds. Finally, we use such an equation to predict the dependence of the damping times on the coarse-graining scale and numerically check it for some one-dimensional models, including the Hamiltonian Mean Field (HMF) model, a scalar field with quartic interaction, a 1-d self-gravitating system, and the Self-Gravitating Ring (SGR).
A generalized zero-range process with a limited number of long-range interactions is studied as an example of a transport process in which particles at a T-junction make a choice of which branch to take based on traffic levels on each branch. The system is analysed with a self-consistent mean-field approximation which allows phase diagrams to be constructed. Agreement between the analysis and simulations is found to be very good.
The study of critical properties of systems with long-range interactions has attracted in the last decades a continuing interest and motivated the development of several analytical and numerical techniques, in particular in connection with spin models. From the point of view of the investigation of their criticality, a special role is played by systems in which the interactions are long-range enough that their universality class is different from the short-range case and, nevertheless, they maintain the extensivity of thermodynamical quantities. Such interactions are often called weak long-range. In this paper we focus on the study of the critical behaviour of spin systems with weak-long range couplings using renormalization group, and we review their remarkable properties. For the sake of clarity and self-consistency, we start from the classical $O(N)$ spin models and we then move to quantum spin systems.
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