Scaling property of flux fluctuations from random walks (original) (raw)
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Physical review, 2017
The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T c the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D SAW f = 4 3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R rms ∼ t 3/4 (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T = T c the model has some aspects compatible with the 2D BTW model (e.g., the 1/ log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T = T c. In the off-critical temperatures in the close vicinity of T c the exponents show some additional power-law behaviors in terms of T − T c with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L 1 2 , which is different from the regular 2D BTW model.
Fluctuations of random walks in critical random environments
Physical Chemistry Chemical Physics, 2018
Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. We here study the influence of the cluster size distribution on diffusion measurements in percolation networks.
All-time dynamics of continuous-time random walks on complex networks
The Journal of Chemical Physics, 2013
The concept of continuous-time random walks (CTRW) is a generalization of ordinary random walk models, and it is a powerful tool for investigating a broad spectrum of phenomena in natural, engineering, social and economic sciences. Recently, several theoretical approaches have been developed that allowed to analyze explicitly dynamics of CTRW at all times, which is critically important for understanding mechanisms of underlying phenomena. However, theoretical analysis has been done mostly for systems with a simple geometry. Here we extend the original method based on generalized master equations to analyze all-time dynamics of CTRW models on complex networks. Specific calculations are performed for models on lattices with branches and for models on coupled parallel-chain lattices. Exact expressions for velocities and dispersions are obtained. Generalized fluctuations theorems for CTRW models on complex networks are discussed.
Interacting random walkers and non-equilibrium fluctuations
The European Physical Journal B, 2008
We introduce a model of interacting Random Walk, whose hopping amplitude depends on the number of walkers/particles on the link. The mesoscopic counterpart of such a microscopic dynamics is a diffusing system whose diffusivity depends on the particle density. A non-equilibrium stationary flux can be induced by suitable boundary conditions, and we show indeed that it is mesoscopically described by a Fourier equation with a density dependent diffusivity. A simple mean-field description predicts a critical diffusivity if the hopping amplitude vanishes for a certain walker density. Actually, we evidence that, even if the density equals this pseudo-critical value, the system does not present any criticality but only a dynamical slowing down. This property is confirmed by the fact that, in spite of interaction, the particle distribution at equilibrium is simply described in terms of a product of Poissonians. For mesoscopic systems with a stationary flux, a very effect of interaction among particles consists in the amplification of fluctuations, which is especially relevant close to the pseudo-critical density. This agrees with analogous results obtained for Ising models, clarifying that larger fluctuations are induced by the dynamical slowing down and not by a genuine criticality. The consistency of this amplification effect with altered coloured noise in time series is also proved.
Single random walker on disordered lattices
Journal of Statistical Physics, 1984
Random walks on square lattice percolating clusters were followed for up to 2 • 10 ~ steps. The mean number of distinct sites visited (SN) gives a spectral dimension of d s = 1.30 5:0.03 consistent with superuniversality (d S = 4/3) but closer to the alternative ds= 182/139, based on the low dimensionality correction. Simulations are also given for walkers on an energetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a "crossover" from fractal-to-Euclidean behavior. Walks on two-and three-dimensional lattices are similar, except that those in three dimensions are more efficient.
Scaling transformation of random walk distributions in a lattice
Physical Review E, 2000
We use a decimation procedure in order to obtain the dynamical renormalization group transformation ͑RGT͒ properties of random walk distribution in a 1ϩ1 lattice. We obtain an equation similar to the Chapman-Kolmogorov equation. First we show the existence of invariants through the RGT. We also show the existence of functions which are semi-invariants through the RGT. Second, we show as well that the distribution R q (x)ϭ͓1ϩb(qϪ1)x 2 ͔ 1/(1Ϫq) (qϾ1), which is an exact solution of a nonlinear Fokker-Planck equation, is a semi-invariant for RGT. We obtain the map qЈϭ f (q) from the RGT and we show that this map has two fixed points: qϭ1, attractor, and qϭ2, repellor, which are the Gaussian and the Lorentzian, respectively. We show the connections between these result and the Levy flights.
Journal of Statistical Physics, 1989
We present new exact results for a one-dimensional asymmetric disordered hopping model. The lattice is taken infinite from the start and we do not resort to the periodization scheme used by Derrida. An explicit resummation allows for the calculation of the velocity V and the diffusion constant D (which are found to coincide with those given by Derrida) and for demonstrating that V is indeed a self-averaging quantity; the same property is established for D in the limiting case of a directed walk.
Scaling properties of random walks on small-world networks
Physical Review E, 2003
Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in d dimensions. At intermediate times, the properties of the walker shows scaling behavior characteristic of an infinite small-world network. Finally, at long times, the finite size of the network becomes important, and many of the properties of the walker saturate. We propose general analytical forms for the scaling properties in all three regimes, and show that these analytical forms are consistent with our numerical simulations.
A mechanism to synchronize fluctuations in scale free networks using growth models
In this paper we study the steady state of the fluctuations of the surface for a model of surface growth with relaxation to any of its lower nearest neighbors (SRAM) [F. Family, J. Phys. A 19, L441 (1986)] in scale free networks. It is known that for Euclidean lattices this model belongs to the same universality class as the model of surface relaxation to the minimum (SRM). For the SRM model, it was found that for scale free networks with broadness λ, the steady state of the fluctuations scales with the system size N as a constant for λ ≥ 3 and has a logarithmic divergence for λ < 3 [Pastore y Piontti et al., Phys. Rev. E 76, 046117 (2007)]. It was also shown [La Rocca et al., Phys. Rev. E 77, 046120 (2008)] that this logarithmic divergence is due to non-linear terms that arises from the topology of the network. In this paper we show that the fluctuations for the SRAM model scale as in the SRM model. We also derive analytically the evolution equation for this model for any kind of complex graphs and find that, as in the SRM model, non-linear terms appear due to the heterogeneity and the lack of symmetry of the network. In spite of that, the two models have the same scaling, but the SRM model is more efficient to synchronize systems.
Transport Properties of Random Walks on Scale-Free/Regular-Lattice Hybrid Networks
Journal of Statistical Physics, 2007
We study numerically the mean access times for random walks on hybrid disordered structures formed by embedding scale-free networks into regular lattices, considering different transition rates for steps across lattice bonds (F ) and across network shortcuts (f ). For fast shortcuts (f /F ≫ 1) and low shortcut densities, traversal time data collapse onto an universal curve, while a crossover behavior that can be related to the percolation threshold of the scale-free network component is identified at higher shortcut densities, in analogy to similar observations reported recently in Newman-Watts small-world networks. Furthermore, we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution, and we discuss access time distributions as functions of the initial and final node degrees. These findings are relevant, in particular, when considering the optimization of existing information networks by the addition of a small number of fast shortcut connections.