on the hull of two dimensional percolation clusters a recently proven equivalence between certain random walks with memory and the hull of percolation clusters in two dimensions is used to estimate the fractal dimension of the latter by means of monte carlo simulations the authors calculate the value dh 1 750 or 0 002 in agreement with a recent conjecture that dh 7 4 exactly

lorentz gas cellular automata on graphs lorentz gas cellular automata lgca are models of parallel many tape turing machines generated by the motion of localized point objects particles signals wave trains etc on a graph in each vertex v of the graph we store a symbol a read write head of the turing machine is represented as an object that hops from one vertex of the graph to another according to a rule symbol stored in the vertex thus the symbols stored in the vertices represent scattering rules or simply scatterers of the lattice gas it is assumed that initially the scatterers are randomly distributed among the vertices of the graph the random environment formed by the scatterers may either be fixed or evolve due to collisions with the moving object the collisions simulate storing of a new symbol in the vertex of the graph we investigate these models with fixed environment in general types of graphs and the models with evolving environments on trees remarkably it occurred that all models with fixed environments and many models with evolving environment behave like a depth first search on the underlying graph observe that such behavior occurs for any initial random distribution of scatterers rather than for some specially prepared initial configurations of scatterers we also give estimates of periods of orbits on finite graphs

equivalence of deterministic walks on regular lattices on the plane abstract we consider deterministic walks on square triangular and hexagonal two dimensional lattices in each case there is a scatterer at every lattice site that can be in one of two states that forces the walker to turn either to his her immediate right or left after the walker is scattered the scatterer changes state a lattice with an arrangement of scatterers is an environment we show that there are only two environments for which the scattering rules are injective mirrors or rotators on the three lattices on hexagonal lattices webb and cohen 2014 proved that if a walker with a given initial position and velocity moves through an environment of mirrors rotators then there is an environment of rotators mirrors through which the walker would move with the same trajectory we refer to these trajectories on mirror and rotator environments as equivalent walks we prove the equivalence of walks on square and triangular lattices and include a proof of the equivalence of walks on hexagonal lattices the proofs are based both on the geometry of the lattice and the structure of the scattering rule

on a conjecture of alley and alder for fluids and lorentz models we discuss a conjecture of alley and alder predicting a relation between the four point and the two point velocity autocorrelation functions for fluids and lorentz models at sufficiently long times if the conjecture is correct a modified burnett coefficient can be defined which has a finite value contrary to the ordinary burnett coefficient which is divergent the conjecture is tested for four classes of models with different methods for three dimensional fluids mode coupling theory yields a negative result the conjecture is confirmed for thed dimensional deterministic lorentz gas d 2 and for a class ofd dimensional stochastic lorentz models d 1 by low density kinetic theory as well as by rigorous results available for one dimension for yet another class of one dimensional stochastic lorentz models which are exactly solvable in one dimension the result is negative again all four classes of models show long time tails in the velocity autocorrelation function and have a finite diffusion coefficient

transport properties of the one dimensional stochastic lorentz model i velocity autocorrelation function point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables stationary renewal process for a fixed configuration of scatterers a particle performs the following random walk the particle starts at the pointx with velocity 1 in between scatterers the particle moves freely at a scatterer the particle is either transmitted or reflected both with probability 1 2 for given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations if this variable is averaged over the distribution of scatterers it decays not faster thant 3 2

tracer diffusion in lattice gases it has been proved that a tracer particle in a reversible lattice gas converges to brownian motion however only in a few particular cases has a strictly positive self diffusion coefficientd been established here we supply the missing piece and show thatd 0 in general the exceptions are one dimensional lattice gases with nearest neighbor jumps only for whichd 0 the proof establishes a variational formula ford which could be used to obtain realistic bounds

a random lorentz model the authors investigate the properties of a model in which a particle moves on a square lattice with a fraction of the sites randomly occupied by stationary scatterers between two successive collisions with the scatterers the particle performs random walk the velocity autocorrelation function measured by the computer moment propagation method has an algebraic long time tail although for some model parameters it decays very slowly at large time they discuss the diffusion process in such a model calculating the diffusion coefficient in the boltzmann and effective medium approximations it is shown that correlated collisions play an important role in the description of diffusion for an intermediate density of scatterers

random walks with short memory a restricted random walk on ad dimensional cubic lattice with different probabilities for forward backward and sideward steps is studied the analytic solution for the generating function exact expressions for the second and fourth moments of displacements and diffusion and burnett coefficients are given as well as a systematic asymptotic expansion for the probability distribution of long walks

generalized boltzmann equation for lattice gas automata in this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi detailed balance starting from a coupled bbgky hierarchy for then particle distribution functions cluster expansion techniques are used to derive approximate kinetic equations in zeroth approximation the standard nonlnear boltzmann equation is obtained the next approximation yields the ring kinetic equation similar to that for hard sphere systems describing the time evolution of pair correlations the ring equation is solved to determine the nonvanishing pair correlation functions in equilibrium for two models that violate semidetailed balance one is a model of interacting random walkers on a line the other one is a two dimensional fluid type model on a triangular lattice the numerical predictions agree very well with computer simulations

lattice lorentz gas in cellular automaton fluids where binary collision laws admit reflections the stosszahl ansatz and the boltzmann equation are no longer valid at low densities this is proven for a class of lorentz gas models defined on cubic lattices for which the diffusion coefficient is calculated from a self consistent ring kinetic equation or effective medium theory the calculated results are in excellent agreement with results from molecular dynamics simulations over the whole density interval

random walk with persistence the exact analytic result is obtained for the fourier transform of the generating functionf r s 0 s n p r n wherep r n is the probability density for the end to end distancer inn steps of a random walk with persistence the moments r 2 n r 4 n and r 6 n are calculated and approximate results forp r n and r 1 n are given

stochastic one dimensional lorentz gas on a lattice we stud ya one dimensional stochastic lorentz gas wher ea light particle moves i na fixed array of nonidentical random scatterers arranged i na lattice each scatterer is characterized b ya random transmissionreflection coefficient we consider the case when the transmission coefficients of the scatterers are independent identically distributed random variables a symbolic program is presented which generates the exact velocity autocorrelation functio n vacf in terms of the moments of the transmission coefficients the vacf is found for different types of disorder for times up to 20 collision times we then conside ra specific type of disorder a two state lorentz gas in which two types of scatterers are arranged randomly i na lattice the na lattice point is occupied b ya scatterer whose transmission coefficient is with probability p or with probability 1 p a perturbation expansion with respect to is derived the 2 term in this expansion shows that the vacf oscillates with time the period of oscillation being twice the time of flight from one scatterer to its nearest neighbor the coarse grained vacf decays for long times like t 32 which is similar to the decay of the vacf of the random lorentz gas wit ha single type of scatterer the perturbation results and the exact one s found up to 20 collision times show good agreement

vanishing theorems for f harmonic forms on smooth metric measure spaces abstract in this paper we first establish a monotonicity formula for vector bundle valued f harmonic p forms on a smooth metric measure space provided f r is less that an explicit constant as applications we get some vanishing theorems for l 2 f harmonic forms on concrete geometric models in the second part for a metric measure space with nonnegative bakry emery ricci curvature and with moderate volume growth we prove that any bounded f harmonic 1 form must be parallel moreover some vanishing theorems under nonnegative m bakry emery ricci curvature assumption are also proved finally we consider smooth metric measure spaces with weighted poincare inequality and show some vanishing theorems for l q f harmonic 1 forms

positive solutions for a class of elliptic equations abstract in this paper the existence and bifurcation of positive solutions for a class of elliptic equations involving the p laplacian are studied by using the sub and super solution method and the topological degree argument

systematic density expansion for random resistor networks the authors present a systematic density expansion for calculating transport properties return probabilities and spectral densities for lattice hopping models with diluted randomness the method is applied to a disordered square lattice where the bonds carry conductances sigma 1 and sigma 2 in concentrations c and 1 c respectively and they calculate explicitly the o c and o c2 contributions to the static conductivity diffusion coefficient and to the coefficient beta c of the long time tail in the velocity autocorrelation function vacf upsilon x 0 upsilon x t approximately beta c t 2 to o c the vacf is calculated for all times and shows negative correlation cage effect for all times

a non intersecting random walk on the manhattan lattice and sle 6 we consider a random walk on the manhattan lattice the walker must follow the orientations of the bonds in this lattice and the walker is not allowed to visit a site more than once when both possible steps are allowed the walker chooses between them with equal probability the walks generated by this model are known to be related to interfaces for bond percolation on a square lattice so it is natural to conjecture that the scaling limit is hbox sle 6 we test this conjecture with monte carlo simulations of the random walk model and find strong support for the conjecture

lattice lorentz gas kinetic theory standard approximation schemes in kinetic theory are compared with the effective medium approximation ema that was developed earlier for the stochastic lorentz gas on cubic lattices in models with reflecting collisions the low density results are in strong disagreement with molecular chaos and in excellent agreement with the simulation data the analysis identifies the diagrams that are taken into account in the different approximations the ema accounts in a self consistent way for all nested ring diagrams

fragmentation of percolation cluster perimeters we introduce a model for the fragmentation of porous random solids under the action of an external agent in our model the solid is represented by a bond percolation cluster on the square lattice and bonds are removed only at the external perimeter or hull of the cluster this model is shown to be related to the self avoiding walk on the manhattan lattice and to the disconnection events at a diffusion front these correspondences are used to predict the leading and the first correction to scaling exponents for several quantities defined for hull fragmentation our numerical results support these predictions in addition the algorithm used to construct the perimeters reveals itself to be a very efficient tool for detecting subtle correlations in the pseudo random number generator used we present a quantitative test of two generators which supports recent results reported in more systematic studies

manhattan lattice theta point exponents from kinetic growth walks and exact results from the nienhuis o n model kinetic growth walks kgw on the manhattan lattice have previously been shown to be equivalent to the static problem of interacting self avoiding walks on that lattice at the theta temperature here we illustrate how a complete set of exponents for the static problem including the crossover exponent phi and surface exponents at the ordinary and special transition points may be obtained from simulations of kinetic walks in the process we find that phi m 0 430 or 0 006 which encompasses the conjectured value 3 7 approximately 0 42857 for the theta point on an isotropic lattice our numerics confirm a predicted set of exponents for both the bulk and surface transitions in addition to results such as the exact internal energy at the bulk transition furthermore we point out that a recently examined variant of the nienhuis o n model can be mapped onto theta point walks on the manhattan lattice which allows identification of the scaling dimensions for that problem and thereby provides a method for proving all the numerical conjectures

motion in a random gyrotropic environment in d dimensions the authors construct a simple model of ballistic motion in a random environment in an arbitrary number of dimensions the motion is determined by a quenched set of random orthogonal matrices on a hypercubic lattice relating incoming and outgoing directions at a particular site they select study and classify a set of these matrices which agree with intuitive notions of isotropy a mean field theory is constructed of the transition between localised and extended trajectories and a qualitative discussion of various features of the phase diagram is made

long time tails in stationary random media ii applications in a previous paper we developed a mode coupling theory to describe the long time properties of diffusion in stationary statistically homogeneous random media here the general theory is applied to deterministic and stochastic lorentz models and several hopping models the mode coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse grained diffusion coefficient and a coarse grained free volume for one dimensional models these amplitudes can be evaluated and the mode coupling theory is shown to agree with exact solutions obtained for these models for higher dimensional lorentz models the formal theory yields expressions which are difficult to evaluate for these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers

diffusion laws for random walks on various heterogeneous lattices diffusion laws and diffusion coefficients are given for random walks on lattices whose scattering probabilities differ from site to site models are considered in which the direction of the particle after scattering depends on its direction before scattering

long time tails in a random diffusion model letw w x x zd be a positive random field with i i d distribution given its realization letxt be the position at timet of a particle starting at the origin and performing a simple random walk with jump rate w 1 xt the processx xt t 0 combined withw on a common probability space is an example of random walk in random environment we consider the quantities t d dt e x t 2 m 1t and t w d dt ew x t 2 m 1t here ew is expectation overx at fixedw and e ew dw is the expectation over bothx andw we prove the following long time tail results 1 limt td 2 t v2md 2 3 d 2 d 2 and 2 limt td 4 st w zs weakly in path space with zs s 0 the gaussian process with ezs 0 and ezrzs v2md 2 4 d d 2 r s d 2 herem and v2 are the mean and variance of w 0 under the main surprise is that fixingw changes the power of the long time tail fromd 2 tod 4 since delta t me mu 0 w 1 x 0 m 1 w 1 x t m 1 with 0 the stationary measure for the environment process our result 1 exhibits a long time tail in an equilibrium autocorrelation function

novel phenomena in lorentz lattice gases abstract in lorentz lattice gases a point particle moves along the bonds of a lattice whose lattice sites are occupied by randomly placed scatterers which scatter the particle according to certain a priori given scattering rules we consider here mainly the simplest case of a square lattice fully or partly covered with two kinds of scatterers which scatter the particle to its right or its left respectively either according to probabilistic or strictly deterministic scattering rules although the global probability of the particle to be scattered to its right or left is equal for the two scattering rules the local probability is very different leading to a completely different diffusive behavior for the two cases while the probabilistic scattering rules lead to gaussian diffusion the deterministic rules lead either to anomalous or no diffusion depending on the concentrations of the scatterers on the lattice in the first case the mean square displacement is still proportional to the time so that a diffusion coefficient can be defined but the diffusion is non gaussian while in the second case the mean square displacement is bounded and all particles are trapped in a finite time in both cases this behavior is due to the existence of closed periodic orbits which in the first case can be of infinite extent extended closed orbits while in the second case they can only be of finite extent the lengths of the extended closed orbits show for a lattice fully occupied equally by right and left scatterers a scaling behavior identical to that of percolation clusters at the bond percolation threshold of a related bond percolation problem for special concentrations on a not fully occupied lattice the same scaling behavior is found leading to two critical lines but no known related percolation like problem exists in this case this critical behavior of the closed orbits in anomalous diffusion appears to obtain for all two dimensional lattices studied so far a number of extensions of the above results are indicated

on the cellular automata approach to the 1d lorentz gas the one dimensional stochastic lorentz gas is studied in continuous space and in the cellular automata approach it is shown that the cellular automata method leads to extra oscillations of velocity autocorrelation the oscillations arise as a result of multiple reflections the differences between the models become unimportant when the product of the concentration of scatterers and the probability of reflection is small

on the blow up of heat flow for conformal 3 harmonic maps using a comparison theorem chang ding and ye 1992 proved a finite time derivative blow up for the heat flow of harmonic maps from d 2 a unit ball in r 2 to s 2 a unit sphere in r 3 under certain initial and boundary conditions we generalize this result to the case of 3 harmonic map heat flow from d 3 to s 3 in contrast to the previous case our governing parabolic equation is quasilinear and degenerate technical issues such as the development of a new comparison theorem have to be resolved

comment on on the complete integrability of the hirota satsuma system the authors show that the aforementioned paper by chowdhury and mukherjee ibid vol 17 p l231 1984 is false

a kinetic theorist s look at lattice gas cellular automata abstract the diffusion behavior of a number of lorentz lattice gases is studied in its dependence on collision rules and its similarity to corresponding lorentz gases that are continuous in space but have the same discrete velocity space the difference in behavior resulting from probabilistic or deterministic collision rules is discussed

percolation and motion in a simple random environment the authors present a particularly simple model of deterministic classical motion in a two dimensional random environment as the parameters of the model are varied a transition occurs from all trajectories being localised to some being extended they construct a mean field theory for this transition and relate the model exactly to percolation models in particular parameter ranges they point out that it is a member of a new class of site percolation analogues of direct bond percolation

deterministic lattice lorentz gas i chiral model the author studies diffusion by using a specific type of cellular automaton where particles move in an environment of scatterers the deterministic collision rules used in the model give rise to certain difficulties related to nonergodicity the class of models introduced by gunn and ortuno 1985 also contains the model studied by gates 1972 previously developed theory including the boltzmann approximation the ring and repeated ring approximations and the effective medium approximation ema is generalized for this model the validity is further investigated using computer simulations in the presence of reflecting scatterers ema yields low density results that show the breakdown of the boltzmann equation while they agree quantitatively with the low density simulations ema also agrees with a phase transition that occurs for some parameter choices

propagating staggered waves in cellular automata fluids the majority of lattice gases have apart from the physical conserved quantities of particle number momentum and energy spurious ones usually staggered in space and time at the level of linear excitations these staggered modes may be purely diffusive or damped propagating waves in the eight and nine bit model on the square lattice one finds a large number of new spurious modes and the authors derive green kubo relations for the diffusivities and damping constants and calculate them in the boltzmann approximation

probability density of the 2d percolation cluster perimeter the authors study the form of the probability density p r t of the end to end distance r of a segment of length t of the perimeter of the 2d percolation cluster at criticality they find that p r t behaves as p r t varies as t 2dh exp r t1dh delta where delta 1 1 1 dh the exponent dh 7 4 is the fractal dimension of the hull the external perimeter of a percolation cluster this result fits very well with recent numerical data

random walks in a one dimensional levy random environment we consider a generalization of a one dimensional stochastic process known in the physical literature as levy lorentz gas the process describes the motion of a particle on the real line in the presence of a random array of marked points whose nearest neighbor distances are i i d and long tailed with finite mean but possibly infinite variance the motion is a continuous time constant speed interpolation of a symmetric random walk on the marked points we first study the quenched random walk on the point process proving the clt and the convergence of all the accordingly rescaled moments then we derive the quenched and annealed clts for the continuous time process

nash estimates and the asymptotic behavior of diffusions afin d analyser le comportement asymptotique d une particule diffusant dans un champ de derive a partir d un potentiel borne lisse on developpe des estimations a priori du type de nash sur la densite de transition du processus

deterministic lattice lorentz gas ii mirror model for pt i see ibid vol 24 p 787 806 1991 as a model for diffusion a lattice version of the lorentz gas is studied using specular collision rules a previously developed theory containing the molecular chaos approximation several ring approximations and the effective medium approximation ema is applied and compared with simulations the correction to the boltzmann equation result resulting from effective medium approximation ema is opposite to the simulation results this indicates that other diagrams than ring diagrams should be considered which is peculiar if one considers the successes of the ema for other models this is investigated by using variants of the model the diffusion tensor is in general nonisotropic the corresponding features are qualitatively described by theory

langton s flies langton s ant is an automaton defined over a two dimensional grid its movement is governed by the environment in a very simple way it turns to the left over white vertices and to the right over black vertices this definition only applies on a two dimensional space we look for generalizations of this automaton to n dimensional lattices remembering the different ways through which the ant was originally defined we consider two approaches the first comes from physics lattice gas and the second from artificial life virtual ants two generalizations are proposed defining two families of dynamical systems from the physics point of view the ant is seen as a particle and hence it has no internal state other than its velocity from the artificial life point of view the ant is viewed as an insect and it has an orientation in space which we represent by an orthogonal basis this constitutes the ant s internal state this formulation allows us to define the ant s behaviour without drawing upon any information relative to the global system of external coordinates each model yields different sets of rules with distinctive behaviours we characterize all the possible rules satisfying some basic restrictions we found that many rules produce trajectories which are restricted to a diagonal plane and are equivalent to a version of langton s ant over a two dimensional grid squared or hexagonal in the particle model only two of them use the whole space and it is shown that such rules do not admit periodical trajectories this result reinforces a previous one reported by leonid bunimovich who states that the skeleton of any bounded trajectory cannot contain any three dimensional polyhedron

hull percolation and standard percolation the author shows that the hull percolation problem considered recently by roux et al 1988 is in one to one correspondence with standard bond percolation their random tiling of the plane by squares with circles is identical with the baxter kelland wu polygon decomposition associated with clusters of the q state potts model which in the q to 1 limit gives percolation

diffusion in lattice lorentz gases nearest scatterers approximation abstract the expressions for diffusion coefficients and for velocity autocorrelation functions of lattice lorentz gases are derived both in the nearest scatterers and boltzmann approximations the results are obtained for linear chain square triangular simple cubic body centred cubic face centred cubic and face centred hyper cubic lattices the diffusion coefficients are compared with those from the effective medium approximation for the square lattice and with computer simulation results for triangular simple cubic and body centred cubic lattices

long time tails in lattice lorentz gases we consider a lorentz gas on a square lattice with a fraction c of scattering sites the collision laws are deterministic fixed mirror model or stochastic with transmission reflection and deflection probabilities and respectively if all mirrors are parallel the mirror model is exactly solvable for the general case a self consistent ring kinetic equation is used to calculate the longtime tails of the velocity correlation function 0 t and the tensor correlation q 0 q t withq x y both functions showt 2 tails as opposed to the continuous lorentz gas where the tails are respectivelyt 2 andt 3 inclusion of the self consistent ring collisions increases the low density coefficient of the tail in 0 t by 30 100 as compared to the simple ring collisions depending on the model parameters

diffusion and propagation in triangular lorentz lattice gas cellular automata the diffusion process of point particles moving on regular triangular and random lattices randomly occupied with stationary scatterers a lorentz lattice gas cellular automaton is studied for strictly deterministic scattering rules as a function of the concentration of the scatterers in addition to the normal and various kinds of retarded diffusion found before on the regular square lattice straight line propagation through the scatterers is observed

crossover dynamics in bond disordered lattices the asymptotic dynamics of the percolation model for a bond disordered lattice is studied the velocity autocorrelation function vacf is investigated for arbitrary concentration of disorder in two and three dimensions using an effective medium approximation ema corrections to the long time tails away from the percolation threshold and to the percolation tails at the threshold are calculated a characteristic time scale for the long time tails is identified and found to diverge at the threshold sufficiently close to the threshold the two types of asymptotic dynamics can be identified clearly for times greater than and less than this characteristic time respectively an approximate scaling of the ema equation is obtained near the threshold for investigation of the crossover region more generally the ema equation is solved numerically for arbitrary concentration in two dimensions to exhibit the complete time dependence of the vacf in all domains near and far from the threshold

kinetic growth walks and trails on oriented square lattices hull percolation and percolation hulls the authors have studied kinetic growth walks on the manhattan lattice and kinetic growth trails on the l lattice they find that the directional properties of these lattices prevent the kinetic growth walks from being trapped except at the origin as a consequence kgws are found to be in the skw universality class for these lattices extensive numerical calculations support this fact moreover it has been shown that the recently introduced hull percolation is equivalent to the kinetic growth walks on the manhattan lattice and the bond percolation hull on the square lattice at the percolation threshold is equivalent to the kinetic growth trails on the l lattice their numerical estimates agree nicely with the original estimates and thus support the equivalence

new universality classes in percolative dynamics we study a site analogue of directed percolation random trajectories are generated and their critical behavior is studied the critical behavior corresponds to that of simple percolation in some of the parameter space but elsewhere the exponents reveal new universality classes as a byproduct we use the model to make an improved estimate of the percolation hull exponents and to calculate the site percolation probability for the square lattice

topological dynamics of flipping lorentz lattice gas models we study the topological dynamics of the flipping mirror model of ruijgrok and cohen with one or an infinite number of particles in particular we prove the topological transitivity and topological mixing up to a natural first integral for the one particle model

cellular automaton fluids 1 basic theory continuum equations are derived for the large scale behavior of a class of cellular automaton models for fluids the cellular automata are discrete analogues of molecular dynamics in which particles with discrete velocities populate the links of a fixed array of sites kinetic equations for microscopic particle distributions are constructed hydrodynamic equations are then derived using the chapman enskog expansion slightly modified navier stokes equations are obtained in two and three dimensions with certain lattices viscosities and other transport coefficients are calculated using the boltzmann transport equation approximation some corrections to the equations of motion for cellular automaton fluids beyond the navier stokes order are given