percolation processes in monomer polyatomic mixtures in this paper the percolation of mixtures of monomers and polyatomic species kk mers on a square lattice is studied by means of a finite size scaling analysis the critical exponents and the scaling collapsing of the fraction of percolating lattices are found a phase diagram separating a percolating from a non percolating region is determined the main features of the phase diagram are discussed in order to predict its evolution for larger kk mer sizes

quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

random sequential addition a distribution function approach random sequential addition rsa of hard objects is an irreversible process defined by three rules objects are introduced on a surface or ad dimensional volume randomly and sequentially two objects cannot overlap and once inserted an object is clamped in its position the configurations generated by an rsa can be characterized in the macroscopic limit by a unique set of distribution functions and a density we show that these nonequilibrium rsa configurations can be described in a manner which in many respects parallels the usual statistical mechanical treatment of equilibrium configurations kirkwood salsburg like hierarchies for the distribution functions zero separation theorems diagrammatic expansions and approximate equations for the pair distribution function approximate descriptions valid for low to intermediate densities can be combined with exact results already derived for higher densities close to the jamming limit of the process similarities and differences between the equilibrium and the rsaconfigurations are emphasized finally the potential application of rsa processes to the study of glassy phases is discussed

a first passage kinetic monte carlo algorithm for complex diffusion reaction systems we develop an asynchronous event driven first passage kinetic monte carlo fpkmc algorithm for continuous time and space systems involving multiple diffusing and reacting species of spherical particles in two and three dimensions the fpkmc algorithm presented here is based on the method introduced in oppelstrup et al 10 and is implemented in a robust and flexible framework unlike standard kmc algorithms such as the n fold algorithm fpkmc is most efficient at low densities where it replaces the many small hops needed for reactants to find each other with large first passage hops sampled from exact time dependent green s functions without sacrificing accuracy we describe in detail the key components of the algorithm including the event loop and the sampling of first passage probability distributions and demonstrate the accuracy of the new method we apply the fpkmc algorithm to the challenging problem of simulation of long term irradiation of metals relevant to the performance and aging of nuclear materials in current and future nuclear power plants the problem of radiation damage spans many decades of time scales from picosecond spikes caused by primary cascades to years of slow damage annealing and microstructure evolution our implementation of the fpkmc algorithm has been able to simulate the irradiation of a metal sample for durations that are orders of magnitude longer than any previous simulations using the standard object kmc or more recent asynchronous algorithms

random cuts in binary mixtures of spheres three dimensional 3d disordered media may be studied by performing random planar cuts through the material and trying to go back to 3d geometry from two dimensional information we have studied from this point of view numerical packings of spheres of two sizes for which the radii of the spheres and the composition of the mixtures have been obtained from the cuts we have also studied the froths generated by the related laguerre voronoi tessellation of the packings and their cuts stereological relations in this specific frame are derived in both cases

edge distribution method for solving elliptic boundary value problems with boundary singularities elliptic boundary value problems are difficult to treat in the vicinity of singularities i e edges and corners of the boundary the concentration of electrical charge on the edges and corners of a conductor is perhaps the simplest example of such problems here we provide a rapid method for accurate treatment of these problems it utilizes a green s function based implementation of last passage monte carlo diffusion methods this is combined with a diffusion algorithm for the scaling of solutions to the laplace equation near a corner singularity to yield the solution of a benchmark problem the charge distribution near the edge and corner of a conducting cube

competitive random sequential adsorption of point and fixed sized particles analytical results we study the kinetics of competitive random sequential adsorption rsa of particles of binary mixture of points and fixed sized particles within the mean field approach the present work is a generalization of the random car parking problem in the sense that it considers the case when either a car of fixed size is parked with probability q or the parking space is partitioned into two smaller spaces with probability 1 q at each time event this allows an interesting interplay between the classical rsa problem at one extreme q 1 and the kinetics of fragmentation processes at the other extreme q 0 we present exact analytical results for coverage for a whole range of q values and physical explanations are given for different aspects of the problem in addition a comprehensive account of the scaling theory emphasizing dimensional analysis is presented and the exact expression for the scaling function and exponents are obtained

first and last passage monte carlo algorithms for the charge density distribution on a conducting surface recent research shows that monte carlo diffusion methods are often the most efficient algorithms for solving certain elliptic boundary value problems in this paper we extend this research by providing two efficient algorithms based on the concept of last passage diffusion these algorithms are qualitatively compared with each other and with the best first passage diffusion algorithm in solving the classical problem of computing the charge distribution on a conducting disk held at unit voltage all three algorithms show detailed agreement with the known analytic solution to this problem

on the application of substochastic semigroup theory to fragmentation models with mass loss a linear integro differential equation modelling multiple fragmentation with inherent mass loss is investigated by means of substochastic semigroup theory the existence of a semigroup is established and under natural conditions on certain coefficients the generator of this semigroup is identified this yields in particular a validation of the formal mass loss rate equation for the model

strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss abstract in the paper we shall present a survey of recent results on substochastic semigroups and provide new methods for determining their honesty these methods are applied to the fragmentation equation with mass loss yielding sufficient conditions for the existence of conservative and shattering solutions our results provide a mathematical framework that clarifies the discussion of phys rev a 43 1991 656 phys rev a 41 1990 5755 j phys a 24 1991 3967 on shattering fragmentation in such models showing among others that there occurs an unexpected mass loss associated with shattering which is not accounted for by the discrete and continuous mass loss contrary to the conjecture of phys rev a 43 1991 656 phys rev a 41 1990 5755

modelling gap size distribution of parked cars using random matrix theory abstract we apply the random matrix theory to the car parking problem for this purpose we adopt a coulomb gas model that associates the coordinates of the gas particles with the eigenvalues of a random matrix the nature of interaction between the particles is consistent with the tendency of the drivers to park their cars near to each other and in the same time keep a distance sufficient for manoeuvring we show that the recently measured gap size distribution of parked cars in a number of roads in central london is well represented by the spacing distribution of a gaussian unitary ensemble

the simulation tabulation method for classical diffusion monte carlo abstract many important classes of problems in materials science and biotechnology require the solution of the laplace or poisson equation in disordered two phase domains in which the phase interface is extensive and convoluted green s function first passage gffp methods solve such problems efficiently by generalizing the walk on spheres wos method to allow first passage fp domains to be not just spheres but a wide variety of geometrical shapes in particular this solves the difficulty of slow convergence with wos by allowing fp domains that contain patches of the phase interface previous studies accomplished this by using geometries for which the green s function was available in quasi analytic form here we extend these studies by using the simulation tabulation st method we simulate and then tabulate surface green s functions that cannot be obtained analytically the st method is applied to the solc stockmayer model with zero potential to the mean trapping rate of a diffusing particle in a domain of nonoverlapping spherical traps and to the effective conductivity for perfectly insulating nonoverlapping spherical inclusions in a matrix of finite conductivity in all cases this class of algorithms provides the most efficient methods known to solve these problems to high accuracy

random sequential adsorption of spheres of different sizes the random sequential adsorption of spheres selected from monodisperse binary and polydisperse distributions on a three dimensional substrate has been studied by computer simulations in the monodisperse case the adsorbed particle density algebraically approaches an asymptotic value of 0 384 0 001 according to feder s law feder j theor biol 87 1980 237 for the case of spheres of two different sizes the large sphere density approaches its asymptotic value exponentially while the small sphere density increases according to feder s law if the sphere radii ri are uniformly distributed over the range ra ri rb before testing for overlaps then the approach to the asymptotic density is algebraic but the effective exponent depends on the model parameter r rb ra

random sequential adsorption relationship to dead leaves and characterization of variability a new construction for the planar unbounded random sequential adsorption rsa model is presented which allows for a direct comparison with matheron s dead leaves model furthermore for the case of disks with random radii the problem of statistical determination of the proposal radius distribution is discussed finally second order characteristics related to the pair correlation function are suggested for describing the variability of the rsa disk systems

kinetics of fragmentation a general discussion of the kinetics of continuous irreversible fragmentation processes is presented for a linear process where particle breakup is driven by an external force the authors develop a scaling theory to describe the evolution of the cluster size distribution they treat the general case where the breakup rate of a cluster of mass x varies as xlambda when lambda 0 corresponding to larger clusters more likely to break up the scaled cluster size distribution phi x decays with the scaled mass x as x 2 exp axlambda as x for small mass phi x has the log normal form exp a ln2 x if the breakup kernel has a small size cutoff while phi x has a power law tail in the absence of a cutoff they also show that a conventional scaling picture applies only for the case lambda 0 for lambda 0 they develop an alternative formulation for the cluster size distribution in which the typical mass scale is determined by the initial condition in this regime they also investigate the nature of a shattering transition where mass is lost to a dust phase of zero mass particles they also study the kinetics of a nonlinear collision induced fragmentation process they analyse the asymptotic behaviour of a simple minded class of models in which a two particle collision results in either 1 both particles splitting into two equal pieces 2 only the larger particle splitting in two or 3 only the smaller particle splitting they map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations scaling is found to hold for different ranges of homogeneity index for the three models

quantitative description and numerical simulation of random microstructures of composites and their effective elastic moduli abstract a digital image processing technique is used for measurement of centroid coordinates of fibers with forthcoming estimation of statistical parameters and functions describing the stochastic structure of a real fiber composite comparative statistical analysis of the real and numerically simulated structure are performed accompanying of known methods of the generation of random configurations by the random shaking procedure allows creating of the most homogenized and mixed structures that do not depend on the initial protocol of particle generation we consider a linearly elastic composite medium which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions the multiparticle effective field method see for references buryachenko appl mech rev 2001a 54 1 47 based on the theory of functions of random variables and green s functions is used for demonstration of the dependence of effective elastic moduli of fiber composites on the radial distribution functions estimated from the real experimental data as well as from the ensembles generated by the method proposed

geometry of random sequential adsorption by sequentially adding line segments to a line or disks to a surface at random positions without overlaps we obtain configurations of the one and two dimensional random sequential adsorption rsa problem we have simulated the one and two dimensional problem with periodic boundary condition the one dimensional simulations are compared with the exact analytical solutions to give an estimate of the accuracy of the simulation in two dimensions the geometrical properties of the rsa configuration are discussed and in addition known results of the rsa process are reproduced various statistical distributions of the voronoi dirichlet vd network corresponding to the rsa disk configuration are analyzed in order to characterize pores in the rsa configuration we introduce circular holes there is a direct correspondence between vertices of the vd network and these holes and also between direct indirect geometrical neighbors and these holes the hole size distribution is found to be a parabola we also find general relations that connect the asymptotic behavior of the surface coverage the correlation function and the hole size distribution

random sequential adsorption with restructuring in two dimensions a modified version of the random sequential adsorption model for the deposition of spherical particles onto a plane is presented successive particles are added via randomly positioned vertical trajectories and if they do not stick to the plane at first contact follow the path of steepest descent on the previously deposited particles before being adsorbed all particles reaching a stable position in contact with three previously adsorbed particles are removed the limiting coverage is found to be 0 610 56 or 0 000 05 larger than that of the standard rsa model and is shown to be reached exponentially with the number of trials the size distribution of connected clusters of spheres is analysed and is found to exhibit an exponential decay for large cluster sizes

random sequential adsorption of polydisperse mixtures asymptotic kinetics and structure the authors study the asymptotic kinetics of the random sequential adsorption rsa of a mixture of particles with a continuous distribution of sizes their results provide further support for the idea that the power law exponent for the approach to the jamming limit is simply related to the number of degrees of freedom of the adsorbing species they also predict the behaviour of the mean radial distribution function in the asymptotic regime

shattering and non uniqueness in fragmentation models an analytic approach abstract fragmentation equations have features which are rather unexpected in such a relatively simple linear framework these are shattering attributed to a phase transition creating a dust of zero size particles and the existence of multiple solutions the existence of these phenomena indicates that the fragmentation equation alone does not offer a full description of the dynamics of the problem in this paper we present a survey of recent results explaining both features from a unified dynamical systems point of view we also use the developed theory to discuss a number of conjectures and observations made in the literature of the subject

random sequential adsorption of lines and ellipses infinitely many lines of zero thickness may be adsorbed onto a plane without overlap and it is shown that the number of lines adsorbed should scale as 13 where is the number of adsorption trials per unit area of surface this is confirmed by numerical simulation the adsorption of ellipses with major and minor axes 2a 2b respectively is also studied the area coverage theta approaches its maximum value theta max as alpha as to infinity where alpha is typically 1 3 as b a decreases from unity theta max first increases to a maximum value 0 58 or 0 01 when b approximately 0 5 and then decreases a feature not noted in previous work

the car parking problem at high densities abstract the radial distribution functions of random 1 d systems of sequential hard rods have been studied in the range of very high densities it is found that as the number of samples rejected before completion increases anomalies in the pairwise distribution functions arise these are discussed using analytical solutions for systems of three rods and numerical simulations with twelve rods the probabilities of different spatial orderings with respect to the sequential order are examined

comment on random sequential packing in square cellular structures based on a paper by nakamura j phys a vol 19 p 2345 1986 the authors correlate two methods for packing squares on a square lattice and find agreement with published computer simulation results

on the solution of the integro differential fragmentation equation with continuous mass loss in this paper we obtain a complete classification of all possible non trivial similarity solutions of the integro differential fragmentation equation with continuous mass loss rate these solutions include the effects of both continuous and discrete mass loss rates the similarity solutions are compared with the solutions found earlier the comparison shows that some previous solutions are obtained as special cases from our solutions the results reported here provide further evidence of the usefulness of the lie group method for obtaining similarity solutions for either differential or integro differential equations

pair correlation function in random sequential adsorption processes an analytical expression of the pair distribution function is derived for the car parking problem and for the random sequential adsorption of k mers onto a one dimensional lattice both on a lattice and in the continuum limit super exponential decay of the distribution function is observed a comparison of the spatial correlations with those at equilibrium demonstrates the influence of the irreversibility of the rsa process

a precise algorithm to detect voids in polydisperse circle packings computer simulations are the primary tool for studying polydisperse particle packings quanti tatively for the problem of packing n unequal circles in a larger container circle nothing is known a priori about the optimal packing i e the packing with the highest packing fraction simulations usually start from a random initial configuration with the aim to finish with a dense final packing unfortunately smaller circles often get stuck in trapped positions and prevent the rest of the packing from growing larger hence the knowledge of the structure of unoccupied areas or holes inside a packing is important to be able to move trapped circles into free circular places or voids a novel algorithm is proposed for detecting such voids in two dimensional arbitrary circle packings by a decomposition of the contact graph combined with a clever object jumping strategy and together with other heuristic methods like swaps and shifts this approach increases the packing fraction significantly its effectiveness for jumping across the maximally random jammed barrier mrj 0 8575 in the large n limit for small benchmark instances as well as for large problem sizes up to n 103 is demonstrated

regional monte carlo solution of elliptic partial differential equations abstract a continuous random walk procedure for solving some elliptic partial differential equations at at a single point is generalized to estimate the solution everywhere the monte carlo method described here is exact except at the boundary in the sense that the only error is the statistical sampling error that tends to zero as the sample size increases a method to estimate the error introduced at the boundary is provided so that the boundary error can always be made less than the statistical error

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

monte carlo methods for calculating some physical properties of large molecules in this paper we describe monte carlo methods for solving some boundary value problems for elliptic partial differential equations arising in the computation of physical properties of large molecules the constructed algorithms are based on walk on spheres green s function first passage walk in subdomains techniques and finite difference approximations of the boundary condition the methods are applied to calculating the diffusion limited reaction rate the electrostatic energy of a molecule and point values of an electrostatic field

generalized random sequential adsorption in one dimension the process of random sequential adsorption in one dimension is generalized so that the rate at which a particle is placed in an empty space is dependent on the size of the space being destroyed and on the sizes of the two spaces created on either side of the particle we write down the equation for this process involving p z the probability of choosing to deposit a particle of size z and f x y mod z the rate at which spaces of length x and y are created from a space of length x y z given that a particle of size z is deposited a scaling theory is given for this process and solutions are presented for two special cases the results of a numerical simulation of a third case are presented

similarity solutions for fragmenting systems with continuous mass loss a linear rate equation describing fragmentation processes which includes both continuous and discrete loss of mass is reduced to a partial differential equation different types of similarity solutions are obtained by applying lie s similarity method

on the similarity solution of the fragmentation equation an analysis of the similarity solution of the linear homogeneous fragmentation equation is given for a general volume conserving daughter fragment distribution for the special case of a polynomial daughter distribution of degree p an exact basic similarity solution for the time evolution of the particle volume distribution is derived the solution is proportional to the meijer g function which may be represented as a linear combination of generalized hypergeometric functions the properties of generalized hypergeometric functions and g functions are given in the special function literature and include the continuation of the solution to large values of the similarity variable which is given here for special cases

on the breakage problem with a homogeneous erosion type kernel the breakage equation with a homogeneous erosion type kernel is studied herein this type of kernel renders the handling of the breakage equation by conventional techniques very difficult necessitating alternative problem solving approaches exploiting the structure of the erosion breakage kernel a new particle erosion equation is derived as the first order term of a formal perturbation expansion with respect to kernel parameters however even this new equation is very difficult to treat because of the multimodality of its solution associated with the developing generations of fragments in order to overcome this difficulty the problem is decomposed into a system of equations for the size distribution of the generations of fragments which admits unimodal solutions the properties and the methods of solution analytical method of moments etc are studied extensively using solution techniques developed in this paper results are reported for some simple cases revealing a very interesting and rather unusual structure of the solutions of the erosion breakage equation pacs number 0590

scaling properties of the number of random sequential adsorption iterations needed to generate saturated random packing the properties of the number of iterations in random sequential adsorption protocol needed to generate finite saturated random packing of spherically symmetric shapes were studied numerical results obtained for one two and three dimensional packings were supported by analytical calculations valid for any dimension d it has been shown that the number of iterations needed to generate finite saturated packing is subject to pareto distribution with exponent 1 1 d and the median of this distribution scales with packing size according to the power law characterized by exponent d obtained results can be used in designing effective random sequential adsorption simulations

packing of spheroids in three dimensional space by random sequential addition packings of spheroidal particles with semi axes of length a b b were generated by random sequential addition rsa simulations as the jamming limit is approached the volume fraction occupied by particles tends towards an asymptote which was determined as a function of aspect ratio this asymptote has a local minimum at spheres with local maxima at prolate spheroids and oblate values of agree with results from rsa simulations of sphere packings but lie below volume fractions obtained in simulations of near spheres packed under gravity volume fractions reported for simulations of spheroids packed under gravity vary widely when the aspect ratio is very large or small differences between these results and the predictions of rsa are discussed

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

pattern formation in size polydisperse adsorption fractal properties and transition to order abstract random sequential adsorption of particles with size polydispersity prsa is investigated for the prsa with a power law small size distribution p r r 1 we reveal the fractal nature of the arising patterns the fractal dimension d is found to decrease from 2 to d a 1 305 the fractal dimension of the apollonian packing as increases from 0 to we examine prsa by a combination of exact scaling mean field and numerical approaches we find that the scaling theory works fairly well for the whole range of while the mean field theory mft is applicable only for small we attribute this failure of the mft to the increasing regularity of the prsa patterns with increasing we confirm this conclusion by the direct measurement of the entropy production rate of the pattern formation process which demonstrates the transition from an irregular regime of the pattern formation to the regular one

monte carlo simulation of the percolation process caused by the random sequential adsorption of k mers on heterogeneous triangular lattices abstract mixed site bond percolation is studied for a random sequential adsorption process of k mers on heterogeneous lattices with variable connectivity by means of monte carlo simulation the percolation phase diagrams are built to render evidence about complex structures critical exponents are also calculated to show that the universality class corresponding to ordinary percolation in two dimensions is preserved

probability distribution function for the random sequential adsorption of hard disks abstract the distribution of the number of hard core objects deposited in a finite region through a random sequential adsorption process is analyzed three idealized model probability distributions for this problem are tested by comparing them with computer simulation distributions for the random sequential adsorption of hard disks on a flat surface a hypergeometric distribution where the effects of mutual exclusion between particles are incorporated by a coverage dependent effective volume per particle is shown to give best agreement with simulation results this effective volume is derived in terms of the fluctuations in the particle number finite size effects in the fluctuations are taken into account

last passage monte carlo algorithm for mutual capacitance we develop and test the last passage diffusion algorithm a charge based monte carlo algorithm for the mutual capacitance of a system of conductors the first passage algorithm is highly efficient because it is charge based and incorporates importance sampling it averages over the properties of brownian paths that initiate outside the conductor and terminate on its surface however this algorithm does not seem to generalize to mutual capacitance problems the last passage algorithm in a sense is the time reversal of the first passage algorithm it involves averages over particles that initiate on an absorbing surface leave that surface and diffuse away to infinity to validate this algorithm we calculate the mutual capacitance matrix of the circular disk parallel plate capacitor and compare with the known numerical results good agreement is obtained

the nonlinear fragmentation equation we study the kinetics of nonlinear irreversible fragmentation here fragmentation is induced by interactions collisions between pairs of particles and modelled by general classes of interaction kernels for several types of breakage models we construct initial value and scaling solutions of the fragmentation equations and apply the non vanishing mass flux criterion for the occurrence of shattering transitions these properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters

lattice free models of directed cell motility directed cell migration often occurs when individual cells move in response to an external chemical stimulus cells can respond by moving in either the direction of increasing chemoattraction or decreasing chemorepulsion concentration many previous models of directed cell migration use a lattice based framework where agents undergo a lattice based random walk and the direction of nearest neighbour motility events is biased in a preferred direction such lattice based models can lead to unrealistic configurations of agents since the agents always move on an artificial lattice structure which is never observed experimentally we present a lattice free model of directed cell migration that incorporates two key features first agents move on a continuous domain with the possibility that there is some preferred direction of motion second to be consistent with experimental observations we enforce a crowding mechanism so that motility events that would lead to agent overlap are not permitted we compare simulation data from the new lattice free model with a more traditional lattice based model to provide additional insight into the lattice free model we construct an approximate conservation statement which corresponds to a nonlinear advection diffusion equation in the continuum limit the solution of this mean field model compares well with averaged data from the individual based model

exact self similar solutions to the fragmentation equation with homogeneous discrete kernel abstract new large time asymptotic self similar solutions of the fragmentation breakage equation are derived in closed form for the case of certain homogeneous discrete fragmentation kernel a fragmentation kernel is called homogeneous and discrete if it allows only certain ratios between the size of the fragments and that of parent particle in this case the fragmentation kernel is not a continuous function the present solutions are the first explicit self similarity distributions for discontinuous breakage functions

theoretical analysis of the steady state particle size distribution in limited breakage processes the steady state particle size distribution is examined resulting from a breakage process with a maximum stable size and a homogeneous continuous kernel the dynamic breakage problem is transformed into one that allows direct solutions for the steady state distribution the latter depends on the breakage kernel and on the ratio of critical to initial size as this ratio goes to zero the steady state distribution approaches its limiting form obtained by the authors previously 7 a general theoretical analysis concerning the steady state distribution is presented herein the asymptotic behaviour is determined with regard to various limits perturbation analysis for nearly uniform kernels reveals several interesting features of the problem for the general continuous kernel the problem can be cast in a matrix form amenable to a conventional theoretical treatment finally comparisons of the new results with existing solutions of the dynamic problem for large times confirm their validity

general solutions and scaling violation for fragmentation with mass loss explicit solutions of a linear rate equation lead to improved understanding of fragmenlatian with discrete and continuous mass loss discrete mass loss provides a general expression for the overall mass 10 rate which accounts for mass loss in the shattering regime where runaway fragmentation for small panicles producer a phase of zero mass panicles an explicit solution far the recession regime where small panicles lose large fractionsoftheir mass to surface recession shows that the total number ofpaniclesincreases with time until the typical panicles become small enough to lose all of their mass and disappear rather than to break a general series solution is presented for fragmentation with continuous and discrete mass loss a proof that continuous mass loss precludes dynamic scaling is presented

a monte carlo method for poisson s equation abstract this investigation presents an analysis of a monte carlo method for estimating local solutions to the dirichlet problem for poisson s equation the probabilistic algorithm consists of a modified walk on spheres that includes the effects from internal sources as part of the random process a new derivation of the asymptotic expressions for the rate of convergence and average runtime of the algorithm is presented these estimates are used to compare the monte carlo method with discrete difference schemes numerical experiments involving some two dimensional problems confirm the efficiency of the probabilistic scheme

equilibrium versus random sequential addition of dimers on a lattice the statistical mechanical properties of particles forming dimers are compared with the results obtained on a two dimensional lattice through a random sequential addition rsa experiment the relation between the most likely fraction of dimers filling the lattice at equilibrium in the disordered phase and the corresponding non equilibrium jamming limit attained through rsa is explored some implications relevant for the physics of the liquid state are also discussed

a multitype random sequential process a random sequential process rsp can be formulated in two different ways i a linear array of n identical compartments is ceaselessly bombarded by particles which occupy 2 contiguous lattice sites these so called bell particles are assumed to make contact in a spatially random manner and to stick only if striking adjacent vacant compartments ii a linear array of n identical compartments is sequentially filled by bell particles the occupation process being selective i e always being directed to vacant sequences of compartments of lengths at least dynamics of rsp s has always been formulated in either way and in the case of lattice spaces all the efforts have been addressed so far to allow in the space filling process just one kind of particle the aim of this paper is to remove that restriction and to consider a lattice space filling problem with various kinds of particles involved the situations of the two above described models making no difference in the one type case

coagulation and fragmentation with discrete mass loss a nonlinear integro differential equation that models a coagulation and multiple fragmentation process in which discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators under the assumptions that the coagulation kernel click to view the mathml source is bounded and the fragmentation rate function a satisfies a linear growth condition global existence and uniqueness of solutions that lose mass in accordance with the model are established in the case when no coagulation is present and the fragmentation process is governed by power law kernels an explicit formula is given for the substochastic semigroup associated with the resulting mass loss fragmentation equation

a study of the nonlinear breakage equation analytical and asymptotic solutions new solutions of the nonlinear collisional breakage equation are given using analytical and asymptotic methods the dynamic nonlinear breakage equation is transformed to a linear one for some simple forms of the collision kernel methods for treating the linear equation are employed to obtain solutions for the nonlinear case furthermore it is shown that under particular conditions the particle size distribution can take asymptotically a self similar form i e the shape of the appropriately normalized distribution is independent of time the self similar distribution is obtained from the solution of a double nonlinear integral equation the latter is solved in closed form and numerically after transformation to a boundary value problem for simple forms of the collision and breakage kernels results for the self similar distribution are presented and discussed

combination of the meshless finite difference approach with the monte carlo random walk technique for solution of elliptic problems abstract this paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations it is based upon the well known monte carlo mc method with a random walk rw technique carried out on the grid of points this method allows for accurate estimation of the solution of the differential equation at selected point s of the domain and or its boundary it extends the standard formulation of the monte carlo random walk mc rw approach by means of its appropriate combination with the meshless version of the finite difference method in this manner the proposed approach may deal with elliptic equations in more general non homogeneous form as well as boundary conditions of both essential and natural types moreover arbitrarily irregular clouds of nodes may be used with no a priori imposed nodes structure therefore the meshless mc rw approach may be applied to the significantly wider class of problems with more complex geometry this concept was examined on variety of 2d boundary value problems selected numerical results are presented and discussed a simple matlab code is included as well

effect of packing method on the randomness of disc packings the randomness of disc packings generated by random sequential adsorption rsa random packing under gravity rpg and mason packing mp which gives a packing density close to that of the rsa packing has been analysed based on the delaunay tessellation and is evaluated at two levels i e the randomness at individual subunit level which relates to the construction of a triangle from a given edge length distribution and the randomness at network level which relates to the connection between triangles from a given triangle frequency distribution the delaunay tessellation itself is also analysed and its almost perfect randomness at the two levels is demonstrated which verifies the proposed approach and provides a random reference system for the present analysis it is found that i the construction of a triangle subunit is not random for the rsa mp and rpg packings with the degree of randomness decreasing from the rsa to mp and then to rpg packing ii the connection of triangular subunits in the network is almost perfectly random for the rsa packing acceptable for the mp packing and not good for the rpg packing packing method is an important factor governing the randomness of disc packings

the random walk on the boundary method for calculating capacitance in this paper we present the random walk on the boundary method for the rapid solution of integral equations that arise in electrostatics and related areas this method is a monte carlo method based on the construction of a markov chain that is readily interpreted as a random walk along the boundary over which integration in the integral equation is taken to illustrate the usefulness of this technique we apply it to the computation of the capacitance of the unit cube obtaining the capacitance of the cube usually requires computing the charge density and this problem has been used as a benchmark by many in the field for algorithms of this kind here the random walk on the boundary method does not require charge density computation and obtains the capacitance of the cube within a statistical error of 2 7 10 7 the most accurate estimate to date

random sequential adsorption series expansion and monte carlo simulation abstract random sequential adsorption is an irreversible surface deposition of extended objects in systems with continuous degrees of freedom coverage follows a power law t j ct where the exponent depends on the geometric shape symmetry of the objects lattice models give typically exponential saturation to jamming coverage we discuss how such function t can be computed by series expansions and analyzed with pade approximations we consider the applications of efficient monte carlo computer simulation method event driven method to random sequential adsorptions with high precision and at very long time scale

kinetics of random sequential parking on a line we study the kinetics of irreversible random sequential parking of intervals of different sizes on an infinite line for the simplest fixed length parking distribution the model reduces to the known car parking problem and we present an alternate solution to this problem we also consider the general homogeneous case when the parking distribution varies asx 1 atx 1 with the lengthx of the filling interval we develop a scaling theory describing such mixture deposition processes and show that the scaled hole size distribution with xtz a scaling variable decays with the scaled mass as exp const 1 as we determine scaling exponentsz and and find that at large times the coverage t has a power law form 1 t t v with nonuniversal exponent 2 1 depending on the homogeneity index

pseudospectral laguerre approximation of transport fragmentation equations we develop a laguerre type pseudo spectral scheme for solving transport fragmentation equations the method converges rapidly for certain type of fragmentation problems and works under mild restrictions on the growth rate of the coefficients of the equation rigorous stability and convergence analyses are provided numerical simulations illustrate the performance of the scheme

modelling the gap size distribution of parked cars abstract we have measured the distribution of distances between parked cars in a number of roads in central london we compare the results with models of random sequential adsorption or random car parking models as they are often called our empirical results do not agree with these models and hence we introduce alternative models for the parking process the first model we propose is one where cars can only park if the space is greater than or equal to a chosen minimum size where the extra space can be used for manoeuvering the results of this model suggest that in addition to a deposition process a re positioning process is required to explain our empirical results hence we introduce a further model where at each time step with probability p a car is randomly placed on a line if the space is empty and with probabilty 1 p is placed in a space but then rolls to the nearest parked car so that it is y distance from the nearest car with probability f y we concentrate on two specific probabilities f y and find that for one case the gap size distribution agrees with the empirical results

non trivial exponents in the zero temperature dynamics of the 1d ising and potts models we consider the glauber dynamics of the q state potts model in one dimension at zero temperature starting with a random initial configuration we measure the density r t of spins which have never flipped from the beginning of the simulation until time t we find that for large t the density r t has a power law decay left r t sim t theta right where the exponent theta varies with q our simulations lead to theta simeq 37 for q 2 theta simeq 53 for q 3 and theta longrightarrow 1 as q longrightarrow infty

similarity solution for fragmentation with volume change the similarity form of the solution of the linear homogeneous equation for fragmentation with volume change is given the solution shows the time evolution of the particle volume distribution when a particle splits into a polynomial distribution of fragment volumes the limits for small and large values of the similarity variable are derived the long time limit for all values of the similarity variable is given and the characteristic time to approach the limit is identified the solution shows the effect of volume change on the particle volume distribution and on the time dependence of the moments of the distribution

random sequential addition of hard spheres in high euclidean dimensions sphere packings in high dimensions have been the subject of recent theoretical interest employing numerical and theoretical methods we investigate the structural characteristics of random sequential addition rsa of congruent spheres in d dimensional euclidean space mathbb r d in the infinite time or saturation limit for the first six space dimensions 1 ensuremath le d ensuremath le 6 specifically we determine the saturation density pair correlation function cumulative coordination number and the structure factor in each of these dimensions we find that for 2 ensuremath le d ensuremath le 6 the saturation density ensuremath phi s scales with dimension as ensuremath phi s c 1 2 d c 2 d 2 d where c 1 0 202 phantom rule 0 2em 0ex 048 and c 2 0 973 phantom rule 0 2em 0ex 872 we also show analytically that the same density scaling is expected to persist in the high dimensional limit albeit with different coefficients a byproduct of this high dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by ensuremath phi s ensuremath ge d 2 1 ensuremath s 0 2 d 1 where s 0 0 1 is the structure factor at k 0 i e infinite wavelength number variance in the high dimensional limit we demonstrate that a pal asti type conjecture the saturation density in mathbb r d is equal to that of the one dimensional problem raised to the d th power cannot be true for rsa hyperspheres we show that the structure factor s k must be analytic at k 0 and that rsa packings for 1 ensuremath le d ensuremath le 6 are nearly hyperuniform consistent with the recent decorrelation principle we find that pair correlations markedly diminish as the space dimension increases up to six we also obtain kissing contact number statistics for saturated rsa configurations on the surface of a d dimensional sphere for dimensions 2 ensuremath le d ensuremath le 5 and compare to the maximal kissing numbers in these dimensions we determine the structure factor exactly for the related ghost rsa packing in mathbb r d and demonstrate that its distance from hyperuniformity increases as the space dimension increases approaching a constant asymptotic value of 1 2 our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions