Movement And Distribution-patterns Of Random Particles In Complex Environments

Soft matter physics is the focus frontier of present condensed matter physics,which is an intersecting subject of physical, chemical and biological sciences.The spaces in which random particles move and distribute are named asexterior environments. Normal environments are isotropic, uniform (regular) andunchangeable. Many physical, chemical, biological processes and even socialphenomena are related to stochastic processes in nature. Some simple stochasticphenomena can be described by the movement and distribution patterns of randomparticles in simple normal environments. For example, the pure random walk (RW)can be used to describe the diffusion of pollens microspores in water (Brownianmotion);the self-avoiding walk (SAW) can be used for investigating the statisticalproperties of polymers in dilute solution;the diffusion-limited aggregation (DLA)model can be used to describe the patterns of Zn-electrochemistry deposits.However, many stochastic phenomena are very complex, which can not bedescribed by the RW, SAW and DLA models mentioned above. In these situations,the system environments are changeable, nonuniform, or anisotropic, andsometimes the interactions between the random walkers and their surroundingsmust be considered. We regard these complex phenomena or processes as a topicof the movements and distribution patterns of random particles in complexenvironments. By using Monte Carlo (MC) methods, we have developed themethods, models and theoretical expressions of the movements and distributionpatterns of random particles in the complex anisotropic, network, and changeableenvironments. The main results of this thesis have been written as 13 papers, 10 ofwhich have been published in the international scientific journals Phys. Rev. Lett.or Phys. Rev. E, etc. They are summarized as follows.1. Movement of random particles in anisotropic environmentsThe standard RW, with isotropic diffusion, is a powerful tool for studyingseveral physical processes such as diffusion, transportation, aggregation, structureformation, and diffusion controlled reactions. Recently, the RW with anisotropy,which results from the effects of anisotropic environments, have extensivelyattracted a great deal of attention. This type of anisotropic systems is verycommon in nature, such as the porous reservoir rocks, the epoxy-graphite diskcomposites, and so forth. Considering these anisotropic environments and thecontinuous walking space, the investigation on diffusion with directed motion in atwo-dimensional continuous space is completed by using the model of thecontinuous directed random walks (CDRW). The average square end-to-enddistance R 2 (t )∝ t2ν is calculated. The results show that this type of walksbelongs asymptotically to the same class (ν =1.0) as the ballistic motions. Forshort time, we observe a crossover from the behavior of purely random walks(ν =0.5) to that of ballistic motions (ν =1.0). The dependence of the crossoveron the direction parameter θ is also studied. There exists a scaling relation of theform < R 2 (t)>～ tf (t /θ?2). The return probability P0 0 (t) is also investigatedand the scaling form similar to < R 2 (t)> is obtained.2. Movement and aggregation of random particles in network environmentsThe regular lattice (environment) has been chosen as the topology ofinnumerable physical models such as Ising model, percolation, SAW, etc. In thisregular lattice, each site is always related to its neighboring ones. Correspondingly,the walker jumps from the current site to its one of nearest neighboring sites in themovement. However, many real environments are more complex, and rarely fallinto the case of simple regular lattice. These complex environments can bedescribed by the network. In fact, any complex system in nature can be modeledas a network, where vertices are the elements of the system and the links representthe interactions or relations between them. Thus, we studied the aggregation anddistribution patterns of random particles in the network environment.A model of particle-cluster aggregation on a small-world network ispresented in two dimensions. The model is characterized by two parameters: theclustering exponent α and the long-range connection rate φ . The results showthat as the parameters α and φ vary there exists a continuous crossover fromthe DLA-like growth to dense growth. Correspondingly, the fractal dimension ofclusters changes from D f≈1.67 to D f≈2.0. We also predict the appearanceof this crossover by means of a theoretical analysis. For a given φ a clusteringtransition is also observed from the dense growth to the DLA-like one at thecritical point α c≈2. The present change of the cluster-patterns results from theappearance of random long-range connections in the network environment, whichcan remove the screening effects during the aggregation of particles. By means ofa simple analysis, we obtain a approximate expression of the fractal dimensionD f as a function of α and φ . The present model first introduces the notion ofsmall-world networks into investigate the aggregation of particles in fractalgrowth, which may be used to other fields in condensed matter physics.A cellular automation model of the ``game of life'' on a two-dimensionalsmall-world network is presented in order to count in long-range interactionsamong living individuals in social or biological systems. The density of life andits fluctuation are calculated, respectively. The present model exhibits anonequilibrium phase transition from a ``inactive-sparse'' state to a ``active-dense''one at a certain intermediate value of the network disorder. Employing thefinite-size scaling analysis, we estimate the location of the critical point withφ c( ∞)≈0.3685. The transition is of the ``second-order'' type with power-lawdiverging length. We obtain the critical exponents 1 /ν=1.70(5), β =0.50(8), andβ / ν=0.85(13). The calculated results indicate that the present model may belongto the universality class of directed percolation. From a practical point of view, thepresent critical transition has a useful guidance for building a network. Since thelong-range connection usually costs more than the local one, it is advantageous toobtain the value of φ c in advance above which the individuals in the system havea fluctuating high density, so that one can establish a high-quality network withleast consumption of resources.3. Movement and distribution of particles in changeable environmentsIn previous interacting walk models, the walking environment of randomwalkers is unchangeable, and the influence of the environment is simplyconsidered as the attractive or repulsive effects of visited sites on the walker, orthe effect on the walker by a phenomenological potential. In general, the walkingenvironment is changeable, and the interaction between the walker andenvironment can be described as that the walker leaves behind a trail at the visitedsite and in turn the trail will affect the movement of the walker around this site.Thus, we studied the movement and distribution patterns of random particles inchangeable environments, and obtain some important and interesting results aboutthis type of system. As a typical example, we studied the movement anddistribution patterns of the random particle(s) in a deformable medium.A model of random walks on a deformable medium is proposed in (2+1)dimensions. The behavior of the walk is characterized by two parameters: thestability parameter β and the stiffness exponent α . The average squareend-to-end distance R 2 (t )∝ t2ν and the average number of visited sitesS (t )∝tk are calculated. As β increases, for each α there exists a criticaltransition point β c from purely random walks (ν =1/2 and k≈1) to compactgrowth ( ν =1/3 and k=2/3). The relationship between β c and α can beexpressed as β c = eα. The landscape generated by a walk is also investigated bymeans of the visit-number distribution N n(β), which is the number of sites withn visits. There exists a scaling relationship of the form N n(β)~ n ? 2 f(n/βz) atβ > βc. The present method is useful for investigating the random walks in whichthere exists interactions between the walker and its environment, such riverformation, heat-seeking missiles, ant swarms, biological evolution, and so forth.Going a step further, multiparticle random walks on a deformable mediumhave been investigated in (2+1) dimensions. The time evolution of the particledistribution is studied. The results show that the randomly distributed particles inthe beginning will be self-organized into a cluster pattern in the intermediate stage,and then return to the random distribution pattern in the late stage. Thedependence of the clustering degree on the stiffness parameter of medium α ,stability parameter of systems β and average particle density ρ 0 is alsoinvestigated. There exists a optimal clustering stability β p, at which the systemhas the strongest clustering ability and corresponds to a maximum clusteringcoefficient Γ p*. The dependence of the optimal clustering coefficient Γ p* on thestiffness α and particle density ρ 0 is obtained, and the landscape of mediumgenerated by particles is also investigated. The present work will be helpful tounderstand the motion behaviors of particles in complex multiparticle systemswith the interactions between elements and their environment.In summary, although the environment is different for different systems, thebehavior of the particle is always determined by both the randomness and theenvironment. With the increase of the effect of the environment, the behavior ofparticle motion shows a transition from a disorder state to a order one: Foranisotropic environments, with the increase of the directed field the behavior ofparticle motion appears the transition from disorder random walks to orderballistic motion;For nonuniform environments, with the increase of the networkdegree the pattern of aggregates exhibits the transition from the disorder DLAstructure to order Eden 'pie' one;For changeable environments, with the increaseof the effect of deformation the behavior of particle motion shows the transitionfrom the disorder delocalization state to order localization one.