Brownian trajectories

A related algorithm can be written also for the Brownian trajectory [10]. However, the essential difference between an algorithm for a Brownian trajectory and equation (4) is that the Brownian algorithm is not deterministic. Due to the existence of the random force, we cannot be satisfied with a single trajectory, even with pre-specified coordinates (and velocities, if relevant). It is necessary to generate an ensemble of trajectories (sampled with different values of the random force) to obtain a complete picture. Instead of working with an ensemble of trajectories we prefer to work with the conditional probability. I.e., we ask what is the probability that a trajectory being at... 

The definition of the above conditional probability for the case of Brownian trajectories can be found in textbooks [12], However, the definition of the conditional probability for the Newton s equations of motion is subtler than that. 

D[X t) is used to denote a path integral. Hence, equation (14) corresponds to a summation of all paths leading from X(0) to X t). The same expression is used for the Brownian trajectories and for Newtonian s trajectories with errors. The action is of course different in both cases. 

In the present review, the main focus of attention will be on general problems concerning the topological properties of polymers presented as Brownian trajectories. We will not investigate the connection of the topology with the concrete chemical structure of polymers. 

The survival probability method is based on calculation of the survival probability of a molecule in each step of a Brownian trajectory. The computational procedure involves initiating a large number of trajectories N) at a point on the initiation surface, with a specified orientation. The Brownian trajectory is generated by a series of translational and rotational steps. The molecule is reflected if it collides with the reaction site, and the trajectory is truncated when the molecule collides with the surface r = q. The survival probability in step k of the Brownian trajectory i is calculated as... 

Generation of the Brownian trajectories for rodlike molecules requires simulation of the anisotropic translational diffusion and rotational diffusion. The rotational and translational diffusion are coupled in this case, however, taking a sufficiently small time step enables the computation of the different components... 

Figure 16 Projection of Brownian trajectories (in cylindrical coordinates) for rodlike molecules for different values of the dimensionless diffusivity perpendicular to the rod axis ()). The initial orientation of the molecule is along the z direction (Gupta and Khakhar [65]).

Critical objects exist in nature. Jean Perrin mentions salted soapy water and Brownian trajectories of particles suspended in a fluid. There are many others, and some of them have aroused the interest of physicists as, for example, the liquid vapour system at the critical point, the magnetic system at the Curie point, and turbulent systems in the inertial range. 

A three-dimensional lattice model (section 3.1.1) was used to simulate aggregation kinetics, in which single particles and intermediate clusters move on Brownian or linear trajectories. Initially, = 50,000 particles are placed randomly in a cubic lattice of size L = 215x 215x 215. A combined cluster is formed whenever a particle or a cluster moves to a lattice point adjacent to another particle or intermediate cluster. This model produces DLCA clusters [77] with fractal dimension around 1.8 (Brownian trajectories) and 2 (linear trajectories). A sequence of two integers is used to describe the... 

We will skip further details of this adventure story. We just need to emphasize one more thing before we get back to polymers. Since a Brownian particle moves due to collisions with molecules, its path breaks into a sequence of many very short flights and turns. In this sense, a Brownian trajectory is pretty similar to the shape of the pol Tner chains which we saw in Section 2.4 (Figure 2.6). Another obvious example of this sort is of a man who is lost in a forest, with no compass, and has no choice but to wander at random. 

Fig. 13.8 A Brownian trajectory is self-similar on average. The random walk (or freely-jointed polymer) of 10 steps was generated computationally. In the main figure, every 10 steps are shown together as a single segment there are 10 /lO = 10 of segments. In the inset, the internal structure of one segment is shown.

The availability of diffusible particles motion Brownian trajectory in reaction process [73] ... 

Meakin, P. Effect of cluster trajectories on cluster-cluster aggregation A comparison of linear and Brownian trajectories in two- and three-dimensional simulations. Ohys. Rev. A, 1984,29(2), 997-999. 

Meakin, R (1984). Computer simulation of cluster-cluster aggregation using linear trajectories results from three-dimensional simulations and a comparison with aggregates formed using Brownian trajectories. J. Colloid Interface Sci., 102, 505-512. 

The hierarchical model with linear trajectories was applied without any modification [52] two clusters are assumed to stick together when they occupy adjacent sites. Such a model is much less time-consuming than cluster-cluster aggregation with Brownian trajectories. The resulting fractal dimension is difficult to measure with precision because of the large statistical fluctuations due to the model itself it was found to be close to 1.9. 

P. Meakin, Effects of cluster trajectories on cluster-cluster aggregation a comparison of linear and Brownian trajectories in two-dimensional and 3-dimensional simulations. Phys. Rev. A 29(2), 997-999 (1984c). doi 10.1103/PhysRevA.29.997 K.-H. Naumann, H. Bunz, Van dta- Waals inbaactions between fiaetal partieles. J. Aerosol SeL 24 (SI), S181-S182 (1993). doi 10.1016AM)21-8502(93)90183-A... 

The results of off-lattice diffusion limited cluster-cluster aggregation (DLCA) simulations in two dimensions are presented in Fig. 9.26. Simulations are limited to two dimensions because it enables one to use large unit cells with many particles. Brownian trajectories are followed. Larger aggregates are generated as a result of bond formation between overlapping aggregates. One finds crossover from fractal to Euclidean behavior when a particular monomer concentration is exceeded. 

The remainder of this section is devoted to the derivation of Eq.[54]. Besides the mathematics we also define the range of applicability of simulations based on the Nernst-Planck equation. The starting point for deriving the Nernst-Planck equation is Langevin s equation (Eq. [45]). A solution of this stochastic differential equation can be obtained by finding the probability that the solution in phase space is r, v at time t, starting from an initial condition ro, Vo at time = 0. This probability is described by the probability density function p r, v, t). The basic idea is to find the phase-space probability density function that is a solution to the appropriate partial differential equation, rather than to track the individual Brownian trajectories in phase space. This last point is important, because it defines the difference between particle-based and flux-based simulation strategies. 

In the DLA model, the individual particles or clusters stochastically diffuse via Brownian trajectories towards one another and every collision between them results in formation of a larger cluster [75]. The clusters formed display a fractal morphology that depends upon the space dimension in the DLA model. In the RLA model, the probability of attachment is small and only a small fraction of collisions between clusters leads to formation of larger clusters. Traditionally, DLA and RLA are called rapid and slow aggregations, respectively. 

The probability of realizing a particular Brownian trajectory is then... 

Monomer-cluster aggregation can occur under diffusion-limited or reaction-limited conditions. Diffusion-limited monomer-cluster aggregation (DLMCA) is simulated by the Witten and Sander model [135]. (See Fig. 55.) In this model monomers are released one by one from sites arbitrarily far from a central cluster. The monomers travel by random walks and stick irreversibly at first contact with the growing cluster. Because of their Brownian trajectories, which simulate diffusion, monomers cannot penetrate deeply into a cluster without intercepting a cluster arm. The arms effectively screen the interior from the flux of incoming monomers therefore growth occurs preferentially at exterior sites resulting in mass fractal objects whose density decreases radially from the center of mass (in three dimensions, df = 2.45). 

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