Random walks trajectories

Particles are transported via random walk trajectories. When a particle is adjacent to the seed or other bound particles it adheres and the next particles is launched Perimeter sites surrounding a seed particle (and subsequently around the aggregate) are filled based on a probability function... 

Particles are transported along linear trajectories Particles adhere irreversibly to the aggregate upon contact N number of particles diffuse along random walk trajectories and aggregate upon collison Simulation ends with a single aggregate composed of N primary particles... 

Solution is typically by finite-difference scheme Approximate Brownian motion by a random walk trajectory... 

Figure 10. The entropy for an uncorrelated Gaussian random process generating random walk trajectories calculated using DEA is graphed versus the natural logarithm of the time. The data indicate a linear relationship between S(t) and In t as predicted by Eq. (91).

Figure 33.2 Two vector models expressing the random walk (trajectory) of diffusion, (a) Sequence of position vectors on each...

Polva s Recurrence Theorem dictates one of the most important differences between 7.B and AB. It states that every random-walk trajectory in one or two dimensions passes through every point in space but that this is not true in three dimensions 17 J.SO. The Recurrence Theorem is the subject of a famous (among mathematicians) joke. 

Fig. 7.3.1. u(0) and u(/) are unit vectors representing the orientation angles of the symmetry axis of a cylindrically symmetric molecule at times 0 and t, respectively. The locus of all the possible vectors u(f) is the surface of a sphere of unit radius (a unit sphere). The reorientation of the molecule can be regarded as a trajectory on the surface of the unit sphere. A random walk trajectory gives rise to rotational diffusion. 

The condition of self-avoidance of a random walk trajectory on //-dimensional lattice demands the step not to fall twice into the same cell. From the point of view of chain link distribution over cells it means that every cell cannot contain more than one chain link. Chain links are inseparable. They cannot be tom off one from another and placed to cells in random order. Consequently, the numbering of chain links corresponding to wandering steps is their significant distinction. That is why the quantity of different variants of iV distinctive chain links placement in Z identical cells under the condition that one cell cannot contain more than one chain link is equal to Z I Z-N) ... 

Probability density w(N)of the fact that random walk trajectory is at the same time SARWstatistics trajectory and at given Z, N, n. will get the last step in one of the two equiprobable cells, which coordinates are set by vectors s = (s, ), differentiated only by the signs of their components s., is equal to... 

This means that for the analysis of SARW statistics of the chain s internal links in the lattice space, a new number of cells T probability density of the random walk trajectory s self-avoiding for the polymier chain with fixed position of the internal link can be described by the Bernoulli distribution in the same form (Eq. 7), but with a new value of cells number ... 

As to reactions in fractal spaces, here the situation is quite opposite. As it is known [21], if to consider a trajectory of oligomer and curing agent molecules diffusive movement as a random walk trajectory, the sites number (s), visited by such walk, is written as follows ... 

The molecular mechanism of particle aggregation is basically the same as that of radical capture or molecular absorption the collision between two different entities after they diffuse toward each other following random walk trajectories. 

In 1981 the diffusion limited aggregation (DLA) model was introduced by Witten and Sanders [96]. In this model particles are added, one at a time, to a cluster or aggregate of particles via random walk trajectories. According to this model, there is competing growth of polymer chains from a surface, which leads to the formation of independent clusters. 

The system treated in this way can be regarded as provided with the dynamics consisting of local vibrations and occasional diffusional steps resulting from the coincidence of attempts of neighboring elements to displace beyond the occupied positions. Within a longer time interval, this kind of dynamics leads to displacements of individual beads along random walk trajectories with steps distributed randomly in time. Small displacements related to vibrations of beads could be considered explicitly but are neglected here, for simplicity. 

The dimension d of a random walk trajectory of reagents (oligomer and curing agent molecules) can be estimated according to the equation [35] ... 

We point out that sometimes one is satisfied with lower bounds on for instance in the statistical test for localization described in Section 9.2. In this case the algorithm can be further speeded up by restricting the computation to a suitable set of random walk trajectories instead of smnming, at the step of the iteration, over y G —M — 1,...,M+1 one can sum over a subset. Our understanding of the poljmier behavior suggests that if the system size is N the endpoint of the polymer is typically at distance 0 y/N) in all regimes (and max 0),... 

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