Lattice, random walk

Lattice models play a central role in the description of polymer solutions as well as adsorbed polymer layers. All of the adsorption models reviewed so far assume a one-to-one correspondence between lattice random-walks and polymer configurations. In particular, the general scheme was to postulate the train-loop or train-loop—tail architecture, formulate the partition function, and then calculate the equilibrium statistics, e.g., bound fraction, average loop... 

Figure 2 shows an example of this 2D shape descriptor. Here, we compare two conformations of a linear polymer model. The polymer chain is an off-lattice random walk with constant bond length and excluded volume interaction between monomers (i.e., a self-avoiding walk). The constant bond length of / = 1.54 A is used to mimic poly methylene. [For a discussion and implementation of this model, see Ref. 56.]... 

For the lattice random walk, there is no energy and the elastic Hamiltonian of Eki. (2) just simulates the entropic effect at non-zero temperatures. One needs to look at the lattice problem in case one is interested in low or zero temperature behavior. A recapitulation of a few properties of polymers is done in Appendix B. 

Accessible porosity and cluster size distributions Effective transport coefficients Diffusion in restricted microgeometries Analytical solutions Lattice random walks Conclusion Glossary References... 

The characteristic pore geometry in porous polymers is frequently too complicated to permit an analytical solution of the conduction equations. Therefore, we have examined the dynamics of lattice random walks in three-dimensional pores (as is Figure 7) [52] by an extension of the lattice walk method [62]. The simulation was performed on an NxNxN lattice each lattice represented a coordinate site for potential molecule occupation. Boundary conditions were Imposed by extending the lattice one unit in each of the six edge directions i.e. if the Interior of the cube was represented by the points [1...N,1...N,1...N], the cube with boundaries was represented by the points [0...N+1,0...N+1,0...N+1j. The extra points were used to describe the edges of the cube. 

Suggestive examples that show the generality of such a model include the case of a general lattice random walk in (1 + 1) dimension, Figure 1.4(A), and the case of a directed walk in 1 + d dimension, that is the process (n, S ) =o,i,..., with S, like before, the partial sums of an IID sequence X, but this time Xi is a discrete random variable taking values in Z , with P(Xi = 0) > 0. Also in these cases we define if ( ) as the distribution of the returns to the origin of course it is very well possible that J if(n) < 1, like for d > 3 or if the walk is asymmetric. 

In a random walk on a square lattice the chain can cross itself. 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

While a single vant performs little more than a pseudo random-walk, multiple-vant evolutions are ripe with many interesting (Conway Life-rule-like) patterns, particularly when the background lattice food color is shown along with the moving... 

Another simple example is the traiditional two-dimensional random-walk on a four-neighbor Euclidean lattice [toff89]. Despite the fact that the underlying lattice is symmetric only with respect to rotations that are multiples of 90 deg, the probability distribution p(s, y) for a particle that begins its random walk at the origin becomes circularly symmetric in the limit as time t —> oo p x,y,t) —> (see figure 12.12). 

Fig. 12.12. A circularly symmetric Gaussian probability distribution p x,y) describing a two-dimensional random walk emerges for large times on the macroscopic level, despite the fact that the underlying Euclidean lattice is anisotropic.

It is easy to invent rules that conserve particle number, energy, momentum and so on, and to smooth out the apparent lack of structural symmetry (although we have cheated a little in our example of a random walk because the circular symmetry in this case is really a statistical phenomenon and not a reflection of the individual particle motion). The more interesting question is whether relativistically correct (i.e. Lorentz invariant) behavior can also be made to emerge on a Cartesian lattice. Toffoli ([toff89], [toffSOb]) showed that this is possible. 

The local conformational preferences of a PE chain are described by more complicated torsion potential energy functions than those in a random walk. The simulation must not only establish the coordinates on the 2nnd lattice of every second carbon atom in the initial configurations of the PE chains, but must also describe the intramolecular short range interactions of these carbon atoms, as well as the contributions to the short-range interactions from that... 

Figure 1 Two examples of random walks 10,000 steps on a cubic lattice. 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

In brief, a discrete quantum random walk comprises of an in general d - regular graph G with n vertices (where lattices are favoured in the literature) and a spin degree of freedom, where the spin can take up d different states. A quantum state at a vertex v with spin i is then... 

---