Hypercubic lattice

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

It will be usefuljto rewrite Eq. (73 ) into the more familiar form. With the equality C = ( 1/f) for the hypercubic lattices together with the definition [Pg.172]

Mean Square End-to-End Distance of a Chain on Hypercubic Lattices. Let us consider a random flight chain on hypercubic lattices. Let rn be a vector from a given site to the nth site. Then we have... 

First, ring formation on lattices can take place on even clusters alone larger than 4, say, for hypercubic lattices. Thus, cyclization is not a random event, but depends on cluster size. One can check this possibility by enumerating the number of rings on lattices. Now pay attention to the Eq. (108) which holds exactly only if the random distribution assumption holds. The enumeration in the two dimensional bond percolation showed that rings are considerably fewer than expected from Eq. (108) that is, D(inter)>l/(f-l). Thus, the random distribution assumption is violated for low dimensional bond percolations. This may be an essential part of the discrepancy observed in low dimensions, which, however, cannot explain the sudden confluence at d = 8, since D(inter) is in nature independent of d in the percolation model (Sect. 5) [84]. 

These theorems have been demonstrated for self-avoiding chains drawn on two-dimensional square lattices. The proof could easily be extended to d-dimensional hypercubic lattices, and one could generalize to all kinds of lattices (see Fig. 3.1). 

Let us consider a self-avoiding chain with N links on a hypercubic lattice. Let Xj be the first coordinate of the point My of order j on the chain (0[Pg.68]

In particular, Kesten5 has shown that, for a hypercubic lattice, there exist numbers A, B, C such that... 

Let us consider a d-dimensional hypercubic lattice and, on this lattice, the selfavoiding chains of origin O. Let PN (r) be the probability distribution of the end-to-end vector, r. This vector r has coordinates qv...,qd which are integers. The parity of r is by definition the parity of qj. 

For the simulations presented in this chapter, a D3Q19 lattice (where D and Q denote the dimension of the lattice and the number of links per lattice node, respectively) was employed that is obtained as 3D projection of a 4D face-centered hypercubic lattice. 

Gefen, Y, Meir, Y, Mandelbrot, B.B. and Aharony, A. (1983). Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices. Phys. Rev. Lett., 50(3), 145-148. 

It is not difficult to show that the growth constant fj, remains unchanged for most obvious geometries, such as the semi-infinite lattice, or indeed any positive wedge angle. A much more subtle result was proved by Hammersley and Whittington [22] who considered SAW confined to a subset Z (/) of the d-dimensional hypercubic lattice, such that the... 

Another interesting constrained SAW problem is that of SAW confined to lie entirely within a square, or rectangle. We will considering in detail the problem of self-avoiding walks on a subset of the square lattice though several of the theorems we give apply to the hypercubic lattice, and so hold for Z", d>2. 

Following the work in [35], Madras in [37] proved a number of theorems. In fact, most of Madras s results were proved for the more general d-dimensional hypercubic lattice, but here we will quote them in the more restricted two-dimensional setting. 

On a mathematically rigorous level, it may be possible to show the equality of the critical points for SAPs and SAWs on quasiperiodic tilings by appropriately modifying the existing proofs for the hypercubic lattice [68]. Furthermore, it would be interesting to carry out an analysis to determine if random walk behaviour can be proved for dimensions greater than four [4]. 

Consider two sites separated by a distance R oa a U-dimensional hypercubic lattice with homogeneous bond energies e. Using the Cartesian metric R is measured as ... 

We now take up the rescaling procedure at the microscopic level. For this purpose we rely heavily on the coarse graining method pioneered by Kadanoff. Consider a set of spins Sj = 1 (in units of h/2) distributed on sites j of a d-dimensional hypercubic lattice, with lattice constant a. These spins interact solely with nearest neighbor spins Sj via a common coupling constant J, and respond to a superposed magnetic field in the manner specified by the reduced Hamiltonian (/J s llksT) ... 

The PAM is studied in the high-dimensional limit mentioned in sect. 4. Since the Kondo effect is independent of lattice dimensionality, working in this limit will not inhibit the study of screening in the lattice. The PAM Hamiltonian on a Ds-dimensional hypercubic lattice is... 

In Fig. 2.1 we show all the possible one-bead moves (on a hypercubic lattice). Move A is a one-bead flip (also called kink-jump ) it is the only one-bead internal move. Moves B and C are end-bond rotations. 

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