Lattices of Effectively Nonintegral Dimensionality II

The achievable values of the dimensionality d using our construction are closely distributed in the interval 1

We use the renormalization procedure to determine the critical behavior of the lattice Coulomb gas problem. In particular, the spatial dimensionality of the truncated tetrahedral lattice according to our definition is found to be 2 log ~ 3 ~ 1.3651 and not. In Section VI, the critical behavior of the spherical model on a d-dimensional grid is outlined, for arbitrary d.

It is found that the sensitivity of the p-vector model varies as (.2.::_ol )+n power of the inverse temperature. In this case, however, no simple expressions exist for the critical exponents as functions of the lattice dimensionality.

The analysis in Section V shows that the effective dimensionality of the mesh is that of a truncated tetrahedron. Using this procedure, it can be shown that the lattice dimensionality of a truncated tetrahedron is log23. Using Nelson and Fisher's definition, the effective dimensionality of the mesh would be 2.

Stillinger's definition of the dimensionality of space is also the same as Nelson and Fisher's discussed above. For n=3 we get the truncated tetrahedron lattice For higher values of n the lattices are non-planar. This deletion of bonds results in a change in the effective dimensionality of the lattice from 2 to 2 ln4/(ln6).

Schematic representation of the graph of the rectangle (r+1) of order (2,3) of the modified rectangular grid. For simplicity, we only discuss the cases of the truncated n-simplex lattice and the (2,1)~ modified rectangular lattice. It is easily seen from equation (A1) of Appendix A that F( )\) has an asymptotic expansion of the form for small A >O.

But from Eq.(37) we know that the singular behavior of /\0 is the same as the singular behavior of the energy.

FIG . 1. (a} A complete graph on 5 points, z eroth order trun- trun-cated 4-simplex lattice

Or for that matter, to find samples that are large enough so that the volume of the experimental sample is comparable to the correlation volume. The properties of these walks are related to some properties of the Ising model [41]. The system exhibits a phase transition in the sense that the generating functions of the random walk become singular as a function of their argument.

By linearizing the recursion equations around the fixed point, we determine the critical exponents from the eigenvalues of the linearized renormalization transformation matrix. For regular lattices, where all lattice points are equivalent, this limiting procedure is unnecessary because the number of self-avoiding walks of length n is independent of the vertex from which the walk starts (as long as the initial vertex is not too close to the boundary of grille). This is not the case for the spatially inhomogeneous mesh studied here, and averaging over all possible starting point positions is necessary.

We use the renormalization group techniques to determine the constants f ,d._ , O' and Y for the truncated tetrahedron lattice by determining the singular behavior of the generating functions C(x) , P(x) , and R(x). The weight of a step of length n is xn For R(x) , there is an additional multiplicative weight factor, depending on the end-to-end distance of the step (Eq. The endpoint of the line can be any of the vertices within the r~b order triangle.

We summarize all possible configurations of the triangle of order r subject to the constraint that one of the endpoints of the walk lies inside it. Perhaps self-avoiding walks are atypical in that the generating functions whose properties define the critical exponents are not given in terms of the partition function of a Hamiltonian. The critical behavior of self-avoiding random walks strongly depends on the connecting properties of grid and not only by dimensionality.

Although the coordination number and. the dimensionality of the lattice is independent of p. whichever definition of dimensionality is used), it is easy to verify that the critical exponents for the self-avoidance walk problem on these networks depend on p. We believe that the fact that there are such simple expressions for the critical exponents in terms of system dimensionality is strong evidence in favor of our definition of dimensionality. We hope that further study of these issues will lead to a better understanding of the dimensionality of the effect on phase transitions in general.

FIG. IL All possible configurations of :in open self :ivoiding w:ilk of order r

PART II

In the second half of this thesis, we propose a model for the melting transition motivated by the above observation. We determine the nature of the transition near the critical point using the renormalization group techniques. The activities of the charges are related to the Fourier coefficients of. exponential of the periodic potential.

In Section IV, we develop the renormalization transformation formalism applied to the lattice Coulomb gas problem. We see that the Fourier transform of the potential is of the form (1/q2 ), where qL (d+l) is the dimensional transfer of momentum. In the high-temperature limit of the melting model, the radius of the cylindrical dimension for the charged gas shrinks to zero.

In the limit of the hardcore radius that goes to zero, they obtained the equation of state. Here (A enR + B ) is the correction to the Coulomb interaction between blocks due to the polarizability of the surrounding medium. Again, the correction due to the polarizability of the medium is proportional to v2 for small v.

Finally, let us determine the behavior of the correlation length as the temperature approaches the critical value from below. Most of the simplifications and approximations of the model are included in the model Hamiltonian.