William Schwalm

Exact solution of linear transport equations in fractal media - I. Renormalization analysis and general theory

We develop in detail a renormalization analysis of transport equations on fractals by considering regular model structures represented by means of families of graphs G(n), each of which is characterized by its adjacency matrix. Particular attention is paid to the correct representation of boundary conditions relevant to specific transport problems. The extension theory for the solution of generic transport problems defined by positive functions in the algebra of a given adjacency matrix is also developed.

Exact solution of linear transport equations in fractal media—II. Diffusion and convection

Renormalization analysis discussed in Giona et al. (1996a, Chem. Engng Sci., 51, 4717–4729) is applied to study diffusion and convection on fractals. Sorption properties and moment analysis are developed on fractals. Extension theory is applied to diffusion in the presence of multiple- length hopping. The case of diffusion in random and heterogeneous structures is also addressed. Finally, renormalization of diffusion with convective bias on fractals is developed.

Exact solution of linear transport equations in fractal media-III. Adsorption and chemical reaction

Absorption kinetics and first-order reaction models on fractal lattices are studied. Adsorption kinetics in macro/microporous fractal solids showing fractal scaling properties and a complex topological pore-network structure are developed by considering the coupling of two fractal graphs. For first-order reactions on fractal substrata, the exact determination of the behavior of the effectiveness factor vs the Thiele modulus is obtained. A model for analyzing the spatial concentration distribution in a fractal catalyst is proposed.

First-order kinetics in fractal catalysts: Renormalization analysis of the effectiveness factor

By applying Input/Output renormalization and the Green function formalism to the discrete formulation of transport phenomena on graphs, we develop exact recursion for the effectiveness factor for first-order reaction in a fractal porous catalyst. The scaling of the effectiveness factor vs the Thiele modulus is obtained in the case of finitely ramified fractals and discussed in detail. Crossover phenomena in hierarchical and heterogeneous fractal pore-network models and renormalization recursion for integral quantities are also analyzed.

Closed formulae for Green functions on fractal lattices

Closed form solutions are found for Schrodinger Green functions x (corner-to-same-corner) and y (corner-to-other-corner) on the 3-simplex, the 4-simplex and two other fractal lattices. It is elementary to derive the well known recursions of the form x → X(x, y), y → Y(x, y) relating generations n and n + 1 of the lattice. We have now obtained an infinite hierarchy of exact solutions to these recursion expressing x, y as closed formulae in terms of initial condition (energy) and generation number n. This amounts to constructing a hierarchy of orbits for the dynamics of the renormalization map. To our knowledge, no other such solutions have been found previously. For each of these solutions, y scales as a power of the lattice size, which is of interest in relation to conductance scaling in Anderson localization. One cannot study power-law scaling numerically using the recursions alone, since the asymptotic behavior of y at large length scale is chaotic with respect to the energy parameter. Thus, the chances of finding any power-law solution are measure zero in the initial conditions, most of which lead to superlocalized, stretched exponential behaviors of y with lattice size. In contrast, the exact solutions each connect a value of energy to an unambiguous asymptotic behavior.

Analysis of linear transport phenomena on fractals

Abstract In this article we apply input/output (I/O) renormalization to linear transport phenomena on fractals. We focus mainly on first-order reaction kinetics, on sorption dynamics and on renormalization of integral quantities in the case of finitely ramified structure. Moreover, care is taken to show how the theory can be extended to thes tudy of infinitely ramified fractals and non-linear model and to the case of an unbounded number of input sites.

Finding Lie groups that reduce the order of discrete dynamical systems

Discrete dynamical systems of the form are considered, where is an n-component vector. Equations X = f(x) define a mapping f from an n-dimensional projective space into itself. Each component of f is a rational function, i.e. a ratio of polynomials in n dynamical variables. Maeda showed that when f commutes with each transformation of a Lie group, a reduction in the order of the dynamical system results. Given a discrete dynamical system, the difficulty is to find its continuous symmetries. We present a way of using f-invariant sets to find these symmetries. The approach taken is to arrange groups in order of increasing complication and to characterize the set of dynamical systems admitting each group. Criteria are given for recognizing and reducing the order of systems admitting subgroups of the projective general linear group in n variables, PGL(n), or certain Lie subgroups of the Cremona group of birational transformations in n variables, . Quispel et al demonstrated the use of canonical group variables for achieving this reduction. We develop canonical coordinates for several groups with elementary Lie algebras and demonstrate reduction of order in each case. Results are used to reduce the order of several examples of recursion formulae taken from the literature on renormalizable lattice models.

Solution of Transport Schemes on Fractals by Means of Green Function Renormalization — Application to Integral Quantities

Green function renormalization is applied to obtain exact recursions for integral quantities and integral transforms of concentration fields, involving a summation over a manifold of lattice sites or over the entire lattice. Definition of the Fourier transform of the Green functions, to some extent equivalent to the dynamic structure factor, is extremely useful to obtain quantitative information on the spatial behavior of concentration profiles and eigenfunctions. Some preliminary results on sorption properties of product lattices are presented. The application of anisotropic lattice models to study the volume-to-surface effects in adsorption and reaction is discussed. The properties of the Green function generating function are also analyzed.

Conic pencils and renormalization dynamics

Abstract Solutions as closed formulas are found for the orbits ( X n , Y n )of certain rational, planar maps which permute curves of a conic pencil and consequently reduce to a semi decoupled form b → B ( b ), u → U ( b , u ). With the solution formulas it is possible to so lve the quantum conductance problem on regular hierarchical lattices in a simple, essentially closed form.

2-D conductance and magnetoconductance in a thin crystal of Bi14Te11S10.

Abstract In a crystal of Bi14Te11S10 the number of scattering centers is manipulated by allowing gases to adsorb/desorb on the surface. An increase in the number of scattering centers causes the temperature dependence of the low temperature conductance to change from logarithmic to exponential and the magnetoresistance to change sign from positive to negative.