Mizuho 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.

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.

GAUGE SIMPLIFICATION OF HAMILTONIAN WITH OFF-DIAGONAL ± 1

Tight-binding Hamiltonians with off-diagonal disorder are studied extensively in connection with localization phenomena. For instance, applying a random magnetic field to a square lattice amounts to assigning phases to the off-diagonal entries of H. A gauge transformation simplifies the resulting H to an equivalent Hamiltonian with apparently less disorder. This paper concerns the special case when non-zero entries of a Hamiltonian H are ± 1 between nearest-neighbor sites. Some thought shows that a one-dimensional chain Hamiltonian with this type of randomness can be transformed to that of an ordered chain by flipping the signs of selected basis functions, i.e. by a unitary transformation. Hence such a chain is not disordered at all. On the other hand, whether or not a similar H for a square lattice implies actual disorder is a topological question. The question can be put this way: Can one find a set of simple closed curves such that reversing the signs of basis functions inside the curves will change al...

DYNAMICS OF CREMONA MAPS FROM PHYSICAL MODELS

A Cremona transformation X=f(x, y), Y=g(x, y) is a rational mapping (meaning that f and g are ratios of polynomials) with rational inverse x=F(X, Y), y=G(X, Y). Discrete dynamical systems defined by such transformations are well studied. They include symmetries of the Yang-Baxter equations and their generalizations. In this paper we comment on two types of dynamical systems based on Cremona transformations. The first is the P1 case of Bellon et al. which pertains to the inversion relation for the matrix of Boltzmann weights of the 4-state chiral Potts model. The resulting dynamical system decouples completely to one in a single variable. The sub case z=x corresponds to the symmetric Ashkin-Teller model. We solve this case explicitly giving orbits as closed formulas in the number n of iterations. The second type of system treated is an extension from the famous example due to McMillan of invariant curves of area preserving maps in two dimensions to the case of invariant curves and surfaces of three dimensional Cremona maps that preserve volume. The trace map of the renormalization of transmission through a Fibonacci chain, first introduced by Kohmoto, Kadanoff and Tang, is considered as an example of such a system.