Yoshitsugu Oono

Chaos, External Noise and Fredholm Theory

Y oshitsugu OONO*l and Y oichiro TAKAHASHI* Research Institute of Industrial Science Kyushu University, Fukuoka 812 *Department of Afathematics, College of General Education University of Tokyo, Komaba, Tokyo 153 (Received December 7, 1979) The Fredholm theory of integral equations is applied to the Pcrron-Frobenius equation which determines the invariant measure of nonlinear difference equations. The resultant Fredholm determinant D is identical to 1/(, where ( is the Artine-Mazur-Ruelle (-function. From the investigation of D we get the observable condition of the invariant measure of chaos, its stability against external noise, etc.

Experimental Study on Chemical Turbulence. II

The chemical turbulence of the Belousov-Zhabotinsky reaction theoretically expected by Kuramoto and Yamada was experimentally studied. The turbulence is caused by the coupling between diffusion and nonlinear reaction, i.e., a diffusion induced chaos. Qualitative features of the turbulence are as follows: (1) the turbulence is characterized by frequent growth and decay of plural modes, (2) the alternation of modes corresponds to the irregular variation of the amplitude, (3) the extent of disorder varies gradually with controllable parameters and there seems to be no discontinuity between turbulent and ordered phases.

Three-Body Intrachain Collisions in a Single Polymer Chain

The effects of global three-body intrachain interactions on the properties of a single polymer chain are studied in detail within the framework of the mean field approach. The usually accepted assumption that the global intrachain interaction can be taken into account only through binary intrachain collisions is not always correct. Hence the theta state cannot generally be the ideal state of a polymer chain.

On the Unperturbed State of a Polymer Chain

The mean square end-to-end distances ∝ N 2ν and the end-to-end vector distribution functions P ( R ) of a polymer chain at (varTheta)-point and in the polymer melt are considered in the framework of the self-consistent approximation developed by Edwards. The following conclusions are drawn. i) At (varTheta)-point 2ν=1, but P ( R ) is not gaussian in three dimensional space. ii) In the melt 2ν=1 and P ( R ) is very close to gaussian in three dimensional space. iii) The mechanisms making 2ν unity in the two cases above are different and hence are generally different in these two cases. iv) The parameter z of the two-parameter theory is not a good parameter for analyzing the data near (varTheta)-point.

On the Divergence of the Perturbation Series for the Excluded-Volume Problem in Polymers

For a lattice polymer on the cubic lattice and for a Gaussian polymer in three dimensional space, it is proved that the square of the expansion factor α is zero for negative real z , the measure of the strength of the intrachain interaction, when the polymer length becomes infinite under the condition that z is kept finite. This fact implies that the perturbation series for α 2 ( z ) is not convergent except at z =0.

Renormalization along the Polymer Chain

A renormalization transformation along the polymer chain, more systematic than that given by Gabay and Garel, is given. The method is closely related to the variation of the cut-off length.

A Renormalization Group Approach and the Phenomenological Theory of a Polymer Chain

An intuitive picture of the renormalization transformation of the monomer density field of a polymer chain is given. The picture is justified by the correspondence between complex spin function ϕ 2 and monomer density field, which allows the field-theoretic calculation of the monomer density profile, the radius of gyration, etc. From the renormalization group approach the following conclusions are drawn: 1) a polymer chain can be phenomenologically described with two parameters, 2) the phenomenological two parameters are, however, complicated functions of many (not two) microscopic parameters, 3) the phenomenological parameter z is proportional to N 1/2 (ln N ) 3/22 near the theta point, where N is the length of the polymer, 4) there exists a class of weighted self-interacting walk models which is, at the theta point, globally gaussian but whose intrachain binary interaction does not vanish and is rather attractive.

Disorder Parameter for Chaos

As a possible dual concept of the order parameter in the theory of phase transition, it is desirable to introduce the disorder parameter for chaos. A mathematically well-defined disorder parameter is proposed, which is presumably identical with the Kolumogorov entropy, though rigorous proof is not yet obtained. Possible potential applications of the disorder parameter to the analyses of time-series are given.

On the Divergence of the Perturbation Series for the Excluded-Volume Problem in Polymers. II. Collapse of a Single Chain in Poor Solvents

For a lattice polymer on the cubic lattice in three dimensional space, it is proved that a single polymer chain is contained in a small volume with probability 1 for negative real z , the measure of the strength of the intrachain interaction, when the polymer length becomes infinite with z being kept finite. The result suggests that the radii of convergence of the perturbation series so far obtained for a chain of finite length N are of order N -1/2 .

A Heuristic Approach to the Excluded-Volume Problem of a Polymer

A heuristic approach to the excluded-volume problem of a polymer chain is given. The universality of the exponent for the mean square end-to-end distance is also intuitively explained. The approach is closely related to Imrys interdimensional scaling theory, but does not utilize the polymer-magnet analogy due to de Gennes.