Kiyoshi Sogo

Eigenstates of Calogero–Sutherland–Moser model and generalized Schur functions

The eigenvalue problem for the Calogero–Sutherland–Moser model is solved exactly. The complete set of eigenfunctions is given by the homogeneous Laurent polynomials. Its subset consisting of polynomial solutions is expressed by the generalized Schur functions which correspond to the λ‐deformed general linear group.

Variational discretization of Euler's Elastica problem

A discrete version of Eulers Elastica problem is formulated by a variational principle. Hirotas bilinear equations, Lax pair formalism and the exact solution are obtained explicitly. Geometrical properties are also discussed such as discrete Frenet–Serret equations.

Inverse Problem and Central Limit Theorem in Chaotic Map Theory

An inverse problem to find chaotic map(s) having a given invariant measure is formulated using a functional equation. A set of exact solutions, termed the Chebyshev hierarchy (maps defined by Chebyshev polynomials), is found for this equation. Then, an analogue of the central limit theorem is formulated and is exemplified by the Chebyshev hierarchy through explicit calculations.

Molecular Dynamics Simulations of One-, Two-, Three-dimensional Hopping Dynamics

K. M. Aoki1,2,3, S. Fujiwara4, K. Sogo5, S. Ohnishi2, and T. Yamamoto6 1 iCFD, 1-16-5 Haramachi, Meguro-ku, Tokyo 152-0011, Japan 2 Faculty of Science, Toho University, 2-2-1 Miyama, Funabashi, Chiba 274-8510, Japan 3 Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Tokyo 169-8555, Japan 4 Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan 5 Kitasato University, 1-15-1 Sagamihara, Kanagawa 228-8555, Japan 6 1-19-8 Shin-Yokohama, Kohoku-ku, Yokohama 222-0033, Japan

Determinant expressions for discrete integrable maps

Explicit formulas for several discrete integrable maps with periodic boundary condition are obtained, which give the sequential time developments in a form of the quotient of successive determinants of tri-diagonal matrices. We can expect that such formulas make the corresponding numerical simulations simple and stable. The cases of discrete Lotka–Volterra and discrete KdV equations are demonstrated by using the common algorithm computing determinants of tri-diagonal matrices.

Methods of computing Campbell-Hausdorff formula

A new method computing Campbell-Hausdorff formula is proposed by using quantum moment-cumulant relations, which is given by Weyl ordering symmetrization of classical moment-cumulant relations. The method enables one to readily use symbolic language software to compute arbitrary terms in the formula, and explicit expressions up to the 6-th order are obtained by the way of illustration. Further the symmetry Codd(A, B) = Codd(B, A), Ceven(A, B) = − Ceven(B, A) is found and proved. The operator differential method by Knapp is also examined for the comparison.

Correlated anomalous diffusion: Random walk and Langevin equation

A random walk model is formulated and examined which gives the correlated anomalous diffusion found in molecular dynamics simulations. The mean square displacement (MSD) shows a logarithmic behavior in one dimension. Corresponding Langevin equation is constructed by solving the inverse problem which gives a procedure to derive random impulse correlation from MSD function.