Statistical Dynamics and Information Geometry

by R. F. Streater; contribution to "Geometry and Nature", Eds. H. Nencka and J.-P. Bourguignon; Contemp. Mathematics, 203, 117-131, Amer. Math. Soc. 1997.

ABSTRACT

We relate the techniques used in statistical dynamics, as described in the article Statistical Dynamics, and the book STATISTICAL DYNAMICS, (Imperial College Press, 1995, by R. F. Streater) to the Fisher metric and Amari's dual affine connections. We show that isolated dynamics is obtained by projecting the Hamiltonian flow onto the exponential family using a geodesic relative to the connection $\nabla^{(-1)}$, cutting the family at right angles. Isothermal dynamics similarly uses geodesics relative to the $\nabla^{(+1)}$ connection.

Finally, we study the "dynamic Ising model", meaning a stochastic dynamics on a lattice of spins, conserving the Ising form for the energy. Further work on this model has been proposed by Benfatto, of the University of Rome 2.