by Matheus R. Grasselli.
Department of Mathematics, King's College London, 2001.We begin with the construction of an infinite dimensional Banach manifold of probability measures using the completion of the set of bounded random variables in the appropriate Orlicz norm as coordinate spaces. The infinite dimensional version of the Fisher metric as well as the exponential and mixture connections are introduced. It is then proved that they form a dualistic structure in the sense of Amari. The interpolating alpha-connections are defined, at the level of covariant derivatives, via embeddings into L_r-spaces and then found to be convex mixtures of the +1 and -1-connections. Several well known parametric results are obtained as finite dimensional restrictions of the nonparametric case.
Next, for finite dimensional quantum systems, we study a manifold of density matrices and explore the concepts of monotone metrics and duality in order to establish that the only monotone metrics with respect to which the exponential and mixture connections are mutually dual are the scalar multiples of the Bogoliubov- Kubo-Mori inner product of quantum statistical mechanics.
For infinite dimensional quantum systems, we present a general construction of a Banach manifold of density operators using the technique of epsilon-bounded perturbations, which contains small perturbations of forms and operators in the sense of Kato as special cases. We then describe how to obtain an affine structure in such a manifold, together with the corresponding exponential connection. The free energy functional is proved to be analytic on small neighbourhoods in the manifold.
We conclude with an application of the methods of Information Geometry and Statistical Dynamics to a concrete problem in fluid dynamics: the derivation of the time evolution equations for the density, energy and momentum fields of a fluid under an external field.