Overview
This paper investigates the relationship between Geometric Thermodynamics and Information Geometry, focusing on the Fisher matrix. It introduces a metric on macroscopic odd-dimensional contact manifolds of thermodynamic variables. The study further develops a Souriau-like Thermodynamics framework utilizing Calabi-Vesentini (CV) manifolds as microscopic event manifolds and Killing moment maps as observable functions. This approach permits the explicit integration for defining the partition function across the CV Tits Satake universality class.
Research Context
The research situates itself within the intersection of Geometric Thermodynamics and Information Geometry, specifically addressing the connection between these two conceptual frameworks. Geometric Thermodynamics has historically dealt with the macroscopic description of thermodynamic systems, while Information Geometry, often employing the Fisher matrix, has typically been associated with statistical manifolds and information-theoretic measures. The paper aims to clarify the structural relationship between these distinct but potentially related geometric approaches.
Approach
The methodology involves several key steps:
- Introduction of a metric on the macroscopic odd-dimensional contact manifold of thermodynamic variables. This metric's pull-back onto isoentropic symplectic submanifolds, transverse to the Reeb field, is defined as Kählerian.
- Observation that the pull-back of this metric on equilibrium states, characterized as Lagrangian submanifolds, corresponds to the Fisher Hessian.
- Application of a Souriau-like Thermodynamics framework. This framework employs Calabi-Vesentini (CV) manifolds as Kählerian microscopic event manifolds.
- Utilization of Killing moment maps as observable functions within this Souriau-like Thermodynamics.
- Systematic application of the theory of compact abelian structures.
- Integration of the setup of Special Kähler Geometry, where CV manifolds are encoded.
- Performance of explicit integration to define the partition function for any entry within the CV Tits Satake universality class.
- Introduction of additional actions completing the abelian structure, described as non-linear Casimir functions of the Killing moment-maps. These functions suggest a generalization of Souriau thermodynamics.
This generalization partially breaks the isometry group symmetry through non-vanishing mean values of the Casimir functions, drawing a parallel to spontaneous magnetization observed in ferromagnetism.
Findings
- A metric was introduced on the macroscopic odd-dimensional contact manifold of thermodynamic variables.
- The pull-back of this macroscopic metric on isoentropic symplectic submanifolds, transverse to the Reeb field, is Kählerian.
- The pull-back of this macroscopic metric on equilibrium states (Lagrangian submanifolds) corresponds to the Fisher Hessian. This finding establishes a direct link between macroscopic thermodynamic geometry and the microscopic information metric.
- Exact partition functions can be explicitly integrated for any entry in the CV Tits Satake universality class within a Souriau-like Thermodynamics framework.
- The generalized Souriau thermodynamics, incorporating non-linear Casimir functions of Killing moment-maps, exhibits a partial breaking of isometry group symmetry due to non-vanishing mean values of these Casimir functions, analogous to spontaneous magnetization.
- New exact Gibbs distributions were developed, serving as an analogue for Cartan Neural Networks, similar to how Gaussian probability distributions are used in conventional Machine Learning in flat space.