The Geometry of Chance: on the theory of non-equilibrium statistical mechanics

The study of Statistical Mechanics goes back to the 1800s and the work of Boltzmann. Since that time the field has been divided into equilibrium theory and non-equilibrium theory, with the formers progression far outpacing the latter. That is until relatively recently. New insights such as the thermodynamic length, fluctuation theorems and spectral methods such as the Observable Representation have given us new tools to deal with large and complex non-equilibrium systems. In this work we will look at two specific tools in depth. The Observable Representation (OR), its irreversible extension the Non-Detailed balance Observable Representation (NOR) and the information length.
The NOR allows one to take the complex and often messy calculations of a systems evolution operator and represent it with a much simpler geometric version. In this version distances correspond to relationships in the original system. We will show how these distances can be used to elucidate the underlying structure of a given system and even to control chaotic systems by forming periodic orbits from said distances.

The second method to be analysed in detail is the thermodynamic length and its non-equilibrium extension the information length. This gives us a measure of distance between probability distribution the system takes in its evolution. Each distribution is represented as a point in statistical space and as the system evolves each point generates a path we can measure the distances of. This abstract space then allows us to often calculate fundamental quantities of systems under study such as the maximum available work or the dissipation as the system evolves.

Both methods may seem abstract and un-necessarily far removed from the actual systems they represent. What we gain from this abstractness far outweighs its added mathematical machinery, for from abstraction we gain generality. These methods allow us to analyse huge classes of system under one umbrella, such as irreversible or chaotic systems which before were out of reach of equilibrium statistical mechanics.