Investigation of the statistical distance to reach stationary distributions

• Define a set theoretic version of the discrete thermodynamic length.
• These sets allow one to analyse systems having zero probabilities in their evolution.
• Numerically analyse the Logistic map using the thermodynamic length.
• Show how the unstable fixed points most efficiently lead the system to equilibrium.

The thermodynamic length gives a Riemannian metric to a system's phase space. Here we extend the traditional thermodynamic length to the information length (LL) out of equilibrium and examine its properties. We utilise LL as a useful methodology of analysing non-equilibrium systems without evoking conventional assumptions such as Gaussian statistics, detailed balance, priori-known constraints, or ergodicity and numerically examine how LL evolves in time for the logistic map in the chaotic regime depending on initial conditions. To this end, we propose a discrete version of LL which is mathematically well defined by taking a set theoretic approach. We identify the areas of phase space where the loss of information of the system takes place most rapidly. In particular, we present an interesting result that the unstable fixed points turn out to most efficiently drive the logistic map towards a stationary distribution through LL.