For a system obeying stochastic dynamics, the slow (left) eigenvectors of the matrix of transition probabilities provide a representation of the system state space. We call such eigenvectors, observables, and this observable representation implicitly possesses a metric reflecting dynamical proximity. For the case of phase transitions, including non-equilibrium and metastable phases, the observable representation puts the states within each phase on the vertices of a simplex, with other states, not in the phase, within that simplex. The barycentric coordinates of those internal points with respect to the extrema give the probabilities that they reach a particular phase under the dynamics. For metastable phases possessing hierarchical structure, this structure can be made pictorially manifest. For many systems (with or without a phase transition), the observable representation can reproduce the system’s coordinate space (as in Brownian motion) or an underlying coordinate space more subtly related to the dynamics (as in spin dynamics for spins in space). When a collection of points is non-dynamical, as in graphs, by providing a fictitious dynamics an observable representation can be produced, which in some cases provides a useful representation of the graph.

Physics Department, Clarkson University et SPhT