Optimal network kinetics: A new significance to the principle of least time

Abstract

The principle of least time is given a precise mathematical formulation in the new context of stochastic processes, where it is referred to as “optimal network kinetics”: In situations influenced by natural selection the efficient processes take place along optimal reaction coordinate paths, which in complex systems may form a network by bifurcations. The optimal paths in the configuration space of the system are defined by monimizing the time scale associated with an individual path in the set of all possible paths, subject to given boundary conditions. In particular it is discussed how optimal paths may be the consequence of detailed balance, which incorporates into the structure of the stochastic matrix of the system a local bias against excessive energy expenditure. Hence the optimal path depends on the configuration space (potential-) energy surface as well as on the temperature, for instance according to the law of Arrhenius at the crossing of a barrier along a reaction coordinate passing through the saddle-points. Exact formulations are given for disordered structures, including the case of variable range hopping conduction, where we obtain an exact derivation of the law of Mott. A variational formulation is given for processes corresponding to classical diffusion on a multidimensional energy surface. The corresponding differential equations defining the optimal path in this space have the form of newtonian equations of motion. However, the situation implies an interesting teleological aspect, which is unlike anything known from conventional dynamics. In order to be capable of crossing energy barriers the point tracing out the optimal path between given endpoints adjusts its mass so that both positive and negative values permit it to go along with, as well as against the locally acting force (the conventional gradient of the energy surface). At any point the appropriate mass, and hence the acceleration that is derived from the force at the current location, depends in a definite way on the entire path that is ultimately going to be completed.