Maximum entropy (maxEnt) inference of state probabilities using state-dependent constraints is popular in the study of complex systems. In stochastic dynamical systems, the effect of state space topology and path-dependent constraints on the inferred
state probabilities is unknown. To that end, we derive the transition probabilities and the stationary distribution of a maximum {it path} entropy Markov process subject to state- and path-dependent constraints. The stationary distribution reflects a competition between path multiplicity and imposed constraints and is significantly different from the Boltzmann distribution. We illustrate our results with a particle diffusing on an energy landscape. Connections with the path integral approach to diffusion are discussed.
There has been interest in finding a general variational principle for non-equilibrium statistical mechanics. We give evidence that Maximum Caliber (Max Cal) is such a principle. Max Cal, a variant of Maximum Entropy, predicts dynamical distribution
functions by maximizing a path entropy subject to dynamical constraints, such as average fluxes. We first show that Max Cal leads to standard near-equilibrium results -including the Green-Kubo relations, Onsagers reciprocal relations of coupled flows, and Prigogines principle of minimum entropy production -in a way that is particularly simple. More importantly, because Max Cal does not require any notion of local equilibrium, or any notion of entropy dissipation, or even any restriction to material physics, it is more general than many traditional approaches. We develop some generalizations of the Onsager and Prigogine results that apply arbitrarily far from equilibrium. Max Cal is not limited to materials and fluids; it also applies, for example, to flows and trafficking on networks more broadly.
Across many fields, a problem of interest is to predict the transition rates between nodes of a network, given limited stationary state and dynamical information. We give a solution using the principle of Maximum Caliber. We find the transition rate
matrix by maximizing the path entropy of a random walker on the network constrained to reproducing a stationary distribution and a few dynamical averages. A main finding here is that when constrained only by the mean jump rate, the rate matrix is given by a square-root dependence of the rate, $omega_{ab} propto sqrt{p_b/p_a}$, on $p_a$ and $p_b$, the stationary state populations at nodes a and b. We give two examples of our approach. First, we show that this method correctly predicts the correlated rates in a biochemical network of two genes, where we know the exact results from prior simulation. Second, we show that it correctly predicts rates of peptide conformational transitions, when compared to molecular dynamics simulations. This method can be used to infer large numbers of rates on known networks where smaller numbers of steady-state node populations are known.
Gibbs and Boltzmann definitions of temperature agree only in the macroscopic limit. The ambiguity in identifying the equilibrium temperature of a finite sized `small system exchanging energy with a bath is usually understood as a limitation of conven
tional statistical mechanics. We interpret this ambiguity as resulting from a stochastically fluctuating temperature coupled with the phase space variables giving rise to a broad temperature distribution. With this ansatz, we develop the equilibrium statistics and dynamics of small systems. Numerical evidence using an analytically tractable model shows that the effects of temperature fluctuations can be detected in equilibrium and dynamical properties of the phase space of the small system. Our theory generalizes statistical mechanics to small systems relevant to biophysics and nanotechnology.
We present a principled approach for estimating the matrix of microscopic rates among states of a Markov process, given only its stationary state population distribution and a single average global kinetic observable. We adapt Maximum Caliber, a vari
ational principle in which a path entropy is maximized over the distribution of all the possible trajectories, subject to basic kinetic constraints and some average dynamical observables. We show that this approach leads, under appropriate conditions, to the continuous-time master equation and a Smoluchowski-like equation that is valid for both equilibrium and non-equilibrium stationary states. We illustrate the method by computing the solvation dynamics of water molecules from molecular dynamics trajectories.
