Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInferring microscopic kinetics of a Markov process using maximum caliber
127
0
0.0
(
0
)
Added by
Purushottam Dixit Mr.
Publication date
2014
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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 variational 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.
rate research
Read More
Rate processes are often modeled using Markov-State Models (MSM). Suppose you know a prior MSM, and then learn that your prediction of some particular observable rate is wrong. What is the best way to correct the whole MSM? For example, molecular dynamics simulations of protein folding may sample many microstates, possibly giving correct pathways through them, while also giving the wrong overall folding rate, when compared to experiment. Here, we describe Caliber Corrected Markov Modeling (C2M2): an approach based on the principle of maximum entropy for updating a Markov model by imposing state- and trajectory- based constraints. We show that such corrections are equivalent to asserting position-dependent diffusion coefficients in continuous-time continuous-space Markov processes modeled by a Smoluchowski equation. We derive the functional form of the diffusion coefficient explicitly in terms of the trajectory-based constraints. We illustrate with examples of 2D particle diffusion and an overdamped harmonic oscillator.
We review here {it Maximum Caliber} (Max Cal), a general variational principle for inferring distributions of paths in dynamical processes and networks. Max Cal is to dynamical trajectories what the principle of {it Maximum Entropy} (Max Ent) is to equilibrium states or stationary populations. In Max Cal, you maximize a path entropy over all possible pathways, subject to dynamical constraints, in order to predict relative path weights. Many well-known relationships of Non-Equilibrium Statistical Physics -- such as the Green-Kubo fluctuation-dissipation relations, Onsagers reciprocal relations, and Prigogines Minimum Entropy Production -- are limited to near-equilibrium processes. Max Cal is more general. While it can readily derive these results under those limits, Max Cal is also applicable far from equilibrium. We give recent examples of MaxCal as a method of inference about trajectory distributions from limited data, finding reaction coordinates in bio-molecular simulations, and modeling the complex dynamics of non-thermal systems such as gene regulatory networks or the collective firing of neurons. We also survey its basis in principle, and some limitations.
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.
The thermodynamic definition of entropy can be extended to nonequilibrium systems based on its relation to information. To apply this definition in practice requires access to the physical systems microstates, which may be prohibitively inefficient to sample or difficult to obtain experimentally. It is beneficial, therefore, to relate the entropy to other integrated properties which are accessible out of equilibrium. We focus on the structure factor, which describes the spatial correlations of density fluctuations and can be directly measured by scattering. The information gained by a given structure factor regarding an otherwise unknown system provides an upper bound for the systems entropy. We find that the maximum-entropy model corresponds to an equilibrium system with an effective pair-interaction. Approximate closed-form relations for the effective pair-potential and the resulting entropy in terms of the structure factor are obtained. As examples, the relations are used to estimate the entropy of an exactly solvable model and two simulated systems out of equilibrium. The focus is on low-dimensional examples, where our method, as well as a recently proposed compression-based one, can be tested against a rigorous direct-sampling technique. The entropy inferred from the structure factor is found to be consistent with the other methods, superior for larger system sizes, and accurate in identifying global transitions. Our approach allows for extensions of the theory to more complex systems and to higher-order correlations.
We report a theoretical study of stochastic processes modeling the growth of first-order Markov copolymers, as well as the reversed reaction of depolymerization. These processes are ruled by kinetic equations describing both the attachment and detachment of monomers. Exact solutions are obtained for these kinetic equations in the steady regimes of multicomponent copolymerization and depolymerization. Thermodynamic equilibrium is identified as the state at which the growth velocity is vanishing on average and where detailed balance is satisfied. Away from equilibrium, the analytical expression of the thermodynamic entropy production is deduced in terms of the Shannon disorder per monomer in the copolymer sequence. The Mayo-Lewis equation is recovered in the fully irreversible growth regime. The theory also applies to Bernoullian chains in the case where the attachment and detachment rates only depend on the reacting monomer.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
