A derivation of the master equation from path entropy maximization

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles.     

Approximate density matrices and Husimi functions using the maximum entropy formulation with constraints

The entropy of an electronic system is defined in terms of the Husimi function, a nonnegative distribution function in phase space. The Husimi function is calculated by maximizing the entropy subject to the constraints that the Husimi function give a Gaussian convolution of the desity when integrating over the momentum coordinates and that its second moment with respect to momentum give a sum of Gaussian convolutions of the density and the kinetic energy density. The result is compared with the Wigner function. Equations are given for calculating the density matrix from the Husimi function. The resulting equation for the exchange energy requires a difficult numerical integration. An alternate method is used to obtain the density matrix from an approximate partially collapsed Husimi matrix that gives the maximum entropy Husimi function as its diagonal. The results are exact for the harmonic oscillator ground state. Exchange energies calculated for H and the He isoelectronic series through C⁺⁴ show slight improvements over those calculated using a maximum entropy Wigner function.     

Markov models of molecular kinetics: generation and validation

Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years. This approach has many appealing characteristics compared to straightforward molecular dynamics simulation and analysis, including the potential to mitigate the sampling problem by extracting long-time kinetic information from short trajectories and the ability to straightforwardly calculate expectation values and statistical uncertainties of various stationary and dynamical molecular observables. In this paper, we summarize the current state of the art in generation and validation of MSMs and give some important new results. We describe an upper bound for the approximation error made by modeling molecular dynamics with a MSM and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast to previous practice, it becomes clear that the best MSM is not obtained by the most metastable discretization, but the MSM can be much improved if non-metastable states are introduced near the transition states. Moreover, we show that it is not necessary to resolve all slow processes by the state space partitioning, but individual dynamical processes of interest can be resolved separately. We also present an efficient estimator for reversible transition matrices and a robust test to validate that a MSM reproduces the kinetics of the molecular dynamics data.     

How the nature of an observation affects single-trajectory entropies

Projection of a Markov process with constant rates of transition to a small number of observable aggregated states can result in complex kinetics with memory. Here, we define the entropy production along a single sequence of aggregated states and show that it obeys detailed and integral fluctuation theorems. More importantly, we prove that projection shifts the distribution of entropy production over the ensemble of paths for a nonequilibrium process toward one characteristic of a system at equilibrium. This statement represents an analog of the second law of thermodynamics for path-dependent entropies and thus a new form of constraint of irreversible systems. Numeric examples are presented to illustrate these ideas.     

Validation of Markov state models using Shannon's entropy

Markov state models are kinetic models built from the dynamics of molecular simulation trajectories by grouping similar configurations into states and examining the transition probabilities between states. Here we present a procedure for validating the underlying Markov assumption in Markov state models based on information theory using Shannon's entropy. This entropy method is applied to a simple system and is compared with the previous eigenvalue method. The entropy method also provides a way to identify states that are least Markovian, which can then be divided into finer states to improve the model.     

关于热力学自发过程及其判据的讨论

文章指出现有各种自发过程的判据都是在指定的约束条件下才能应用,缺乏普适性是自发过程定义多样化的引发原因。在无约束条件下将热力学第一定律代入总熵判据得出并分析讨论了总熵判据的另一种形式,结合自发过程的特点总结出了热力学变化过程中能量变化的本质,给出了自发过程的通用定义。进一步指出原总熵判据只能分辨可逆与不可逆,不能分辨自发与非自发。文章给出的总熵判据的另一种形式——封闭系统任意过程的做功能力判据具有分辨自发与非自发的能力。通过理论研讨和实际应用表明,做功能力判据与总熵判据完全等价,在相应约束条件下可还原为当前热力学中各类方向判据。填补了常见的变温过程和变压过程在以前的教科书中无自发和非自发判据的空白。以前教科书中由于自发过程定义和解释的混乱而出现的一些疑难问题,在通用定义和做功能力判据面前都能得到满意的解答。     

Information theoretical analysis of the hydrogen atom

This is an analysis of the statistical nature of the time-independent Schrödinger equation through the use of the information entropy concept. We first study the Schrödinger equation in a general way and then by actually computing entropies of various states of the hydrogen atom for a re-examination of the problem. It is found that there exists a variational procedure involving maximizing entropy for obtaining all solutions once one solution is known. Based on certain observations of the particular single system, some general conclusions can be deduced. First of all, we can safely say that the Schrödinger equation, among many other interpretations, is but the consequence of a principle of minimum potential energy expectation with certain proper constraints imposed. In addition, the ensemble concept in statistical thermodynamics is also useful in understanding microscopic quantum systems and many quantum mechanical concepts such as energy quantization and wave nodal properties can be discussed in the light of information theory and statistics in general.     

Probing molecular kinetics with Markov models: metastable states, transition pathways and spectroscopic observables

Markov (state) models (MSMs) have attracted a lot of interest recently as they (1) can probe long-term molecular kinetics based on short-time simulations, (2) offer a way to analyze great amounts of simulation data with relatively little subjectivity of the analyst, (3) provide insight into microscopic quantities such as the ensemble of transition pathways, and (4) allow simulation data to be reconciled with measurement data in a rigorous and explicit way. Here we sketch our current perspective of Markov models and explain in short their theoretical basis and assumptions. We describe transition path theory which allows the entire ensemble of protein folding pathways to be investigated and that combines naturally with Markov models. Experimental observations can be naturally linked to Markov models with the dynamical fingerprint theory, by which experimentally observable timescales can be equipped with an understanding of the structural rearrangement processes that take place at these timescales. The concepts of this paper are illustrated by a simple kinetic model of protein folding.     

Constrained dynamics of localized excitations causes a non-equilibrium phase transition in an atomistic model of glass formers

Recent progress has demonstrated that trajectory space for both kinetically constrained lattice models and atomistic models can be partitioned into a liquid-like and an inactive basin with a non-equilibrium phase transition separating these behaviors. Recent work has also established that excitations in atomistic models have statistics and dynamics like those in a specific class of kinetically constrained models. But it has not been known whether the non-equilibrium phase transitions occurring in the two classes of models have similar origins. Here, we show that the origin is indeed similar. In particular, we show that the number of excitations identified in an atomistic model serves as the order parameter for the inactive-active phase transition for that model. In this way, we show that the mechanism by which excitations are correlated in an atomistic model - by dynamical facilitation - is the mechanism from which the active-inactive phase transition emerges. We study properties of the inactive phase and show that it is amorphous lacking long-range order. We also discuss the choice of dynamical order parameters.     

Dynamical constraints and adiabatic invariants in chemical reactions

For long-range electrostatic potentials and, more generally, when the topography of the potential energy surface is locally simple, the reaction path coordinate is adiabatically separable from the perpendicular degrees of freedom. For the ion-permanent dipole and ion-quadrupole interactions, the Poisson bracket of the adiabatic invariant decreases with the interfragment distance more rapidly than the electrostatic potential. The smaller the translational momentum, the moment of inertia of the neutral fragment, and the dipole or quadrupole moments are, the more reliable the adiabatic approximation is, as expected from the usual argumentation. Closed-form expressions for an effective one-dimensional potential in an adiabatic Hamiltonian are given. Connection with a model where the decoupling is exact is obtained in the limit of an infinitely heavy dipole. The dynamics is also constrained by adiabatic invariance for a harmonic valley about a curved reaction path, as shown by the reaction path Hamiltonian method. The maximum entropy method reveals that, as a result of the invariance properties of the entropy, constraints whose validity has been demonstrated locally only subsist in all parts of phase space. However, their form varies continuously, and they are not necessarily expressed in simple terms as they are in the asymptotic region. Therefore, although the influence of adiabatic invariance has been demonstrated at asymptotically large values of the reaction coordinate only, it persists in more interesting ranges.     

Ultrafast dynamical localization of photoexcited states in conformationally disordered poly(p-phenylenevinylene)

We consider two types of ultrafast dynamical localization of photoexcited states in conformationally disordered poly(p-phenylenevinylene). First, we discuss nonadiabatic interconversion from higher energy extended exciton states to lower energy more localized local exciton ground states. Second, we calculate the dynamics of local exciton ground states on their Born-Oppenheimer potential energy surfaces. We show that within the first C-C bond oscillation following photoexcitation (～35 fs) the exciton becomes self-trapped and localized over approximately eight monomers. This process is associated with a Calderia-Leggett type loss of phase coherence owing to the coupling of the polymer to a dissipative environment. Subsequent torsional relaxation (on a time scale of approximately picoseconds) has little effect on the localization. We conclude from this that the initial torsional disorder determines the spatial distribution and localization length of vertical excitations but that electron-phonon coupling is largely responsible for the localization length of self-trapped excitons. We next consider the effect of dynamical localization on fluorescence depolarization. We show that exciting higher energy states causes a larger fluorescence depolarization, because these states have a larger initial delocalization. Using the observation that fluorescence depolarization is a function of excitation wavelength and polymer conformation, we show how the models of exciton localization discussed here can be experimentally investigated.     

Ring connectivity: measuring network connectivity in network covalent solids

In atomistic models of amorphous materials, ring statistics provide a measure of medium-range order. However, while ring statistics tell us the number of rings present in the model, they do not give us any information about the arrangement of rings, e.g., whether the rings are clustered and how big the cluster is. In this work we present a method to calculate the ring connectivity, or clustering, of rings. We first calculate the rings present in the model using the shortest path criteria of Franzblau and then find the rings that are connected together and group them into clusters. We apply our method to a set of models of disordered carbons, obtained using a reverse Monte Carlo procedure developed in a recent work. We found that in these carbon models the five-, six-, and seven-membered rings are connected together, forming clusters. After isolating the clusters, we found that they resemble defective graphene segments twisted in a complex way. The clusters give further insight about the arrangement of carbon atoms in microporous carbons at a larger length scale. Moreover, the method can be applied to any network covalent solid that contains rings and thus gives information about the ring connectivity present in such materials.     

Statistical mechanics of coarse graining: estimating dynamical speedups from excess entropies

The excess entropy of fluids has been shown to play a decisive role in the determination of dynamical properties [Y. Rosenfeld, Phys. Rev. A 15, 2545 (1977)]. We argue that it could play an equally important role in connecting dynamical properties of atomistic and coarse-grained models of molecular fluid systems. Molecular dynamics simulations for an atomistic and a coarse-grained model of water confirm the validity of this conjecture, showing that the sizable enhancement of the diffusion rate upon coarse-graining is a simple function of the difference in the excess entropy of the two models. This empirical observation could ease the way to a first-principles prediction of the relation of dynamical properties estimated from models at different resolution.     

Measurement of nonequilibrium entropy from space-time thermodynamic integration

The entropy of a system transiently driven out of equilibrium by a time-inhomogeneous stochastic dynamics is first expressed as a transient response function generalizing the nonlinear Kawasaki-Crooks response. This function is then reformulated into three statistical averages defined over ensembles of nonequilibrium trajectories. The first average corresponds to a space-time thermodynamic perturbation relation, while the two following ones correspond to space-time thermodynamic integration relations. Provided that trajectories are initiated starting from a distribution of states that is analytically known, the ensemble averages are computationally amenable to Markov chain Monte Carlo methods. The relevance of importance sampling in path ensembles is confirmed in practice by computing the nonequilibrium entropy of a driven toy system. We finally study a situation where the dynamics produces entropy. In this case, we observe that space-time thermodynamic integration still yields converged estimates, while space-time thermodynamic perturbation turns out to converge very slowly.     

The kinetics of homogeneous and two-step nucleation during protein crystal growth from solution

Many experimental reports for the kinetics of crystal nucleation and growth, from an isothermal solution, point to a sigmoidal-like behavior for the process. Here we consider three different nucleation models from the literature and show that all lead to sigmoidal or sigmoidal-like behavior for the kinetics of nucleation. A two-step nucleation process is known to occur within certain supersaturated protein solutions, and it is demonstrated in this report how the sigmoidal law yields kinetic information for the two-step and homogeneous nucleation modes. We propose here that two-step solute-rich associates form in the solution around seed nuclei that are already present at or near the point in time when the solution is prepared. Using this hypothesis, we are able to model the time-dependent volume of the two-step phase per unit volume of solution and show that this compares well with reported experimental data. A kinetic model is given for the proposed process, which leads to a sigmoidal rate law. Additionally, a relation between the initial and final nuclei densities and the induction time is derived. As a result of this study, the conclusion is that two-step activity increases with increasing initial supersaturation or increasing salt concentration.     

Systematic coarse-graining of potential energy landscapes and dynamics in liquids

Recent efforts have shown that the dynamic properties of a wide class of liquids can be mapped onto semi-universal scaling laws and constitutive relations that are motivated by thermodynamic analyses of much simpler models. In particular, it has been found that many systems exhibit dynamics whose behavior in state space closely follows that of soft-sphere particles interacting through an inverse power repulsion. In the present work, we show that a recently developed coarse-graining theory provides a natural way to understand how arbitrary liquids can be mapped onto effective soft-sphere models and hence how one might potentially be able to extract underlying dynamical scaling laws. The theory is based on the relative entropy, an information metric that quantifies how well a soft-sphere approximation to a liquid's multidimensional potential energy landscape performs. We show that optimization of the relative entropy not only enables one to extract effective soft-sphere potentials that suggest an inherent scaling of thermodynamic and dynamic properties in temperature-density space, but that also has rather interesting connections to excess entropy based theories of liquid dynamics. We apply the approach to a binary mixture of Lennard-Jones particles, and show that it gives effective soft-sphere scaling laws that well-describe the behavior of the diffusion constants. Our results suggest that the relative entropy formalism may be useful for "perturbative" type theories of dynamics, offering a general strategy for systematically connecting complex energy landscapes to simpler reference ones with better understood dynamic behavior.     

Tracing the minimum-energy path on the free-energy surface

The free-energy profile of a reaction can be estimated in a molecular-dynamics approach by imposing a mechanical constraint along a reaction coordinate (RC). Many recent studies have shown that the temperature can greatly influence the path followed by the reactants. Here, we propose a practical way to construct the minimum-energy path directly on the free-energy surface at a given temperature. First, we follow the blue-moon ensemble method to derive the expression of the free-energy gradient for a given RC. These derivatives are then used to find the actual minimum-energy reaction path at finite temperature, in a way similar to the intrinsic reaction path of Fukui on the potential-energy surface. [K. Fukui, J. Phys. Chem. 74, 4161 (1970)]. Once the path is known, one can calculate the free-energy profile using thermodynamic integration. We also show that the mass-metric correction cancels for many types of constraints, making the procedure easy to use. Finally, the minimum-free-energy path at 300 K for the addition of CCl2 to ethylene is compared with a path based on a simple one-dimensional reaction coordinate. A comparison is also given with the reaction path at 0 K.     

Markov chain modeling of polymer translocation through pores

We solve the Chapman-Kolmogorov equation and study the exact splitting probabilities of the general stochastic process which describes polymer translocation through membrane pores within the broad class of Markov chains. Transition probabilities, which satisfy a specific balance constraint, provide a refinement of the Chuang-Kantor-Kardar relaxation picture of translocation, allowing us to investigate finite size effects in the evaluation of dynamical scaling exponents. We find that (i) previous Langevin simulation results can be recovered only if corrections to the polymer mobility exponent are taken into account and (ii) the dynamical scaling exponents have a slow approach to their predicted asymptotic values as the polymer's length increases. We also address, along with strong support from additional numerical simulations, a critical discussion which points in a clear way the viability of the Markov chain approach put forward in this work.