Kramers Equation Research Articles

PFDB: A standardized protein folding database with temperature correction

We constructed a standardized protein folding kinetics database (PFDB) in which the logarithmic rate constants of all listed proteins are calculated at the standard temperature (25 °C). A temperature correction based on the Eyring–Kramers equation was introduced for proteins whose folding kinetics were originally measured at temperatures other than 25 °C. We verified the temperature correction by comparing the logarithmic rate constants predicted and experimentally observed at 25 °C for 14 different proteins, and the results demonstrated improvement of the quality of the database. PFDB consists of 141 (89 two-state and 52 non-two-state) single-domain globular proteins, which has the largest number among the currently available databases of protein folding kinetics. PFDB is thus intended to be used as a standard for developing and testing future predictive and theoretical studies of protein folding. PFDB can be accessed from the following link: http://lee.kias.re.kr/~bala/PFDB.

An Analytical Study of Fractional Klein–Kramers Approximations for Describing Anomalous Diffusion of Energetic Particles

In this article, anomalous transport models of energetic particles in space plasmas are developed by deriving the fractional force-less Fokker–Planck equation and the fractional diffusion-advection equation from the Klein–Kramers equation. Analytical solutions of the space-time fractional equations are obtained by use of the Caputo and the Riesz fractional derivatives with the Laplace–Fourier technique. The solutions are given in terms of Fox’s H-function and the profiles of the particles densities are discussed in each case for different values of fractional orders.

Tempered fractional Langevin–Brownian motion with inverse β-stable subordinator

Time-changed stochastic processes have attracted much attention and wide interest due to their extensive applications, especially in financial time series, biology and physics. This paper pays attention to a special stochastic process, tempered fractional Langevin motion, which is non-Markovian and undergoes ballistic diffusion for long times. The corresponding time-changed Langevin system with an inverse β-stable subordinator is discussed in detail, including its diffusion type, moments, Klein–Kramers equation, and the correlation structure. Interestingly, this subordination could result in both subdiffusion and superdiffusion, depending on the value of β. The difference between the subordinated tempered fractional Langevin equation and the subordinated Langevin equation with external biasing force is studied for a deeper understanding of the subordinator. The time-changed tempered fractional Brownian motion by an inverse β-stable subordinator is also considered, as well as the correlation structure of its increments. Some properties of the statistical quantities of the time-changed process are discussed, displaying striking differences compared with the original process.

Quantifying the validity and breakdown of the overdamped approximation in stochastic thermodynamics: Theory and experiment

Stochastic thermodynamics provides an important framework to explore small physical systems where thermal fluctuations are inevitable. In the studies of stochastic thermodynamics, some thermodynamic quantities, such as the trajectory work, associated with the complete Langevin equation (the Kramers equation) are often assumed to converge to those associated with the overdamped Langevin equation (the Smoluchowski equation) in the overdamped limit under the overdamped approximation. Nevertheless, a rigorous mathematical proof of the convergence of the work distributions to our knowledge has not been reported so far. Here we study the convergence of the work distributions explicitly. In the overdamped limit, we rigorously prove the convergence of the extended Fokker-Planck equations including work using a multiple timescale expansion approach. By taking the linearly dragged harmonic oscillator as an exactly solvable example, we analytically calculate the work distribution associated with the Kramers equation, and verify its convergence to that associated with the Smoluchowski equation in the overdamped limit. We quantify the accuracy of the overdamped approximation as a function of the damping coefficient. In addition, we experimentally demonstrate that the data of the work distribution of a levitated silica nanosphere agrees with the overdamped approximation in the overdamped limit, but deviates from the overdamped approximation in the low-damping case. Our work fills a gap between the stochastic thermodynamics based on the complete Langevin equation (the Kramers equation) and the overdamped Langevin equation (the Smoluchowski equation), and deepens our understanding of the overdamped approximation in stochastic thermodynamics.

One- and Two-dimensional Solitary Wave States in the Nonlinear Kramers Equation with Movement Direction as a Variable

We study self-propelled particles by direct numerical simulation of the nonlinear Kramers equation for self-propelled particles. In our previous paper, we studied self-propelled particles with velocity variables in one dimension. In this paper, we consider another model in which each particle exhibits directional motion. The movement direction is expressed with a variable $\phi$. We show that one-dimensional solitary wave states appear in direct numerical simulations of the nonlinear Kramers equation in one- and two-dimensional systems, which is a generalization of our previous result. Furthermore, we find two-dimensionally localized states in the case that each self-propelled particle exhibits rotational motion. The center of mass of the two-dimensionally localized state exhibits circular motion, which implies collective rotating motion. Finally, we consider a simple one-dimensional model equation to qualitatively understand the formation of the solitary wave state.

Nonequilibrium work relation from Schrödinger’s unrecognized probability theory

Jarzynski’s nonequilibrium work relation can be understood as the realization of the (hidden) time-generator reciprocal symmetry satisfied for the conditional probability function. To show this, we introduce the reciprocal process where the classical probability theory is expressed with real wave functions, and derive a mathematical relation using the symmetry. We further discuss that the descriptions by the standard Markov process from an initial equilibrium state are indistinguishable from those by the reciprocal process. Then the Jarzynski relation is obtained from the mathematical relation for the Markov processes described by the Fokker–Planck, Kramers and relativistic Kramers equations.

On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type

In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previously known results for the Fokker-Planck equation and the Kramers equation.

Solitary Wave State in the Nonlinear Kramers Equation for Self-Propelled Particles

We study collective phenomena of self-propagating particles using the nonlinear Kramers equation. A solitary wave state appears from an instability of the spatially uniform ordered state with nonzero average velocity. Two solitary waves with different heights merge into a larger solitary wave. An approximate solution of the solitary wave is constructed using a self-consistent method. The phase transition to the solitary wave state is either first-order or second-order, depending on the control parameters.

Brownian motion of a classical particle in quantum environment

The Klein–Kramers equation, governing the Brownian motion of a classical particle in a quantum environment under the action of an arbitrary external potential, is derived. Quantum temperature and friction operators are introduced and at large friction the corresponding Smoluchowski equation is obtained. Introducing the Bohm quantum potential, this Smoluchowski equation is extended to describe the Brownian motion of a quantum particle in quantum environment.

Classical cumulant dynamics for statistical chemical physics

We developed classical cumulant dynamics for statistical mechanics in order to evaluate thermal equilibrium properties of a given system. The equations of motion (EOMs) for momentum and position were formulated together with those for second-order cumulant variables, which are functions of second-order moments. From the Kramers equation, and simplified EOMs were obtained by assuming a stationary state limit. The present method combined with the umbrella integration method was applied to evaluate free energy surface of a seven-particle Morse cluster. With low computational costs, the present approach gave almost equivalent free energy barrier those by conventional classical molecular dynamics.

Effective long wavelength scalar dynamics in de Sitter

We discuss the effective infrared theory governing a light scalar's long wavelength dynamics in de Sitter spacetime. We show how the separation of scales around the physical curvature radius k/a ∼ H can be performed consistently with a window function and how short wavelengths can be integrated out in the Schwinger-Keldysh path integral formalism. At leading order, and for time scales Δ t ≫ H−1, this results in the well-known Starobinsky stochastic evolution. However, our approach allows for the computation of quantum UV corrections, generating an effective potential on which the stochastic dynamics takes place. The long wavelength stochastic dynamical equations are now second order in time, incorporating temporal scales Δ t ∼ H−1 and resulting in a Kramers equation for the probability distribution—more precisely the Wigner function—in contrast to the more usual Fokker-Planck equation. This feature allows us to non-perturbatively evaluate, within the stochastic formalism, not only expectation values of field correlators, but also the stress-energy tensor of ϕ.

Heat, temperature and Clausius inequality in a model for active Brownian particles

Methods of stochastic thermodynamics and hydrodynamics are applied to a recently introduced model of active particles. The model consists of an overdamped particle subject to Gaussian coloured noise. Inspired by stochastic thermodynamics, we derive from the system’s Fokker-Planck equation the average exchanges of heat and work with the active bath and the associated entropy production. We show that a Clausius inequality holds, with the local (non-uniform) temperature of the active bath replacing the uniform temperature usually encountered in equilibrium systems. Furthermore, by restricting the dynamical space to the first velocity moments of the local distribution function we derive a hydrodynamic description where local pressure, kinetic temperature and internal heat fluxes appear and are consistent with the previous thermodynamic analysis. The procedure also shows under which conditions one obtains the unified coloured noise approximation (UCNA): such an approximation neglects the fast relaxation to the active bath and therefore yields detailed balance and zero entropy production. In the last part, by using multiple time-scale analysis, we provide a constructive method (alternative to UCNA) to determine the solution of the Kramers equation and go beyond the detailed balance condition determining negative entropy production.

Variational Derivation of Dissipative Equations

A new variational principle is formulated to derive various dissipative equations. Model equations considered are the damping equation, Bloch equation, diffusion equation, Fokker–Planck equation, Kramers equation and Smoluchowski equation. Each equation and its time reversal equation are simultaneously obtained in our variational principle.

Insights into Active Matter from Coupled Oscillators

Researchers have studied collective phenomena of self-propelled particles using nonlinear Kramers equations, often employed for analyzing coupled oscillator systems. In addition, they have discussed the transitions between the spatially-uniform disordered, spatially-uniform ordered, and solitary wave states.

Yield, Esterification Degree and Molecular Weight Evaluation of Pectins Isolated from Orange and Grapefruit Peels under Different Conditions.

Orange (Citrus sinensis) and grapefruit (Citrus paradise) peels were used as a source of pectin, which was extracted under different conditions. The peels are used under two states: fresh and residual (after essential oil extraction). Organic acid (citric acid) and mineral acid (sulfuric acid) were used in the pectin extraction. The aim of this study is the evaluation the effect of extraction conditions on pectin yield, degree of esterification “DE” and on molecular weight “Mw”. Results showed that the pectin yield was higher using the residual peels. Moreover, both peels allow the obtainment of a high methoxyl pectin with DE >50%. The molecular weight was calculated using Mark-Houwink-Sakurada equation which describes its relationship with intrinsic viscosity. This later was determined using four equations; Huggins equation, kramer, Schulz-Blaschke and Martin equation. The molecular weight varied from 1.538 x1005 to 2.47x1005 g/mol for grapefruit pectin and from 1.639 x1005 to 2.471 x1005 g/mol for orange pectin.

Lie Symmetries and Fundamental Solutions of the Linear Kramers Equation

Проблематика. В статье проведен теоретико-групповой анализ фундаментальных решений одномерного линейного уравнения Крамерса. Цель исследования. Используя метод Аксенова–Береста, найти алгебру инвариантности фундаментальных решений исследуемого уравнения, а также построить фундаментальное решение уравнения в явном виде, используя найденную алгебру симметрий. Методика реализации. Используются методы теоретико-группового анализа дифференциальных уравнений с частными производными, в частности метод Аксенова–Береста построения в явном виде фундаментальных решений линейных дифференциальных уравнений. Результаты исследования. Найдена алгебра Ли нетривиальных симметрий исследуемого одномерного линейного уравнения Крамерса. Построено в явном виде в элементарных функциях фундаментальное решение этого уравнения. Показана эффективность использования симметрийных методов для исследования фундаментальных решений линейных уравнений Колмогорова–Фоккера–Планка. Выводы. С использованием метода Аксенова–Береста найдена алгебра инвариантности фундаментальных решений одного одномерного линейного уравнения Крамерса, операторы которой были использованы для построения инвариантных фундаментальных решений этого уравнения. Показано, что фундаментальное решение этого уравнения, найденное ранее С . Чандрасекаром без использования методов симметрийного анализа дифференциальных уравнений, является инвариантным фундаментальным решением.

Probabilistic density function method for nonlinear dynamical systems driven by colored noise.

We present a probability density function (PDF) method for a system of nonlinear stochastic ordinary differential equations driven by colored noise. The method provides an integrodifferential equation for the temporal evolution of the joint PDF of the system's state, which we close by means of a modified large-eddy-diffusivity (LED) closure. In contrast to the classical LED closure, the proposed closure accounts for advective transport of the PDF in the approximate temporal deconvolution of the integrodifferential equation. In addition, we introduce the generalized local linearization approximation for deriving a computable PDF equation in the form of a second-order partial differential equation. We demonstrate that the proposed closure and localization accurately describe the dynamics of the PDF in phase space for systems driven by noise with arbitrary autocorrelation time. We apply the proposed PDF method to analyze a set of Kramers equations driven by exponentially autocorrelated Gaussian colored noise to study nonlinear oscillators and the dynamics and stability of a power grid. Numerical experiments show the PDF method is accurate when the noise autocorrelation time is either much shorter or longer than the system's relaxation time, while the accuracy decreases as the ratio of the two timescales approaches unity. Similarly, the PDF method accuracy decreases with increasing standard deviation of the noise.

Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics

A popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions. One common model is the Smoluchowski partial adsorption condition, which uses a Robin boundary condition in the separation coordinate between two possible reactants. This boundary condition can be interpreted as an idealization of a reactive interaction potential model, in which a potential barrier must be surmounted before reactions can occur. In this work we show how the reactive boundary condition arises as the limit of an interaction potential encoding a steep barrier within a shrinking region in the particle separation, where molecules react instantly upon reaching the peak of the barrier. The limiting boundary condition is derived by the method of matched asymptotic expansions, and is shown to depend critically on the relative rate of increase of the barrier height as the width of the potential is decreased. Limiting boundary conditions for the same interaction potential in both the overdamped Fokker--Planck equation (Brownian dynamics) and the Kramers equation (Langevin dynamics) are investigated. It is shown that different scalings are required in the two models to recover reactive boundary conditions that are consistent in the high friction limit (where the Kramers equation solution converges to the solution of the Fokker--Planck equation).

Stochastic entropy production arising from nonstationary thermal transport.

We compute statistical properties of the stochastic entropy production associated with the nonstationary transport of heat through a system coupled to a time dependent nonisothermal heat bath. We study the one-dimensional stochastic evolution of a bound particle in such an environment by solving the appropriate Langevin equation numerically, and by using an approximate analytic solution to the Kramers equation to determine the behavior of an ensemble of systems. We express the total stochastic entropy production in terms of a relaxational or nonadiabatic part together with two components of housekeeping entropy production and determine the distributions for each, demonstrating the importance of all three contributions for this system. We compare the results with an approximate analytic model of the mean behavior and we further demonstrate that the total entropy production and the relaxational component approximately satisfy detailed fluctuation relations for certain time intervals. Finally, we comment on the resemblance between the procedure for solving the Kramers equation and a constrained extremization, with respect to the probability density function, of the spatial density of the mean rate of production of stochastic entropy.

Some physico-chemical properties of Prunus armeniaca L. gum exudates

The objectives of this paper were to investigate some physicochemical properties of Prunus armeniaca L. gum exudates (PAGE). PAGE had, on average, 66.89% carbohydrate, 10.47% uronic acids, 6.9% moisture (w.b.), 2.91% protein, 4% ash and 1.59% fat. PAGE was composed of monosaccharides including l-arabinose, d-galactose, xylose, mannose and rhamnose in molar percentages of 41.52%, 23.72%, 17.82%, 14.40% and 2.54%, respectively. Elemental analysis showed that PAGE had high values of nutrients. FTIR analysis demonstrated the presence of carboxyl, hydroxyl and methyl groups and glycoside bonds. The weight average molecular weight, number average molecular weight and polydispersity index were found to be approximately 5.69×105g/mol, 4.33g/mol and 1.31, respectively. Rheological measurement of PAGE solutions as a function of concentration (8, 10 and 12% (w/w)) and temperature (10, 20, 30 and 40°C) demonstrated that the gum solutions had a non Newtonian shear thinning behaviour. Intrinsic viscosity for PAGE in deionized water was 3.438dl/g based on Kramer equation.