Classical Kramers Research Articles

Classical and quantum thermodynamics in a non-equilibrium regime: Application to thermostatic Stirling engine

We have developed a thermodynamic theory in the non-equilibrium regime, which we describe as a thermodynamic system–bath model [Koyanagi and Tanimura, J. Chem. Phys. 160, 234112 (2024)]. Based on the dimensionless (DL) minimum work principle, non-equilibrium thermodynamic potentials are expressed in terms of non-equilibrium extensive and intensive variables in time derivative form. This is made possible by incorporating the entropy production rate into the definition of non-equilibrium thermodynamic potentials. These potentials can be evaluated from the DL non-equilibrium-to-equilibrium minimum work principle, which is derived from the principle of DL minimum work and is equivalent to the second law of thermodynamics. We thus obtain the non-equilibrium Massieu–Planck potentials as entropic potentials and the non-equilibrium Helmholtz–Gibbs potentials as free energies. Unlike the fluctuation theorem and stochastic thermodynamics theory, this theory does not require the assumption of a factorized initial condition and is valid in the full quantum regime, where the system and bath are quantum mechanically entangled. Our results are numerically verified by simulating a thermostatic Stirling engine consisting of two isothermal processes and two thermostatic processes using the quantum hierarchical Fokker–Planck equations and the classical Kramers equation derived from the thermodynamic system–bath model. We then show that, from weak to strong system–bath interactions, the thermodynamic process can be analyzed using a non-equilibrium work diagram analogous to the equilibrium one for given time-dependent intensive variables. The results can be used to develop efficient heat machines in non-equilibrium regimes.

Proton tunneling in a two-dimensional potential energy surface with a non-linear system-bath interaction: Thermal suppression of reaction rate.

We consider a proton transfer (PT) system described by a proton transfer reaction (PTR) coordinate and a rate promoting vibrational (RPV) coordinate interacting with a non-Markovian heat bath. While dynamics of PT processes has been widely discussed using two-dimensional potential energy surfaces, the role of the heat bath, in particular, in a realistic form of the system-bath interaction has not been well explored. Previous studies are largely based on a one-dimensional model and linear-linear system-bath interaction. In the present study, we introduce an exponential-linear (EL) system-bath interaction, which is derived from the analysis of a PTR-RPV system in a realistic situation. This interaction mainly causes vibrational dephasing in the PTR mode and population relaxation in the RPV mode. Numerical simulations were carried out using the hierarchical equations of motion approach. We analyze the role of the heat bath interaction in the chemical reaction rate as a function of the system-bath coupling strength at different temperatures and for different values of the bath correlation time. A prominent feature of the present result is that while the reaction rate predicted from classical and quantum Kramers theory increases as the temperature increases, the present EL interaction model exhibits opposite temperature dependence. The Kramers turn-over profile of the reaction rate as a function of the system-bath coupling is also suppressed in the present EL model, turning into a plateau-like curve for larger system-bath interaction strength. Such features arise from the interplay of the vibrational dephasing process in the PTR mode and the population relaxation process in the RPV mode.

Quantum Kramers model: Corrections to the linear response theory for continuous bath spectrum.

Decay of the metastable state is analyzed within the quantum Kramers model in the weak-to-intermediate dissipation regime. The decay kinetics in this regime is determined by energy exchange between the unstable mode and the stable modes of thermal bath. In our previous paper [Phys. Rev. A 42, 4427 (1990)PLRAAN1050-294710.1103/PhysRevA.42.4427], Grabert's perturbative approach to well dynamics in the case of the discrete bath [Phys. Rev. Lett. 61, 1683 (1988)PRLTAO0031-900710.1103/PhysRevLett.61.1683] has been extended to account for the second order terms in the classical equations of motion (EOM) for the stable modes. Account of the secular terms reduces EOM for the stable modes to those of the forced oscillator with the time-dependent frequency (TDF oscillator). Analytic expression for the characteristic function of energy loss of the unstable mode has been derived in terms of the generating function of the transition probabilities for the quantum forced TDF oscillator. In this paper, the approach is further developed and applied to the case of the continuous frequency spectrum of the bath. The spectral density functions of the bath of stable modes are expressed in terms of the dissipative properties (the friction function) of the original bath. They simplify considerably for the one-dimensional systems, when the density of phonon states is constant. Explicit expressions for the fourth order corrections to the linear response theory result for the characteristic function of the energy loss and its cumulants are obtained for the particular case of the cubic potential with Ohmic (Markovian) dissipation. The range of validity of the perturbative approach in this case is determined (γ/ω_{b}<0.26), which includes the turnover region. The dominant correction to the linear response theory result is associated with the "work function" and leads to reduction of the average energy loss and its dispersion. This reduction increases with the increasing dissipation strength (up to ∼10%) within the range of validity of the approach. We have also calculated corrections to the depopulation factor and the escape rate for the quantum and for the classical Kramers models. Results for the classical escape rate are in very good agreement with the numerical simulations for high barriers. The results can serve as an additional proof of the robustness and accuracy of the linear response theory.

A Protein Nanopore-Based Approach for Bacteria Sensing.

We present herein a first proof of concept demonstrating the potential of a protein nanopore-based technique for real-time detection of selected Gram-negative bacteria (Pseudomonas aeruginosa or Escherichia coli) at a concentration of 1.2 × 108 cfu/mL. The anionic charge on the bacterial outer membrane promotes the electrophoretically driven migration of bacteria towards a single α-hemolysin nanopore isolated in a lipid bilayer, clamped at a negative electric potential, and followed by capture at the nanopore’s mouth, which we found to be described according to the classical Kramers’ theory. By using a specific antimicrobial peptide as a putative molecular biorecognition element for the bacteria used herein, we suggest that the detection system can combine the natural sensitivity of the nanopore-based sensing techniques with selective biological recognition, in aqueous samples, and highlight the feasibility of the nanopore-based platform to provide portable, sensitive analysis and monitoring of bacterial pathogens.Electronic supplementary materialThe online version of this article (doi:10.1186/s11671-016-1715-z) contains supplementary material, which is available to authorized users.

Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead–spring chain models for dilute polymers: The two-dimensional case

We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead–spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier–Stokes system in a bounded domain Ω in Rd, d=2, for the density ρ, the velocity u˜ and the pressure p of the fluid, with an equation of state of the form p(ρ)=cpργ, where cp is a positive constant and γ>1. The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the classical Kramers expression. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. This extends the result in our paper J.W. Barrett and E. Süli (2016) [9], which established the existence of global-in-time weak solutions to the system for d∈{2,3} and γ>32, but the elastic extra-stress tensor required there the addition of a quadratic interaction term to the classical Kramers expression to complete the compactness argument on which the proof was based. We show here that in the case of d=2 and γ>1 the existence of global-in-time weak solutions can be proved in the absence of the quadratic interaction term. Our results require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a nonnegative initial density ρ0∈L∞(Ω) for the continuity equation; a square-integrable initial velocity datum u˜0 for the Navier–Stokes momentum equation; and a nonnegative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M associated with the spring potential in the model, we prove, via a limiting procedure on a pressure regularization parameter, the existence of a global-in-time bounded-energy weak solution t↦(ρ(t),u˜(t),ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ρ(0),u˜(0),ψ(0))=(ρ0,u˜0,ψ0).

Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier–Stokes system in a bounded domain [Formula: see text] in [Formula: see text], [Formula: see text] or [Formula: see text], for the density [Formula: see text], the velocity [Formula: see text] and the pressure [Formula: see text] of the fluid, with an equation of state of the form [Formula: see text], where [Formula: see text] is a positive constant and [Formula: see text]. The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the sum of the classical Kramers expression and a quadratic interaction term. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term.

Theoretical approach to modelling the low-barrier chemical reactions initiated by pulsed electron beam

A possibility to analyze low-barrier chemical reactions induced by electron beam is proposed within the framework of generalization of classical Kramers approach. A relationship for calculation of chemical reaction rate is received for potential barriers comparable with energy of system. It has been shown that results of our approach agree with results of dynamical modeling within 2% not only in the area of applicability of classical (Kramers) approach but also more widely.

Characterization of early-age hydration process of cement pastes based on impedance measurement

This paper investigates the hydration process of cement pastes using an isothermal calorimetry, Fourier transforms infrared spectroscopy (FTIR), X-ray diffraction (XRD) analysis and non-contact impedance measurement (NCIM). The reliability of impedance data from NCIM is checked by classical Kramers–Kronig transformation. Hydration behaviors of pure cement paste in different stages (dissolution, competition and acceleration stages) are interpreted by FTIR, XRD and NCIM. Influences of water to cement ratios, fly ash, slag, silica fume and curing temperature are investigated by compressive strength test and NCIM. It is found that the compressive strength and impedance data have a good linear relationship.

Bistable stochastic biochemical networks: large chemical networks and systems with many molecules

In this paper we continue the program started in Hwang and Velazquez (J Math Chem, to appear). We describe some chemical systems exhibiting bistable behavior with reaction constants of order one, but where bistability is due to the presence of a large number of chemical species or a large number of molecules of some of the species. We derive generalizations of the classical Kramers’ formula that gives the switching times for some particular systems exhibiting a large number of species.

Bistable stochastic biochemical networks: highly specific systems with few chemicals

In this paper we describe a class of stochastic biochemical systems exhibiting bistable behavior, in the sense that the invariant measure associated to the system is concentrated near two different classes of states of the system. We develop methods that allow us to describe the behavior of the invariant measure in some suitable asymptotic limits, as well as the switching times for the transitions between the states close to each of the states with high probability. Due to the discrete character of the problem, switching times cannot be computed using the classical Kramers’ formula, and alternative methods are required.

Escape of a driven particle from a metastable state: A semiclassical approach

In this article we explore the dynamics of escape of a particle in the semiclassical regime by driving the particle externally. We demonstrate that under suitable approximations the semiclassical escape rate essentially assumes the structure of classical Kramers rate. Both internal (due to thermal bath) as well as external noises (due to driving) are being considered. The noises are stationary, Gaussian, and are characterized by arbitrary decaying memory kernel. Finally, we subject our formulation to rigorous numerical test under variedly changing conditions of the parameters.

Generalization of the classical Kramers rate for non-Markovian open systems out of equilibrium

We analyze the behavior of a Brownian particle moving in a double-well potential. The escape probability of this particle over the potential barrier from a metastable state toward another state is known as the Kramers problem. In this work, we generalize Kramers’ rate theory to the case of an environment always out of thermodynamic equilibrium reckoning with non-Markovian effects.

Energy diffusion controlled reaction rate of reacting particle driven by broad-band noise

The energy diffusion controlled reaction rate of a reacting particle with linear weak damping and broad-band noise excitation is studied by using the stochastic averaging method. First, the stochastic averaging method for strongly nonlinear oscillators under broad-band noise excitation using generalized harmonic functions is briefly introduced. Then, the reaction rate of the classical Kramers' reacting model with linear weak damping and broad-band noise excitation is investigated by using the stochastic averaging method. The averaged Ito stochastic differential equation describing the energy diffusion and the Pontryagin equation governing the mean first-passage time (MFPT) are established. The energy diffusion controlled reaction rate is obtained as the inverse of the MFPT by solving the Pontryagin equation. The results of two special cases of broad-band noises, i.e. the harmonic noise and the exponentially corrected noise, are discussed in details. It is demonstrated that the general expression of reaction rate derived by the authors can be reduced to the classical ones via linear approximation and high potential barrier approximation. The good agreement with the results of the Monte Carlo simulation verifies that the reaction rate can be well predicted using the stochastic averaging method.

Kramers’ law for a bistable system with time-delayed noise

We demonstrate that the classical Kramers' escape problem can be extended to describe a bistable system under the influence of noise consisting of the superposition of a white Gaussian noise with the same noise delayed by time tau . The distribution of times between two consecutive switches decays piecewise exponentially, and the switching rates for 0<t<tau and tau<t<2tau are calculated analytically using the Langevin equation. These rates are different since, for the particles remaining in one well for longer than tau, the delayed noise acquires a nonzero mean value and becomes negatively autocorrelated. To account for these effects we define an effective potential and an effective diffusion coefficient of the delayed noise.

An exact formulation of hyperdynamics simulations

We introduce a new formula for the acceleration weight factor in the hyperdynamics simulation method, the use of which correctly provides an exact simulation of the true dynamics of a system. This new form of hyperdynamics is valid and applicable where the transition state theory (TST) is applicable and also where the TST is not applicable. To illustrate this new formulation, we perform hyperdynamics simulations for four systems ranging from one degree of freedom to 591 degrees of freedom: (1) We first analyze free diffusion having one degree of freedom. This system does not have a transition state. The TST and the original form of hyperdynamics are not applicable. Using the new form of hyperdynamics, we compute mean square displacement for a range of time. The results obtained agree perfectly with the analytical formula. (2) Then we examine the classical Kramers escape rate problem. The rate computed is in perfect agreement with the Kramers formula over a broad range of temperature. (3) We also study another classical problem: Computing the rate of effusion out of a cubic box through a tiny hole. This problem does not involve an energy barrier. Thus, the original form of hyperdynamics excludes the possibility of using a nonzero bias and is inappropriate. However, with the new weight factor formula, our new form of hyperdynamics can be easily implemented and it produces the exact results. (4) To illustrate applicability to systems of many degrees of freedom, we analyze diffusion of an atom adsorbed on the (001) surface of an fcc crystal. The system is modeled by an atom on top of a slab of six atomic layers. Each layer has 49 atoms. With the bottom two layers of atoms fixed, this system has 591 degrees of freedom. With very modest computing effort, we are able to characterize its diffusion pathways in the exchange-with-the-substrate and hop-over-the-bridge mechanisms.

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first-passage time (MFPT) of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kramers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.

GL(2, Bbb R) dualities in generalised Z(2) gauge theories and Ising models

We study a class of duality transformations in generalised Z(2) gauge theories and Ising models on two- and three-dimensional compact lattices. The theories are interpreted algebraically in terms of the structure constants of a bidimensional vector space H with algebra and coalgebra structures, and it is shown that for any change of basis in H there is a related symmetry between such models. The classical Kramers and Wannier dualities are described as special cases of these transformations. We derive explicit expressions for the relation between partition functions on general finite triangulations for these cases, extending results known for square and cubic lattices in the thermodynamical limit. A class of symmetry transformations in which the gauge coupling changes continuously is also studied in two dimensions.

Classical q-deformed dynamics

On the basis of the quantum q-oscillator algebra in the framework of quantum groups and non-commutative q-differential calculus, we investigate a possible q-deformation of the classical Poisson bracket in order to extend a generalized q-deformed dynamics in the classical regime. In this framework, classical q-deformed kinetic equations, Kramers and Fokker-Planck equations, are also studied. Pacs: 05.20.Dd, 45.20.-d, 02.20.Uw Keywords: Kinetic theory, q-deformed classical mechanics, quantum groups, quantum algebras

Radiative recombination processes of Bi80+

The Dirac_Slater method is used to calculate the photoionization cross sections of Bi79+ from low to high energy. The numerical results are compared with those calculated by the quasi_classical Kramers formula, and the validity of Kra mers formula is analyzed. The relativistic effects and multipole contributions o f the photoionization processes are also discussed. For the comparison with the merged_beam rate coefficients, the theoretical rate coefficients were evaluated with the experimental electron_velocity distribution function(characterized by t he temperatures) parallel and perpendicular to the electron_beam direction. Thou gh the effects of relativity are considered in the present work, the measured ra te coefficients exceed our results in the energy range close to the threshold.

Quantum phase-space function formulation of reactive flux theory

On the basis of a coherent-state representation of the quantum noise operator and an ensemble averaging procedure a scheme for quantum Brownian motion has been proposed recently [Banerjee et al., Phys. Rev. E 65, 021109 (2002); 66, 051105 (2002)]. We extend this approach to formulate reactive flux theory in terms of quantum phase space distribution functions and to derive a time-dependent quantum transmission coefficient—a quantum analog of the classical Kramers–Grote–Hynes coefficient in the spirit of Kohen and Tannor’s classical formulation. The theory is valid for arbitrary noise correlation and temperature. The specific forms of this coefficient in the Markovian as well as in the non-Markovian limits have been worked out in detail for the intermediate to strong damping regimes with an analysis of quantum effects. While the classical transmission coefficient is independent of temperature, its quantum counterpart has significant temperature dependence particularly in the low-temperature regime.