Phenomenological Langevin Equation Research Articles

Mechanism of Proton Pumping in Complex I of the Mitochondrial Respiratory Chain

We propose a physical mechanism of conformation-induced proton pumping in mitochondrial Complex I. The structural conformations of this protein are modeled as the motion of a piston having positive charges on both sides. A negatively charged electron attracts the piston, moving the other end away from the proton site, thereby reducing its energy and allowing a proton to populate the site. When the electron escapes, elastic forces assist the return of the piston, increasing proton site energy and facilitating proton transfer. We derive the Heisenberg equations of motion for electron and proton operators and rewrite them in the form of rate equations coupled to the phenomenological Langevin equation describing piston dynamics. This set of coupled equations is solved numerically. We show that proton pumping can be achieved within this model for a reasonable set of parameters. The dependencies of proton current on geometry, temperature, and other parameters are examined.

Comment on "Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass".

In a recent paper by B.G. da Costa et al. [Phys. Rev. E 102, 062105 (2020)2470-004510.1103/PhysRevE.102.062105], the phenomenological Langevin equation and the corresponding Fokker-Planck equation for an inhomogeneous medium with a position-dependent particle mass and position-dependent damping coefficient have been studied. The aim of this comment is to present a microscopic derivation of the Langevin equation for such a system. It is not equivalent to that in the commented paper.

Physical model of proton-pumping Q-cycle in respiratory and photosynthetic electron transport chains

We examine proton pumping in bc-type complexes within respiratory and photosynthetic electron transport chains. Our simple physical model includes a process of population of a first quinone molecule by two electrons and two protons at the N-side of the membrane which is followed by the diffusion of this quinone toward the unloading sites of the bc-complex. Then translocation of the protons to the P-side of the membrane occurs. One of the electrons proceeds to the drain reservoir and the second one populates a second quinone at the N-side to pump more protons. We derive the equations of motion for the proton and electron operators, rewrite them as a set of the rate equations, and couple them to phenomenological Langevin equations governing the motion of the shuttles. We solve the resulting equations numerically and determine the time dependencies of the shuttles positions and electron populations, as well as the quantum yield of the proton pumping and its thermodynamic efficiency.

Stochastic resonance in a proton pumping Complex I of mitochondria membranes

We make use of the physical mechanism of proton pumping in the so-called Complex I within mitochondria membranes. Our model is based on sequential charge transfer assisted by conformational changes which facilitate the indirect electron-proton coupling. The equations of motion for the proton operators are derived and solved numerically in combination with the phenomenological Langevin equation describing the periodic conformational changes. We show that with an appropriate set of parameters, protons can be transferred against an applied voltage. In addition, we demonstrate that only the joint action of the periodic energy modulation and thermal noise leads to efficient uphill proton transfer, being a manifestation of stochastic resonance.

A combined quasi-continuum/Langevin equation approach to study the self-diffusion dynamics of confined fluids

In this work, we combine our earlier proposed empirical potential based quasi-continuum theory, (EQT) [A. V. Raghunathan, J. H. Park, and N. R. Aluru, J. Chem. Phys. 127, 174701 (2007)], which is a coarse-grained multiscale framework to predict the static structure of confined fluids, with a phenomenological Langevin equation to simulate the dynamics of confined fluids in thermal equilibrium. An attractive feature of this approach is that all the input parameters to the Langevin equation (mean force profile of the confined fluid and the static friction coefficient) can be determined using the outputs of the EQT and the self-diffusivity data of the corresponding bulk fluid. The potential of mean force profile, which is a direct output from EQT is used to compute the mean force profile of the confined fluid. The density profile, which is also a direct output from EQT, along with the self-diffusivity data of the bulk fluid is used to determine the static friction coefficient of the confined fluid. We use this approach to compute the mean square displacement and survival probabilities of some important fluids such as carbon-dioxide, water, and Lennard-Jones argon confined inside slit pores. The predictions from the model are compared with those obtained using molecular dynamics simulations. This approach of combining EQT with a phenomenological Langevin equation provides a mathematically simple and computationally efficient means to study the impact of structural inhomogeneity on the self-diffusion dynamics of confined fluids.

Solvent-Induced Acceleration of the Rate of Activation of a Molecular Reaction

An increase in the rates of activated processes with the coupling to the solvent has long been predicted through the phenomenological Langevin equation in the weak coupling regime. However, its direct observation in particle-based models has been elusive because the coupling typically places the processes in the spacial-diffusion limited regime wherein rates decrease with increasing friction. In this work, the forward and backward reaction rates of the LiNC<==>LiCN isomerization reaction in a bath of argon atoms at various densities have been calculated directly using molecular dynamics trajectories. The so-called Kramers turnover in the rate with microscopic friction is clearly visible, thus providing direct and unambiguous evidence for the energy-diffusion regime in which rates increase with friction.

Dressed‐particle formulation of Brownian motion

AbstractWe consider a simple model of a classical harmonic oscillator coupled to a field. In standard approaches Langevin‐type equations for bare particles are derived from Hamiltonian dynamics. These equations contain memory terms and are time‐reversal invariant. In contrast, the phenomenological Langevin equations have no memory terms (they are Markovian equations) and give a time evolution split in two branches (semigroups), each of which breaks time symmetry. A standard approach to bridge dynamics with phenomenology is to consider the Markovian approximation of the former. In the current article, we present a formulation in terms of dressed particles, which gives exact Markovian equations. We formulate dressed particles for Poincaré nonintegrable systems, through an invertible transformation operator Λ introduced by Prigogine and collaborators. Λ is obtained by an extension of the canonical (unitary) transformation operator U that eliminates interactions for integrable systems. Our extension is based on the removal of divergences due to Poincaré resonances, which breaks time symmetry. The unitarity of U is extended to “star unitarity” for Λ. We show that Λ‐transformed variables have the same time evolution as stochastic variables obeying Langevin equations and that Λ‐transformed distribution functions satisfy an exact Fokker–Planck equation. The effects of Gaussian white noise are obtained by the nondistributive property of Λ with respect to products of dynamical variables. Therefore, our method leads to a direct link between dynamics of Poincaré nonintegrable systems, probability, and stochasticity. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Star-unitary transformations: from dynamics to irreversibility and stochastic behavior.

We consider a simple model of a classical harmonic oscillator coupled to a field. In standard approaches, Langevin-type equations for bare particles are derived from Hamiltonian dynamics. These equations contain memory terms and are time-reversal invariant. In contrast, the phenomenological Langevin equations have no memory terms (they are Markovian equations) and give a time-evolution split in two branches (semigroups), each of which breaks time symmetry. A standard approach to bridge dynamics with phenomenology is to consider the Markovian approximation of the former. In this paper, we present a formulation in terms of dressed particles, which gives exact Markovian equations. We formulate dressed particles for Poincaré nonintegrable systems, through an invertible transformation operator Lambda introduced by Prigogine and co-workers. Lambda is obtained by an extension of the canonical (unitary) transformation operator U that eliminates interactions for integrable systems. Our extension is based on the removal of divergences due to Poincaré resonances, which breaks time symmetry. The unitarity of U is extended to "star unitarity" for Lambda. We show that Lambda-transformed variables have the same time evolution as stochastic variables obeying Langevin equations, and that Lambda-transformed distribution functions satisfy exact Fokker-Planck equations. The effects of Gaussian white noise are obtained by the nondistributive property of Lambda with respect to products of dynamical variables.

Stochastic motion of 7 × 7 kinks at monoatomic step edges on the Si(1 1 1) surface

An offset of a straight step, called a kink, is occasionally formed on semiconductor surfaces. The motion of the kink on the Si(1 1 1) 7×7 surface in the [ 1 ̄ 1 ̄ 2 ] step was studied in detail by high-temperature scanning tunneling microscopy (STM), and thermal fluctuations of the kink displacement along the step edges was observed. The kink displacement did not diverge with time, suggesting that a restoring force acts on the kink. The displacement, however, could be clearly represented by the gaussian distribution and it was therefore considered to be a Brownian particle. The temperature dependence of the mean square displacement of the kink position showed that the displacement is a thermal activation process with an apparent activation energy of 1.54±0.1 eV. From the equation of motion on the kink displacement including an incoming and outgoing flux as a fluctuation source, the phenomenological Langevin equation was derived. The activation energy of the kink displacement is related to the diffusion coefficient of the two-dimensional adatom gas and the latent heat of the atoms from the kink site to the surface adatom.

Optical four wave mixing spectroscopy for multilevel systems coupled to multimode Brownian oscillators

We generalize the multimode Brownian oscillator model for a two-level system to optical four wave mixing spectroscopy of a multilevel system, based on the recent work of Sung and Silbey [J. Chem. Phys. 115, 9266 (2001)]. For the Ohmic limit where the phenomenological Langevin equation becomes correct, we present exact expressions for energy-gap correlation functions in the strong damping case as well as in the weak damping case.

NUMERICAL ASPECTS OF BUBBLE NUCLEATION

Bubble nucleation has been studied on lattices using phenomenological Langevin equations. Recently there have been theoretical motivations for using these equations. These studies also conclude that the simple Langevin description requires some modification. We study bubble nucleation on a lattice and determine effects of the modified Langevin equations.

Diffusion and electromigration on disordered surfaces

We study a one-dimensional disordered solid-on-solid model in which neighboring columns are shifted by quenched random phases. The static height-difference correlation function displays a minimum at a nonzero temperature. The model is equipped with volume-conserving surface diffusion dynamics, including a possible bias due to an electromigration force. In the case of Arrhenius jump rates a continuum equation for the evolution of macroscopic profiles is derived and confirmed by direct simulation. Dynamic surface fluctuations are investigated using simulations and phenomenological Langevin equations. In these equations the quenched disorder appears in the form of time-independent random forces. The disorder does not qualitatively change the roughening dynamics of near-equilibrium surfaces, but in the case of biased surface diffusion with Metropolis rates it induces a new roughening mechanism, which leads to an increase of the surface width as \(\).

Microscopic Basis for Onsager-Machlup Theory

The phenomenological Langevin equation in the theory of Onsager-Machlup (OM) is derived from a microscopic theory based on the closed time path generating functional formalism. Following the theory of Wakou-Koide-Fukuda (WKF) we derive the general form of the Langevin equation and carry out the adiabatic expansion. Then, at first order in the adiabatic expansion, the Langevin equation of OM type is naturally derived. In the course of the derivation, the theory of WKF is extended to the multi-macrovariable case, which is defined as the space average of arbitrary operators. The method of adiabatic expansion is made more systematic by using the inversion formula. We proceed to second order in the adiabatic expansion and show that the Langevin equation extended by Machlup-Onsager is obtained.

Numerical approach to the onset of the electroweak phase transition

We investigate whether the universe was homogeneously in the false vacuum state at the critical temperature of a weakly first-order phase transition such as the electroweak phase transition in terms of a series of numerical simulations of a phenomenological Langevin equation, whose noise term is derived from the effective action but the dissipative term is set so that the fluctuation-dissipation relation is met. The correlation function of the noise terms given by a non-equilibrium field theory has a distinct feature if it originates from interactions with a boson or with a fermion. The spatial correlation function of noises from a massless boson damps with a power-law, while the fermionic noises always damp exponentially above the inverse-temperature scale. In the simulation with one-loop effective potential of the Higgs field, the latter turns out to be more effective to disturb the homogeneous field configuration. Since noises of the both types are present in the electroweak phase transition, our results suggest that conventional picture of a phase transition, namely, nucleation of critical bubbles in a homogeneous background does not apply or the one-loop approximation breaks down in the standard model.

A stochastic approach to thermal fluctuations during a first order electroweak phase transition

We investigate the role played by subcritical bubbles at the onset of the electroweak phase transition. Treating the configuration modelling the thermal fluctuations around the homogeneous zero configuration of the Higgs field as a stochastic variable, we describe its dynamics by a phenomenological Langevin equation. This approach allows to properly take into account both the effects of the thermal bath on the system: a systematic dissipative force, which tends to erase out any initial subcritical configuration, and a random stochastic force responsible for the fluctuations. We show that the contribution to the variance ≪ 2≫ ν in a given volume V from any initial subcritical configuration is quickly damped away and that, in the limit of long times, ≪ 2( t)≫ ν approaches its equilibrium value provided by the stochastic force and independent from the viscosity coefficient, as predicted by the fluctuation-dissipation theorem. In agreement with some recent claims, we conclude that thermal fluctuations do not affect the nucleation of critical bubbles at the onset of the electroweak phase transition making electroweak baryogenesis scenarios still a viable possibility to explain the primordial baryon asymmetry in the Universe.

Algebraic spatial correlations in lattice gas automata violating detailed balance

In this paper we discuss the existence of generic long-range correlations in spatially homogeneous and stable equilibrium states of closed lattice gas automata whose stochastic collision rules violate the symmetry conditions of detailed balance and in addition satisfy local conservation laws. Such correlations occur even though the collision rules are strictly local and invariant under all symmetries of the lattice. First a phenomenological (Langevin equation) approach is discussed. Next we present a theoretical analysis on the basis of an approximate microscopic (ring kinetic) theory. This theory is used to calculate the amplitude ofr−α tails in the spatial correlations, and the result is compared with computer simulations.

Viscosity effect on nonadiabatic isomerization and electronic relaxation of molecules in liquids

It is the aim of the present work to analyze the effect of solvent viscosity on nonadiabatic isomerization or electronic relaxation of molecules in liquids. Only the case with a sizable internal barrier is studied here. A quantum model is proposed to evaluate the rate constant of these processes in the overdamped and underdamped cases. The viscosity is introduced through the correlation function for the degrees of freedom that are coupled to the solvent and the correlation function is evaluated classically by using the phenomenological Langevin equation. In addition, the present approach takes into account the role played by other intramolecular degrees of freedom.

Critical fluctuations around non-equilibrium steady states

Scaling and renormalisation-group concepts are used to derive phenomenological Langevin equations for non-equilibrium steady states. For the Schlogl model the authors unambiguously identify the critical Langevin equation in 4- epsilon dimensions simply by requiring that the correct mean-field limit be reproduced. The approach is applicable to models for which only the mean-field behaviour is well established.

On the exact and phenomenological Langevin equations for a harmonic oscillator in a fluid

We have derived, starting from the Liouville equation and using projection operators, the three-dimensional Langevin and Fokker-Planck equations for the time dependence of the momenta and position of two Brownian particles of mass M interacting with a harmonic potential (i.e., a harmonic oscillator) in a fluid of particles with mass m, with M ⪢ m. The resulting Langevin equation has a very complicated structure in that the friction coefficients, x, due to the hydrodynamic interaction between the oscillator particles, are functions of R(t), the time-dependent seperation of the oscillator particles, and the noise terms, though Gaussian, are non-stationary and of a generalized form, i.e., neither “additive” nor “multiplicative”. In addition, there is a term involving the mean force exerted by the fluid on the oscillating Brownian particle. We then investigated the various approximations which must be made to reduce this “exact” Langevin equation to the frequently used one-dimensional, phenomenological Langevin equations (and corresponding Fokker-Planck equations) with purely “additive” and “multiplicative” noise. These approximations are of three types: (a) one must neglect the term arising from the rotational motion of the oscillator in the fluid; (b) one must neglect the R(t) dependence of X[R(t)], leading to purely “additive” noise, or approximate X[R(t)] by a Taylor series expansion to quadratic order in R(t) - R0 (where R0 is the equilibrium separation of the oscillator particles), leading to “additive” and “multiplicative” noise; and (c) one must either neglect the mean force term or approximate it by a term linear in R(t) - R0. We conclude from these results that one-dimensional phenomenological Langevin equations with simple noise structure are of doubtful validity for the dynamical description of the relative momenta (or vibrational energy) of oscillating molecules in a fluid.

Renormalization of the vortex diffusion constant in superfluid films

We study the dynamics of vortices in superfluid helium films, using a phenomenological Langevin equation to account for diffusive as well as convective motion of vortices. We show that in the long-wavelength limit, dynamic screening effects give rise to a renormalization of the diffusion constant, while the convective parameter remains unchanged. The critical behavior is analyzed with renormalization-group methods. In the limit T..-->..T/sub c//sup -/, we find a universal cusp singularity for the diffusion constant, reaching a finite value at T/sub c/.