Brownian motion of a classical particle through potential barriers. Application to the helix-coil transitions of heteropolymers

We discuss some properties of the one-dimensional Fokker-Planck equation, and in particular the time required to go through a potential barrier of arbitrary size and shape. We apply the resulting formulas to the melting of helical polymers made of two types of monomers (A and B) with different melting temperatures: We consider a restricted problem in irreversible melting, where onesingle boundary (separating a coil region from a helical region) moves through the chemical sequence. In a crude approximation the distribution function for the boundary point is then ruled by a simple Fokker-Planck equation. When the temperatureT is equal to or higher than the average melting point¯T, the boundary tends to move over macroscopic distances, thus extending the size of the coil regions. In an interval¯T ^* the progression is predicted to be slow (logarithmic). At higher temperatureT>T ^* essentially all barriers are overcome and the progression is fast.     

Distribution function for systems far from thermal equilibrium

We present the general stationary solution of the Fokker-Planck equation for a system of interacting subsystems weakly coupled to reservoirs at different temperatures, T_j. For T_j = T, it reduces to the Boltzman distribution function.     

Solution of a laser Fokker-Planck equation for intensity and inversion

In the commonly used laser model (two levels, homogeneously broadened, one running mode, no detuning) the polarization can be eliminated adiabatically, if the dephasing time is much shorter than the other time constants. We thus arrive at a Fokker-Planck equation for inversion and light intensity, which does not fulfill the detailed balance condition. By a suitable expansion of the distribution function it is shown that the Fokker-Planck equation is equivalent to a tridiagonal recurrence relation for the coefficient vectors, which can be solved by matrix continued fractions. In this way all moments and two-time correlations can be obtained. Especially the correlation coefficients between inversion and intensity, the photon counting coefficient [<(n–<n>)²>–<n>]/<n>² for the stationary state as well as the eigenvalues of the Fokker-Planck equation are presented in the threshold region.     

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker-Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t₀ to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t₀, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker-Planck equation are also performed in a unified way in the present method.     

Quantum mechanical solutions of the laser masterequation

We treat a laser consisting of one mode described by a running wave and a set of atoms with two optically active levels which are homogeneously broadened. We start from the laser density matrix equations ofWeidlich andHaake and define a distribution functionf for lightfield and atomic variables, where we use for the lightfield the coherent state representation and for the atomic system a modified version of the distribution function used bySchmid andRisken in a previous paper. We derive a partial differential equation forf which is completely exact and is of the type of a generalized Fokker-Planck equation, i.e. it contains higher derivatives. Using a recently stated theorem ofHaken andWeidlich we show that this distribution function allows to calculate single-time as well as multitime quantum mechanical correlation functions. If the leading terms of the generalized Fokker-Planck equation are retained we find the semiclassical Fokker-Planck equation ofRisken,Schmid andWeidlich. Our treatment can be extended to several modes connected with standing waves and multilevel atoms.     

Vector coherence initiation of multi-level superfluorescence

An SU(n) coherent state representation is developed for the theory of multi-level cooperative spontaneous emission. A general Fokker-Planck equation is derived, whose characteristics are equivalent to the semiclassical equations for large numbers of atoms. The characteristics equations have non-classical initial values described by a vector coherence probability distribution. The theory can be used for the multilevel case with coherent pumping, to calculate the effects of pump interactions.     

Stochastic quantization and path integral formulation of Fokker-Planck equation

The operator formalism (Fokker-Planck dynamics) associated to a general n-dimensional, non-linear drift, non-constant diffusion Fokker-Planck equation, is derived by a stochastic quantization from the corresponding path integral formulation in phase space.     

Two-dimensional Brownian vortices

We introduce a stochastic model of 2D Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other (“plasma” case) and at negative temperatures, like-sign vortices attract each other (“gravity” case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N-body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N, where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N→+∞. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-Hückel approximation valid at the order O(1/N). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-Hückel model of electrolytes.     

Microscopic theory of brownian motion: III. The nonlinear Fokker-Planck equation

The nonlinear Fokker-Planck equation for the momentum distribution of a brownian particle of mass M in a bath of particles of mass m is derived. The contribution to this equation arising from initial deviation from bath equilibrium is analysed. This contribution is free of slow M-dependent decays and with certain restrictions leads to an effective shift in the initial value of the B particle momentum. The nonlinear Fokker-Planck equation for an initial bath equilibrium state is analyzed in terms of its predictions for momentum relaxation and mode coupling effects. It is found that in addition to nonlinear renormalization of the type previously found for the momentum correlation function, mode coupling leads to long-lived memory of the initial momentum state.     

Mouvement brownien dans une assemblée de bâtonnets minces dans la phase isotrope

Using the kinetic methods of the Brussels School, we establish the equation (to the 2nd order in the perturbation) for the return to equilibrium of the one particle energy distribution function in an isotropic fluid made of thin slabs interacting through a P₂-type potential. We show that for a very heavy brownian particle in a bath of identical light particles in equilibrium, this equation reduces to a Fokker-Planck equation, the coefficients of which are explicitly calculated. In the same manner we study the return to equilibrium of an angular excitation.     

The Fokker-Planck equation of a laser with many modes and multi-level atoms

Using the Glauber-SudarshanP-representation for the field modes and a quasi-distribution function recently presented for arbitrary quantum systems we derive an exact generalized Fokker-Planck equation for a multi-mode laser containing a set of multi-level atoms with homogeneous and inhomogeneous level broadening. By introduction of suitable collective atomic coordinates this generalized Fokker-Planck equation is reduced to an ordinary one which may serve as a basis for the adequate treatment of laser light statistics.     

Quasi-exactly solvable Fokker-Planck equations

We consider exact and quasi-exact solvability of the one-dimensional Fokker-Planck equation based on the connection between the Fokker-Planck equation and the Schrödinger equation. A unified consideration of these two types of solvability is given from the viewpoint of prepotential together with Bethe ansatz equations. Quasi-exactly solvable Fokker-Planck equations related to the sl(2)-based systems in Turbiner’s classification are listed. We also present one sl(2)-based example which is not listed in Turbiner’s scheme.     

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

In the classical theory of Brownian motion we can consider the Langevin equation as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, we can reinterpret the Langevin equation as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta we can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. We may associate, according to Weyl functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions we can express the streaming as a continuity equation of the Wigner function. We find that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation.     

On the dynamics of statistical fluctuations in heavy ion collisions

We show how statistical fluctuations can be treated within the collective approach to heavy ion reactions. In the classical limit, the equation of motion for the distribution d in the collective variables Q_μ and their conjugate momenta P_μ turns out to be a Fokker-Planck equation. We briefly describe the connection of this equation to one of the Smoluchowski type for a distribution in Q_μ only, often used in heavy ion physics. For anharmonic motion our general Fokker-Planck equation is simplified to be linear in the deviations of the Q_μ mand P_μ from their mean values. The solution of this equation is discussed in terms of a simple Gaussian. The parameters of this Gaussian are determined completely by the first and second moments in Q_μ mand P_μ. The equations for the first moments are identical to the Newton equations including frictional forces. Those for the second moments are linear differential equations of first order and hence easily solvable. The whole derivation is completely analogous to that for the Newton equation reported recently. Here the starting point is the quantum mechanical von Neumann equation rather than the Heisenberg equations. As an intermediate result we obtain and discuss briefly a quantal equation for the reduced density operator d which includes frictional effects.     

A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker-Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker-Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was offered that a Chapman-Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker-Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker-Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker-Planck pdes and Chapman-Kolmogorov equations for stochastic processes with finite memory.     

Derivation of an Improved Hodgkin-Huxley Model for Potassium Channel by Means of the Fokker-Planck Equation

This paper demonstrates the derivation of Hodgkin-Huxley-like equations from the Fokker-Planck equation. The primary result is that instead of the familiar equation expressing the potassium conductance as a function of the variablen which obeys a first order differential equation, the expression , whereL = 2.7, is to be used. This form is obtained by solving analytically an approximate solution to a Fokker-Planck partial difference equation. Instead of the Hodgkin-Huxley interpretation as the probability of occupying the conducting state, the parameter n(t) is now interpreted as the position of the “peak” of the population distribution function P(N, t), which changes in time described by the Fokker-Planck equation. This new approach enables close fitting of the experimental voltage clamp data for potassium conductance. In addition, the Cole-Moore shift paradox can be quantitatively explained in terms of the shift of the distribution function P(N,t) by the initial clamped transmembrane potentialV _i before the final clamped transmembrane potentialV _f is applied, thus increasing the time necessary for the establishment of equilibrium.     

On the application of truncated generalized Fokker-Planck equations

The Pawula theorem states that the generalized Fokker-Planck equation with finite derivatives greater than two leads to a contradiction to the positivity of the distribution function. Though negative values are inconsistent from a logical point of view, we show that such distribution functions with negative values can be very useful from a practical point of view. For a Poisson-process, where the exact solution is known, we compare the solution of the second order Fokker-Planck equation to the solutions of Fokker-Planck equations of finite order. It turns out that for certain parameters the approximations of the distribution function and the moments are much better for some higher order and that the magnitude of negative values may be very small in the relevant region of variables.     

Fick's law and Fokker-Planck equation in inhomogeneous environments

In inhomogeneous environments, the correct expression of the diffusive flux is not always given by the Fick's law Γ=−D∇n. The most general hydrodynamic equation modelling diffusion is indeed the Fokker-Planck equation (FPE). The microscopic dynamics of each specific system may affect the form of the FPE, either establishing connections between the diffusion and the convection term, as well as providing supplementary terms. In particular, the Fick's form for the diffusion equation may arise only in consequence of a specific kind of microscopic dynamics. It is also shown how, in the presence of sharp inhomogeneities, even the hydrodynamic FPE limit may becomes inaccurate and mask some features of the true solution, as computed from the Master equation.     

A comparative study of the P and Q representations of a feedback controlled two-qubit system

In this Letter the Master equation of a two qubit system is transformed into Fokker-Planck equations in order to find the Glauber-Sudarshan P function representation. For the two qubit system examined in this Letter, the P representation is ill defined, which indicates the system is non-classical. A qualitative measure of the non-classical nature of the system is found by taking the semi-classical limit of the Fokker-Planck equation and obtaining a simplified Glauber-Sudarshan P representation. The agreement between the simplified P representation and the Q representation as well as the system stability are discussed when feedback is present and absent.