Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t₀=t is naturally understood where t₀ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.     

Exact local in time evolution equations for macroobservables

By introducing an evolution equation for the generalized Gibbs state σ(t), the N-particle distribution function is expressed as a linear functional of σ(t). Exact equations local in time for the time evolution of macroobservables are obtained. The kinetic coefficients appear as the fixed point of a conveniently defined microscopic expression which may be considered as a natural extension of the Kubo formula.     

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.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

Asymptotic solutions of Fokker-Planck equations for nonlinear irreversible processes and generalized Onsager-Machlup theory

Fokker-Planck equations for nonlinear processes are solved asymptotically in the limit k→0 where k is the Boltzmann constant. It is shown that the leading asymptotic solutions for conditional (two-gate) distribution functions simply correspond to generalizations of the Onsager-Machlup theory to nonlinear processes. The asumptotic solution method used in the paper is similar to the well-known W.K.B. method in quantum mechanics. A stability criterion of nonlinear irreversible processes is also considered and compared with the Glansdorff-Prigogine stability criterion.     

Non-Markovian diffusion equations and processes: Analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.     

Brownian particles with long- and short-range interactions

We develop the kinetic theory of Brownian particles with long- and short-range interactions. Since the particles are in contact with a thermal bath fixing the temperature T, they are described by the canonical ensemble. We consider both overdamped and inertial models. In the overdamped limit, the evolution of the spatial density is governed by the generalized mean field Smoluchowski equation including a mean field potential due to long-range interactions and a generically nonlinear barotropic pressure due to short-range interactions. This equation describes various physical systems such as self-gravitating Brownian particles (Smoluchowski-Poisson system), bacterial populations experiencing chemotaxis (Keller-Segel model) and colloidal particles with capillary interactions. We also take into account the inertia of the particles and derive corresponding kinetic and hydrodynamic equations generalizing the usual Kramers, Jeans, Euler and Cattaneo equations. For each model, we provide the corresponding form of free energy and establish the H-theorem and the virial theorem. Finally, we show that the same hydrodynamic equations are obtained in the context of nonlinear mean field Fokker-Planck equations associated with generalized thermodynamics. However, in that case, the nonlinear pressure is due to the bias in the transition probabilities from one state to the other leading to non-Boltzmannian distributions while in the former case the distribution is Boltzmannian but the nonlinear pressure arises from the two-body correlation function induced by the short-range potential of interaction. As a whole, our paper develops connections between the topics of long-range interactions, short-range interactions, nonlinear mean field Fokker-Planck equations and generalized thermodynamics. It also justifies from a kinetic theory based on microscopic processes, the basic equations that were introduced phenomenologically to describe self-gravitating Brownian particles, chemotaxis and colloidal suspensions with attractive interactions.     

Das Fluktuationsspektrum von Elektronen in einem stationären Nichtgleichgewichtszustand

The Boltzmann equation is used to calculate the time correlation function and the fluctuation spectrum for electrons obeying classical statistics. The stationary joint distribution for one electron to be initially ink ₀=k(0) and finally ink=k(t) is given by the product of the conditional probability and the stationary distribution. These quantities can be found from the Boltzmann equation if there exists, for any initial distribution, a unique solution which satisfies the Markov equation and tends to a stationary solution for large times under stationary conditions. It is proved that these conditions hold for linear collision operators and in the relaxation approximation. General operator expressions for the fluctuation spectrum and the differential conductivity in a stationary electric field are given, which can be evaluated within the usual approximation schemes known for the stationary, nonequilibrium solutions of the Boltzmann equation. In equilibrium they reproduce the classical fluctuation dissipation theorem. In a nonequilibrium state they define a noise temperature depending on the field. In the relaxation approximation and for polynomial band structure the exact solution can be found. For parabolic and biparabolic spherical bands the result is discussed explicitly.     

The mutual interaction of molecular rotation and translation

The dynamics of molecular rototranslation are treated with an equation of motion with a non-Markovian, stochastic force/torque. It is shown that this Mori/Kubo/Zwanzig representation is equivalent to a multidimensional Markov equation which may be identified with analytical models of the molecular motion. Langevin and Fokker-Planck equations for two such models are derived from the general equations of motion. The analytical results are compared with a computer simulation of the velocity/angular velocity mixed autocorrelation function, C _vω(t) = <v(0) . ω(t)> for a triatomic of C _2v symmetry.     

Coarse-grained kinetic equations for quantum systems

The nonequilibrium density matrix method is employed to derive a master equation for the averaged state populations of an open quantum system subjected to an external high frequency stochastic field. It is shown that if the characteristic time τ_stoch of the stochastic process is much lower than the characteristic time τ_steady of the establishment of the system steady state populations, then on the time scale Δt ～ τ_steady, the evolution of the system populations can be described by the coarse-grained kinetic equations with the averaged transition rates. As an example, the exact averaging is carried out for the dichotomous Markov process of the kangaroo type.     

The transient solution of the laser Fokker-Planck equation

The transient solution of the laser Fokker-Planck equation is investigated in the threshold region. Especially for the initial condition, that no photons are present att=0, the transient of the laser distribution function, of the mean intensity and of the mean squared deviation (variance) into the stationary values are shown. Finally we give a physical explanation of the effects involved and we discuss some features of the transient photon counting probability.     

Time-local gaussian processes,path integrals and nonequilibrium nonlinear diffusion

In this paper we discuss the concept of time-local gaussian processes. These are processes for which the state variable at time t + τ is gaussian distributed around its most probable value at that time, for a specified realization a small time interval τ earlier. On one hand it will be shown that these processes are related to a very simple path sum. On the other hand the associated stochastic differential equation is derived by means of the Kramers-Moyal method, and will be seen to be the most general nonlinear Fokker-Planck equation. The significance of the present formulation for nonequilibrium processes and the comprehension of critical phenomena will be evaluated.     

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.     

Ghost-matter mixing and feigenbaum universality in string theory

Branelike vertex operators, defining backgrounds with ghost-matter mixing in Neveu-Schwarz-Ramond superstring theory, play an important role in a world-sheet formulation of D branes and M theory, being creation operators for extended objects in the second quantized formalism. We show that the dilaton beta function in ghost-matter mixing backgrounds becomes stochastic. The renormalization group (RG) equations in ghost-matter mixing backgrounds lead to non-Markovian Fokker-Planck equations whose solutions describe superstrings in curved spacetimes with branelike metrics. We show that the Feigenbaum universality constant δ=4.669 ..., describing transitions from order to chaos in a huge variety of dynamical systems, appears analytically in these RG equations. We find that the appearance of this constant is related to the scaling of relative spacetime curvatures at fixed points of the RG flow. In this picture, the fixed points correspond to the period doubling of Feigenbaum iterational schemes.

     

A criticism of the standard treatment of phonon drag

By the study of a simple example, namely the evolution in timet of an electron-phonon system with fixed, total momentum, it is shown that the “standard” treatment of “phonon drag”, which involves solving the (linearized and spatially homogeneous) coupled electron and phonon Boltzmann equations by an iteration procedure, is not always correct. In the asymptotic limit (t→∞), the iteration or “standard” procedure does not give the “correct” (i.e. the equilibrium statistical mechanical) result for the distribution of momentum between electrons and phonons. However, a proper treatment of the Boltzmann equations does lead to the “correct” sharing of momentum between electrons and phonons fort→∞. All the calculations in this paper are performed for metals at high temperatures (i.e.,T>Θ_D, the Debye temperature).     

Mouvement Brownien dans une assemblee de batonnets minces dans la phase nematique

Using the kinetic methods of the Brussels school, we establish the equation (to the second order in the perturbation) for the return to equilibrium of the one-particle energy distribution function in the nematic phase of a fluid made of thin slabs interacting through a P₂-type potential. On the basis of the mean field equilibrium theory developed by Maier and Saupe for such a fluid, we show that for a very heavy brownian particle, this equation reduces to a Fokker-Planck type equation; the friction coefficient thus obtained is compared with the friction coefficient obtained for the isotropic phase and we show that they are equal for the transition temperature.     

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.     

Calculating coefficients for a system of moment equations used to describe the magnetization kinetics of a superparamagnetic particle in a fluctuating field

A system of equations is derived for moments [averages of spherical harmonics 〈Y _l,m 〉(t)] that determine the dynamics of the magnetization M of a superparamagnetic particle in a fluctuating field. The system is derived by representing the Gilbert equation in a fluctuating field, and the corresponding Fokker-Planck equation for the distribution function of M, in terms of angular momentum operators, which in turn makes it possible to express the coefficients of the system of moment equations in terms of Clebsch-Gordan coefficients. Fiz. Tverd. Tela (St. Petersburg) 41, 2020–2027 (November 1999)     

The quantum Fokker-Planck equation for certain boson systems

Quantum relations between a class of boson Langevin equations and the associated Fokker-Planck equations are derived. The Fokker-Planck equations for the Wigner distribution Φ_sym related with symmetric ordering of the boson operators, the distribution Φ_A related with antinormal ordering, and the distribution Φ_N related with normal ordering (P-representation) are given.     

Solutions of the Fokker-Planck equation for a double-well potential in terms of matrix continued fractions

Solutions of the Fokker-Planck (Kramers) equation in position-velocity space for the double-well potentiald ₂x²/2+d₄x⁴/4 in terms of matrix continued fractions are derived. It is shown that the method is also applicable to a Boltzmann equation with a BGK collision operator. Results of eigenvalues and of the Fourier transform of correlation functions are presented explicitly. The lowest nonzero eigenvalue is compared with the escape rate in the weak noise limit for various damping constants and the susceptibility is compared with the zero-friction-limit result.