Takeo Nishigori

Memory function approach to nonlinear deterministic systems: An exact linear equation

A set of nonlinear evolution equations is cast into an exact linear non‐Markovian equation with the memory kernel reflecting the nonlinearity and coupling with irrelevant variables. The equation is deterministic in contrast to the generalized Langevin equation derived in a similar way. The solution to the nonlinear equations is expressed by a sum of exponential functions. A simple illustrative example is treated to show the effectiveness of the present approach.

Short‐time behavior of electron thermalization in gases

A simple power series expansion in time is shown to complement nicely the eigenfunction expansion in electron thermalization problems. With the hot‐electron zero‐field mobility as an example, a recursion relation is derived to facilitate the calculation of higher‐order short‐time expansion coefficients. The electron–atom momentum transfer cross section is modeled by a velocity polynomial. Numerical results for the mobility in helium are discussed to see the usefulness of the present expansion.

Memory function approach to electron thermalization

A linear closed equation of the form e(t)=ωe(t) +∫t0Λ(t−t’) e (t’)dt’ is proposed to describe the nonlinear relaxation of electron energy e(t) in the Lorentz gas. Higher‐order memory functions associated with Λ(t) decay rapidly, and are easily amenable to approximation. Good numerical results for e(t) suggest a useful extension of our memory function approach to many nonlinear problems for calculating various integral quantities.

A uniform WKB approach to electron thermalization in gases

Abstract The relaxation of a nonequilibrium distribution of electrons in gases is governed by a linear Fokker-Planck equation. The time variation of the electron transport properties can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker-Planck operator. This eigenvalue problem can be transformed to a Schrodinger equation with a potential function parametrized by the electron-atom cross section and the gas temperature. In the present paper, the eigenvalues and eigenfunctions are calculated with a WKB semiclassical approximation and compared with exact results. The application of the WKB approximation provides a useful interpretation of the results and permits the use of initial delta function distributions at much higher initial energies than previously possible. The specific applications in this paper are for the relaxation of electrons in helium and neon.

Adiabatic Switching-Off of Interactions and Foundations of the Boltzmann Equation

The Boltzmann equation for a monatomic gas is derived with the aid of the adiabatic switching-off of interactions, starting with the Heisenberg equation of motion for a number operator in phase space introduced by Ono. It is shown that neither Kirkwoods time-averaging procedure nor the random phase approximation is necessary by virtue of the switching-off of interactions and a new method of time-differentiation. The effectiveness of the present formalism is demonstrated in an analysis of higher order interactions, and the Boltzmann equation is corrected so as to include multiple scattering effects.

Space–time memory functions and solution of nonlinear evolution equations

A new approach is presented for solving a certain class of nonlinear partial differential equations. A space–time memory function Λ(r,t) is introduced to exactly convert a given nonlinear evolution equation into the following linear form: (∂/∂t) f(r,t)=Ω(r) f(r,t) +∫t0dt′∫dr′Λ(r−r′,t−t′)f(r′,t′). A Markovian integro‐differential operator Ω(r) and the memory function Λ(r,t) reflect the nonlinearity, and are determined depending on a given initial condition. The approach is useful if higher‐order memory functions associated with Λ are insensitive to approximation. The Korteweg–de Vries equation is treated as an example. For certain initial profiles the memory function is shown to be identically zero, and we find exact linear partial differential equations leading to the single‐ and the two‐soliton solution. In the case of the three‐soliton solution, the second‐order memory function vanishes exactly, and Λ(r,t) is found to be a single exponential function of t.A new approach is presented for solving a certain class of nonlinear partial differential equations. A space–time memory function Λ(r,t) is introduced to exactly convert a given nonlinear evolution equation into the following linear form: (∂/∂t) f(r,t)=Ω(r) f(r,t) +∫t0dt′∫dr′Λ(r−r′,t−t′)f(r′,t′). A Markovian integro‐differential operator Ω(r) and the memory function Λ(r,t) reflect the nonlinearity, and are determined depending on a given initial condition. The approach is useful if higher‐order memory functions associated with Λ are insensitive to approximation. The Korteweg–de Vries equation is treated as an example. For certain initial profiles the memory function is shown to be identically zero, and we find exact linear partial differential equations leading to the single‐ and the two‐soliton solution. In the case of the three‐soliton solution, the second‐order memory function vanishes exactly, and Λ(r,t) is found to be a single exponential function of t.

On the memory functions in simple classical liquids

The relation between the two memory function formalisms for correlation functions in classical liquids is discussed. It is found that the kinetic equation formalism of Duderstadt and Akcasu with a simple exponential memory function can account for the double Gaussian form of the memory function in the generalized-hydrodynamics approach. The former therefore gives reasonably good results for the coherent scattering function S(k, ω), as is shown for the case of liquid Rb at 315 K in the range 1.25 ⩽k ⩽ 5.5 Å−.

Microscopic theory of memory functions

We define a sequence of microscopic dynamical variables by decomposing a Hilbert space into orthogonal subspaces, and construct for them a new hierarchy of equations which is particularly useful for highly correlated systems. A formal solution is shown to give a microscopic expression of Moris generalized Langevin equation. With a classical liquid as an example, we demonstrate that the theory facilitates a first-principles calculation of memory functions.

Theory of Kinetic Equations for Systems with Production and Absorption of Particles

This paper is concerned with systems which are not in thermal equilibrium because of production and absorption of particles. On the basis of a new Hamiltonian describing such a nonequilibrium system, we develop a method for deriving kinetic equations for singlet density, time correlations of density, etc., including all the higher order interactions necessary to describe the production and absorption of particles. The foundations of the Boltzmann and Langevin equations for neutron distributions are studied. The time-correlation function is shown to obey a kinetic equation identical to that for the singlet density. It is also shown that the description of density fluctuations based on the Langevin equation is equivalent to the simplest decoupling of the rigorous hierarchy of equations for correlation functions.

Memory function approach to a nonlinear oscillator

An exact linear equation of motion of the form x(t) + ω2x(t) + ∫t0Λ(t − t′)x(t′)dt′ = 0 proposed for an undamped anharmonic oscillator. A renormalized frequency ω and a memory function Λ(t) reflect the nonlinearity. The laplace transform Λ(z) of the memory function is given by a combination of infinite continued fractions in z2. With a cubic anharmonic oscillator as an example, we show that higher-order memory functions Λn(t), which are associated with Λ(t), oscillate rapidly so that Λn(t) ≡ 0 is a good approximation (cf. the instantaneous decay approximation Λn(t) ∝ δ (t) in dissipative systems).