Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form e−λnt where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ 0/x p , where x=mev2/2kBTb is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.


Abstract
The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form l e t n where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.

Introduction
The current paper considers the relaxation of an ensemble of electrons dilutely dispersed in a background of an inert gas, for either Argon, Krypton or Xenon for which the electron momentum transfer cross sections have a deep Ramsauer-Townsend minima. It has been demonstrated that the steady state electron distribution functions in these moderators are bimodal over a small range of external electric field strengths [1][2][3][4]. In this paper, we report on the time evolution of the electron distribution functions as determined from the numerical solution of the appropriate Fokker-Planck equation. We show the time dependent formation of a bimodal steady distribution function from an initial Maxwellian for the appropriate electric field strengths for Argon, Krypton or Xenon.
The present work is directed towards the study of the time evolution of the velocity distribution of electrons, initially Maxwellian, to a nonequilibrium steady state under the influence of an external electric field. The theoretical treatment is based on previous works [2,12,18,31]. For a small range of electric field strengths, the steady nonequilibrium electron energy distribution functions develop bimodal structures [1][2][3][4]. This phenomenon occurs for the heavier inert gases, Ar, Kr and Xe with deep Ramsauer-Townsend minima in the momentum transfer cross sections. We follow the methodology of previous works [12,18,31] which did not consider the low E/n values that yield the steady state bimodal distributions.
The current work considers the time evolution of the distribution of electrons initially Maxwellian and uniformly dispersed in a large excess of an inert gas at equilibrium at temperature T b and subject to an external electric field. A finite difference method of solution of the time dependent Fokker-Planck equation is employed [32] with concern to the time dependence of the electron distributions in the external electric field. The steady state distributions can be bimodal as dependent on the electric field strength and there exists a direct relationship between the maxima in the distribution functions with the minima in the Ramsauer-Townsend momentum transfer cross sections [4].
The statistical mechanics of nonequilibrium distributions such as the bimodal distributions studied in this paper is an important research field, particularly in space physics, where the electron distributions are often reported as Kappa distributions of the form, as the thermal speed, m is the particle mass, T b is the bath temperature and k B is the Boltzmann constant. The three dimensional Kappa distribution is normalized according to ò The Kappa distribution, analogous to the bimodal distributions of interest here, arises from the nature of the steady solutions of a particular Fokker-Planck equation given by a Pearson ordinary differential equation [4,33,34]. These nonequilibrium distributions are the result of the dynamical aspects of the Fokker-Planck equation. However, much of the validation of the Kappa distribution is based on the extremum of the Tsallis nonextensive entropy which gives uniquely the Kappa distribution [35][36][37][38]. We show instead the monotonic increase of the Kullback-Leibler entropy as the system approaches a nonequilibrium steady state.
Thus we also consider in this paper a model system characterized by a power law momentum transfer cross section, that is, σ(x) = σ 0 /x p . For p = 2, this permits the demonstration of the physical origin of the Kappa distribution that features prominently in space physics [35] and the physics of optical lattices [39] which is generally rationalized in terms of the Tsallis nonextensive entropy [36][37][38]. This is an important issue in nonequilibrium statistical mechanics which includes a large number of non-Boltzmann distributions of great variety [40][41][42] such as the bimodal electron distributions reported in this paper that are not the Kappa distribution which is the unique solution of the extremum of the Tsallis entropy functional [35][36][37][38]43]. It should be very clear that there exists an abundance of different nonequilibrium distributions in nature that cannot be rationalized with the Tsallis nonextensive entropy functional.

Time dependent Fokker-Planck equation
The current paper is directed towards the solution of the time dependent Fokker-Planck equation defined for the electron distribution function with the reduced speed, x = v/v th with v the electron speed and the thermal speed where k B is the Boltzmann constant, T b is the temperature of the background gas and m e is the electron mass. The dependence of the electron distribution function on reduced speed and time, f (x, t), is given by the Fokker-Planck equation [4,12,18,31], where σ(x) is the cross section for electron-inert gas collisions and, The electric field strength parameter, α, is defined by, where e is the electron charge, E is the electric field strength, M is the atom mass and n is the background gas number density [12,18,31].
The linear Fokker-Planck operator, L, is defined with equation (2). The steady state, nonequilibrium distribution of the FPE is given by We consider a solution of the time-dependent Fokker-Planck equation with a spectral method with the representation of the speed dependence of the distribution on a discrete grid of quadrature points in x [12,18]. As the Fokker-Planck equation involves a linear self-adjoint operator, the time-dependent solution can be expressed in terms of the eigenfunctions, ψ n (x), and eigenvalues, λ n of L, defined by equation (2), that is, and the solution of equation (2) is of the form The coefficients c n in equation (7) are defined with the initial distribution which is chosen as the Maxwellian, ( ) and the discrete representation of the Fokker-Planck operator is [12], where D ij is the discrete representation of the derivative operator [44] and

Schrödinger equation isospectral with the Fokker-Planck equation
The convergence of the eigenvalues of the Fokker-Planck operator is determined from the equivalent Schrödinger equation with a nonclassical basis set as discussed at length elsewhere [31].  (2)  70 4.6293 (2) and for Kr the cross section reported by England and Elford [48] was used. We show in figure 1 the bimodal potential function in the Schrödinger equation for Ar for different electric field strengths with the energy levels shown by the horizontal lines. Figure 2 shows the variation of the eigenvalues versus E/n for (A) Ar and (B) Xe. The striking feature is the minimum of λ n versus E/n for both moderators. This is indirectly a reflection of the bimodal nature of the stationary distributions versus E/n.

Time evolution of the electron distribution function and the Kullback-Leibler entropy
Although the spectral method provides an accurate description of the spectrum of the Fokker-Planck operator, in practice it is often difficult to obtain a converged time dependent solution for all times, especially for short times when many eigenfunctions contribute. We employ instead a finite difference scheme introduced by Chang and Cooper [32] and previously used for the electron Fokker-Planck equation [31]. This is modified here for the application to the 1D Fokker-Planck equations. We set t n = nΔt and use a backward Euler difference for the time derivative, that is, ¶ ¶ = - where V is a tridiagonal matrix, that is, ( ) The discretized version of equation ( [4], the bimodality arises owing to the Ramsauer-Townsend minima in the momentum transfer cross sections for these systems, and is very sensitive to E/n over a small range of values. These are unusual steady nonequilibrium distributions. In addition to the intrinsic interest in the time-evolution of the electron distribution to a bimodal steady state distribution as dependent on the external electric field, we are also interested in illustrating the dynamics for the evolution of the electron energy distribution. The different heating of the electrons in different moderators is shown in figure 4 corresponding to the distributions in figure 3. The temperature, T(t), is defined in terms of the average energy, that is, ). An important aspect of these results is that these nonequilibrium distributions cannot be determined on the basis of the Gibbs-Boltzmann statistical mechanics nor with the application of the Tsallis nonextensive entropy formalism [43]. The approach to a nonequilibrium steady state is a dynamical process as determined by the Fokker-Planck equation. We can add an additional understanding by showing the approach to this non-equilibrium state in terms of the Kullback-Leibler shown in figure 5 corresponding to the distributions in figure 3. The Kullback-Leibler entropy is defined by and measures the departure of the nonequilibrium distribution, f (x, t), from the local Maxwellian, f SS (x). This entropy functional serves only to measure the departure from the local equilibrium and does not provide any information regarding the distribution function which is used in its definition. This entropy functional serves to measure the 'distance' from the nonequilibrium steady state. We show a selected number of time dependent nonequilibrium distributions starting from a Maxwellian at temperature T b and evolving in time to bimodal steady state distributions. There are a vast number of situations that can be considered for different atomic moderators and with different electric field strengths. We chose a small select number of physical situations to report here.

Model systems that yield Kappa distributions and distributions which resemble the electron runaway effect
We consider the form of the electron distribution functions for a model system with a power law momentumtransfer cross section, introduced briefly in a previous publication [33]. For p = 0, this is a hard sphere cross section for which the eigenvalues of the Fokker-Planck operator are known [49].
which is a Kappa distribution with κ = (1 − α 2 )/α 2 [33,[50][51][52] and given by It is this Kappa distribution that has attracted considerable attention in space physics [35,36,38,37] and the Tsallis nonextensive formalism [43] has been used to validate this distribution on entropic principles. A very different behaviour is the distribution function for p = 3, for which which increases without bound, analogous to a runaway electron effect [53][54][55]. In figure 7, we show the behavior of this model cross section for values of p between 2 and 3. In figure 7(A), we show the distributions for p = 2.5 and different values of E/n. The increase of the high speed portion of the distribution without bound is evident and decreases with a decrease in E/n from curve a to curve d. In figure 7(B), we show the distributions for E/n = 0.001Td and different values of p with distributions increasing without bound especially for the larger p values. These nonequilibrium distributions result from the dynamical information in the Fokker-Planck equation and they cannot be derived from entropy functionals [33,52]. In figure 8, we show the behaviour of f x ln ss ( ( ))for E/n equal to (A) 0.001 Td and (B) 0.002 Td for decreasing values of p. The distribution function exhibit long 'tails' perhaps analogous to a Kappa distribution and there is no entropic principle for their calculation.

Summary
In this paper, we have reported on the time evolution of the electron distribution in different inert gas moderators under the influence of an external electric field expressed as E/n in Townsends, Td. Owing to the Ramsauer-Townsend minima in the electron-atom momentum transfer cross sections, the electron distributions achieve bimodal steady state distributions. Concomitant with the approach to these steady states,  the temperature of the electrons increases. The approach to these nonequilibrium steady states is noted with the Kullback-Leibler entropy which increases in time monotonically from some initial value to zero. This entropy functional does not serve to determine these distributions but is a measure of the time dependent approach to the nonequilibrium steady state. We note that neither the Boltzmann-Gibbs nor the Tsallis entropy functions can be used to validate these nonequilibrium distributions. We study further this aspect of nonequilibrium statistical mechanics with a simple power law momentum transfer cross section, σ(x) = σ 0 /x p , which yields the Kappa distribution for p = 2 of considerable interest in space physics and optical lattices. For other values of p and electric field strengths, we report distribution functions analogous to electron runaway effects. Although the current work is theoretical, we can anticipate potential experimental verification of the bimodal distributions analogous to the experimental verification of the negative mobility effect [17,19,20,26] predicted earlier [18].