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On the Correct Implementation of Fermi–Dirac Statistics and Electron On the correct implementation of Fermi–Dirac statistics and electron trapping in nonlinear electrostatic plane wave propagation in collisionless plasmas Hans Schamel and Bengt Eliasson Citation: Physics of Plasmas 23, 052114 (2016); doi: 10.1063/1.4949341 View online: http://dx.doi.org/10.1063/1.4949341 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/23/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in How electronic dynamics with Pauli exclusion produces Fermi-Dirac statistics J. Chem. Phys. 142, 134113 (2015); 10.1063/1.4916822 Linear electrostatic waves in a three-component electron-positron-ion plasma Phys. Plasmas 21, 122119 (2014); 10.1063/1.4905067 Small amplitude nonlinear electrostatic waves in a collisional complex plasma with positively charged dust Phys. 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Downloaded to IP: 130.159.82.198 On: Thu, 02 Jun 2016 12:31:32 PHYSICS OF PLASMAS 23, 052114 (2016) On the correct implementation of Fermi–Dirac statistics and electron trapping in nonlinear electrostatic plane wave propagation in collisionless plasmas Hans Schamel1,a) and Bengt Eliasson2,b) 1Physikalisches Institut, Universitat€ Bayreuth, D-95440 Bayreuth, Germany 2SUPA, Physics Department, University of Strathclyde, John Anderson Building, Glasgow G4 0NG, Scotland, United Kingdom (Received 17 March 2016; accepted 1 April 2016; published online 19 May 2016) Quantum statistics and electron trapping have a decisive influence on the propagation characteristics of coherent stationary electrostatic waves. The description of these strictly nonlinear structures, which are of electron hole type and violate linear Vlasov theory due to the particle trapping at any excitation amplitude, is obtained by a correct reduction of the three-dimensional Fermi-Dirac distribution function to one dimension and by a proper incorporation of trapping. For small but finite amplitudes, the holes become of cnoidal wave type and the electron density is shown to be described by a /ðxÞ1=2 rather than a /ðxÞ expansion, where /ðxÞ is the electrostatic potential. The general coefficients are presented for a degenerate plasma as well as the quantum statistical analogue to these steady state coherent structures, including the shape of /ðxÞ and the nonlinear dispersion relation, which describes their phase velocity. VC 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4949341] I. INTRODUCTION tunneling effects in the description of electron holes,17,18 but in this paper, we will neglect these effects. We deal with the Plasmas support a large number of nonlinear structures general framework of structures within the Vlasov-Poisson such as shocks, double layers, solitons, envelop solitons, and (VP) description for which Bernstein, Greene, and Kruskal vortices. A particular class of nonlinear structures occurs due (BGK)20 presented the first mathematically correct solution. to particle trapping, where ions or electrons are trapped in Physically, however, another method, the pseudo-potential the self-consistent potential wells in the plasma. Such struc- method19 has proven preferential, as it is mathematically as tures arise naturally in streaming instabilities or other not general and correct but allows from the outset the selection necessarily large amplitude perturbations. Typically, the of additional physically relevant solutions that are not possi- trapping leads to excavated regions in the phase (velocity, ble to obtain within BGK theory.20 Hence, our model is coordinate) space, and hence, they are often referred to as 1–7 semi-classical in that it takes into account the modification ion and electron holes. Electron and ion holes have been of the equilibrium electron distribution function to a Fermi- 8–11 observed in the laboratory and by satellite observations Dirac distribution, while we consider structures large enough 12–14 in space. At much smaller scales and in dense plasmas, that electron tunneling effects can be neglected. the electron density is so high that quantum effects must be 15,16 taken into account. The reason is that the inter-particle II. CONSERVED QUANTITIES AND REDUCTION distance is small enough that the wave-functions of the elec- OF DIMENSIONALITY trons start to overlap, and Pauli’s exclusion principle dictates For pedagogic reasons, we here briefly review some ba- that two electrons, which are fermions, cannot occupy the sic properties of the Vlasov-Poisson system which will be of same quantum state. This changes the equilibrium distribu- importance in Sections III–VII. For brevity, we restrict the tion of the electrons from a Maxwell-Boltzmann distribution discussion to electrons, but the results are easily extended to to a more general Fermi-Dirac (FD) distribution that takes include one or more ion species. The Vlasov equation into account electron degeneracy effects. In the limiting case of zero temperature, the equilibrium distribution for the e @t þ v rþ r/ rv fe ¼ 0 (1) Fermi gas is a flat-topped distribution in momentum space me (or in velocity space for non-relativistic particles). At very small scales, there are also effects due to electron tunneling, describes the evolution of the electron distribution function and in this case, the dynamics of a single electron is fe in 3 spatial and 3 velocity dimensions, plus time, where e described by the Schrodinger€ or Pauli equation, and statisti- is the magnitude of the electron charge, and me is the elec- cal models are based on the Wigner equation. Some works tron mass. The electrostatic potential / is obtained from have taken into account the first-order electron quantum Poisson’s equation ð a) 2 e 3 [email protected] r / ¼ ni À fed v ; (2) b)[email protected] 0 1070-664X/2016/23(5)/052114/7 23, 052114-1 VC Author(s) 2016. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.159.82.198 On: Thu, 02 Jun 2016 12:31:32 052114-2 H. Schamel and B. Eliasson Phys. Plasmas 23, 052114 (2016) where ni is assumed to be an immobile and homogeneous with an equilibrium number density n0 andp anffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi electron tem- 2 neutralizing ion background density. Using that the Vlasov perature Te. The plasmap frequencyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xpe ¼ n0e =me0, the and Poisson equations are Galilei invariant, stationary waves thermal velocity vthe ¼ kBT0=me, and the Debye screening in an inertial frame moving with the phase velocity of the length kDe ¼ vthe=xpe will be used for normalization of the wave can be obtained by setting the time-derivative to zero dependent variables, here temporarily being denoted by tilde, in Eq. (3), giving time-independent Vlasov equation into their dimensionless form through t ¼ xpe~t; x ¼ x~=kDe; v ¼ ~v=v ; / ¼ e/~ =k T , and f ðvÞ¼ðv =n Þf~ ð~vÞ, which e the B 0 0 the 0 0 v rþ r/ rv fe ¼ 0: (3) we will use below. me Consider next the underlying 1D Vlasov-Poisson (VP) system (5) and (6), which is perturbed by a coherent plane Solutions of Eq. (3) are constant along particle trajectories in wave structure /ðx; tÞ¼/ðx À v tÞ that propagates with ve- ðr; vÞ space, given by E¼m v2=2 À e/ ¼ constant, where 0 e locity v in the lab frame. It is then constant in time in a v2 ¼ v2 þ v2 þ v2, and hence a solution can be expressed as 0 x y z frame moving with the wave. In this wave frame, the normal- fe ¼ f0ðEÞ,wheref0ðEÞ is any function of one argument. (f0 can also be a multi-valued, as discussed below.) The con- ized Vlasov equation hence becomes served energy E is essentially due to reversal and translational ½v@x þ /ðxÞ@vfeðx; vÞ¼0; (7) symmetries of the Vlasov equation in time. Depending on the shape of the potential /, there may exist additional symme- where we, for brevity, have dropped the superscript “1D” on tries leading to other conserved quantities. The most important fe and have written vx ¼ v. For our purposes, we write the so- case for our purposes is when / depends only one spatial vari- lution of Eq. (7) as able, which we will take to be in the x-direction. In this case, there is a translational symmetry along the y and z directions, feðx; vÞ¼f0ðnÞhðÞþftðnÞhðÞ; (8) which leads to that the y and z components of the particle mo- where ¼ v2=2 À /ðxÞ is the single particle energy and hðxÞ mentum (and velocity) are conserved (vy ¼ constant and is the Heaviside step function. The separatrix, which divides vz ¼ constant along the particle’s trajectory) and can be used to eliminate these quantities in E, so that a “one-dimensional” the phase space into a trapped and untrapped particle region, 2 is given by ¼ 0. The unperturbed distribution function is conserved energy can be found, Ex ¼ mevx=2 À e/ ¼ constant. Hence, the solution of the time-independent thereby extended, as a consequence of /ðxÞ, to the inhomo- geneous distribution f ðnÞ, which represents the untrapped Vlasov equation for this case can be written fe ¼ f0ðEx; vy; vzÞ, 0 where f is a function of three arguments. A last step to reduce (or free) particles. The new argument n is a generalized ve- 0 27 the two velocity dimensions perpendicular to the x axis is to locity and is defined by integrate the distribution function as pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ðð n ¼ hðÞsgnðvÞ 2 þ hðÞ À2: (9) 1D fe ðExÞ¼ f0ðEx; vy; vzÞ dvydvz: (4) The solution involves a two-parametric (sgnðvÞ, ) constant of motion of (7), such that (8) solves (7).
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