Barghouthi, Imad A

Qatanani, Naji A

IACS Institutional Repository

Jiaii I phys. 77B (2). 241 -245 (2003)i  U PMonte Carlo simulation of Maxwell molecule interactions in space plasmaImad A Barghouthi'* and Naji A Qatanani^' Department of Physics, Faculty of Science, Al-Quds University, P.O, Box 2(K)02, Jerusalem, PalestineDepartment of Mathematics, Faculty of Science, Al-Quds University, P.O. Box 2()()()2, Jerusalem. PalestineE-mail barghouihi2C^ yahoo comReceived 13 Aii^ iust 2002, (uiepted 17 December 2002Vhstruet A Maxwell molecule interactions model by the Monte Curio method is proposed for space plasma simulations The model describesj >»llismn bciweiMi a minor ion with u background neutral. As a result of a Maxwell molecule collisions, the magnitude of the relative velocity is iii)Lh.inin d but its direction is altered However, the velocity of the center of gravity remains the same both in magnitude and direction before and after i|i^ collision In the simulation, pairs of particles are generated at random, the changes in the velocities due to Maxwell molecule interactions arc. liLiil.ili ilKr>words Monte Carlo simulation, Maxwell-molecule interactions, space plasma.rU'S Nos. 52 65 Pp, 52 25 YaIiulcaling with gas mixtures, il is convenient to describe each spec ICS in llic mixture by a separate velocity distribution function / i;;, »\, /) The velocity distribution function is defined such i dv^dr^ represents the number of particles ofspa ICS s which at time f, have velocities between and + 1 </v .ind positions between and + dr^. The evolution in time  ^s IS determined by the net effect of collisions k1 the How in phase space (r^ , , v ,^) of particles under the 'lliicncc of external forces (gravitational, electric, polarization, niagnciic), 'Fhe mathematical description of this evolution cn hy the well-known Boltzmann equation [ 1 ]V /;+ |G  + - ^ f £  + ‘' ' , V „  / = ^St - d )\oUh^erc E IS the electric field, B  is the magnetic field, d / dt is the •^crivaitve, V is the coordinate space gradient, is the space gradient, c is the speed of light, and and Jlic charge and massi of species s. The quantity (5/^. / S t)  '^ prc^ iis the rate of change of /v (r  ^, v^ ., r) in a given region Authorof phase space (r, v) as a result of collisions. As far as the collision term is concerned, the appropriate expression for binary elastic collision between ions and neutrals is the Boltzmann collision integral [2 ]^  (2)where dv^  ^is the volume element in velocity space, d£2 is an element of solid angle in the center-of-mass reference frame, X is the center-of-mass scattering angle, is the relative velocity of the colliding particles .S’and/?, is the differentialscattering cross section, and the primes denote quantities evaluated after a collision. For partially ionized plasmas (Boltzmann's equation can be solved for each test species independently of the other species), there arc essentially three collision terms that have been extensively used to describe the relevant collision processes (ion-ion Coulomb interactions, ion- neutral polarization interactions and ion-neutral resonant charge exchange interactions).In this paper, however, our main concern is for ion-neutral collision processes dominated by the long-range polarization©2003IACS242 hmid A Barffhoulhi and Naji A Qatananiinieractlon [3]. With this so called Maxwell molecule interaction, the lon-nculral collision frequency is independent of velocity, and as a consequence, the calculation of ion velocity distribution function is significantly simplified This interaction model is often used to simulate non-resonant lon-neuiral interactions [4f For this model, the differential scattering cross section IS independent of scattering angle ^  and inversely proportional to the relative speed i,e.constant-------------- (3)/.>(v,) = c x p ( -» i ,v ; / 2 AT,). (4)As mentioned earlier, J f , ( y ) d v  gives the number density(n)  o f the test ions. Integrating over the entire velocity space the velocity distribution function is then/ ,( v ,)  = «vw,2 k  kT>^3/2 rexp' /.2 ^2kT, (5)In this case, the piobabihly ol collision between two particles (,s and h) is independent of their velocities (v^  and ), and we consider the escape ol a minor ions ,v through a background neutrals h { we assumed one neutral species). In the present study, we confine our attention to the Monte Carlo simulation of the Maxwell molecule interactions, i.c . we are interested in computing the velocity of the test ion (minor) after Maxwell molecule collision with the neutral hIn order to simulate the efleet of Maxwell molecule interactions in a plasma, we intioduce a Maxwell molecule model by a Monte Carlo method for a particle simulation. The Monte Carlo method is used to approximate the solution of physical problem by using random sampling The standard procedure of the Monte Carlo simulation is to lollow the motion of the minor ion (one at a time) fora short period of time, at the end of which the change in the minor ion velocity due to Maxwell molecule interactions is determined. The test ion ,s is injected into the simulation region with a random initial velocity that is consistent with the assumed non-drifting Maxwellian immediately below the injection region. The minor (lest) ion .v is considered to move fora short interval of time At under the inlluence of the cxteinal (orces 7'he lime interval between two successive collisions IS randomly generated using a properly weight random number generator 15-7). When a collision occurred, the minor ion velocity after collision was determined by using another set ol random numbers having statistical properties determined according to the chosen Maxwell molecule collision model. Now, we u.se Monte Carlo technique to generate the initial velocity rtf the minor ion ,v. Wo describe the Maxwell molecule interactions and generate the period At between successive collisions and scattering angle, we generate the random velocity of the background neutrals h. Finally, we obtained the minor ion velocity aftercollisioji.Outside the simulation region, the minor (lest) ions ,y arc assumed to be in static equilibrium with some thennal velocity( IkT Jm ,)  " . their velocity distribution function f\ can be written as a non-drifting Maxwellian, offset in velocity.In most of space plasma investigation, the ratio of ion-ncutrul collision frequency to the ion cyclotron frequency very small. As -> 0 . the test ion velocity distribuiuinfunction in velocity space becomes symmetric about an axis that is parallel to the magnetic field direction [8]. Because of the cylindrical symmetry, it is convenient to introduce a cylindrical coordinate system with its axis along the magnetic field. In this coordinate, the ion velocity components in the parallel and perpendicular directions of the magnetic field arc denoted b\ V|| and .IIn the cylindrical coordinate system and in terms of and I'j , the test velocity distribution function / , (wj cq'\(5i becomeswhere1/22 k  kTexp2kTnK exp -{ 2 k  kT, 2kTi(ij'(f)hiand represent the normalized velocihdistributions of the minor (test) ions in the parallel and perpendicular directions of the magnetic field, respectivelyNow, we will use random numbers generator to obtain a sd of tesi particles distributed according to some given distributions (6a) and (6b). We need this to create the test particles and lo inject ihem into the simulation region.Generation o f  ,■For test ions distributed according to /v(vj^J =exp-m ,v 2 ^2 n k l ;9 . 7 - , the probability o f the test ion o f being in somtV yrange of values dv^, is then = 2k vwhich leads to [9]Monte Carlo simulation of Maxwell molecule interactions in space plasma 243values of are given by j  = ra n d ( ) ,with(8).V2InkT,exc2 k nvhcrc rand ( ) is a pseudo-random number with uniform irobabiliiy between the limits 0.0 and 1 .0 , that i.s, taking each aiuc of randO in turn, we must solve eq. (8) and find the ^^ rr^ -sponding value of . which is shown to be22 k n(12b)( 1 2 c)(9)The generation of Vj^ ,^ is similar to generation of , therefore - '.2kT,,I Ills 1 ,, IS the perpendicular velocity of the randomly injected csi ion into the simulation region.WiM’iv t ion  o f  'V'iimsi notice that we are interested in test ions that are crossing he boiiom boundary of the simulation region {ie. > 0 ). riu probability of finding a particle with parallel velocity \^^ cpl across the boundai7  is proportional to the flux of suchnthlog[rand O - l ] . (13)However, the generation of is different from the generation of Vjv. In the case of v'jv, we were interested in those test particles that can reach the boundary of the simulation region, while in case, the background neutrals exist in the simulation region.The probability of picking up a neutral from the neutral ..riK Ics t . i y , .  )vi, , i.e. P (% ) = cv,Js{v,s) , where c is the background that arc distributed according tofh{yvih) along the direction of the magnetic field, is proportionalim.ili/alion constant obtained from J  P(v\^ = IKnowing wc gel2kT, Jexp—niA'i2kT (10)to i.e., = , where c is theproportionality constant. The integral of the probability over the whole intervals (-*»,*») is equal to I :I  (^v||/,)c/V||/, = 1 , this leads toSimilar to eq. (9), the value of the test ion parallel velocity is'.M\cn byp , M =IkT"‘b2 n k T jexp ■h^VfIkTh (14)n\.-  I log [ran J  ( ) - l ] . ( 11 )Hic numerical values of and arc different from each "ilici because of rand ( ). At this stage, we have created, at the ^^ niindary level of the simulation region, a test ion from the ‘^“iicsponding test velocity distribution.Ihc background neutrals are assumed to be in static ^^ luilihrium and, consequently, their distribution function is '^ umc(.l to have a Maxwellian distributionGeneration of a neutral particle from the above probability distribution, is given by the following equationrandO = J  P{vif^)civ'^ . The integral is computed(15)randO = ;  ^ 1 + erf\ i 7 t k n )  2kT,) can be written in terms of and2kTh( 12) where erf is the error function.Hence, we obtain an explicit formula forvl/20 2 a)Verf"' (2 r a n d ( ) - l ). (16)244 hnad A Barghouthi and Naji A QarananiEqs. (9 ). ( 1 1 ), ( 13) and (J 6) give Ihe vckx:itics of the colliding particles (test ion s and background neutral h). Now, we will investigate the cITeci of Maxwell molecule interactions on the above velocities and consequently, the velocities of the colliding particles after collision.Our aim is to simulate ihc interaction of a test ion moving under the influence of an external forces through a background of neutrals atoms, these neutrals are distributed according to Maxwell distribution function with background temperature 7)^ . The simulation steps must provide us the time between each pair ol collisions (interactions) and the change in the test ion velocity due to each collision, l^or a Maxwell molecule interaction, the collision f requency \v is dependent of the relativevclouly jf,,, = V,-V,, .Consider the lest particle with velocity , the probability that this test particle survives at time / without suffering a collision IS P{t), while the quantity wdt is the probability that the test particle collide with the background neutrals between lime / and / + dt.The probability that a particle will suffer no collision in the interval d i is given by Pit -\-dr) = P(t){ \ -  wdf) , using Titylor expansion and the condition /^ (O) = I, the probability P(t) will be/ f^O = cxp(-‘U7). (17)In order to determine the time interval r  between collisions, we .set up the cquatum ;rrando =je '" '( li .Computing Ihc integral on Ihc right, wc gel the relation Iand{) = I / I — e , and henceT = -log(l - (u ')ra n d o )/ vr. (18)The mean distance travelled by the lest particle between collisions IS called mean I'ree path A and given by^ ~ ~ (' iT. +^’l ) log[l -  n rand()]|. (19)M IS the collision Irequency for Maxwell molecule interaction and given by Schur.A | 1 ).Thus, wc can generate pair of particles (test and background neutral) at random. Next we calculate the changes in the velocitiesdue to binary collisions in Ihe lime interval t . A sarcsullofabinary collision, Ihe magnitude of the relative velocity g,,, is unchanged but its direction is altered by the scattering angleX However, the velocity of the center of gravity givenby»'ro = m, +w*remains the same both in magnitude and direction before - after the collision.The relative velocity and the velocity o f the center of gravn after the collision are given by 'g'<h =Ti - KandVcT- =(20a,(20biwhere prime denotes after collision.From eqs. (20a) and (20b), the velocity of test pamclq alter collision isand similarlyni, + nih (?()ci(2(VliThe parallel and perpendicular components of the rdaiiu velocity g,,, arcgM  =Sshl = » ( 21 )where (p is the angJe between and v^jl in the perpendicular plane.Similarly, the parallel and perpendicular components ofarewi.. ntu-'^CGl w,(m, +nif,)m + ntfjvl/2(221After collision, we are interested to determine the ve lo c it)of the test particle v ; however, the velocity of the backgroundis generated randomly for each collision as shown earlierMonte Carlo simulation of Maxwell molecule interactions in space plasma 245Prom eq. (20c), we have= ^CGt ." m,+i7v,J /2nh, , 2niu ,n, ^CClS.hi OOStpni  ^ -rnifj(23)wlicrc (p' is the angle between and in theperpendicular plane, glht = g cos0 " , q ' is the angle between ihc relative velocity g '  and the parallel direction and=[H -g.M)  .Now, the question arises as to how to select the random diieeiions,/. e. the angles and 9 ' .  The angles (p and q>' lake on values from 0 to 27t and are chosen randomly with a unilorm distribution between 0 and 2jt ■rhe  formulas for construction of <p and <jo' will be writtenand(/) = 27T rand( )tp' -  I t: rand( ). (24)However, the cosine of the angle 0 '(cos0 ') , be uniformly diMiibutcd over the interval [-1, IJ. The formula for constructingcosO ' lollows:cos6 ' = 2 rand() -  1 . (25)The values of rand() in these formulas should of course, bedillercnlWc have presented a Monte Carlo method for a Maxwell niolccule interaction. The major steps of the Monte Carloi^inulalum arc(0 Test ion (minor) is randomly generated (/. e. '^ iv and are detentiined) from a non-drifting Maxwellian distribution (of temperature )(ii) The time interval between collisions is randomlygenerated such that it has an exponential probability density function (i.e. P(r) = )(iii) The final velocity of the test ion due to the external forces is computed.(iv) A neutral particle from the background is randomly chosen from a Maxwellian distribution (of temperature(v) The relative velocity g  and the velocity of the center of gravity between the colliding particles (test ion s and neutrals h) are computed. While is the same before and after collision both in magnitude and direction, the magnitude of ^  remaining same, its direction after collision is changed.(vi) and g  after collision are used to calculate the velocity of the test ion (v ') after collision.Steps 2-6 arc repeated using this velocity (v[) as the initial test ion velocity.The result of such procedure gives information about the test ion (minor) for a long period of time.Time average of various kinds can be computed from such data; this time, averages have been set equal to the instantaneous averages over the assembly, in accordance with the Ergodic theory.R eferencesft] R W Schunk Rev Geophyx Space Phys. 15 429 (1977)[2] A R Baraka! and R W Schunk Plasma Phys 24 389 (1982)[3] A Dalgamo, M R C McDowell and A Williams Phil. Trans. RoySoc. (London), Ser. A 250 411 (1958)[4] A R Barakat and J Lemaire Phys Rev A42 3291 (1990)[5J 1 A Barghouthi, A R Barakai and R W Schunk J. Oeophys Res.98 17.583 (1993)[6J 1 A Barghouthi, A R Barakat and R W Schunk Ann. Geophysicae12 1, 076 (1994)[71 1 A Barghouthi, V Pierrard, A R Barakat and J Lemaire Astrophys.SpaieSn 277 427 ( 2001)[8] J P St-Maurice and R W Schunk Rev Geophys. Space Phys. 1799 (1979)[91 C H Aldrich Space Sd Rev 42 131 (1985)

Monte Carlo Simulation of Maxwell Molecule Interactions in Space Plasma

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Monte Carlo Simulation of Maxwell Molecule Interactions in Space Plasma

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