Collisions and Nonlinear Effects in Plasmas A_ f - by Adrian Anatol Dolinsky N65 25 265 (ACCr'SSION NUMBER) - (PAGES) INASA CR OR TNX OR AD NUMBER) (THRU) (CODE) GPO PRICE $ OTS PRICE(S) $ ._ Hard copy (HC) " " Microfiche (MF) https://ntrs.nasa.gov/search.jsp?R=19650015664 2020-03-17T01:45:09+00:00Z APPROVAL SHEET Title of Thesis_ Collisions and Nonlinear Effects in Plasmas Name of Candidate_ Adrian Anatol Dolinsky Doctor of Philosophy_ 1965 Thesis and Abstract Approved_ Derek Ao Tidman Research Professor Institute for Fluid Dynamics and Applied Mathematics Date Approved_ May 7, 1965 VITA Name_ Adrian Anatol Dolinskyo PermanentAddrezs_ 9018 Piney Branch Roa@ Silver Spring_ Marylando Degree and date to be conferred_ Pho D0, 1965o_ Date of birtho ° September 30, 1935o" Place of birth_ WoJnilow (Voinilovl_ Ukraine° Secondary educatlon_ Seton Hall Preparatory Sch6ol South Orange, New Jersey, 195ho Collegiate institutions attended Fordham College University of Maryland Dates Degree Date of Degree 195_58 Bo So 1958 1958=65 PhoDo 1965 Ma_or_ Plasma Physics°. Minors_ Mathematics_ Nuclear Physics° J Publications_ "Numerical Integration of Kinetic Equations", Physics of Fluids _. _36 (1965)o Positions held_ Research Assistant in the Institute of Fluid Dynamics and Applied Mathematics of the University of Maryland_ 1961-1965o ABSTRACT Title of Thesis_ Collisions and Nonlinear Effects in Plasmas Adrian Anatol Dolinsky, Doctor of Philosophy, 1965 Thesis directed byoo Professor Derek Ao Tidman Nonlinear and collision effects in the behavior of plasmas are investigated for an electron gas embedded in a neutralizing0uniformly smeared out background of positive charge° Nonlinearity enters into the description of the behavior of a plasma through the collision term (arising from interparticle correlations) and the self=consistent electric field term (ioeo the ensemble average of the sum of C_ulomb fields of all of the plasma particles) in an exact kinetic equation o It is impossible (at the present time) to treat both nonlinear terms simultan= eouslyo For this reason the investigation is divided into two separate parts o In PART ONE the effect of the collision term on the behavior of a spatially homogeneous plasma is investigated; in PART TWO the self= consistent electric field term is treated under conditions which enable us to drop the collision termo ....._tj____] In PART ONE the problem of relaxation of the exact Balescu- Lenard kinetic equation is solved numerically as an initial value problem for isotropic velocity distribution functions° Several different forms of the initial distribution fUnction are selected_ a Gaussian, peaked at about 0028 of the electron thermal velocity; a resonance function; and a Maxwellian coexisting with a sharply peaked Gaussian (the peak of the Gaussian being located at 2o0 electron thermal velocities)° The Fokker_ Planck kinetic equation is also solved numerically under the same restric_ tions and with the same initial distribution functions° A comparison of the solutions of the two kinetic equations shows very small difference between them, and a probable reason for this is advanced° In addition_ a relaxation time is defined, and the long time behavior of the distri© bution functions is studiedo In PART TWO the problem of light©by-light scattering in a plasma is investigated° Two coherent_ monochromatic_plane®polarized_ plane electromagnetic waves (produced by two lasers) pass through a large volume of a quiescent electron plasma and are scatteredo When the frequencies of the impinging waves are tuned so that their difference is approximately equal to the frequency of the natural longitudinal plasma oscillations, these oscillations are excitedo they are limited by the action of several physical mechanisms_ damping 0 the collisional damping, and the nonlinear effectso However o the Landau We are interested in the nature of the nonlinear effects° For this reason, the plasma is assumed to be describable by means of the collisionless plasma moment equations coupled with the Maxwell equations° The amount of nonlinearity is assumed to be small, and the equations are handled by the method of multiple time and spatial scales, a generalization due to Frieman and Sandri of a perturbation scheme developed for nonlinear mechanics by Krylov, Bogoliubov, and Mitropolskyo The results show that there is a slow rotation and/or change in magnitude of the amplitudes of the two impinging electromagnetic waves (as they pass through the plasma)° The rotation is both in space and in timeo At the same time, a longitudinal electric field is built up slowly inside the plasma, and its amplitude changes slowly in space and in timeo All of the above variations in space and in time proceed at rates which are proportional to the strength of the impinging radiation° Furthermore, the strength of the longitudinal field is at most of the order of magnitude of the strength s of the incident electromagnetic waveso This indicates the effectiveness of nonlinearity in limiting the longitudinal plsmma oscillations o COLLISIONSANDNONLINEAR EFFECTSIN PLASMAS by Adrian Anatol Dolinsky Dissertation submitted to the Faculty of the Graduate School of the University of Maryland in partial fulfillment of the requirements for the degree of Doctor of Philosophy 1965 PREFACE Weshall be concerned with the behavior of fully ionized plasmas, ioeo gaseous mixtures of several species of charged particles at sufficiently high temperatures and low densities to assure complete ionization for all times° Under such circumstances quantum effects can be neglected provided the De Broglie wavelengths of particles are much smaller than the average interparticle distanceso At the same time, we shall assume that particle thermal velocities and macroscopic streaming velocities are small compared to the velocity of lighto Consequently relativistic effects are also negligible° Such plasmas- can therefore be described by the laws of classicsl physics for a many=body system of particles interacting through Coulomb forces° A complete statistical description of a plasma would be by means of a probability distribution function in the phase sp_ce of all of the particles° This probability distribution function must obey Liouville's equation° However, a solution of Liouville°s equation is generally imposs- ibleo Besides, a description by means of a probability distribution function in the phase space of all particles yields more information than is necessary for many _urposeso Many physical properties of a plasma can, however, be determined from a knowledge of a one-particle distribution function for each species of particles° By a one-particle distribution function we mean the average (ioeo ensemble average) particle number density of a given species in the six-dimensional positlon-velocity space° ii iii Wewould like to write downa differential equation from which a one-particle distribution function can be determined for all times if it is known at some initial time, say t = 0 o Such an equation ought to contain no more detailed information than is contained in one-particle distribution functions; i oeo, only terms containing one-particle distribution functions should be presento Such an equation (also called a kinetic equation) can be derived from the BBGKY (Born-Bogoliubov_Green-Kirkwood_Yvon) hierarchy of equations - which are derived from Liouville_ equation - if some assump- tions are made about the correlation functions for particles° The first assumption is that the correlation functions are in some sense small compared to the order of magnitude of the one-particle distribution functions° This is usually true throughout most of the phase space of a many-body system of particles interacting through Coulomb forces° If it is also assumed that one is dealing with phenomena that vary slowly in space and time (compared to the plasma period _ and Debye length), P then the appropriate kinetic equation for the one-particle distribution function f (_,_,t) for the species o becomes _f _f e _f ---- +v o --- + ÷ - v ×B)o ---- - c( ) _t -- _x m c -- -- 3v where ea and ma are, respectively,the charge and mass of a particle of species o ; _(_,t) is the electric field, which includes both an externally produced field and the self-consistent field of plasma particles (ioeo the iv sum of Coulomb fields of all particles, averaged over the ensemble); B(x,t) is an externally produced magnetic field; and C(fa_f T) is a collision term (of the order of magnitude of the pair correlation functions)_ arising from correlations between particles° The derivation of an appropriate expression for C(f ,fT) is impossible without the introduction of additional assumptions° Some problems, however, can be treated, to a good approximationr by neglecting the c:_llision termo The resulting kinetic equation is somet_..mes called the collisionless Boltzmann equation, or the Vlasov equation° It can be used, for example, to describe reasonably well the behavior of a plasma at very high temperatures and very low densities° In general_ however_ the collision term is important and should be retained° Two different forms of C(fa,f v) are widely used in plasma theory° One of them is called the Fokker=Planck collision term, or the Rosenbluth_MacDonald=Judd collision term; the other_ a more exact collision term, is called the Balescu_Lenard collision term° The Fokker-Planck collision term is derived in the same way and under the same assumptions as the collision term for a gas in which particles interact through strong, short-range forces° It can be obtained, for example, by making a Taylor expansion of the Boltzmann collision integral to treat distant collisions° Here, however, an additional assumption has to be made that only those two-particle collisions are to be counted for which Vthe impact parameter for colliding particles is less than some character- istic length, which is chosen to be the Debye radius° The assumptions under which the Fokker=Planck collision term is derived have many questionable features° First, it is assumed that a plasma particle collides with only one other plasma particle at any one time_ ioeo, only two-body collisions are assumed to existo However_ because of the long range nature of Coulomb forces_ a particle will collide with many other part_cles simultaneously° Second_ the time between two collisions is assumed to be much greater than the time duration of a collisiono This is also incorrect for the same reasono Third_ the screening of the charge of a particle by oppositely charged particles does not appear naturally, but has to be added in as an extra assumptiono We may summarize by saying that the Fokker_Flanck collision term treats collective effects improperly° The more exact expression for the collision term, which is used in plasma theory, is the Balescu_Lenard collision termo It can be derived from the BBGKY hierarchy of equations by making the so-called Bogoliubov adiabatic hypothesis° This is that the higher interparticle correlation functions relax to their asymptotic long=time forms rapidly over the time scale in which the one-particle distribution functions are changing° (The Bogoliubov adiabatic hypothesis cannot be made for high frequency phenomena llke electron plasma oscillationso In such phenomena the one-particle distribution functions change on a time scale comparable to the time scale vi of change of the interparticle correlation functionso) The resulting Balescu_Lenard collision term treats collective effects properly, taking into account automatically the screening of charged particles and the many-bodycollisions° The kinetic equation is generally nonlinearo The nonlinear terms in the equation are the self®consistent electric field term and the collision termo Both nonlinear terms are important in the behavior of a plasma_ and w_ _,all _ l_e_ted in both of them in this work° We shall be interested in the self-consistent field term,because its nonlinearity has not been studied sufficiently° However, this nonlinearity_ even if small, is capable of limiting plasma oscillations effectively° We shall also be interested in the collision term, because it has not been investigated sufficiently_ Only the Fokker-Planck kinetic equation has been studied so far to any great extent, whereas only the llnearlzed version of _he Balescu_Lenard kinetic equation has been integratedo To simplify the mathematics, we shall limit ourselves to plasmas composed of only one species of particles, electrons, embedded in a uniformly smeared out background of positive charge to ensure charge neutrality on the average° It is not possible to treat the self-consistent field term and the collision term simultaneously° Further_the Balescu_Lenard collision term we use is valid only for a spatially homogeneous ,field-free plasma; where_s the simultaneous presence of both the self-consistent field term and the vii Fokker=Planck collision term makesthe problem generally intractable (except whenthe kinetic equation is linearized)o For this reason we divide our investigation into two separate parts and select two particular problems° In PARTONEthe effect, of collision terms on the behavior of a spatially homogeneous plasma is investigated° In PART TWO the self-consistent field term is treated under conditions which enable us to drop the collisional terms for the problem of light-light scattering in a plasma° In PART ONE_ to make the problem mathematically _ractable_ we limit ourselves to one®particle distribution funo_ions which are isotropic in velocity space° The exact Balescu=Lenard equation is solved numerically as an ihi_ial valUe_pr6blem for suohdistribution func$iOnSo Several initial distribution functions are chosen_ a Gaussian, peaked at 0028 of the electron thermal velocity; a resonance function; and a very sharp Gau_slan_ peaked at 200 electron thermal velocities_ coexisting with a Maxwellfamo The exact Fokker=Planck equation is also solved numerically for _he same initial distri- bution 'functions° The values of the plasma parameters are chosen such that differences between the solutions of the two kinetic e_uations - if there b_ _j _ _iii b_ _oticeableo Only small differences (a few percent) between the solutions of the two kinetic equations were obeerv@d for the initial distribution functions selected, and a possible explanation for this is advanced° The difference between the solutions of the two kinetic equations for the test particle viii problem is also analyzed_ and a reason for this difference is giveno In addition, a relaxation time is defined, and the long time behavior of the three initial distribution functions is investigated by means of a numerical integration of the Fokker_P_anck equation° In PART TWO we treat the problem of light-by=light scattering in a plasma: Two coherent, monochromatic_ plane_polarized_ plane electro- magnetic waves impinge on a quiescent electron plasma and are scattered° _.._ _:._ fr_._i_s of the two incident wav_ are tuned so tha_ their difference is approximately equal to the frequency of the longitudinal plasma oscillations_ those oscillations are excited° However_ they do not grow linearly with time because of the iimi_ing effect of _eve_al physical mechanisms° We are interested in the nature of the mechanism of nonlinearity onlyo Therefore we assume the plasma to be describable by the collisionless moment and Maxwell equations° We also assume the nonlinear terms in these equations to be small compared to the linear'_termso The equations can then be handled by the method of multiple time scales and spatial scales, a generalization due to Frieman and S_ndri of a perturbation scheme developed by Krylov, Bogoliubov, and Mitropolsky for nonlinear mechanics° The results show that there is a slow rotation and/or change in the magnitudes of the amplitudes of the two impinging electromagnetic waves as they pass through the plasma° The rate of rotation is proportional to ix the strength of the impinging radiation° At the same time_ a longitudinal electrostatic oscillation is built up slowly inside the plasma° The rate of bu±id_up of this oscillation is proportional to the rate of change of the amplitudes of the transverse fields; the strength of the amplitude of this oscillation is at most of the order of magnitude of the strengths of the transverse fields° All of these effects are due to a proper treat_ merit of the small nonlinear terms_ in the equations of motion_ and cannot b_ _i_d _ _pl_ carrying conventional p_rturbatlon throaty _o second order° ACKNOWLEDGEMENTS The author is deeply indebted to Professor Do Ao Tidman for suggesting these topics and for his guidance and interest along the Wayo The author is also gratefUl to Dro Ro Goldman for many helpful discussions of PART TWOe especially since PART TWO forms a part of a problem on which Dro Goldman and the author have been cooperating° Many thanks are due to Mrs° Lana Einschlag for her wonderful typing of the manuscript° The work was supported in part by the National Aeronautics and Space Administration under Grant NsG 220=62o A lot of the computing time, on the IBM 7090 computer_ was financed by the Computer Science Center of the University of Maryland° X TABLE OF CONTENTS Section Page PREFACE ooooooooooooooooooooooo ....... ooooooooooooooooooooooooooo ii ACKNOWLEDGMENTS ooooooooooooooooooooooooooooooooooooooooooooooooo PART ONEs NUMERICAL INTEGRATION OF KINETIC EQUATIONS Io INTRODUCTION ooooooooooooooo .... oo ......... oooooooooooooooo 1 Iio KINETIC EQUATIONS ooooooooooooooooooooooooooooooooooooooooo 4 Ao BALESCU-LENARD (BL) EQUATION oooo ...... oooooooooooooooo h B0 THE FOKKER.PLANCK EQUATION WITH THE ROSENBLUTH- MACDONALD-JUDD COLLISION TERM (RMJ EQUATION) oooooooo_o 8 Co RELATIONSHIP BETWEEN THE BL AND THE RMJ EQUATIONS ooooo I0 Do DEPENDENCE OF THE KINETIC EQUATIONS ON k oo_ooooooooo 12 O IIio RELAXATION TIME ooooooooooo ..... ooooooooooooooooooooooooooo 15 IVo NUMERICAL INTEGRATION oooooooooooooooooo ........ ooooooooooo 16 Ao BL EQUATION oo ...... ooooo ....... oooo ..... oooooooooooooo 16 Bo RHJ EQUATION ooooooooooooooooo ..... o ........ ooooooooooo 19 Co INITIAL DISTRIBUTION FUNCTIONS oooooooooooooooooooooooo 21 Do RELAXATION TIME, ¢(v,T) ooooooooooooo ..... ooooo00o0oooo 2_ Vo RESULTS OF NUMERICAL CALCULATIONS ooooooooooooooooooooooooo 25 Ao COLLECTIVE EFFECTS oooooooooooo ..... ooooooooooooooooooo 25 Bo RELAXATION TO A MAXWELLIAN oooooooooooooooooooooooooooo 30 Co RELAXATION TIME oo ..... 0ooooooooooooooooooooooooooooooo 32 Vlo DISCUSSION oooooooooooooooooooooooooooooooooooooooooooooooo 3_ TABLE OF CONTENTS _continued) Section Page APPENDIX Ao DERIVATION OF THE ISOTROPIC BL EQUATION 000°0°0°°00° 39 APPENDIX Bo THE TEST PARTICLE PROBLEM oooooooooooooooooooooooooo _7 APPENDIX Co THE EXPONENTIAL DECAY oooo .... oooooooooooo ..... ooooo 53 REFERENCES oooooooooooooooooooo ..... 000000000000000000000000°0000 58 PART TWO: NONLINEAR EFFECTS IN THE LIGHT= BY=LIGHT SCATTERING IN A PLASMA Io INTRODUCTION oooooooo .......... o...... ooooooooooooooooooooo 69 Iio ELECTRON-PLASMA EQUATIONS ooooooooooooooooo ..... ooooooooooo 75 Ao MOMENT AND MAXWELL EQUATIONS oooooooooooooooooooooooooo 75 Bo WAVE EQUATIONS oooooo ....... ooooooooooo ..... ooooooooooo 77 III0 PERTURBATION EXPANSION oooooooooooooooooooooooooooooooooooo 81 Ao THE NATURE OF THE PERTURBATION SCHEME o oooo oo..... o oo oo 81 Bo THE O(E) AND 0(¢ 2 ) PLASMA EQUATIONS o.... o.... ooooooo 83 IVo THE 0(¢) SOLUTIONS AND SECULAR TERMS IN 0(¢ 2 ) oooooooooooo 88 Ao INITIAL AND BOUNDARY CONDITIONS oooooooooooooo,ooooooooo 88 Bo TRANSVERSE COMPONENTS OF 0(E) SOLUTIONS ooooooooooooooo 91 Co SECULAR BEHAVIOR IN 0(¢ 2 ) EQUATIONS oooooooooooooooooooo 92 Do LONGITUDINAL COMPONENTS OF O(z) SOLUTIONS 0000000°000°0 95 Vo REMOVAL OF SECULAR BEHAVIOR FROM 0(¢ 2) EQUATIONS oooooooooo 97 Ao EVALUATION OF NON-LINEAR TERMS oooooooooooooooooooooooo i00 Bo ELIMINATION OF SECULAR TERMS FROM LONGITUDINAL FIELDS oo 102 Co ELIMINATION OF SECULAR TERMS FROM TRANSVERSE FIELDS ooo 10h TABLEOFCONTENTS(continued) Section Page VIo PROPERTIESOFTHERELATIONSWHICHREMOVESECULARBEHAVIORoo 107 Ao THEPUREINITIAL VALUEPROBLEM..... ooooooooooooooooooo107 Bo THEPUREBOUNDARYVALUEPROBLEMooooooooooooooooooooooo116 Co MIXEDINITIAL_VALUE-BOUNDARYVALUEPROBLEM00000000o000 118 VII_ THESCATTERING CROSS SECTION oooooooooooooooooooooooooooooo 120 VIIIo DISCUSSION oo ..... ooooooooooooooooooooooooooooooooooooooooo 125 APPENDIXo DERIVATION OF WAVE EQUATIONS oooooooooooooooooooooooooo 128 REFERENCES oooooooooooooooo_ooooooooooooo ..... oooooooooooooooooooo 132 LIST OFFIGURES Figure IIio IVo VIo Vllo VIIIo IXo Xo Page Io BL Solution for the Initial Gaussian with kolko 1 _178 lO8 59m O X O0000oo0000000000OO0000000000000OO000 Iio BL Solution for the Initial Gaussian with ko/_ o- 300 oooooooo...... o...... oooooooooooooooooooooooo 60 BL Solution for the Initial Resonance Function oooooooooo 61 BL Solution for the Initial Gaussian Coexisting with a Maxwellian ....... 0ooooo .... o...... o..... oooooooooooooooo 62 Vo RMJ Solution for the Initial Gaussianwith ko/k D = 300 oooooooooooooooooooooo ..... ooooooooooooo ..... 63 RMJ Solution for the Initial Resonance Function ooooo0oooo 6_ RMJ Solution for the Initial Gaussian coexisting with a Maxwellian ooooooo ..... oooooooooooooooooooooooooooooooooo 65 E(v,_) for the Initial Gaussian with ko/k D = 300 ooooooo 66 C(v_x) for the Initial Resonance Function ooooooooooooooo 67 ¢(v,T) for the Initial High=Energy Gaussian Peak Coexisting with a Maxwellian oo .... ooooo .... oooooooooooooo 68 XIo Schematic Drawing of Light=Li6ht Scattering in a Plasma 00 70 PARTONE NUMERICALINTEGRATIONOFKINETICEQUATIONS Io INTRODUCTION The problem of the relaxation to equilibrium of a fully ionized non-equilibrium plasma has been of interest for some time° In the absence of analytic solutions to the appropriate kinetic equations_ which are non-linear 9 various authors have investigated problems that do not require the solution of a non-linear kinetic equation° As an example of such problems_one may mention the case of the relaxation to equilibrium of t_ ,_ri_ ......_tion of a test particle in_ec_ed into quiescent plasma_ In these problems the appropriate kinetic equation can be linearizedo Up to date the only investigations of the relaxation of a test=particle distribution function have been _arried out by means of the Fokker=Planck kinetic equation° Thus Kran_er I stuSied the _hermal= ization of a fast ion in a plasma by means of a n_merical _olution of the Fokker_Planck equation° Frisch 2 defined certain charaateristic times which he called time lags in the thermalization of a fa_t ion injected into a plasma, and proceeded to calculate them without actually solving the Fokker®Planck equationo Ree and Kidder 3 obtained an analytic solution f_r the !i_ma!iz_tio._ of a fast test ion injected into a plasma by approximating the friction and dispersion coefficients in the Fokker- Planck equation° Their solution is valid only when the speed of the test ion is less than the average speed of the plasma electrons_ but large enough so that the plasma electrons interact more strongly with the test ion than do the plasma ions° Attempts at an actual solution of a non=linear kinetic equation have up to now been confined only to the Fokker_Planck equation° Further- more, they have been numerical solutions° In fact, the only investigation of the relaxation of a non-linear kinetic equation up to date is that of MacDonald, Rosenbluth and Chuck _, who solved numerlcally_as an initial value problem, the non®linear Fokker-Planck equation for an electron-posltron plasma which is spatially homogeneous and is,tropic in velocity° It would be interesting to carry out a similar investigation for the non=linear B_:-,_l,ez_,_ _tion_ This would be espec_.ally interestiz_ in view of the fact that the Balescu-Lenard equation® by treating collective effects properly, gives a more general description of the behavior of a spatially homogeneous plasma than does the Fokker-Planck equation, which does not treat collective effects properly° The only attempt so far at a solution of the Balescu_Lenard (B/L} equation is the _oz_tiono a_ an initlal value problem, of the linearlzed BL equation by Rosenberg and WuJo These two authors took a multicomponent plasma and perturbed _he distribution _nct±on_of each species of particles slightly from the equilibrium Maxwellian distribution° Then they proceeded _ to investigate the decay of this small perturbation in th_ lin_r _rcxizationo This paper presents a numerical integration of the exact Balescu- Lenard (BL) kinetic equation for different initial distributions of an electron plasma embedded in a neutralizing, uniformly smeared out, positive- charge background° The Fokker_Planck equation with the Rosenbluth, MacDonald, Judd collision term (RMJ equation) is also integrated with the same initial distributionso By comparing the solutions of the two kinetic equations one hopes to arrive at an estimate of the importance of collective effects in the relaxation of these distribntionfun_tionBo The simplifying assumption made,ln these calculations_i8 that the distribution functions are isotropic in velocity space° For a limited class of such distribution functions _ for example_ for distribution functions which are monotomically decreasing functions of Iv ! _ our results indicate that for most purposes there is a negligible difference _a few percent) between the predictions of the BL and RMJkinetic equationso This is because these isotropic distributions are sufficiently stable that the v and k integrals in the BL equation ((AI) and (A2)) do not approach a zero D+of the Landau denominator _ _ anywhere in the range of integration° Thus collective effecSs_ which are treated properly in the BL mquation_ but not in the RMJ equation, are of little importance for such distributions° We also define numerically a relaxation time in section (III)_by considering how close all portions of a given initial distribution function will get to the final Maxwellian after a certain time, and whether or not they will stay close to the Maxwellian for all times after that time° Our conclusion is that a, distribution f_mction often oscillates about the final Maxwellian at certain points in velocity spaceo These points depend on the form of the initial distribution function° This behavior points out that the relaxation to the final Maxwellian cannot in general be taken to be an exponential decay (with the possible exception of the high=energy tail)o This conclusion _ agrees with the solution of the linearized Balescu_Lenard equation of Rosenberg and wuS_ which is a superposition of exponential decays° Iio KINETIC E_UATIONS Ao BALESCU-LENARD (BL) E_UATION Let f(vl,t) be the one-particle distribution function for a spatially uniform electron plasma embedded in a _uiformly smeared out background of positive charge° f(vl_t) has two normalization condi- tions f(vl,t) dv I = (i) and I 2 2vI f(v l,t) dvI • v° (2) where v is the thermal speed of electrons o o For the purposes of numerical integration i% is convenient to choose a set of dimensionless variableso Therefore we shall define three dimensionless variables YI' _ _ and g(Vl,_) by Y1 0 _ t ' (_) and 0 '_ (5) where _ (v) is the Spitzer deflection time 6 for electrons moving with D o velocity v° , given by the expression where 2 3 m v 2 1 v 8, n o e _n^ L- ° _ v2 er_\_j - v e erf is the error functiono From equations(l) and (2), the two dimensionless normalization conditions for g(yi,T) are ]g(vl,T) dVl = i (6) (7) 6sad I _I g(Vl' ) dVl = i (8) For isotropic velocity distributions, the BL equation can be written in dimensionless form as v _T -- _ _'i © i V_I dV1 G(VI_ ) #_Vi_T ) v,_. ,v ,.aV_ V _V o +_ Iv_ dVlg(vl,.)_(vl,.) 8V o + v _ o(v,T)¢(v_z) + v2 [g(v_)]2 _(v,_)} (9) where the functions G(V,_) and ¢(V_T) occurring in (9) are defined by and @o G(V,T) - V_dV _ g(ViT) V k 4tn-_ ° k D _n H(V_T) - 32 (lo) , (11) '7 where the functions r(v,T) , H(V_v) _ and L(Vov) are defined by _ i® v2 gCv_) dV_r(v,_):7 P o - w-°2-_v - (12) denoting the principal value integral)_ {13) and 2 2 L(V,_) m tan=l _ =_ 2_2 .......v__!._,_,_._)..... _= ta.u /k \^ , ---, + r(v_) \,.kD_' The quantity k is the upper limit on the o Appendix Ao Its value was taken as 7 kI integration mentioned in KT k = a o 2 ' e where K is the Boltzmann constant, T is the temperature, and e electronic chargeo k D is the Debye wave number_ given by 1 k D = where n o is the electron particle densityo a is defined by 4 16_2- n e k o (vo) 2 3 _D kD m v o where m is the electron masso is the (16) (_7) The isotropic_ spatially homogeneous RMJ equation for an electron plasma embedded in a uniform background of positive charge is _'7 3_= 16_ 2 n o e ¢._ k° I S2f v_f(v_t) J'tj-- dv *t m2 _v 2 v s Is2 _f+ 1 v V "_ f(v_t) dv + 3v _v3 o o v_f(v_t) dv _ v v _)2 (1 + v-L)]- v_dv_f(v;'t) (i = v 2v O + [f(vit) ]2 _ _ (18) Transforming to the same dimensionless variables defined in (3)-(5), eqo (18) becomes 9R awm I" l "V;g(V_) dV,O * _l V V i$ g(V_T) dV; V V 3 o If" iv G+ .&2 _g V'g(v;_)dV': V" dV'g(V_T) ® V.3V _V ;o o v-I-_ + [g(V,'t ) ]2 o (1 + 2V)j , (_9) where all of the symbols have the same meaning they had in the dimensionless BL equationo The two normalizations given by (7) and (8) hold also in the case of the RMJ equation° Co RELATIO_N_SH_I__P_B_E_N_THE BL AND THE RMJ_E_UATI_O_NS eqo (19)o k In the limit of ._o ÷ ® kD eqo (9) tends asymptotically to This can be seen from the following considerations_ F(V_) _ i and Vg(V_) _- 1 _, For D i- << i 0 L(V _) _ tan=l 2 2T _ and £n H(V_I:) -_ £n/!^(lff'\4 V-D) Therefore t by (20) _ (23), (20) (21) (22) (23) IvThe double Integral _l dVl G(VI_T) o single integrals in the following wayo° can be reduced to (2h) f fvV VI2 dVI G(VI,T) = _i dVl V2 dV2 g (V2'T) o o V I i0 ll IV I 2 V2 g(V2,T)+ dV2 V2 dV1 V1 0 0 . _v_f_v._v_,__v_ ÷ 1 Iv V_h-- g(V_,T) dV _ 3 o (25) Substituting first (24) and then (25) into (9) we obtain (19)0 Do DEPENDENCE OF THE KINETIC EQUATIONS ON k o It is shown in Appendix A that k is the upper limit imposed o on Ikll in (A2) to make the _l ® integral convergent° Its value is more or less arbitrary, except that it must satisfy the condition kD --_" << 1 k o (26) We have, somewhat arbitrarily, fixed its value by eq0 (15)o This choice indeed satisfies (26), because in this case kD l m mm _ uk n o o (27) 12 13 but the right-hand side of (27) is << i under the conditions under which the BL equation is assumed to holdo Let us test the sensitivity of the BL equation to variations in k° Since k° enters only in the form [k°_ into eqo (9), let us ° \kD] take (9) at T = 0 and differentiate it with respect to _D/ o This boils down to evaluating the quantity 8 _ ¢(V_0) in (9)_ whose value, by (ll), (13), (lh)_ and (17)_ is a ,_(v,o) = k.% _n kD EIkD / + F(V®0 + Vg(V,_ When k ._o ÷ . (28) reduces to kD o (28) _D_ _D e k'_ ¢(V,0) = ko _n _D° (29) Applying to (9) and substituting (29), we obtain 82 i ......k g(v,o) - k k _D -K° _n-Z°_( ) kD kD [ _ g(V,0)]R_ (30) g(V_0)SRM J is really the RMJ limit of (9)where the expression [_ KT at T = 0r except that k° has not been restricted to the value -_ o gm e By (26), we obtain the condition _2 g(V,0) << KT even if k = e _!because of (26)!o (31) % r(v_) _"_ Vg(V_T ) 3 (32) and (28) is well approximated by (29)° From the above considerations_ we conclude that the relaxation of _a_ one-particle distribution function is not very sensitive to changes in the value of k , at least for v < _k_ o This is of course consistent o \kD/ _h t%c ._:=_iy iog_rlthmic dependence of the BL equation (similar to the equation>forlargevaluesof (Thereasonableinsensitivity kkD/ o to the cut-off value k was also noted by Rosenberg and Wu 5 in the case 0 of the linearized BL equation)o IIio RELAXATION TIME One may try to define a relaxation time as a function of velocity for the one-particle distribution function° For this purposeo let us restrict ourselves to isotropic distributions and write all of the expressions in terms of dimensionless variable_o We define a function ¢(V_) by ITM v21g vo l= ¾ax(Vll E(V_) iV+_ v2 gmax (v) dV JV-6 where gmax(V) is the final Max-_elli_n distribution_ and 6 is a small number° A relaxation time _R may then be defined to be that value of after which c(V,_) is l_ss than some preassigned small positive number, It is of course possible that _(V_T) as a function of decreases for a while to less thsm. a and then increases again before f:na/].2 appro_chlng zeroo These occurrences are easily recugnized in the program, and the relaxation time is that value of x , say TR such that c(V,_) < A for x > _R o (33) A o i5 IVo NUMERICAL INTEGRATION Ao The principal value integral in the expression for P(V) was approximated by the first two non=vanishing terms of a series expansion about the singular pointo Thus we obtained O ÷ V+h V _2 = V2 +(V_v+++)h+_(v_+_ v_ +_ 32V _V - _ gj h3 (34) where h is a small numbero _e numerical integration of the BL equation was carried out by using the difference equation n on+l n mAT I 1 .....gi+l _ 2gi + gi-l_, gi = gi + _i Vi (AV)2 16 17 ivi v2 dV G(V)_(V) 0 + ova 0 / n _ n (gi+l _ gi_l _ Gncn +vi k. 2AV J • (35) for all Vies , except Vo: = V I = 0 and Vo! = VM _ where VM is the maximum value of V usedo At Vo = VM the difference equation was1 n+l gM I( n n n nn _A_ I 1 2gM _ 5gM=l + 4gM=2 = gM=3_ n n n _Ii 3gM- 4gM=l+ gM-2 -_ _v ........J_l iVM v2 dV G(V)_(V)O ^n ,.n n ,,_- '+%-1+ gM-2._IvM v2+ 2 _ V "_ o _v g(V)i(v) n n n 2) 3gM _ 4gM_l + gM_ n n + VM k- ..... £-A _ - GM CM (36) 18 In the above equations superscripts refer to time points_ and subscripts_ to space points o h was chosen to be equal to AV o The quantities n n ' @i ' and Fi are define by the equations Gn - ) (37)l G(Vi _n n _ ¢(vi_n) _ (38)¢i _ r(vi_ n) o (39) The values of integrals were approximated by finite sums° The size of subintervals in the range of integration was chosen to be AV in all caseso Whenever the number of subintervals was even_ the integrals were evaluated by using Simpson_s ruleo Whenever the number of sub= intervals was odd_ a combination of SimpsonVs rule and Newton-Cotes three-eighthsquadrature formula was used° Whenever only one subinterval was available, the trapezoidal rule was usedo n+l At V i = VI = 0 _ the value of gl was determined by the equation n+l n+l gl = g2 ° (40) This was based on the fact that if the BL equation is to hold at V = 0 for all times_ Bo RMJ E_UATION The numerical integration was carried out by using the difference e _uat ion n+n \ I i+l " gi-i (G i n n + 3_i _v = _ si + Qi) (_2) n n where the quantities Gi , Si , and Qi are defined by the e_uations VMf n _= | v g(v) dV , Gi ; V i (_3) 19 2O and _in _ 2__13 fVlo _ g(V) dV (44) (45) Eqo (42) was used for all point_ except V i = VI = 0 and Vi - V M o At V i = V1 = 0 eqo (40) was used_ as in the case of the BL equationo At V i = VM the difference equation was n+l n I_2_4 = 5gM®l + 44=2 " gM-3 _= _ + _ \ 3_v>_ - - o (G_÷ in(o_- _-sM ÷ _) + (46) Co INITIAL DISTRIBUTION FUNCTIONS The following different initial distribution functions and lip _ different values of (=o_ used_ \kD/ were io Initial Gaussian Function g(V,0) = 0o2289 e=2°03(v=0°28)2 (_7) with and VM = 5o0 AV = 0005 o (h8) (49) g(V,_) was computed from T = 0 to _ = 2oh T - 0 to _ = 702 for the RMJ case at intervals AT = 00004 , for two different values of (k°_ kD / Srom the BL case_ and from AT _ where (50) a) k o --- = io4178 x 108 kD (51) and b) k o m k D 300 (52) 21 22 20 Initial Reson an_£e___Fun_c_tio_n- with g(V,O) = -_8 1 (53) VM = 20o0 (5h) AV = 0oi , (55) and k 0 ---= 50 k D (56) g(VzT) was computed from T = 0 to _ = 0o4 for the BL case_ and from T = 0 to T = 3o4 for tne RMJ case at intervals AT _ where AT = 0o01 o (57) 3o I_!ti_Maxwelli_an_ F,unction_Coexistln _ with a High-Energy Ga.uaaJan Function 3 = O o8936 2_ eg(V,O) _ ( )3/2 =lO0(V=2) 2 + 0o01192 e (58) 23 with VM=4oO ® (59) AV m 0002 , (60) and k o --- = i00 (6z) g(V,_) was computed from T = 0 to T = 0o19 for the_ BL case, and from T = 0 to T = lo3 for the RMJ case at intervals AT _ where aT - ooooo5 o (62) The calculations were performed on the IBM 7090 electronic computer o Do RELAXATION_TIME, E(V_T) The number 6 in eqo (33) was set equal to AV o The integrals were performed using Simpson_s ruleo k ¢(V,T) was evaluated for the initial Gaussian with o 300 k D k (the Gaussian with ._o = 1o4178 × 108 was not doner because it is k equivalent to the Gaussian with 9= 300 but with AT increased slightly), the initial resonance function_ and the high-energy Gaussian coexisting with a Maxwellian0 The quantities e(V,_) were computed from the solutions of the RMJ equation only_ because earlier calculations showed the BL mud the RMJ solutions to be almost identical for the above initial distribution functions° ¢(V,T) was computed for the initial Gaussian for values of in the range 0 S +T S 702 ; for the initial resonance function_ in the range 0 _ _ _ Sob ; and for the initial high-energy Gaussian coexisting with a Maxwellian, in the range 0 -< T _< io3 o 2h Vo RESULTS OF NUMERICAL CALCULATIONS Ao COLLECTIVE EFFECTS The most important result of the numerical integrations was that no significant difference was found between the solutions of the BL equation and the solutions of the RMJ equationo For the initial k Gaussian function with _= 104178 x 108 _ the results were essentially identical for the two kinetic equationso The difference was at most 1%0 This result was expected because KuI_) 4 >>i o The same kind of behavior was found in the case of the initial Gaussian function k k with __o 300 , and the initial resonance function D with -°= 500 kD kD This seems to be an 25 26 interesting result 9 because in these two cases quantities of the order Pk°_h Table I shows of unity cannot be neglected relative to An _\kD _ o the values of g(V_T) at T = 204 _ calculated from both the BL and k the RMJ equations_ for the initial Gaussian with o=_= 300 _ for several kD values of V o Table II shows the values of g(Vv) at _ = 0o4 calculated from both the BL and the RMJ equationso for the initial resonance functiono Table I V 0 O025 005 0075 io0 _-o_ lo5 lo75 200 2025 205 3=0 g(v,o) 0°2055 002285 0o2076 0o1464 0008006 0oC3397 0001118 00002856 0°0005658 OoOOO087 0o000010 = oo I _v) J_nax _ 0o3299 0o3004 oo2268 001419 0007362 00O3166 0001129 0oO03337 OoOOO8178 0o0001661 0o000028 _=2oh P_4J 0°3257 0o2979 0o2256 0o1419 0°07409 0003205 0001140 00003274 0o0OO7334 000001233 00000015 _=2o4 g(V_) BL 003243 0°2973 002253 001418 0007410 0°03206 0o01141 0o003273 000007328 0o0001232 0°000015 27 Table II V 0 0o4 008 lo2 lo6 2o0 2o4 2o8 3o2 3o6 ho0 T=0 g(V,O) 0o7789 0ohh77 0o1121 0o02287 00005046 OoOO!297 0o0003882 0o0001327 T " @0 gmax(v) 0o3299 Oo2595 0o1263 0o03805 00007091 OOOOO8!78 OoOOOO58h OOO0O0026 _=Oo h g(V_T) RMJ 0o5979 004067 Oo1241 0o02320 0°004667 0o001192 0o0003622 0o0001258 OOOO0O508 0o0000213 OOOOOOO97 0o0 0o0 OoO 000000488 000000207 000000095 T=0oh g(v_T) BL 005934 004066 0oi,234 0002320 0°004692 00001198 0oOO03633 0o0001261 000000488 000000207 0o0000095 Perhaps the most interesting case was that of the initial Maxwellian coexisting with a sharp high-energy Gaussian peaked at V = 2 o This case is similar to the test particle problem° But the behavior D_ this case was very similar to the behavior of the previous caseso The relaxation of the Maxwellian part of the initial distribution 28 proceed@d withoutreally exhibiting collective effectSo This is not surprising any more in view of the behavior of all of the previous cases o However_ even the peak of the Gaussian _ailed to exhibit collective effectso The difference between the BL and the RMJ solutions for the peak was less than 2%0 A difference of about _% was observed to the right of the peak at velocities which were bet-_e_n 202 and 203 thermal sp_dso However, these differences are too small to show unmistakably the ._:.. t_-_e _:f :.::_le_tive effectso Table III _hows the va!u_-_ cf g(V_) at T = 0ol9 in the vicinity of the Gausalan peak for the solutions of the BL and the RMJ equationso Table III V lo70 lo74 lo78 io 62 lo86 io9o lo94 lo98 0o00066010 0o00049089 0000432260 0o003516h7 _=0o19 RMJ 0o00163051 0o00205418 OoOOOh3724 0o00071164 0000185265 000045O775 0o008_0283 0o01151361 0000284698 0o00229391 0o00183944 0000146794 0o00116586 0o00092151 0o00279051 0000380954 0000499052 0o00611075 0000688563 0o00706345 :=0o19 g(V_) BL 0o00162568 0o00204975 0o00278595 0o003802h3 0o00497691 00006O878O 0o0o685531 0o00703384 29 TABLE III= continued V 200 2002 2006 _-oi0 2o14 2o18 2022 2026 2o 30 _=0 g(%o) 0001197109 0o01149523 oooo83h55o OoO044c_6o 0000169192 0o000h7530 OOOOOO9982 0o000017_6 0o00000384 gmax (V ) 0000081779 0o00072_88 0000056748 0o000h4213 0o0003h282 0o0002645h 0000020316 0o00015528 OoO0011811 _=0o19 g(V_T) RMJ 0o0068873h 0000653863 0o005h1764 0o00398559 0o00258245 0000146207 0o0007178h 0000030369 0o00011028 _=0o19 g(v,T) BL 0o0068622h 0o00652063 oooo5_1858 0000400460 0o00261129 OoOOlhg051 0o00073915 Oo0O031639 0o000116h3 Graphs i, 2, 3, and 4 are_respectively_the plots of the solutions of the BL equation (or the RMJ equation_ since the two give almost identical r_'_!ts) fo_ the four cases mentioned above_ 3O Bo RELAXATION TO A MAXWELLIAN Graphs 5_ 6_ and 7 are_respectivelyothe plots of the solutions k of the RMJ equation for the initial Gaussian with _o = 300 _ the initial kD resonance function_ and the initial high=energy Gaussian coexisting with a Maxwelliano Graphs 5 and 6 agree with the earlier c_Iculazions of Rosenbluth eto alo _ as well as with the findings of Ree eto alo _ that the high®energy tail of an initial distribution function relaxes much slower than the low-energy portionso Graph 5 shows that the point V = 0 which at T = 0 is below the final Maxwellian_ approaches the final Maxwellian and then overshoots ito However_ in the time interval 0 _ T • 7o2 the point V = 0 was not found to start descending toward the final Maxwelliano The two normalization conditions_ equations (7) and (8)9 remained good throughout the whole time interval° The error at _ = 7o2 in the particle=number normalization was less than 0o013%, while the error in the energy normalization was less than 1o8%0 We think that the distribution was not followed long enough in time to permit the point V = 0 to start descending toward the final Maxwelliano The fact that it seems to take a very long time for this to occur is not surprising, since the initial distribution function is very broad and its gradients in velocity space are smallo Graph 6 shows the distribution function for T > 0 dipping below the initial distribution and moving farther away from the final 31 Maxwellian in the higher energy portion of the graph° This tendency to dip seemsto increase with time and to movedownthe high=energy tall o However_what happens to the dip after a very long time can only be guessed_because the distribution function was not followed long enough in time° Oneof the reasons for not following the distribution function longer in time was the large error creeping into the energy normalization° Particle normalization remained good 007 are monotonically decreasing with tlmeo In fact_ for large values of _ they approximate straight lines on the semi=log paper° But we should not conclude from this fact that for these 32 33 curves the decay becomesexponential after a certain time° A look at Graphs 6 and 7 discloses that these curves may cease decreasing and start increasing after a while° The curves in Graphs 9 and l0 were not followed long enough in time to exhibit this behavior° Weconclude from the above discussion that it is impossible to define a relaxation time_ VR_as explained in (III) 9 within the time limits used in the calculations° We also suspect _hat_in general_ an initial distribution fUnction does not decay to a final Maxwelllan exponentially_ even if the exponential decay is assumed to set in after some time_ and not immediately° This suspicion applies to finite V o As for the high=energy tail of a distributlon_ it is still possible to visualize an exponential decay there° For example_ in the case of the initial Gaussian_ 6(V_T=0) _ 1 as V _ _ o If we make use of the fact that the hlgh_energy tail of a distribution function relaxes very slowly toward the final Maxwellian_ ¢(V,_) ÷ 1 as V ÷ _ _ even for large T's o This would give us almost a straight line when plotted on the semi-log papero Therefore it is possible for the relaxation to assume the form of an exponential decay in the high- energy tailo This argument would also be valid for other initial distribution functions which approach zero faster than the final Maxwellian as V ÷ ® (like the Gaussian above)° In Appendix C we present a mathematical proof of the impossibility of an exponential decay of an initial distribution function to a final Maxweliian_ 3_ VIo DISCUSSION The lack of any significant difference between the solutions of the BL and the RMJ equations for the cases treated in this paper has to be taken as a matter of facto It is somewhat surprising in k o cases in which £n _ is of the order of unityo The greatest puzzle is presented by the case of a Maxwellian coexisting with a sharp_ high® _nergy Gs__'_ _ b_r_,,_e of its sim_,larity with the test particle problem° The solution of the BL equation for the test particle problem indicates that collective effects may become important when the test- k o _ 0(i) o By means of arguments particle velocities are high and £n kD analogous to those based on the solution of the RMJ equation_ we obtain some characteristic times for the test particlem_,such as the "slowing down time", T , given by s Mtu3 T = ,,, (63) 2--2 k et _p • £n __o + £n Mt_J kD mp ] the "deflection time", TD _ given by _t u5 TD = =' 2 2 --_ "_ £n o - £n =-2 mp p B5 and the "energy exchange time"_ XW_ given by MtuB n (65) Here et and Mt are the test_partic!e charge and mass respectively_ u is the test particle velocity; 8 , defined by e -zKT _ is the field p_l;ic!_ _alpe_a_e, _ is the plasma freqaency of the field particles_ P and kD is the _bye wave n_ber of the field particleso In equations (6k), (65), _d (66) the te_ containing n _ is the s_e as . kD the one obtained from the solution of the _ equation for the test particle problem° _e te_ c_t_ning _n _ derives from the collective effectso p On the other h_d, if we ass'_e _n isotropic velocity _stribution for test particles and, by analogy with the treatment of MacDonald_ Rosenbluth_ and Chuck4, write the test-particle distribution function in the form 2 Mtv 2e ft(v_t) = g(v_t) e , (66) we can define a characteristic time it takes the inflection point of g(v,t) to diffuse into the high=energy tail of the distribution by Mt et _p n kD (6_) 36 where Win f is the velocity at the inflection point of g(v_t) o (The derivations of characteristic times are given in Appendix Bo) The reason for the disparity between the test particle problem and the numerical solutions of problems discussed in this paper has to be sought in the behavior of the Landau denominator_ D+(_kloikl°_l ) defined in (A6) of Appendix A_ which appears on the right_hand side of tbo BL eau_t_o_ (eqo (A2))o In the RMJ equation D+ = 1 _ because the collective effects are neglectedo In the BL equation_ the value of the Landau denominator varies and may even assume the value zero° When this happenso the integrand of the integral_on the right=hand side of eqo (A2)$ may contain a singularity if the zero of D+ is not canc=AeC by a zero of the numerator of the integrando We shall see that in the problems _h_owhich were solved numerically in paper the integrand has no singularities_ while in the test particle problem the integrand does have singularities° Let us confine ourselves to isotropic distributions° By (Ag), (A13), andl(A17)= we see that Im(D +) -- 0 only when uI - kl ° v I = 0 or u I = _ ° vI ÷ _ o When uI = 0 _ we see by (A16)_ (A13)_ and (Ag) that Re(D _) # 0 o Therefore D+ _ 0 ° _2When u I _ _ _ Im(D +) ÷ 0 , since f(lull) ÷ 0 _ and Re(D + ) *l _ _2 2 ° It is possible to find kI uI 37 a ,_Av-and a _kI in the range 0 < ._,,Ikll_ k • O 2 k I This choice will yield D+ = 0 o such that Suppose we look at the problems which were solved numerically° When IVll ÷ _ by (All} and (A12)_ f(vl I _ 0 faster than F(u]) o Therefore the numerator in the integrand of the integral_on the right side of eqo (A2), is of the order of' _2 _ and the Landau denominator also of the order If(vl)_2 o Hence_ for IVli _ ® _ theby (A17) _ zero of D+ is canceled by the zero of the numerator of the integrand_ and the integrand does not get too close to any of its singularities° Let us now look at the test particle problem° Here_ on account of the tenuity of the test particle distribution_ only the field particle distribution enters into the evaluation of the Landau denominator° D+ For Iyll ÷ ® , is of the order of the square of the field particle _tr_._-_i._f_ f_'_ctio:_o In the numerator of the integrand of _qo (A2), F(u l) and _F • Bu-_ refer to the field partlcles_ while f(v l) and BulBf refer to the test particles° There exists a high velocity range in which the test particle distribution is still finite while the field particle distribution is already approaching zeroo Therefore the Landau denominator 38 will vanish faster than the numerator of the integrand, and the integrand will get very close to a singularity° The preceding arguments confirm the fact that collective effects become significant in the solution of the BL equation only when the integrand in eqo (A2) gets very close to a singularity, at which D+ • 0 , in the range of integration° Such a situation may be realized, for example, in the anisotropic case of two contrastreaming electron _.sm_ .... _J _:L_i.l_ iS certainly worth a more thorough investigation° APPENDIXA DERIVATION OF THE ISOTROPIC BL E_UATION The general anisotropic BL equation for a spatially uniform electron plasma embedded in a uniformly smeared out background of positive charge has the form 7 Bf " - _ (vl_t (A1)s-_ ---o J ) , _vI where f(Yl,t) is the one=particle _istribution function with the two normalization conditions given by equations (i) and (2)0 J(Yl,t) is u_ine_ by the expression 4 2n e J(vl't) --_2 m kI d_ I (Vl) S_u_ _ F(Ul) _ " 2 , (A2) F(u) uI and is defined by i Bu I F(U) ' /k \ are defined by dv 2 , k1 I o uI - kl Vl and u=gom SUl kl SYI D+(.kl,ikl o Vl ) , the Landau denominator, is given by the expression 2 D+('kl,ik I o vl) m i - _2 k I i SF duSu (A3) (Ah) (A5) (A6) 39 4O with Up , the plasma frequency, given by i and ¢ being a small positive numbero (A7) Let us also define the quantity T by the expression _ SF d_u Thus (A6) can be rewritten as 2 D+ = i ÷-_2 _ o kI (A8) (Ag) Let us now specialize all of the above formulas to the case of isotropic velocity distributions0 We can write f(vl,t) = f(vl,t) o (AIO) SF in _cb _se_ (i3) shows that both F(u) and S-_ do not depend on kI o 41 To simplify integrations_ choosethe z=axis in the direction of YI ° Let (kl_e_¢) be the polar=spherical coordinates of the vector kl o We ca_ now perform two integrations _n _A3_ a_ follows We can now perform two integrations in (A3) as follows F(u) sin ede 6(v ! cos e _ de JO O Il= 2w v 2 dv f(v) du 6(v U _ u) f® = 2w I v f(v)dv J|ul (All) From (All) we get SU Let us now simplify the expression for W o If Or and _i }_r_ def_.._ to b%_'espectively_he real and the imaginary parts of W _ ioeo = _r + i_i ' (AI3) we obtain the following expressions for @r and @i 42 and _r = _P _ _F du CJu_) _F ¢± = _ _(v I cos e) (A15) With the aid of (A12)_ equations (A14) and (A15) can be written in the form Pj =m U _ U 1 = _ _® u2 f(u} du P] 2 2 o u = uI (Al6) and = -2 25_ f(luiI) (AZT) where we have made use of eqo (A4)o does not depend on kI or ¢ o on $ o We can see from (Alh) and (A15) that Consequently D+ does not depend Let us now try to simplify the expression for J(Yl_t) o Eqo (A2) can be rewritten in the form 43 _F I2no f(vl ) J(Vl,t) - _ _ D+I2 _ m k I t _f o I _kl kl cl1_l _F(Ul) _ _vl k_ ID+l2 In eqo (A18) the ,integrations over ¢ can be performed immediatel_:o If we also change the variable of integration over e from e to u1 , such that (_8) du I = ®v I sin 8 de (A_9) and then interchange the order of integrations over k I and u I _ we obtain 4 h_noe Y1J(Yl_t) = --_ -_ f(v l) m vI vI k L Io° - _i _VlJ-Vluldul;(ull_oIk°-_ _i o(A20) We have chosen an upper cut-off k on the integral over Ik I in eqoo 1 (A20)o Its meaning will become clearer when the integral over Ikll is evaluated° The integral over Ikll is an even function of uI o Conse- quently the integrands in the two integrals over uI are even functions of uI o obtain If we make use of (AI2) and substitute (A20) into (AI)o we S__f 8_ne= I f(vI12_ ) i f(ulJ _..2_o l_ _ Vl u2 duI _ t m2 v12 Sv I o ko dkl i -o h ID÷I2 ._i 8f f+ v I v_ Vlo ul% F(ull k--T ID+j---'_ (A21) Let us now perform ....._ the integration over kI o k kl ID+I2 --Jo kl _ 2 i 2i Im V Ik° kl dk I ---_ m2 * _0 k I + P r k2 + (02 _ Im _ £n _I 2 Im_ _5 1 i ik2 ÷ 2,12 o_ l=p _I 2 _r --_-__= 2 h tan-I -_- + 2 0i tan=l k2o + _; 0r 0i Or T:_....,_s d.ef_n_: the quantities H(u I) _ L(u I) _ and ¢(u I) by i_2 + 2 012 H(ul) _ an °--2---2--L- 2 012 (A2B) _2 Oo Oi L(Ul ) --tan _l _ _ _an =I k2 + m2 O r o p r (A2_) and _( )_ lUI k ..Zo an kD (_5) As one can see, _o has:to be:,a finite number if (A22) = (A25) are to remain finite° We shall define its value somewhat arbitrarily by (15)o We shall now introduce the dimensionless variables defined by (3) - (5)0 By (A16) and (12), we obtain _6 v o Substituting (A26) and (A17) into (A23) and (A24) and making use of (3) = (5), we obtain HCnI) = H(V) (A27) and L(Ul) = L(V) (A28) where H(V) and L(V) are given_respectivelygby (13) and (14)o A look at (A25) shows that _(uz) = _ (A29) where #(V) is given by (ii)o Further 9 substituting (3) = (5) into (All), we obtain F(u) = 2_ G(V) v o (A30) where G(V) is given by (i0)o Substituting (B) _ (5), (A25)_ (A29), and (A30) into (A21) we obtain eqo (9)° APPENDIXB THE TEST PARTICLE PROBLEM The BL equation for the test particle problem can be written in the form 7 _2 Bft [F(v) + 2 By _v _t _v ...... (B1) where ft(v) given by is the test panicle distribution function° F(v) is with FI(V) and F2(v) £(y) = £1(y) + F2CZ) being defined by 8 2 / k _' et _p2 e 2_/ o \! F1¢v)= - M_j ( .,.,_ kD__\._nkD/ vB 8 being defined by 2 et (_ (B3) (B_) T(v) is given by with TI(Y) T(v) = T1(v) + T2(v) 8 being defined by (BS) _7 _8 " _2 , '_-- v3 ,,,.up.l ./ v2 ½ - 3v and T2(v ) being defined by 8 TI(V) = _Sln kD_V)f v2 _I _ 3 Y v'_ (_) (ST) -When we substitute (B2) = (B7) into (BI)_ perform all of the differentiations_ v and drop terms of the order ._o v , where v is the thermal velocity ofO field particles_we ubtain-.the diff_ren_ia_.-equa_ion (=t " e kl\ 'i_", ko_ v- ftMt / n ----- kD/ v__ o -V-- 4_ Y _ft L Mt v3 _Y i S2ft ) +F T2 : __ o (s8) _9 The terms inside the first pair of braces on the right side of (B8) are identical with the ones obtained from the solution of the RMJ equation_ The terms inside the second pair of braces are due to the collective effects° If we now confine ourselves to isotropic distributions, the following relations holds and S2ft B2f 2 Sft i ---- _- _v. _v_ _v v _v S2ft 2 S2ft V V _ -'_m- = V _ _)v_ _)v. _)v2 (Bg) (BZ0) Substituting (B9) and (BIO) into (B8), and dropping terms of the form ft as well as terms of order one relative to terms of order _n _pD_V_, we obtain the differential equation et m Mtv2 kD Sv P (Bil) 5O Let us now define a function gt(v_t) by Mt v2 ® 2-'F= ft(v,t) - gt(v_t) e o (BI2) gt(v_ t) satisfies the diffusion equation Mt ?--2 _t p _-_= M_v2 n kD If gt(v,t) has an inflection point at Vin f _ the speed with which this point diffuses into the high-speed region is given by gt =cOnst MtV_nf n kD kDVin fq + En m------ -jUp 4 We can define a characteristic time T by O (BI3) (B14) T O Vinf @t Jgt=cons t Mt V3nf 2 --2 ILu k kDVin f-et Up _D + _n _m P (BIS) 51 Let us now go to the anisotropic case° Let us assume that ft(vgt=O) = 6(v _ u) o (Bi6) We shall take velocity moments of eqo (B8) at t = 0 o First, let us multiply (B8) through by y , and then integrate over y o Making use of (BI6) and integrating by parts_ we get _ e_ _ k _ m _ _M t L kk, + Mt _p2J £n __OkD+ £n ---_!, _uB up J We can define a "slowing down time ''6 by (BZT) T _ - u = , _Mt u3 . °\St/ et2 mp + Mt--_p2/! Kn _D + £n _I Let us now multiply (B8) through by v y , and then integrate over v o We obtain (818) (uu) --et-_ n e B"T - - Mt kD,/ - _tt u 5 ek_ + m 52 We can def_e_B"deflection time ''6by 2 u TD = .._.z.=_ _ 2 --(u_)It 2 5 Mtu 2ele. i V =._-_m J (B20) where UiL is the component of velocity perpendicular to u o We can also define an "energy exchange time ''6by u2 Mt u3 , n=o"= Vl (B21) APPENDIX C THE EXPONENTIAL DECAY Here we shall present a proof of the impossibility of an exponential decay of a distribution function° It is based on an adaptation and general- ization of the method used by Rosenberg and Wu 5 to solve the linearized Balescu-Lenard equation ° Let us write the kinetic equation for a spatially homogeneous -_,_asma,_ _ '_h_ _ynb_l_ form _-_f- c(f,f) (ci) _t t where C(f,f) is a collision operator which has not yet been specified° Thus we have not yet limited ourselved to any particular kinetic equation° Let us restrict ourselves to collision operators which are bilinear functions of f(v,t) o The collision operator of the Fokker_Planck equation satisfies this requlrement_ but the collision operator of the Balescu-Lenard equation does not° If fo(V) is the Maxwellian distribution to which f([,t) will relax, we define a function fl(v,t) by the expression fl(Y,t) _ f(y,t) - fo(V) o (C2) Substituting (C2) into (C1), (C1) can be rewritten in th e form _fl _- = C(fo,fl) + C(fl,fo) + C(fl,fl) o (c3) 53 54 The term C(fo,f o) = 0 and therefore was not written explicitly in (C3)o If fo(V) is incorporated into the definitions of the operators C(fo,fl) and C(fl,f o) , the right side of (C3) can be said to consist of two linear functions of fl and one bilinear function of fl ° When fl is small compared to fo and eqo (C3) is linearized, the term C(fl,f I) is simply droppped from the equationo If the Balescu-Lenard equation is linearized, it also satisfies the linearized eqo (C3)o Let us further restrict ourselves to flgs which are isotropic in velocity space° Suppose a complete orthonormal set of real functions of Ivl , I$n(V) } , has been selected, and fl is expanded in terms of the members of this set, so that fl(v,t) = _ an(t ) _bn(V) nmO Then eqo (C3) can be written in the form ---- = + _ Bkm n a a8t _" Akn an m n n mln where Akn and Bkm n are defined by (c_) (c5) Akn- (¢k' C(fo'¢n)) + (¢k' C(_n' fo )) (c6) and Bknm _ (¢k' C(_m'¢n)) o (C7) 55 The symbol C(fo,_n) , etCoo of Ivl, (¢k' C(fo'$n)) denotes the scalar product of Sk(V) and If we select a different orthonormal set of real functions {_n(V)} , and define the matrix element (c8) I then the matrix A can be obtained from the matrix A by an orthonormal transformationo Let us restrict ourselves now to the Fokker_Planck _:,'_ion, !_+ us also assume that the set {_n} is the same complete set used by Rosenberg and WuSo Rosenberg and Wu 5 showed that _ has real, non-positive eigenvalues in the case of the linearized Balescu-Lenard e_uationo This must also be true in the case of the linearized Fokker- Planck e_uation_ and since the matrix A , or A , is not changed when the Fokker-Planck e_uation is linearized_ A must have real, non- positive eigenvalues in the case of the non-linear Fokker-Planck e_uationo Hence also the matrix A must have real, non-positive eigenvalueso Let us denote a particular eigenvalue by (-7 (_)) and the corresponding eigenvector by X (_) , so that the equation AX (9) = -V (_) X (_) is _a.tisf_,_:_ We shall now expand the function vectors of A o Thus we obtain a (t) n - in terms of the eigen- a (t) = _ b(_)(t) X (_) (C9) n n o 56 Substituting into (C5), we obtain ab(_I .v(_lb(_) St where DX_ v is defined by b(_) b (_) (cio) Dl_v-: _ B_n _) X(_)X(_) kzm,n m n o Since Dk_ v # 0 , (ClO) does not have any solutions of the form "_ .!constl tb _ _.._i_) _ o Hence an initial distribution function cannot relax to a final Maxw_llian via the Fokker_Planck equation by means of a simple exponential decay° (cn) In the linear approximation eqo (Cl0) reduces to the equation = -_(_) b(_) o (C12) Eqo (C12) has the solution where C (v) b (_) . C (_) e-y(v)t is some constant, determined by initial conditions° Substituting (C13) into (C9), and subsequently into (C4), we obtain (in the linear approximation) (ci3) fi(v,t)= _ _ c(_)e-V(v)tx(_)n Cn (v) ° n (cih) 57 Since fo(V) satisfies the normalization conditions (i) and (2), we must have f"v2 fl(v,t) dv = v4 fl(v,t) dv = 0 o 0 Let us define the n_Ambers a (u) and B (_) by the following equations_ n ¢n(V) dv n o and 8(_)--n _ C(v) X(V)n ]_ vh Cn (v) dv o (c15) (c16) (c17) Then, eq0 (C15) yields the following two equations_ and =0 8 (_ e_y(_)t m 0 Eqso (C18) must hold for s_.l times 0 including t - 0 o (C18) become When t = 0 , a(_)- [s(_)=o o V c(_) If we had only one exponential decay in fl(v,t) , by (C14), _ 0 when _ - W_ and C (_') - 0 when _ # _o Consequently a(_) W 0 and B (_) # 0 when _ = _, while _(_) • B (_) m 0 when _ # _o It would then follow from eRo (C19) that a (u) = 8 (W) - 0 o Therefore fl(v,t) has to contain more than one exponential decay even in the linear (cl8) (Cl9) approximation° REFERENCES io Kranzer, Physo Fluids _, 212 (1961)o 2o Frisch, Physo Fluids _, 1167 (1961)o 3o Ree and Kidder_ PhySo Fluids _, 857 (1963)o _o MacDonald_ Rosenbluth, and Chuck_ Physo Revo 107, 350 (1957)o 5o Rosenberg and Wu, Can° Jo Phys0 _2, 5_8 (196h)o 60 Lo Spitzer_ Jro D Physics of Fully Ionized Gases_ Intersclence Publishers, Inco, New York_ 1956o 7o Do Montgomery and Do Tidman_ Plasma Kinetic Theor_., McGraw=Hill (1965)o 8o Tidman_ Montgomery_ and Guernsey_ Physo Fluids _ 1089 (1964)o 58 Figure I 59 N T 0 0 d 4------6 0 d 0 q 0 0 Figure II 6O o r,_ N T 0 0 0 +---B q 0 0 0 o. 0 Figure III 61 o 0 z _1 J x D 0 0 b o II o 0 N 0 5 0 0 0 T Figure IV 62 T 0 0 I o o o o o O o o o Figure V 63 0 T D, u-h 0 o 0 8 o 0 0 0 0 0 0 Figure VI 64 O 0 0 6 8 q 0 0 8 0 0 0 m kO Figure VII 65 d M I T b- 0 0 0 6 0 0 6 0 0 0 Figure VIII 66 .001 .0001 I I I I I ' I 2 3 4 5 7 Graph 8 _±_ IX 67 I0 2.5 h > I % 0.5 2.0 •I 1.0 ,01 .001 I I I I I I I 0 .5 I 1.5 2 2-.5 :3 3.5 -g Graph 9 Figure X 68 I0 v .I .01 .001 I I I I I I I I I I I I 0 .I .2_ .3 .4 .5 .6 .7 .8 .9 1.0 I.I 1.2 2.2 1.8 .02 .5 2.0 1.0 1.3 T Graph l0 PARTTWO NONLINEAREFFECTSIN THELIGHT-BY-LIGHT SCATTERINGIN A PLASMA Io INTRODUCTION Recently there has been considerable interest in the scattering 1 of light by light inside a plasma° Platzman_ Buchsbaum_and Tzoar calculated_ using quantum mechanics_ the incoherent cross section for the scattering of light by light in the presence of a plasma to lowest order in the plasma parameter_ ioeo to lowest order in the reciprocal of the number of particles in the Debye sphere° Kroll_ Ron_ and Rostoker _ calculated, by solving the Vlasov equatlon_ the scattering cross section for two plane electromagnetic waves each one of which is monochromatic and coherent° With the pre_ent-day state of laser technology such a light= bymlight scattering experiment is feasible° This scattering process is of practical interest_ because it can be used® among other things, as a density probe for plasmas,° It has advantages over the process of incoherent scattering of a single light beam incident on a pla_ma_ because the scattered energy flux per unit incident energy flux is much larger for the light-llght scattering process than it is for the scattering of a single incident light beam (as was pointed out by Platzman et alo I and by Kroll et0 alo'2)0 The reason for this fact is that a single light beam passing through a quiescent plasma is only scattered by the thermal density fluctuations, which are small° On the other hand, the presence of two incident light beams enables us to tune their frequencies so that their difference is equal to the natural frequency of longitudinal oscillatlons0 69 7O The two light beamsare then able to excite coherent plasma density oscillations_ and are in tu_ scattered by these oscillations° These density oscillations are much larger than the thermal density fluc%ua%ions® and therefore enhance the scattering process° Weshall makethe following model for the scattering processo Twoinfinite plane waveso with wave vectors _l and _2 _ smd frequencies ml and _2 respectivelyo i_pinge on a quiescent plasmao confined in a large volume V o A detector is placed very far from the plasma _ud measures the scattered energy flux over a long period of time T o quiescent electron plasma JJ Figure XI 71 For the sake of simplicity we sh_l assumethe volume V to be a rectangular box and the plasma to _onsl_t ::.t' one species of particles_ electrons_ with average particle density !:,_o To ensure charge neutrality) the electron plasma is embedded in a _._iifc:_y smeared out background of positive charge of charge density N e o 0 The differentia& _::1C'_ _ section per unit frequency :tn'_e_:va_i for 2_3 the scattering of light b_ _ plasma i_ 2 de r ,, sin2 G 2 e the c_'_c._.l ele_,_n radius; 0 is the a_.gge ofwhere r 0 _ ) mc scattering_ ioeo the angJ_e betw_en the incident energy flux and the scattered energy flui; k is the wave vector of incident light; _ is .= the frequency of incid_nt, ligh_; _' is the frequency cf _ca_tered _ight i is a unit vector poir_tin_ in the direction of the " "_ ....; _" ..... S(k_) is the spectral density_ defined by * Jim V _ T_ 0 where n(k_) is the Fourier t_an_form of n(_t) which is defined to be the fluctuation of the electron density about the equilibrium density N 0 ° 72 The calculation of n(x,t) is difficult_ because the equations describing the behavior of _ plasma are non_,!inearo To make _he problem tractable_ one usually rest_icts oneself to incident light beams whose amplitudes are small in the _onse that the changes they produce in the plasma variables are small cc_ared with the values of these variables for the quiescent plasma (ioeo !n'_i_iasma variable which is zero at all spatial points for all timeso This c_ be seen from eqo (27)_ which requires _(1) the existence of _L (cha_.ging on the slow spatial and/or time scale) to eliminate the secular terms in the nonlinear driving force° " emBy eqSo (23) IJ - I - k2 'c,os{ I + b(_Xl,_tl)(ki _ k2) sin(_ l _ _2 ) 0 However_ we can show that b(¢_l_et !) - 0 identically° The argument goes as follows° if E(LI)m b sin(_l-_2) , n(1)_ c(,_(_l- _I,2) The pre_ence of n ('I) introduces an 0(¢ 2)then, by ego (19), o transverse current of the form n (I) v_ I) into eqo (26)0 Taking into 'Iaccount the form of vTl vTas given by eqo (3&)_ we see that n (I) (!) contains terms proportional to (A2 cos ¢i ) and (A1 cos _2) o The first of these is polarized paral,lei _,o A 2 but oscilla_es w_th phase _i _ the second one is polarized parailel to A1 _ but osciiia_es with phase @2 ° Both terms will consequently be in phase with the natural trans® verse plasma oscillations_ and will drive these oscillations° A slow spatial and/or _emporal drift of the phase angles ¢i and ¢2 may not be fast enough to get the natural transverse oscillations and the transverse current (n(1) v$1))_ sufficiently quickly out of phase with each other to limit the oscillationso We therefore require_ in general_ an additional relative rotation of the directions of polar- n(1)ization of the current ( v ) and the transverse plasma oscillationeo 94 A glance at egso (26) and (31) shows that the nonlinear terms oscilatlng with 91 or 92 are proportional to sin _i or sin 92 o On the other hand_ the slow variation of E_l)- in eg0 (26), contain= ins sin @l or sin 92 _ will be proportional to _ and/or or _2 with respect to _t I and/or eXl o Therefore there is no provision for the rotation of the directions of polarization to remove secular terms from ego (26)° The presence of E_I) = b(k 1 = k2 ) sin(gl=92 ) , as we can see_ creates secular terms in ego _26) which cannot be removed° We shall therefore set b(¢xi_£t I) = 0 o The presence of E_I) = a(kl = k2 ) cos($i=_2) , on the other n (I)_ sin(_l=92) Hencehand, creates no such problems° By ego (!9)_ o the current (n(i) y_l)) will contain terms proportional to (_2 sin _i ) and (_i sin _2)0 The nonlinear terms in ego (26) oscillating with phases 91 or 92 will be proportional to cos 91 or cos _2 o But the slow variation of E_I) in ego (26), containing cos 91 or cos @2' will be proportional to the derivatives of _ or _2 ' with respect to cxI and/or at I o Therefore slow rotation of the amplitudes will he possible° Do LONGITUDINAL COMPONENTS OF 0(c ) SOLUTIONS Let us define the quantities and @4 by the following expressions o _3 - _l + _2 '"4 - _i = _2 ¢3 - ¢i + ¢2 ¢4 _ ¢1 = ¢2 @3 =- @I + ¢2 @4 = @i = _2 where a(¢Xl,Ctl) (36) We shall take E_I) to be of the form (1) = a(cxl_t_) kh cos @4 (37) is an unknown function to be determined by the solution of 0(¢ 2 ) equationso The initial condition that El1) be zero at t - 0 everywhere, yields the initial condition on a(¢Xl_Ctl)_ a(eXl,¢t l) • 0 at t • 0 _ for all x , inside the plasma and on the boundary° The condition that E_I) boundary condition on all t o be zero on the boundary for all times_ yields the a(Exl,_tl)_ a(asi,Etl)_= 0 on the boundary for 95 96 An inspection of eqo (16) to (22) discloses immediately that their solutions have the following longitudinal components_ (i) a k_ n - _ sin _4 (38) yL(1) . eam_4 _h sin _4 2 2 0 o Similarly, if at t = 0 _ Aly = A2y - 0 _ then Aly - A2y - 0 for t > 0 (and consequently A2x = 0) o Therefore, if _i and _2 are at t-0 in the plane of _l amd k2 _ they wall remain in that plane for t • 0o On the other hand_ if _i and A_are perpendicular_to the plane, of_kl and _2 at t • 0 _ they will remaln, so for t > 0 o These two results are not surprising, because in both cases the component of _2 which is perpendicular to _i is either p_rallel or antiparallel to _i ; and Ii0 likewise, the component of parallel or antiparallel to and _2 can changeo _i which is perpendicular to _2 is _2 o Therefore only magnitudes of We shall now show that solutions to eqSo (57) _ (59) can be obtained for some specialized cases of physical interest and that those solutions are periodic° For this purpose let us introduce new variables_ defined by_ - ctI Yl _AIy Alz Zl _ _ A2z Z2 - ----- V --''-'-'m'-- a 2 _ - cos 16m 2 _i_2 (63) iii become Then eqSo (61), when expressed in terms of the new variables_ dY 1 m i Vd_ =Y2 dY 2 "" =By vdT i dz I m _Z 2 Vdr dz 2 dT -_ d_v [] Y(Yl Y2 + Zl z2)d_ o (6_) The initial conditions can also be written in terms of the new varialbeso They are Ylo [] Yl (T•O) Y20 = Y2 (T=O) Zl0 " Zl(_•0) z20 = z2(_=0) v == 0 when T _ 0 (65) 112 Let us confine ourselves to the special case when _i and _2 both lie in the plane of _i and _2 ° Eqso (6_) then reduce to the three equations_ dY 1 dT ®Y2 ' dY 2 m By I VdT and dv aEumm dT = 7 YlY2 Eqso (66) have the properties of the derivatives of elliptic functions_ defined as follows_ If u • __d$ _ --' is an elliptic integral of the first kindo then sn(u,k) _ sin $ , cn(u,k) cos $ , and dn(u_k) _ _ o From these definitions we (66) d d obtain _u sn(u.k) • cn(u,k) dn(u,k) . _u cn(u,k) = = sn(u,k) dn(u.k) , d and _u dn(uok) m =k 2 sn(u_k) cn(u,k) o Therefore we make the ansatz that Yl = YI0 cn(_T_K) F2 • F20 dn(_T,_) , and v = c sn(_T_K) , where _, _, and c are unkown constants to be determined° Substituting the ansatz into eqso (66) 0 we obtain yl = Ylo cn I/_7 Y20 T i_ YlO 1Y20 y2 g Y20 dn [_7 Y20 T i/_ YI--O1 ' Y20 v = /_7 YlO sn[ _7 Y20 T i_ YlO 1 ' Y20 ° The elliptic functions [i/2_u with a period equal to 4 __ o oo Jl.k2sin 2 solutions (67) are periodic functions of _ o i sn(u_k) z cn(u,k) , and dn(u,k) are periodic in Therefore the 113 (67) The other special case, when A1 and 82 are both perpendicular to the plane of kI and k2 , can be solved in an identical way° In this case eqSo (64) reduce to dz I dT _z2 dz 2 ---- - zI v (68)dr dv = 7 zI z2 0dv The solutions of e_so (68) are identical with the solutions (67) of eqSo (66) when B • i , and zI , z2 , zlO , and z20 replace Yl ' Y2 ' YlO ' and Y20 ' respectively, _4 We will now cite an example where the solutions of eqSo (64) are periodic elliptic functions although the component of A2 which is perpendicular to kI is not parallel or antiparallel to A1 o Let us, first of all, derive some conservation laws applying to the components of _i and _2 o Multiplying the first of eqSo (64) by Yl ' the second by Y2 ' then adding the two equations and iht_gratinj_ we mbtain 2 2 2 Y2 2 Y20 Yl + ----8• Ylo + B (69) Performing identical manipulations with the third and fourth equations of the set (6_), we obtain 2 2 2 2 zI + z2 = zlO + z20 (70) Let us now divide the first equation of the set (64) by the thirdo We obtain _l Y2 m 8 a 0 dzI z2 Making use of eqso (69) and (70), we can write 2 Y20 2_i/2 _1 _l/2 (Ylo + _ " Yz j dZl 2 2 _2_i/2 (zlO + z20 - _i j (Tz) ,115 Let us define two new variables, 81 and 82 , by means of the expressions 2 2 )1/2 Yl = (Ylo + Y20/6 sin eI 2 _2 _i/2 sin 82zI = (zlO + _20 _ (72) From eqso (72) and (71) we obtain or with c defined by de I = de2 eI _ c = e2 (73) c = sin =I I io I_YlO y2 1/2 2 ' 2-)1/22 + _ (zlO ÷ z20 Then dY I m on substituting for dT in the first of eqSo (64), we obtain _1/2 _ del ._112 - • ---- • dt 0 E (2 2 2 2 2 ) sin2k_ /YlO ÷ zlO " YlO + sin2 el = (zlo + z20 (7_) If we now select the special case in which _i and _2 are parallelz B!/2 - i_ and eqo (7_) is the differential of an elliptic integral of the first kind° Hence eI is an elliptic function of t o Bo THE PURE BOUNDARY VALUE PROBLEM We shall now neglect the initial conditions but retain the boundary conditions° Thus we can study the case of no time dependence of a , _l ' and A2 o Eqs, (50), (55), and (56) now reduce to 2 2 _4 _ _a e_Vo _ " (_ o4) ÷0(Po)_e_ , (75) ] a j k_2 kI _ A2 + O(P O) terms_ i (T6) and k2 ° " A2" hm c2 ml r(_l ) aL k_ b2®Ai_ O(P°) te_ o (77) As can be seen from eqo (75)_ a pure boundary value problem is an impossibility when Po = 0 0 Therefore we must assume a finite tempera_ ture for the quiescent plasma° Let us take, for the sake of simplicity, the y _ z plane to be the boundary between the plasma and the vacuum,with the plasma on the positive side of the y _ z .plane° Let us also assume _l and _2 to be parallel to one another, for the time being, and to point in the direction of the positive x-axiso We can obtain some conservation laws from eqSo (75) (77)° For example, multiplying ego (75) by _ and dotting eqo (76) with i16 117 _i ' then adding and integrating_ we obtain 2 _i k4 k_ 2 24 ov a Al(eXl=0) c p p The conservation equation (78) shows that a(¢xl) is boundedo that the longitudinal field E (1) and hence the density n (I) _L bounded in spaCeo (78) This means 8Lre a comparison between the values of E_l)(etl)(_We would like to make (1)(gx I) Since _,2(¢xi) _ Ai,2(et I) , we obtain from eqo (78)and _L o and the first one of eqSo (62) that a c c 2 _ik4 a2(etl ) vo But ml _ c kI , c k 4 o Therefore or and _h = ml = _2 _ _p ° 2 2(¢_I) c a2 " _ -_, (¢tl,) vo Furthermore mh * C(kl ® k2) " la(¢_z)l c V o Vo always, )>aver (c2/_ml/kl))t the instant_ _l Will be independent of x o For x < (_2/(ml/kl))t ,8.neous value of on the other hand_ AI will generally differ from one spatial point to another° Thus an observer located at a point x , with x > (c2/(_i/kl))t (c2/(m2/k2))t , or with x _ (v_/(_4/kh))t- _ depending on which=with X ever convective velocity is the fastest D will not have yet experienced the effects of the boundary for the first t secondso As far as he is concerned, he sees only an initial=value problemo On the other hand, an observer located at a point with the coordinate x less than the product of the fastest convective velocity and the time_ will have already experienced the influence of the boundaryo The reason for this behavior is the finite velocities of propagation of the changes in a z _ _ and _2 ' respectively° Vllo THE SCATTERING CROSS SECTION ___ - , __ We shall now estimate the scattering cross section for the llght= by-llght scattering process, The differential cross section per unit frequency interval is given by eq0 (1)o We have to calculate the spectral density S(k,_) , which is defined by eqo (2)0 To lowest order in where S(k,=) - lim V_ T_ 21_n(1)(k,_)I2 NVT o n(1)(k,_) is defined by (80) n(1)(x,t) - n(1) (k,m) ei_ °x+_t) Since we are considering the resonance process_ sin _4 o Therefore n(1) (x_t) • (81) I I T/2 e_i(ko x+mt )n(1)(k,m) = dx dt n(1)(X,t) V ~ =T/2 I {T/2 )dx dt, e=i(_°5+_t a(¢x,¢t) sin _h v J-T/2 k_ • i¢4 i e=i(k.k4)o xm dx IT/2dt e=i(_+uh)t a(¢5_¢t) =TI2 120 121 "_'¢4I-e dx V e=i(k+k4) ox IT/2 )}- _ dt e=i(_=_4)t a(¢x_¢t J=T/2 ==i% ] S(k®_) can now be written in the form 21n(l)(k,_) 12 S(k_m) . ¢2 lira -- N VT V_ o T_ 2 k_ i --- I32_t2 N V-_ 0 T-_ , la(_,_4,,_=%)12 = e2i% a*(_*_,_-%) a(_®_4o=._41 =2i% ] The spectral density (82) Let us take a closer look at a(k_m) o Since a(¢x,¢t) is a very slowly varying function of x and t , its Fourier transform a(k,_) is sharply peaked at k = 0 and m = 0 and has a small spread in k and m about this peak° Consequently the cross terms in eqo (82) are very small compared to the other terms_ and we shall neglect themo Let us also neglect the spread in k and _ o Thus a(k-k4,m+u4) will be approximated by the quantity (VT _ 6v k 6 ) where _ is the _-_4 _C_4 ' value of a(x,t) at the peak° The value of S(k_m) will then be approximately 2 a k 32_ 2 N e _ _ o 122 (83) where 6(k) and _(_) are the Dirac delta functions° We shall not 2 write _ in the expression for S(k_) from now on_ because the presence of _ 2 _ which is of 0(¢ 2 ) _ is sufficient to indicate that s(_,_) is of o(E2) o We may note that if we were not dealing with a resonance process_ n(1)(x_t) = 0 _ and the first non_vanlshlng term in the expansion of n(x_t) would be n(2)(_t) (which is of 0(e2))o Consequently S(k,m) would be of 0(_ h) o Therefore the resonance process enhances the scattering cross .section significantly° We would like to compare our cross section with that obtained by Kroll, Ron, and Rostoker2o The cross sections will differ only because of the differences in the spectral densities° The spectral density S RR of Kroll, Ron_ and Rostoker 2 is, when expressed in our notatlon_ S_R(_,_). ___ 1 k4(Al°A2)2 128_ 2 N 2 _'_ om _i _2 I_(_,_)I [6(k=k4)_(_÷_) + 6(k+_4) _(_=_4)] , (8_) 123 where ¢(k,m) , the longitudinal dielectric functlon_ is approximated by Here F is the Landau damping decrement or the collisional damping decrement_ whichever is larger° Let us define R to be the ratio of _RR(k_) to our S(k,_) o Then 2 k_ (_l o A2}2 2 2 _ F2o _i _ {_ k } When R < i ® SkRR(k,m) < S(k,m) , and the density n(x_C) of Kroll 2 et alo is smaller than our densityo This means that Landau damping and/or colllslonal damping is more effective in limiting the density oscillations than are the nonlinear effects° On the other hand, when R > 1 , the nonlinear effects are more effective than the damping mechanisms° Thus, given the numerical values of the plasma parameters and the electric fields produced by the two lasers, we can determine by means of the expression (85) which physical mechanism is the dominant one in limiting the longitudinal plasma oscillations° (85) Let us estimate the ratio R by using a set of typical numerical values of the plasma and the incident electric field parameters° We shall use the set selected by Kroll, Ron, and Rostoker 2 in their calculations of the damping effects° Therefore we choose N - I0 I_ cm=3 0 _T " I0 eV , m - 5o6_ x i0 II see ®IP , I (of incident electric field) 007 x i0 _4 cm, E (amplitude of the incident electric field) _ 108 V/cm and F (collisional damping decrement) m lol x lO ®3 o With this C choice of parametersz according to Kroll et alo2, the collisional damping dominatesover theLandaudamping° SinceIAII, IA21,and I_I are of o(_), I_21 and I_I arebothof 0(Al) .whichinturnisof O(E incident)° Substituting the above numerical values of the plasma and the incident electric field parameters into eq0 (85)9 we obtain 10 =8 < R < 10 ©7 o (This estimate was made under the assumption that E1 and _2 are parallel°) Therefore for this choice of parameters the damping effects limit the longitudinal plasma oscillations more effectivel M than do the nonlinear effectsQ VIII, DISCUSSION Wehave shown that the presence of even a small amount of non- linearity, an the equations used to describe the behavior of a plasma, can effectively limit the amplitude of plasma oscillations driven by two light beams, In fact_ under some circumstances_ the nonlinear effects limit these oscillations more strongly than the Landau damping and the collislonal damping mechanisms, The nonlinear effects are always accompanied by a non=vanishing longitudinal electric field of0(_)_ E_l) _ whenever theylimi_.plasm_oscill- ationso This is a very interesting fact_.because,_ E_I)• . _'as_well am the _(i) satisfythe 0(_ plasma e_uations_ which are._transverse field _T ' linear, snd_therefore keepthe, O_e_.tr_nsverae'_md._t.he 0(_)_longitudinal components of fields completely separated from one another, The transverse fields are determined by the output of the two lasers° But there is no experimental device which sets up a longitudinal _(i) field_ _L o All that is done is to make sure the plasma is an a _uiescent state at the beginning of the experiment, The experimental set-ups for the case when the difference in frequencies of the two laser beams approximately equals to the natural frequency of longitudinal plasma oscillations® and for the case when it is not_ are identical° When the difference in frequencies of the impinging beams is not equal to the frequency of plasma oscillations, no secular terms arise in the equations of motion, and _E_l) remains identically zero - there is only a second=order field E_2)- o However, when the frequency of one 125 126 of the impinging waves is varied until it differs from the frequency of the other impinging wave by the frequency of plasma oscillations_ a resonance process results_ Longitudinal plasma oscillations are excitedo are simultaneously limited by nonlinear effects_ and E(LI) appears spontaneouslyo This longitudinal oscillation in turn scatters the light beams o It is also interesting to note what happens when the amplitudes of the two impinging waves are varied while keeping everything else constant° Let us assume for the moment that the electric fields ET(1) and E(Ll) and the damping decrement F have been made dimensionless_* The density fluctuation n(x_t) which is limited by nonlinear effects is O(E(LI)) 0 Since EL(1) is of O(E(TI)) _ n(x_t)_ is also of 0(_i)) o As _T-(1) increases or decreases, n(x_t)_ will also increase or decrease d respectively° On the other hand_ _he density fluctuations_ n(x_t) which are limited by Landau and/or collisional damping are of _(1)2 0<__o They will also increase or decrease as _i)increase. or decreases, respectively° The damping decrement r , however, does not depend on _i) _(i) is varied° , and will not change when ":T J .... *r is made dimensionless by dividing it by the plasma frequency (1) _(1) '"p o ET and "_L may be made dimensionless by dividing them by (ll2N my 2)1/2 o o- _ the square root of the thermal ener_ densityo E I)2(i) <_ P _< (E_I) and the density n(x_t)When ET _ _ which is llmitedby a damping mechanism is smaller than the density n(x,t) which is limited by nonlinear effects° On the other handp when E_ I)2 (1) >_ r _ _> (i) and the situation is reversed° We conclude ET ' F ET _(i) is very weak, andfrom this that damping effects dominate when _T the nonlinear effects dominate when E_I)- is strong° ,The conclusion is borne out by the numerical calculations in Section VII0 This is not unreasonable_ because E_ I) is a measure of (i) does not affect F nonlinearity in the eguations of motion_ but ET _,(1) very small results in very the damping decrement° Keeping _T small nonlinear terms_ without affecting the damping° An increase in E(1) T , on the other hand_ increases the magnitude of nonlinear %erms_ while still keeping the damping decrement unchanged° Therefore an increase in E$ I) results in the increasing importance of nonlinearity as compared with the damping effects° 127 APPENDIX DERIVATION OF WAVE E_UATIONS We shall derive the wave equation for E (e_o (i0) an the text )o Taking the curl of eq. (8) and substituting into e_o (9)_ we obtain / E_'] @2E 4_N e _v h_e ___ _ _ R -_ (nv) @x x _x ' c St 2 c c _v If we now substitute for _-_ from eqo (4) into eqo (AI)_ there results (_) 2 _ne 2 _x P mc _ h_N e o _ e .._ (vo_ v+--- vxB) _ mc _ c ]¢_e ;) # • ÷- "_- _nv;2 _t c "_ n * V ° Sx V. O _ 4_e 2 mc n(E + _ v x B) (_) @ I The term @x p can be obtained from eqo (5) in the following wa_: 128 129 From eqo (3) we have @ 1 @n 1 @ @x _ N @t N Bx O O o (nv) Substituting ego CA3) into eqo (5)_we obtain 0 " _" - _- 1 -_,_ o (nv) - _-_ o v = v o ,_x_ p i) _'' e = ° ,,_ v = o v =,--= (p x B + p x B)o Applying the operator _" @xx . to eqo (A4) and then substituting (A3) into it_ we obtain (A3) @2 @ 2Po @2 @n V2 @v .--=. .m o -_ + p ..,.@t2 @_ __= NO _ _x_ o _t • "--" @t _ o n @t @x v o p NO _ . ~ _ ,,I, P +_.e xB+pxB mc _ = (AS) 130 We can obtain the term P V2 Sv o _ from eqo (4), which yields V2 _v Po V2 _ P = P e V2==.____ _ . o _ EPo _t N m 3x m O _ = PO v ° _x v + v x O r_ _ _ mc _ NO Po V2 i +vo_ v =_ n(_.+ v×B) " 0 (A6) Also, from eqo (6), we obtain n = = w_e .-- o E _x (AT) Substituting ,(A6) and (A?) into (A5)j we obtain P v2 E " Po "_xV+ v x I '" Inv2 _ . _ v2 ÷ n _ ax _ Nomo \a_ +vo-- v ÷ (E +i vx +No _ \c - St _x _x 131 +e pxB+l:,x 3P where we have defined the thermal velocity v2 -= O N m o o If we now apply the operator --=_ I 2 V to eC.o _t 2 = _= vo • and then substitute (A8) into it, we obtain eqo (i0) in the text° (A2) (A8) REFERENCES io Po Mo Platzman, So Jo ,Buchsbaum, and No Tzoar, Physo Revo Letters 12, 573 (196 )° 20 No Mo Kroll, Ao Ron, and No Rostoker, Physo Revo Letters I_3, 83 (196_)o 3° Do Co Montgomery and Do Ao Tidman, PlasmaKinetic Theor_ McGraw® Hill Book Coo, New York, 19640 40 Eo Ao Frieman, Jo Math° Physo _, 410 (1963)o 5o No No Bogoliubov and Yo Ao Mitropolskyz As totic Methods ia the Theor_ of Non=LinearmOscillations p Hindustan Publishing Corporatiom (India), Delhi_6_ 1961; translated from the Russian° 132

Collisions and nonlinear effects in plasmas

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