Volume 140, number 7,8 PHYSICS LETTERS A 9 October 1989

INTEGRAL METHODS IN KINETIC PROBLEMS

Mario SOLER Departamento de Fistca Atomtca y Astrofistca, Umverstdad Complutense, 28040 Madrid, Spare

Received 28 April 1989, accepted for publication 19 June 1989 Communicated by A R Bishop

Integrating the Chapman-Kolmogorov equation is proposed as an alternative to the usual Fokker-Planck procedure, when the final answer in kinetic problems must be obtained numencally The plasma case is formulated in terms of Langevin's equation to obtain the general integral soluuon The method is implemented for a simple example, and compared with the Fokker-Planck results

The solution of kinetic problems in plasma phys- ics, as well as in many other fields where collisions play an important role, is usually attempted through a Fokker-Planck formulation of the collisional terms. In practice, unfortunately, the solution of this equa- tion is so complicated that except for very simple cases numerical computation is used at some stage of the problem.

This work takes for granted that numerical com- putation constitutes in practice an essential tool for the solution of kinetic problems. A natural question is then to look for the best way to arrive at a com- puted solution, and the correct position should be to pose "ab initto" the physical problem. This may dif- fer from the frequent procedure in which computa- tion starts when the algebrmc treatment becomes hopelessly complex.

In our problem it is well known that the physical basis for the Fokker-Planck theory starts at the Chapmann-Kolmogorov equation,

f ( v , t + A t ) = f P ( v , t + A t l v ' , t ) f ( v ' , t ) d v ' , ( I )

where the norm of the transition probability P in- tegrated over dr ' is 1 as At--,0. The conditions for the derivation are well known.

It ts at this stage that the above question should be posed. Is the Fokker-Planck equation the best route for an essentially computational determination off(v, t)?

Some very general arguments suggest that while differential methods are expedient when looking for analytical solutions of physical problems, for com- puted solutions integration should in principle be preferred. This is because much less information is lost when adding than when subtracting two close numbers.

We concentrate in the following on the use of in- tegral methods in a typical problem for which the Fokker-Planck solution can be found only by com- putation: the collisional evolution of distribution functions in a plasma. Out of the many possible routes we will choose the one based on Langevin's equation because of the physical transparency it pro- vides at all the steps of the problem.

We will start with the usual calculation of the mo- mentum transfer between an incident beam of par- ticles and a scattering center, along the traditional lines presented in refs. [1] and [2]. We will add however to this consideration the statistical nature of the process. This leads directly to a Langevm equation whose physical content is identical to that of Fokker-Planck. It should be mentioned that the importance of Langevin's formulation in this [3,4] as in other fields of physics [ 5 ] has been the subject of much recent mterest.

Let us consider a monochromatic beam of pani- cles of class a and momentum m~'v '. Assume they are randomly distributed with axial symmetry, and that a particle of class 13 travels with velooty v along the

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symmetry axis. We need not detail here the kine- matics of colhsion in the Cartesian frame moving with the mass center. The relevant quantities de- scribing the scattering process are the components m~aAu, = (0, O, rn~ Au:) ( m~ is the reduced mass; u,=v, -v; ) of the average momentum transfer in rime At andthe quantities m~a[ Au, 2 - (A-u,) 2 ] J/2 ex- pressing the statistical deviations of the momentum transfer from the mean.

In the chosen reference frame we can express the process by means of the Langevin equatmn,

d v ' - m " ~ ( [ O0 ]+au fJ ) m, ~ : (2)

where

F (Au2')'/2 ] O"J= L / (Au2)I/2 [ AU2- (~--~z)2] 1/2

and the fj are random Gaussian variables satisfying

<fj(t)fk(s) > =J, kJ(t--s) . (3)

Since, as is well known, a = = 0 in the mentioned reference frame (in the approximation in which quantities not containing the Coulomb logarithm are neglected), the matrix w,j =a, ktrkj is a tensor of the second rank whose general expression in an arbitrary reference frame is found to be, after evaluating the Coulomb scattering momentum transfer,

~,j =L=/~ J'J - u ' u j / u 2 (4) 4xu

where L ~/~=2( 4ne~e~/m~) 2, u= I v - v ' l . Therefore, since the matrix contained in the r.h.s, of (4) is a projector,

cr,j = ( L ~/f~) i/2 6,j - u, uj/u 2 ( 5 ) (4nu) '/2

The Langevm equatmn can thus be written for a monochromatic incident beam of particles,

u, + tr,jfj. (6) dr, = _ ( 1 + m~/ml~)L ~/1~ 4nu3 dt

We must now generalize th~s equation so that it can account for the time evolution of a monochro- manc set of particles of velocity v when a "gener-

alized beam" of particles ~ with velocity distribution funcUon f ( v ' ) scatters against it. A simple proce- dure is as follows.

Let the general Langevin equation be written, as is usual,

dr, =B,+auf j . (7)

dt

The expectation values of the quantities < Av, > and < Av, vj > can be obtained from this equation. The ma- trix a v will depend now on v, because the incident beam is no longer monochromatic as before. The fol- lowing expressions are then valid [6 ] m the ap- proximaUon in which first order contributions in At and f ( t ) are retained, as well as contributions sec- ond order in f ( t ) , but products f ( t )A t and terms second order an At are neglected:

. Oajk <Av, > =~, . a , , ~ ,

< Av, Avj > = 2tr, k trkj. ( 8 )

It is also well known, on the other hand, that these quantities can be easily obtained from the distribu- tion functions once we have in (4) the contribution from each monochromatic component:

l <Av,>'/l~=-(l+rnJmlOL~/O-4-~ I u~fa(v' ) dr ' ,

1 < Av, Avj > "/13=L"/13-~ f (¢~,j-u, uj/u2)ff~(v ' ) dr' .

(9)

These can be expressed in terms of the Rosenbluth potentials.

The matrix < Av,Avj> is symmetric, so a,j always exists.

It is important to realize the physical sigmficance of ~,j(v) also in this general case. For each value of v, a unitary transformation can always be found which makes xt diagonal in a certain reference frame. In such frame the quantities a,,(At),/2 represent the half width of the expected Gaussian dispersion of v m the three axial directions.

Let us finally comment that we have assumed, as is done m the Fokker-Planck formulation, that products of more than three factors Av have zero ex- pectation value. This is of course a consequence of

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our choice of Gaussian random variables in (3). Our general Langevin equation (7) contains now phys- ical information identical to the equation of Fokker- Planck.

In order to use Langevin's equation for the actual solution of (1), our first step is to consider a dis- tribution function as a collection of monochromatic "test particle" packets, each in interaction with the rest. The individual evolution of each packet can be described by one Langevin equation, and that of the total plasma by the evolution of the collection itself.

Secondly, the evolution of each packet in time At will consist in the displacement of the velocity by the amount (Av)At and the dispersion of the initial 6- function like distribution to a Gaussian of half widths tr,,(At) 1/2 in a certain frame. We may assume that in practice each monochromatic packet has such a high number of particles that its stochastic transition to a smooth Gaussian can be considered. In the men- tioned frame we would thus have after time At,

8(v-v ' ) ----' f(v) At

In a general frame, taking into account the proper normalization,

f ( v ) = ~P(e , t+At lv ' , t )~ (v -v ' )dv ' , (10)

where

P(v, t+ At lv', t) = ( 4nAtD)-1/2

( l Xexp - ~-~ [v, - (v', + ( Av, )At) ]

× [ v , - ( g + ( Av, )At) ] ( D - t ) u )

( D = IDI, the determinant of D U =a , kakj). We can now reduce an arbitrary dis tr ibut ionf(v ' )

to the addition of 6-function monochromatic com- ponents, so that in the general case, using the same transition probability P of (10), eq. (1) is imple- mented. In fact, our transition probability is but a generalization for the present case of well known expressions, and could have been introduced di- rectly. Its physical meaning, and the obvious way to

make use of it in a computational scheme is partic- ularly clear in the context of Langevin's equation. This was in fact the approach followed in this work to arrive at (10).

We will now describe the use of ( 1 ) with the tran- sition probability in (10) to describe the selfcolli- sional evolution of a distribution function in spher- ical symmetry. The vector v can be assumed fixed along the z axis, since for any other orientation the result would be the same in view of the mentioned symmetry.

The transition probability is in this case:

v; P(v, t+ Atlv' ,At)~exp 4DxxAt 4DvyAt

[ v ~ - ( v ' - ( AVz)At) ] 2) - 4Dzz At " ( 11 )

Here spherical coordinates are conveniently used for the Cartesian components of the velocities. We then have Dxx =Dyy =D±, D==Dll, and D±, DiM can easily be obtained from the spherical Rosenbluth poten- tials. Integrating the exponential function over d~ is trivial. Integration over sin 0 d0 is easy enough to do numerically. A faster alternative is to use interpo- lation from error and Dawson integrals. Let us call F(v, v') the appropriately normalized resulting function.

For the integration over dv' we assume v' to be a discretized variable. We can then reducef (v ' ) to an addition ofNcomponentsf(v' )a(v ' - v,) with I run- ning say from 1 to N. The functionf(v, t+ At) is thus an addition of N functions F( v, v,) v, 2 . The functions F(v, vl) are considerably more peaked than Gaus- sians, behaving approximately as exp [ - x ( v - v' - ( A v ) At) s ] (x > 0) except for very small exponen- tial arguments. A reasonable approximation is there- fore to assume they are zero except for the grid point closest to the maximum o f F ( v , vl), and to its two neighboring points. This approximation consider- ably shortens the computation and is in fact similar to the "three grid point schemes" always used in the differential one-dimensional numerical computa- tions. For larger At contributions from more than three grid points might be contemplated. Yet as we will see one is not free to take arbitrary time inter- vals. Actually the Markovian nature of eq. (7) holds strictly only for times At- ,0 (an idealization even

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V o l u m e 1 4 0 , n u m b e r 7 , 8 P H Y S I C S L E T T E R S A 9 O c t o b e r 1 9 8 9

for physical systems consisting of large but finite number of scattering processes). Existence of small but finite correlation times in (3), because of the fi- nite time step used in the computation, has some consequences that may actually be invoked to im- prove the computational scheme. We turn to this point below. One advantage of the integral scheme is that norm conservation can be imposed exactly by demanding that the norm of each monochromatic component at point 14- ½ and time t be the same as the addition of norms at points J - ½ , J + ½, J + 3 where the initial packet has expanded at time t+ At.

Energy conservation, which lS only approximate in differential schemes where norm is exactly con- served is also not exactly conserved in this integral approach. However it is not hard to introduce ap- proximate corrective actions that ensure exact con- servation without undesirable consequences. This is due to the total stabdlty exhibited by the scheme, al- lowing considerable flexibility for approximations. One possibility is to adjust a "correlation parame- ter" C, as a factor to the term (Av)At. According to whether C> 1 or C< 1, energy increases or decreases at each time step, without any visible change in the colhs~onal evolution of the system towards equihbrium.

A comparison between the time evolution with differential and with integral schemes is shown in fig. 1. The differential method used is a predictor-cor- rector Cranck-Nicholson scheme with exact norm conservation. In order to minimtze the boundary in- fluence, more grid points than shown must be taken. Th~s also improves the energy conservation (about 20% is lost here). Energy is exactly conserved in the integral scheme with slight adjustment of the men- uoned correlation parameter C at each time step.

There is one more feature in the integral scheme which makes it physically attractive. This is the pres- ervation of the stochastic nature of the process at each of the cells m velocity space. In the Cranck-Ni- cholson scheme, the implicit part of the iteration is responsible for introducing at the first time step an unphysical change in all the initially empty cells of the distribution. (This would not be the case in ex- plicit schemes, but of course they cannot be used in practice because of their poor stability properties. ) The unphysical time evolution of the tails intro- duced by the implicit advance is responsible for the visible difference between both solutions. Also the differential method does not quite converge to a Gaussian (straight line) because of the energy losses. Improvement can only be obtained if more grid

0 O0

-10 O0

"0 O0

0~-30 O0 _o t

1 : - 4 0 O0 :

-50 O0 -

1 - 60 O0

0 CO

1 t " 75

t - 50

t = 25

t = 1

0 O0 -

- I 0 0 0 ;

)[ - 4 0 O0 ~

- 50 O0

t - 75 t * 50

a b . . . . . , . . . . . . . . . . . . . . . . . . . . . . - 6 0 O0 , ~ . . . . . . . .

20 O0 40 O0 60 O0 80 O0 0 CO ' '20 O0 . . . . 4 0 00' 60 O0 80 O0 X ~ W "~ W X ~-~ W x W

Fig. 1. Evolution of step distribution in sphertcal symmetry, (a) using integral formulation; (b) using &fferentlal (Cranck-Nicholson) scheme Dimensionless velocity is w= v/ur, Vr= (2Te/rn,)~/2. Time units t , are t, = (L'/env~). Time step Is At=0.5tu. In (b) the partly ~mphclt advance produces instantaneous unphyslcal tail change for first step and any At. This distorts t ime evolution also at subsequent t ime steps. Not shown is a mdd mstab]hty producing some nega t lvef (v) values from grid point 20 to the last grid point (35) taken in th~s case. No problem shows up m (a) where uniform tail evolution to a strict Gaussian is seen for the 20 grid points taken

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points and smaller time increments are taken. The great advantage of the integral method lies in that very few grid points provide a completely satisfac- tory time evolution. Therefore, even if for similar grids the computation time may be cut just by a fac- tor of two, the possibility of achieving satisfactory answers in terms of conserved quantities with coarse grained grids, puts computation of these problems by integral methods in a new class of "exact" solutions.

When more than one species is present, and ex- ternal forces exist, the corresponding Langevin equa- tions are obviously generalized, with (Av,) ~/~ in- cluding the additive friction from the different species, as well as the external forces. Since the sto- chastic terms for the different species are indepen- dent, we will have D o =O'tkO'kj = Zl~( a °tfj) 2

Finally, the space dependence has to be intro- duced. We must assume that a different distribution function exists at each space point. It only rests to take into account the local influence of the different distribution functions among themselves. Argu- ments similar to the ones used in the differential case lead to the equation

f ( x , v, t+At )

= f P ( v , t +Atlv ' , t ) f ( x , v', t) d v ' + v V f A t . J

(12)

In conclusion, an integral method has been de- veloped for the computation of kinetic problems. In well known simple cases it is absolutely stable and is amenable to yield exact conservation of the physical laws. Programming is considerably simpler than in differential schemes and takes less computer time. Applications of the method for functions depending on more than one variable will be considered else- where. In developing the method a crucial guiding line was to formulate the kinetic problem in terms of a general Langevin equation.

References

[ 1 ] M.N. Rosenbluth, W.M. MacDonald and D.L. Judd, Phys. Rev. 107 (1957) 1.

[2] B.A. Trubmkov, Reviews of plasma physics 1, Vol I (Consultants Bureau, New York, 1965) p. 105.

[3] N.J. Flsch and C.F.F. Karney, Phys. Rev. Lett. 54 (1985) 857.

[4] N.J. Flsch, Rev. Mod. Phys. 59 (1987) 175 [5] S. Adler, Phys. Rev. Left. 60 (1988) 1243. [6] M Lax, Rev. Mod. Phys. 38 (1966) 541.

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