Referat - Padurariu George-Octavian

7/29/2019 Referat - Padurariu George-Octavian

http://slidepdf.com/reader/full/referat-padurariu-george-octavian 1/20

Department of Plasma Physics,

Spectroscopy and Self-Organization

Fokker-Planck Theoryof Collisions

Coordinator:

Prof. Dr. Lucel Sîrghi

Master Student: Pădurariu George-Octavian



Fokker-Planck theory of collisions

1. Introduction

I begin the analysis by reviewing collision between two particles. Suppose a test particle

having mass m T and charge q T collides with a field particle having mass mF and charge qF. Since

the electric field associated with a charge q is^

2

0/ 4 E r q r where r is the distance from charge,

the equations of motion for the respective test and field particles are:

..

3

04

T F T T T F

T F

q qm r r r

r r

..

3

04

T F F F F T

T F

q qm r r r

r r

.

The center of mass vector is defined to be :

T T F F

T F

m r m r R

m m

.

Adding equations (1) and equation (2) gives:

.. .. ..

0T F T T F F

m m R m r m r

showing that the center of mass velocity.

R is a constant of the motion and so does not change as

a result of the collision. If .

T r and.

F r are defined as the change in respective velocities of two

particles during a collision, then it is seen that these changes are not independent, but are related

by:

. . .

0T F T T F F

m m R m r m r



It is useful to define the relative position vector:

T F r r r

and reduced mass:

1 1 1

T F m m

.

Then, dividing equations (1) by their respective masses and taking the difference between

the resulting equations gives an equation of motion for the relative velocity:

.. ^

2

04

T F q q

r r r

.

Solving equations (2) and (5) for T r and

F r gives:

F

F

r R r m

T

T

r R r m

and so the respective test and field velocities are:

. .

F

T

r R r m

. .

T

T

r R r m

.

Since.

0 R the change of the test and field particle velocities as measured in the lab frame can

be related to the change in the relative velocity by:

. .

F

F

r r m



. .

T

T

r r m

.

The collision problem is first solved in the center of mass frame to find the change in the

relative velocity and then the center of mass result is transformed to the lab-frame to determinethe change in the lab-frame velocity of the particles.

Let us now consider a many-particle point of view. Suppose a mono-energetic beam of

particles impinges upon background plasma. Several effects are expected to occur due to

collisions between the beam and the background plasma. First, there will be a slowing down of

the beam as it loses momentum due to collisions. Second, there should be a spreading out of the

velocity distribution of the particles in the beam since collision will tend to diffuse the velocity

of the particles in the beam. Meantime, the background plasma should be heated and also should

gain momentum due to the collision. Eventually, the beam should be so slowed down and so

spread out that it becomes indistinguishable from the background plasma which will be warmer

because of the energy transferred from the beam.

2. Statistical argument for the development of the Fokker-

Planck equation

The Fokker-Planck theory is based on a logical argument on how a distribution function

attains its present form due to collisions at some earlier time. Consider a particle with velocity v

at time t that is subject to random collisions which change its velocity. We define , F v v as

the conditional probability that if a particle has a velocity v at time t , then at some later time

t t , collisions will have caused the particle to have velocity v v .

Clearly at time t t the particle must have some velocity, so the sum of all the

conditional probabilities must be unity , F must be normalized such that:

, 1 F v v d v .



This definition of conditional probability can be used to show how a present distribution

function , f v t comes to be the way it is because of the way it was at time t t . This can be

expressed as:

, , , f v t f v v t t F v v v d v ,

which sums up all the possible ways for obtaining a given present velocity weighted by the

probability of each of these ways occurring. This analysis presumes that the present status

depends only on what happened during the previous collision and is independent of all events

before that previous collision. A partial differential equation for f can be constructed by Taylor

expanding the integrand as follows:

1

, , , , , , , : , , .2

f f v v t t F v v v f v t F v v t F v v v f v t F v v v v f v t F v v

t v v v

Substitution of equation (15) into equation (14) gives:

1

, , , , , , : , ,2

f f v t f v t F v v d v t F v v d v v f v t F v v d v v v f v t F v v d v

t v v v

where in the top of line advantage has been taken of , f v t not depending on v . Upon

invoking equation (13) this can be recast as:

, ,1, : ,

2

vF v v d v v vF v v d v f f v t f v t

t v t v v t

.

By defining

,vF v v d vv

t t

and

,v vF v v d vv v

t t



the standard form of the Fokker-Planck equation is obtained,

1

, : , .2

f v v v f v t f v t

t v t v v t

The first term gives the slowing down of a beam and is called the frictional term while

the second term gives the spreading out of a beam and is called the diffusive term.

The goal now is to compute /v t and /v v t . To do this, it is necessary to

consider all possible ways there can be a change in velocity v in time t and then average all

values of v and v v weighted according to their respective probability of occurrence. We

will first evaluate /v t and then later use the same method to evaluate /v v t . The

problem will first be solved in the center of mass frame and so involves a particle with reduced

mass , speedrel T F

v v v , and impact parameter b colliding with a stationary scattering

center at the origin. The deflection angle for small angle scattering is:

2

02

T F

rel

q q

b v

.

The averaging procedure is done in two stages:

The effect of collisions in time t is calculated. The functional dependence of v on the

scattering angle is determined and then used to calculate the weighted average v for

all possible impact parameters and all possible azimuthal angles of incidence. This result

is then used to calculate the weighted average change in target particle velocity in the lab

frame.

An averaging is then performed over all possible field particle velocities weighted

according to their probability, weighted according to the particle distribution function

F F f v .

Energy is conserved in the center of mass frame and since the energy before and after the

collision is entirely composed of kinetic energy, the magnitude of 2

rel v must be the same before

and after the collision. The collision thus has the effect of causing a simple rotation of the



relative velocity vector by the angle . Let z be the direction of the relative velocity before the

collision and let^

b be the direction of the impact parameter. If 1 2

,rel rel

v v are the respective

relative velocities before and after the collision, then:

^

1rel rel v v z

^ ^

2cos sinrel rel rel v v z v b .

Thus

2^ ^ ^ ^ ^

2 1cos 1 sin cos sin

2

rel rel rel rel rel rel rel

vv v v v z v b z v x y

where is the angle between the impact parameter and the x axis and the scattering angle is

assumed to be small. The fact that the z component is negative reflects the slowing down of the

particle in its initial direction of motion.

The cross-section associated with impact parameter b and the range of azimuthal impact

angle d is bd db . In time t the incident particle moves a distance v t and so the volume

swept out for this cross-section is bd dbv t . The number of relevant field particles encountered

in time t will be the density F F F f v dv of relevant field particles multiplied by this volume,

F F F f v dv bd dbv t and so the change in relative velocity for all possible impact parameters

and all possible azimuthal angles for a given F v will be:

, F F F v b v tf v dv d vbdb .

The limits of integration for the azimuthal angle are from 0 to 2π and the limits of

integration of the impact parameter are from the 900 impact parameter of the Debye length. On

writing v in component form, this becomes:

/2

22

0, cos , sin , / 2 .

D

rel rel F F F rel b

v b v tf v dv d v bdb



The x and y integrals vanish upon integration over and so :

/2 /2

22 2^ ^ ^

2 2 2

2 2 2

0 0

, ln2 2

D D F T T F rel rel F F F rel F F F F F F

b brel rel

q q q qv b z v tf v dv bdb z v tf v dv bdb z tf v dv

b v v

where/2

/ D b .

Using equation (12) the above expresion can be transformed to give the test particle

velocity change to be:

2 2 2 2^

32 2 2

0 0

ln, ln

4 4

T F T F T F T F F F F F F

T rel T T F

v vq q q qv b z tf v dv t f v dv

m v m v v

.

This result was for a particular value of F v and summing over all possible F v weighted

according to their probability, gives:

2 2

2 3

0

ln, ,

4

relk T F T F F F F

T rel

vq qv b v t f v dv

m v

The integrand can be expressed in a simpler form by considering that:

^ ^ ^2 2 2 k

k

r r x y z x y z

r x y z r

and alsoT rel

r

v v

. Thus:

rel relk

T rel

v v

v v

and:

2

1 1 relk

T rel rel rel

v

v v v v

.

Equation (28) can therefore be written as:



2 2

2

0

ln 1, ,

4

T F T F F F F

T T rel

q qv b v t f v dv

m v v

.

TheT v has been factored out of the integral because

T v is independent of

F v , the

variable of integration. Since F v is a dummy variable, it can be replaced by '

v and sinceT

v is

the velocity of the given incident particle, the subscript T can be dropped. After making these re-

arrangements we obtain the desired result:

'2 2

'

2 '

0

ln

4

F T F

T

f vq qvdv

t m v v v

.

The averaging procedure for v v is performed in a similar manner, starting by noting

that:

cos ,sin , 2v v ,

and so equation (25) is replaced by:

/2

22 2

0, cos , sin , / 2

D

rel rel rel F F F rel b

v v b v tf v dv d bdb v

.

The terms linear in sin and cos vanish upon performing the integration. Also, the

^ ^

z z term scales as 4 and so may be neglected relative to the

^ ^

x x and^ ^

y y terms which scale as

2 . Thus, we obtain:

/2

/2

2 2 2^ ^ ^ ^ ^ ^ ̂ ^ ^ ̂3 2 2 2 2 2

2 2

00

ln, cos sin

4

D D T F

rel rel rel F F F rel F F F F F b

rel b

q qv v b v tf v dv d x x y y bdb v tf v dv x x y y bdb tf v dv x x y y

v

Sincerel v defines the z direction, we may write:

^ ^ ^ ^

2

rel rel

rel

v v x x y y I

v ,

where I is the unit tensor. Thus,



22 2

2 2 3

0

ln,

4

rel relk relk T F T T F F F

rel

v I v vq qv v b t f v dv

v

.

Using equation (12) this can be transformed to give:

22 2

2 2 3

0

ln,

4

rel relk relk T F T T F F F

T rel

v I v vq qv v b t f v dv

m v

.

The tensor may be expressed in a simpler form by noting:

2

2 3

rel relk relk relk relk relk

T T T rel rel rel rel rel

v v v v v I v v I

v v v v v v v v

.

Inserting equation (40) in equation (39) and then integrating over all the field particles

shows that:

2 2

2 2

0

ln

4

T T T F F F F T F

T T T

v v q q f v dv v v

t m v v

.

However,T v is constant in the integrand and so

T v may be factored from the integral.

Dropping the subscript T from the velocity and changing F v to be the integration variable '

v this

can be re-written as:

2 2

' '

2 2

0

ln'

4

T T T F F

T

v v q qv v f v dv

t m v v

.

It is convenient to define Rosenbluth potentials:

' ' '

F F g v v v f v dv

'

'

'

F T F

f vmh v dv

v v

in which case:



2 2

2 2

0

ln

4

T F F

T

q q hv

t m v

2 2

2 2

0

ln

4

T F F

T

q q g v v

t m v v

.

The Fockker-Planck equation, equation (20) thus becomes:

2 2 2

2 2, 0

ln 1:

4 2

T T F F F T T

F i e T

f q q h g f f

t m v v v v v v

The first term on the right hand side is a friction term which slows down thea mean velocity

associated withT f while the second term is an isotropization term which spreads out the velocity

distribution described by T f . The right hand side of equation (46) gives the rate of change of the

distribution function due to collisions and so is the correct quantity to put on the right hand side

of equation: f

vf af C f t x v

.

3. Slowing down

Mean velocity is defined by

vfdvu

n

where n fdv . The rate of change of the mean velocity of species T is thus found by taking

the first velocity momentum of equation (46). Integration by parts on the right hand side terms

respectively give:

F F T T

h hv f dv f dv

v v v

and



2 2

: 0 F F T T

g g v f dv f dv

v v v v v v v

where Gauss\s theorem has been used in equation (49). The first velocity moment of equation

(46) is therefore:

2 2

2 2, 0

ln

4

T T F F T

F i e T T

u q q h f dv

t n m v

.

Let us now suppose that the test particles consist of a mono-energetic beam so that:

0.T T f v n v u ,

In this case equation (50) becomes:

0

2 2

2 2, 0

ln

4

T T F F

F i e T v u

u q q h

t m v

Let us further suppose that the field particles have a Maxwellian distribution so that:

3 2

2exp 2

2

F F F F

F

m f v n m v kT

kT

and so, using equation (44):

3 2 '2

'

'

exp 2

2

F F F T F F

F

m v kT n m mh v dv

kT v v

.

The velocity integral in equation (54) can be evaluated using standard means to obtain:

2 F T F

F

F

n m mh v erf vkT

where 2

0

2exp

x

erf x w dw

is the Error Function.

Thus, equation (52) becomes:



0 0

2 2 2 2

1 1

2 2 2 2

0 0

ln ln

4 2 4 2

i T i i e T e eT T T

T i i T e ev u v u

n q q m n q q mu m mv erf v v erf v

t m v kT m v kT

where 1 1 1

, ,i e i e T m m

.

This can be further simplified by nothing (i) quasi-neutrality implies:

0i i e en Zq n q ,

where Z is the charge of the ions, (ii) the masses are related by:

, ,

1T T

i e i e

m m

m ,

and (iii) the velocity gradient of the Error Function must be in the direction of 0

u because that is

the only direction there is in the problem. Using these relationships and realizing that both the

left and right sides are in the direction of u , equation (57) becomes:

2 2

2 2

0

2 2ln1 1

4

i e

i ee T T T

T i e

m merf u erf u

kT kT n e q m mu d d Z

t m m du u m du u

.

Let us define:

,

,

,2

i e

i e

i e

mu

kT

which is the ratio of the beam velocity to the thermal velocity of the ions or electrons. The

slowing down relationship can then be expressed as:

2 2

2 2

0

ln1 1

4 2 2

i ee i eT T T

T i i i i e e e e

erf erf n e m mq m mu d d Z

t m m kT d m kT d

.



Figure above plots '

erf x x as a function of x . This quantity has a maximum when

0.9 x indicating that the friction increases with respect to x when x is less than 0.9 but

decreases with respect to x when x exceeds this value. Examination of this figure suggests that

the function has a linear dependence for x well to the left of the maximum and varies as some

inverse power of x well to the right of the maximum.

This conjecture is verified by nothing that equation (56) has the limiting values:

3

2

01

2 2lim 1

3

x

x

xerf x w dw x

1

lim 1 x

erf x

and so

1

4lim3 x

erf xd xdx x

21

1lim x

erf xd

dx x x



which is consistent with the figure.

The existance of these two different asymptotic limits according to the ratio of the test

particle velocity to the thermal velocity indicates three distinct regimes can occur, namely:

a. Test particle is much faster than both ion and electron thermal velocities, that is

, 1i e

.

b. Test particle is much slower than both ion and electron thermal velocities, that is

, 1i e

.

c. Test particle is much faster than the ion thermal velocity, but much slower than the

electron thermal velocity, 1i

and 1i

.

A nominal slowing down time can be defined by writing the generic slowing down

equation:

s

u u

t

so

s

u

u t

this can be used to compare slowing down of test particles in the three situations listed above.

a. Test particle faster than both electrons, ions:

Here the limit given by equation (64b) is used for both electrons and ions so that the

slowing down becomes:

2 2

2 2 2

0

1 1ln

4

T T

i ee T

T

m m Z m mn e qu

t m u

and so:



2 2 3

0

2 2

4

ln1

T s

T e T

e

m u

mn e q Z

m

since 1 1e im m . Equation (67) shows that if the test particle beam is composed of electrons,

the ion friction scales as 2 Z Z of the total friction while the electron friction scales as

2 2 Z of the total friction. In contrast, if the test particle is an ion, the friction is almost

entirely from collisions with electrons. The slowing down time is intensitive to the plasma

temperature except for the weak ln dependence. The slowing down time for ions is of the

order of i e

m m times longer than the slowing down time for electrons having the same velocity.

b.

Test particle slower than both electrons, ions:

Here the limit given by equation (64b) is used for both electrons and ions so that the

slowing down equation becomes:

3 2 3 22 2

2 2

0

ln1 1

2 23

e i eT T T

T i i e e

n e m mq m mu u Z

t m m kT m kT

and the slowing down time becomes:

2 2

0

2 2 3 2 3 2

3 1

ln

1 12 2

T s

e T i eT T

i i e e

m

n e q m mm m Z

m kT m kT

.

The temperature terms scale as the inverse cube of the thermal velocity and so if the ion

and electron temperatures are the same order of magnitude, then the ion contribution dominates.

Thus, the slowing down is mainly done by collisions with ions and the slowing down time is:

3 2

2 2

0

2 2

23

ln1

i iT s

e T T

i

kT mm

n e q m Z

m

.

The beam takes a longer time to slow down in hotter plasmas.



c. Intermediate case, when we have beam faster than ions, slower than electrons:

In this case, the slowing down equation becomes:

3 22 2

2 2 2

0

ln 1 41 1

4 2 3

e eT T T

T i e e

n e mq m mu u Z t m m u m kT

and the slowing down time is:

2 2

0

3 2

2 2

3

4

1 4ln 1 1

23

T s

eT T e T

i e e

m

mm mn q e Z

m u m kT

.

4. Electrical resistivity

If a uniform steady electric field is imposed on a plasma this electric field will accelerate

the ions and electrons in opposite directions. The accelerated particles will collide with other

particles and this frictional drag will oppose the acceleration so that a steady state might be

achieved where the accelerating force balances the drag force. The electric field causes equal and

opposite momentum gains by the electrons and ions and therefore does not change the net

momentum of the entire plasma. Furthermode, electron-electron collisions cannot change the

total momentum of all the electrons and ion-ion collisions cannot change the momentum of all

the ions. The only wave for the entirety of electrons to lose momentum is by collisions with ions.

Let us assume that an equilibrium is attained where the electrons and ions have shifted

Maxwellian distribution functions:

2

3 23 2exp 2

2i

i i i i

i i

n f v m v u kT kT m

2

3 23 2exp 2

2

ee e e e

e e

n f v m v u kT

kT m .



The current density will be:

i i i e e e J n q u n q u .

It is convenient to transform to a frame moving with the electrons, with the velocitye

u . In this

frame the electron mean velocity will be zero and the ion mean velocity will berel i e

u u u .

The current density remains the same because it is proportional torel

u which is frame-

independent. The distribution functions in this frame will be:

2

3 23 2exp 2

2

ii i rel i

i i

n f v m v u kT

kT m

2

3 23 2exp 2

2

ee e e

e e

n f v m v kT kT m

.

Since the ion thermal velocity is much smaller than the electron thermal velocity, the ion

distribution function is much narrower than the electron distribution function and the ions can be

considered as a mono-energetic beam impinging upon the electrons. Using equation (57), the net

force on this ion beam due to the combination of frictional drag on electrons and acceleration due

to the electrical field is:

2 2

1

2

0

ln

4 2rel

rel e i e ei i

e ev u

u n q q mm v erf v q E

t v kT

.

In steady-state the two terms on the right hand side balance each other in which case:

2 2

1

2

0

ln

4 2rel

e i e

e ev u

n q e m E v erf v

v kT

.

If rel

u is much smaller than the electron thermal velocity, then equation (64a) can be used to

obtain:



2 2 1 2

3 2 3 22 2

0 0

ln ln

3 32 2

e i rel e

e e e

n q e u Ze m J E

m kTe m kT

whererel eu J n e and 1

1 1 1e i e em m m have been used. Equations can be used to define

the electrical resistivity:

2 1 2

3 22

0

ln

3 2

e

e

Ze m

kT

so that:

E J .

If the temperature is expressed in terms of electrons volts k e and the resistivity:

4

3 2

ln1.03 10

e

Z Ohm m

T

.

5. Runaway electric field

The evaluation of the error function in equation (78) involved the assumption thatrel

u ,

the relative drift between the electron and ion mean velocities, is much smaller than the electron

thermal velocity. This assumption implies that '

14 3 x erf x x

, so that x lies well to

the left of 0.9 in the curve plotted in the picture above. Consideration of picture above shows

that '

1 x erf x has a maximum value of 0.43 which occurs when 0.9 x so that equation

(78) cannot be satisfied if:

2 31

2 2

0 0

ln lnmax 0.43

4 2 8

e i e e

e e e

n q e m n Ze E v erf v

m v kT kT

.



If the electron temperature is expressed in terms of electron volts, attaining a balance between

the acceleration due to the electric field and frictional drag due becomes impossible if:

Dreicer E E ,

where the Dreicer electric field is defined by:

3

18

2

0

ln ln0.43 5.6 10 /

8

e Dreicer e

e e

n Ze E n Z V m

kT T

.

If the electric field exceeds Dreicer

E then the frictional drag lies to the right of the

maximum in figure above. No equilibrium is possible in this case as can be seen by considering

the sequence of collisions of a nominal particle. The acceleration due to E between collisions

causes the particle to go faster, but since it is to the right of the maximum, the particle has less

drag when it goes faster. If the particle has less drag, then it will have a longer mean free path

between collisions and so be accelerated to an even higher velocity. The particle velocity will

therefore increase without bound if the system is infinite and uniform. In reality, the particle

might exit the system if the system is infinite or it might radiate energy. Very high, even

relativistic velocities can easily develop in runaway situations.

Documents

Referat - Padurariu George-Octavian