On the solution of the semiclassical Boltzmann equation

Lukes,

J?hyslcd

‘r.

31

1761-1773

1965

ON THE SOLUTION OF THE SEMICLASSICAL BOLTZMANN EQUATION by T. LUKES

*)

Northampton College of Advanced Technology,

London, England.

synopsis A Green’s

function

solution

dependent

inelastic

differential

and integral

it is shown

that

integral.

Explicit

Green’s

function.

Green’s

function

scattering

the

in the presence

equations

Green’s

results For

of the semiclassical obeyed

function

are obtained

small

angle

Boltzmann

of an electric

by the Green’s may

functions

be expressed

in certain

scattering

cases

a partial

equation field

by by

series

for time Integro-

are considered

means

differential

valid

is derived.

and

of a functional

expansions

of the

equation

for the

is derived.

1. Ilztrodztction. The evaluation of the transport coefficients of metals and of non-metals depends on the solution of the Boltzmann equation for the distribution function; even for steady, space independent fields this is a difficult problem which leads in general to an integral equation which has no general solution in closed form (see, for example, Zimanl). For the case of elastic scattering on a spherical energy surface the solution has a well known, simple form. A general method for elastic scattering on an arbitrary energy surface has been given by Sondheimerz). An expansion of the solution has been given by Prices) and a numerical solution for elastic scattering has been considered The present work is based on equation. By the “Boltzmann approach usual in the practical of the validity of the equation

by Taylora). a Green’s function solution of the Boltzmann equation” is understood the semi-classical evaluation of transport properties. Questions will not be discussed.

2. The solution of the Boltzmann equation. The Boltzmann equation a spatially uniform field E(t) may be written in the linearised form

[$ fdu,

t)lOOU+ v*[eE(t)

$

f&)]

+ -& fl(v, t) = 0

for

(1)

where fl(v, t) + fO(E) is the distribution function of the electrons at time t; in the absence of a field this is taken to have the form f&). *) Present address: Department of Applied Mathematics and Mathematical Physics, University College, Cardiff. -

1761 -

In order to write down the solution of (I) explicitly,

consider the equation

or

where the effect of collision is represented by a collision operator sponding to (5) onct can construct the inhomogeneous equation

9’. Corre-

i‘ ,$

[K(v’, f’; v, lt-1 + Y/‘k[v’, f’; v, t- = h(v -- v’) h(t --- 2’)

/?(V’,t’; V, t) = 0

for

t .:: f’

= fb(V ~ II’) and, for any phvsical

(4n)

for

2 = t’

(4b)

process, vve must ha\.c lim k[v’, t’; v, t] -F 0 I /‘--cc

(4c)

/r(u’, t’; v, t) can be regarded as the matrix element of the operator /z(~,r’) which, if the operator in (3) is not explicitly time dependent, will onl!depend on the time through the interlral t - t’ = s. Equation (3) for the operator h: can then be written

with the initial

conditions /i(s) = 0

s -1 0

=l The solution

of (5) and (6) is j? = c Y’s

S>O s (

=o so that,

(6)

s = 0

for transitions

between

discrete

states

0 IV>,

:u //
(7)

(This satisfies (hb), since = 6(v - v’)). For negative times, (v jkl v’> is given by the principle reversibility; = iv’ lk( 4) / Vl k(v’, v, s) (defined for a continuum k(V’, where

LB(v)

is the density

21, S)

=

of Iv) states) (V

Ih(

V’>

of states in velocity

of microscopic

is then given bJ

.9(V)

space.

(8)

ON THE

SOLUTION

It is convenient

OF THE

in equation

SEMICLASSICAL

which, in the linear approximation,

fl@>

of (1) may be formally

4 = [;

+

1763

[

eE

afo x

1

is a function

of energy only, so that the

written

i-J-’u(t, E), v =

=l/dt’dv’[; However,

EQUATION

(1) to define a field vector u(t) by

u(t) = -

solution

BOLTZMANN

+L?]-’

6(v -

v’) S(f -

t’) u(t’. v’) *v’.

from (3)

(10) so that fr(v, t) = J;dt’ --oo (where the time integration =

dv’k(v’, t’; v, t) u(t’) .v’

(11)

limits are imposed by (4~))

/“rds dv’k(v’,

v, s) u(t -

s) .v’

(12)

0

where the change of variable s = t - t’ has been made. For a time independent field ue this equation leads to a time independent steady state distribution function /i(v) = fJds

dv’k(v’, v, s) uo.v’

(13)

0

The

function k(v’, v, s) is the Green’s function of the time-dependent Boltzmann equation (4). Equation (13), however, shows that if the steady state solution only is required this can be expressed in terms of the function W(v’, v) = j%(v’,

v, s) ds.

(14)

0

This is the Green’s function obtained by putting

of the steady

state

solution

of (1) i.e.,

that

$ fl(V, q= 0. In order to study the time dependent

behaviour

of fi(v, t) it is convenient

1764

T. LITKES

to consider

a time dependent

field u(t’) given by u(t’) = ueS(t’) .S(f’) =

where

1 t < t’

(15)

xot>t For t < t’ the distribution

function

f’(v, t) then reaches

value; for t > t’ it decays due to the effect of collisions We introduce the fourier transforms

K(v’, v, W) = &v’, Co U(W) = j?(t) cc Equation

the steady

state

only.

v, t) eiwt dt

(16b)

eimt c1t

(16c)

(1 1) can then be written co) = Jdv’K(v’,

Fl(v,

v, W) U(W) .v’

(17)

The boundary condition (4a) is then seen to embody a causality condition; it precludes the “response” fi(v, t) preceding the “force” u(t’). It follows that the function K(v’, v, o) (considered as a function of the complex variable 0)) can have no singularities in the upper part of the complex plane. For the particular choice (15) for u(t’) one obtains

(18) (where P denotes becomes Fl(v,

that

co) =

the principal

s

dv’[K(v’,

value

v, a,)] uo. v’

is to be taken),

z~(w) + I’

so that

(17)

1

(--~->I (‘9)

io)

and one obtains

dv’ e-tot K(v’, v, (0) u. .v’

= .;

s

dv’K(v’,

v, 0) uo. v’ + s dv ’ [&/du

2R) >I

XS(W) + 11 A (

e-iwt “(“‘;,:’

=

“) } 1~0.v’

(20)

This equation holds for all t. Since function K(v’, v, w) has no poles in the upper half of the complex

ON THE

SOLUTION

OF THE

SEMICLASSICAL

BOLTZMANN

EQUATION

1765

plane it is possible to apply Cauchy’s theorem to the function K(v’, u, U) efut where u lies on the real axis and obtain

K(V’,

V,

u) eiut =

$- I K(v’, y>w)

eiwt

t>o

dw

W-U

K(v’,

v, u) e-iut =

KP’SVI 4 _fI_ f w ni

e_gmt

t
dw

(214

(214

(where, in evaluating the integrals, the contour of ingration is along the real axis and closed by an infinite semi circle in the upper half of the complex plane). In particular for u = 0

Substituting fl@,

into (20) from (22b) t < 0) =JK(

v’, Y, 0) uo.v’

dw’ =/I+‘,

v) uo.v’

dv’

(23)

where (14) and (16b) have been used; this is an agreement with (12) and gives the steady state solution. On the other hand for t > 0, substituting from (22a) into (20)

fl(v, t > 0) = /dY’

[-&-

+ 1 dv’ [&

$ K(v’~Ov’ O) ei@t do,] $ K(v’z’

1~g.v’ +

Lo) e-gut dw] u0.v’

so that

$ fl(v, t >

0) = /dv’ [ JJ g $

K(v’,

V, co) et,,]

K(v? v, 0) e-got

u. .v’ -

1

u. .v’.

(24)

The integral over w for the first term in (24) can be done by means of a contour along the real axis and closed by an infinite semicircle in the upper half of the complex plane and therefore vanishes. The integral in the second term gives, by means of (16b), just k(v’, v, s), so that +

fi(v, t > 0) = -

/dv’k(v’,

v, s) uo.v’

(25)

?‘. LUKES

1766

This shows that the function h(v’, v, s) determines the relaxation of the distribution function in the absence of a field. For t = 0 we obtain, using (2) and (4b) L?/(v) = uo*v which checks that at t = 0 the function Boltzmann equation. From equation of the Boltzmann functions K(v’, v, energy, for elastic

(26)

fr(v, 0) satisfies

the steady

state

(13) it is seen that in order to write down the solution equation explicitly, it is necessary to known either of the s) or W(v’, v). However, since uo only depends on the scattering it is possible to write (11) in the form

(276) This last equation

can be used to define a relaxation

s

ds’ -jgrad;;,i

Iv =

time tensor z:

IT(V’J v) v’

This is a general definition of a tensor relaxation time in an elastic scattering process. For the particular case in which ug(t.) is assumed to be sharply peaked at e = FJP (27b) may be further written in the form /r(v) = UO(&F).?v.

(29)

3. General relations for the Green’s fmxtions. In this section three general equations for the Green’s functions K(v’, v, s) and W(u’, v) will be considered. (a) The operator L?’ defined by (2) can be expressed as an integral operator 5,

L-1 = afl

at

toll

__Yfl(V,

.

t) = /dl+(v,v’)

where p(v, v’) is the scattering in velocity space. It is convenient to write

[

fl(V’,

fo(v’Nl -

probability

t) ____ fo(v’)l

-

per unit time into a volume dv’

P(VJv’)

q,(d)[l 1 fo(v)U

-

fo(v)l

s

--I (30)

fdv, 4 ~__ fo(v)P - fo(41

fo(v’)]

= p(v’p v,

$(v’, v) dv’ = G(V).

(31)

(32)

ON THE

SOLUTION

OF THE

SEMICLASSICAL

BOLTZMANN

EQUATION

I767

Using (3) one obtains $

k(v’, u; t’, t) -

s

dv”[k(v’, v”; t’, t) p(v, v”) .a(~“) S(v -

(with the initial conditions (4)) which is a general integro-differential Green’s

function. 7;

Writing

v”)] dv” = 0

for the time

(33)

dependent

this as

K(v’, v, s) =

dv”K(v”, v, s)[p(v, v”) -

s

and using (14) one obtains j”dv”W(v’,

equation

K(v’, v”; t’, t) .

on integration,

v”)[p(v’, v”) -

I

o(v”) 6(v -

v”)]

(34)

using (4~) 6(v -

v”)] = --6(v

-

v’)

(35)

which is an integral equation for the Green’s function W(v’, v) required in the steady state solution. (b) For formal purposes it is convenient instead of (30) to consider again discrete states Iv) rather than a continuum. For this purpose we define operators

p, (r, by the equations c

= (v Ipj v’) B(v’)

dv’ = p(v, v’) dv’

(36a)

dv’ z (v dv’

Equation

101 v’>

=

o(v)

6(v

-

v’)

=9(v)

dv’

(36b)

(33) can thus be written ;

= Iz [ -

v’


lul v’>l[l

(37)

where cr is diagonal in the Iv> representation. Similarly k(v’, v, s) is replaced by (v ]e-bLysjv’> (see (7)). Comparing (37) and (2) it follows that .Y=cT-p. The exponential

matrix

(v ]e-zsI converges

(38)

series

v’) = (v

I[

1-

9s

92 + -9 2!

for any .9 and s and may therefore

+ ...

v’ I/

be summed

) explicitly

(39) if 9

is known: = -

s
C
1768

T. LUKES

In terms of (38), going over to a continuous

k(v’, v, s) = S(v - v’) - s[u(v) b(v -

+$:/d v”p(v’,

v”) p(v”, v) - p(v, v’)La(v) -t U(V’)’

7

(41)

the series for W(v’, v) is W(v’, v) = o-1(2)‘) 6(v + / a-l(v’)

(c)

p(v, v’)] +

u(v) a(v’) 6(v - v’)] + . . .

+

Similarly

2)‘) -

representation

v’) ‘+ o-1(1.“) p(v’, v) o-i(v)

p(v’, v”) p(v”, v) a-l(v)

By the group of the operator e-Y(t-l”

one obtains tr . . . t,_]

on dividing


9

dv” + .

i.e..

= ,-2z(t-P,

the interval

(42)

{

t -

e -*c(f”--l’) t’ into ‘n intervals

by time points

v’) = xv je-2(fcfnm1) ,~IP(t”-l~-f”-e) . . e- Z’(fI 4lj v! i

LV’,l

Inserting complete one obtains

sets of Iv) states


2; c ... c
v’) _

i.e. using the fact that C lv’ic;v’j = 1 Y’ vn_l)(vn_l/

e Y/‘(f,,I -I,, 2) ,..

. . . c;vl le -wt,-“)I

,,1 ,

(43)

We now consider (43) in the limit n --f co. In this case all the factors in (43) can be expanded to first order in the time intervals. Using (38) one can then write

+

O(ti -

Vi-l> =
(4

-

ti-l)[o(%-l)

qti qvi

-

&-I) + O(& vz-1)

-

&i)Yj

]

*d(v -

I =

i

(44)

t&1)2.

Substituting into (43) and going over to a continuous means of (36) we obtain k(v’, v, t -

vi-1

t’) = f dvi dvs . . . dv,_r[d(v

-

q-r)

-

(t -

representation

by

tn_-l){u(vn_-l) *

vn-1) -

-

p(v, vn-I)}]

. . . [d(V$ -

Vi-l)

-

p(vz, VH))]

. . . [6(Vl -

v’) -

-

p(v1,

v',>l.

This is a general expression integral.

-

(ti (tl -

ti_1)(c7(v&1) qvi t’){a(v’)

6(v -

-

Vi--l) -

VI) (45a)

for the Green’s function by means of a functional

ON THE

SOLUTION

OF THE

SEMICLASSICAL

moreover,

1769

EQUATION

The form of (45) does not lend itself

4. Small angle approximation. easy evaluation;

BOLTZMANN

it differs somewhat

from the general

to

form of

functional integrals in quantum mechanics, about which some information is availablea). However, for small angle scattering it is possible to derive a partial differential equation for k(v’, v, s) by a method similar to Feynman’s derivation of the Schrijdinger equationv). Integrating over all the VZ’Sin (45) except vn-i k(v’, v, s) = _/-dvn-lk(v’, vn-1; t’, tn_l)[b(v -

(t -

t?4{+)

qv

-

vn-1)

-

vnq)

p(v,

-

(46)

v,-l)}]

this can be written

{v -

k(v’, v, s) = f k(v', v -

d+J(v)qv -

v,q);

vn-1)

s -

-

p(v,

in which each term which depends on v In this way one obtains

s[ dvn-1

+

&C

k(v’, v, s) -

(fh -

-& k(v’, v, s) As -

-$

Q-l),i12

Q',

i=j

1 dslp(v3 v?&-l)

i

v~_~) -

bl)}]

vn-r can be expanded.

C (vi - z+,_&

$

v, 4

-

5

+

“(n-l),iJ2

-

- %2-l) + I[d(v

a2 k(v’, v, s) + . . . avzavj

~

V(,-l),i)

(vi -

k(v’, v, s) + i

i

+ c (Vi- "(n-1)),&-

+ 4 C (% -

-

i

i

+

ds)[d(v

Vn-*,J

&

a2 (v,G-1) av; p

P(v,

v?&-l)

+

-t

a2 p(v, avtav5

qv -v,)

(45b)

If the scattering probability is confined to small angles the higher terms may be neglected. If further, we define

order

+ C (% i#i

vn-l,i)(vJ

-

‘n-l,i)

-

dii = / [vt dv&,

= f [vi -

and neglect terms of order dv~dv&~

vi]p(v, v;][v, -

VW1)

+

. *.

-a(v)

v’) dv’ v;] p(v, v’) dv’

(45) reduces to

1

(464 (4W

Passing

to thcb limit

Is

,I 0 \vc obtain

i’ -- /i(v’, v, s) = k(v’, TJ,slip(v) 2s

~- rr(v)i

‘-p(V)

(1x1integration

-=

1‘ (V,

V’)

over s a partial differential

ClV’

1

equation

for

Jt’(V’,

-

V’)

z

It’(V’,

V)lp(V)

--

O(V)_

~

2; i

.illi

is obtained:

--

i -C)(V

V)

ITT(V’,

V)

.J?!i_

i

(48)

5. .Swwnatio7z of tlzilGrecu’s fuuctioH swics iu particzllar ruses. (a) Elastic The integral in (27) can scattering on a spherical energy surface. br written in the discrete representation as C (V !e-9fesi v’>
of the form

pi v’> =
If the surface is taken to be spherical

(50)

and the transition

ON THE

SOLUTION

OF THE

SEMICLASSICAL

BOLTZMANN

EQUATION

177 1

probability is taken to be a function of the angle of scattering between v and v’ only, then the sum in (49) is the one which occurs in the usual theory in determining the relaxation time; this is given by the expression 1 -=-

7 j-g

[I -

cos 01 $(k, k’)

(5’)

7

In terms of T we have s If the series expansion of (52) gives

= L
(40) is substituted

into (49) then repeated

application

(v leCSsj v’)(v’ [1) = epsi’(v 11) = eesiT u which shows having been the energy). switched off

(53)

that the average velocity at time s conditional on the velocity v’ at s = 0 decays exponentially (T may, of course depend on The relaxation of the distribution function when the field is is given by (25) as i

fl(v, s) = -

which, for the particular

j

ds u&) e-“‘@).v

case of a ~(8) sharply peaked

= -fl(v, This is the exponential heuristically postulated. (27) as

at &

0) e-“i’(“‘(using

=

FF

gives

26)

(54)

decay of the distribution function sometimes The distribution function can be obtained from /r(v) = f de UO(E)*V?(E)

(55)

which, for a sharply peaked Q(E) at E = &Fgives the usual result /l(V)

=

ILO

*f-(w).

(56)

(b) A transition probability independent of angle. We consider the case of a transition probability which is independent of angle but depends on the energy, so that $(v, w’) = p(&, 8’). This is a case which in fact occurs in the theory of lattice scattering in semi-conductors, and has been solved by making the assumption that the collisions can be treated as elastics). We shall show that an exact solution is possible without making this assumption. Using the series (42) in (13) one obtains

/i(v) = /dv’[o-r(u’) +/+(v’)

6(v - v’) + o-l@‘) &J’, w) u-l(v) + p(v’, u”) p(v”, v) cr-i(v) dv” + . ..] ZL~.Y’.

(57)

1772

T. LUKES

Since the energy is even in v all the integrals except

the first vanish and

one obtains /i(Y) = o-i(&) UO(&). w.

(58)

(c) Case of a degenerate kernel. It is shown in the theory of integral equations (Courant and Hilberts) that any kernelp(v, Y’) can be expressed as an infinite sum of the form p(v, v’) = Z Xi@‘) yr(v). i

(59)

If the series in (59) is finite, the kernel is said to be degenerate; in this case, a solution of the Boltzmann equation for elastic scattering has been obtained by Son d h ei m e r 2). We shall show that a solution can be obtained for elastic and inelastic scattering for the simple case $(V, V’) = x(u, E) y(v’, E’).

(60)

By making use of the detailed balancing condition P(v, v’) fo(a)[l -

fo(41= P(v"4 fO(E')[l - fob)1

(61)

all the terms in the series (57) except the first vanish on integration obtain /i(V) = a-l(v) UO’U.

and we

For elastic scattering (60) is a particular case to which Sondheimer’s applies and (61) is in agreement with Sondheimer’s (38).

method

(62)

Discussion. The object of the present work has been to exploit the consequences of writing the solution of the Boltzmann equation in terms of a time-dependent Green’s function. It has been shown that a number of results can be derived, essentially by treating the operator e-Z(t-“) analogously to the operator eeiHitPt’), in quantum mechanics. Equation (11) for the distribution function shows that this expression depends on a conditional average of the electron velocity. In this respect the expression is similar to the trajectory solution of Chamberslo) and the conductivity equation of Kub o ii). The similarity with Chamber’s equation is again displayed in the functional integral expression (45); here the electron is carried over the various sections of its path essentially by means of first order perturbation theory, whereas in Chamber’s expression the path integral is performed by means of a time-dependent relaxation time. The introduction of the Green’s function would appear to put the time dependence of the Boltzmann equation on a more satisfactory basis. Its ultimate usefulness will depend on whether the general expressions derived for the Green’s function will turn out to be easier to evaluate than the original Boltzmann equation. This question can only be answered by considering the evaluation of particular models for the scattering process. It is hoped to consider this question elsewhere. Received 3-3-65

ON THE

SOLUTION

OF THE

SEMICLASSICAL

BOLTZMANN

EQUATION

1773

REFERENCES Ziman,

J. M., Electrons

1) 2)

Sondheimer,

3)

Price,

P. J., I.B.M.J.R.

4) Taylor, 5) Wilson,

and Phonons

(Clarendon

E. H., Proc. roy. Sot. 268 (1962) 1 (1957)

Press, Oxford,

147.

P. L., Proc. roy. Sot. “75 (1963) ZOO. A. H., The theory of metals (Cambridge University

6) 7)

Gelfand, Feynman,

I. M. and Yaglom, R. P., Rev. mod.

8)

Radcliffe, Courant,

J. R., Proc. Phys. Sot. A 68 (1955) 675. R. and Hilbert, D., Methods of Mathematical

1953, vol.

1, p. 114).

9)

Press, Cambridge.

1953, par.

8.1).

A. M., J. math. Phys. 1 (1960) 48. Phys. 20 (1948) 367.

R. G., Proc. Phys. Sot. A 85 (1952) 358. 10) Chambers, Ku bo, R., Canad. J. Phys. 34 (1956) 1274.

11)

1960).

100.

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(Interscience,

New York,