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Scattering and the Boltzmann relaxation equation
Guenault,
Physica
A. M.
MacDonald,
30
309-323
D. K. C.
1964
SCATTERING
AND THE BOLTZMANN EQUATION
by A. M. GUENAULT
RELAXATION
D. K. C. MACDONALD
and
*)
Division of Pure Physics, National Research Council Ottawa, Canada
Synopsis The “Boltzmann able
popularity,
change
possible
as specific previous
examples,
assumptions, equation
function
solving
statistical
we cannot relaxation
is valid
study
-(P
problems.
piston
expect
scattering
to be represented
Moreover
when,
assumption
rate
function, We
of
P,
consider
and also,
as we are not aware
of it. The results show that, without
consistent
the
here how far
model in both the “Brownian
in some detail
and
has consider-
We examine
equation.
the only
Peq.)/T,
describing
distribution
transport
this latter
-
for
for an arbitrary
of the Rayleigh
limits,
=
physicists,
due to scattering.
steady-state
the behaviour
and “Lorentz” detailed
Boltzmann
for
kP/at],,,,t. metal
can have general validity utility
movement”
strictly
equation”, amongst
in time of a distribution
this equation its
relaxation particularly
if, the
is that
rather
at all generally Boltzmann
7, the “relaxation
of a
unusual by the
relaxation time”,
is
constant.
1. Introduction. In non-equilibrium statistical mechanics, but particularly in the field of electron transport theory in metals and alloys, it is sometimes suggested that it is a good approximation to write
p _ pea.
--I 8P
at scatt. =-
(1)
7
for the I ate of change with time of the distribution function due to scattering. In this equation Peq. denotes specifically the distribution function in true thermal equilibrium, i.e. usually the Maxwell-Boltzmann for gas molecules, and the Fermi-Dirac distribution for electrons in metals; it is assumed that T, the so-called “relaxation time”, may itself be a function of the velocity, I’, or energy, E, of the particles (i.e. 7 = T(V) or T(E)) but is otherwise to be treated as a constant. Equation (1) as it stands may be solved quite generally for arty arbitrary initial distribution, however caused, to yield P(V, t) = P( V’, 0) ectit + Peq.( V) (1 -
e-t/r)
@a)
or
P(V, t) *j
Dr. MacDonald
Peq.( V) = [P(V, 0) -
died July
28, 1962. -
309
-
Peqv)]
e-t/t.
(2b)
310
A. M. GUl?NAULT
AND
D. K. C. MACDONALD
This particular form of equation (1) for the change in P due to scattering (of one kind or another) seems to have different terminology depending on the field of work involved. For the sake of definiteness let us in what follows here call an equation of the type indicated in equation (1) a “BR equation” (i.e. a Boltzmann relaxation equation). It is true that for certain specific types of perturbation (e.g. an electric field applied to a typical metal at “high” temperatures (T 2 0); cf. e.g. Wilsonl) a steady state is maintained in which the scattering to a first approximation can be considered to obey equation (l), but we find ourselves asking the following question. Is there, in fact, a model of scattering which leads to equation (1) for the return to equilibrium for entirely arbitrary initial disturbance of the distribution function? Starting from the so-called master equation we can give what we believe is a rather general answer to this question by showing that the BR equation will follow for an arbitrary initial disturbance if the transition probabilities can be assumed to follow an unusual and somewhat remarkable form. We have then proceeded further to examine in particular the relaxation due to scattering of the Rayleigh piston model in two limiting situations, namely the familiar Brownian movement approximation governed by the FokkerPlanck equation, and the somewhat less familiar Lorentz limit (this latter being examined in some detail). We conclude that a BR equation will not be found for the return to equilibrium due to scattering*) on either of these models. Consequently it is our opinion that it is improbable that any normal scattering mechanism would yield equation (1) for an airbtrary perturbation. It seems worthwhile to draw attention to this situation, and to invite possible rebuttal, because, in metal physics at least, it appears to be commonly argued that the BR equation should be a good first approximation in many scattering situations for all types of perturbation; indeed “hand-waving” arguments (which we shall outline and criticize below) are sometimes adduced to support this thesis. 2. The general validity of the BR equation. We take as our starting point in the argument the “master equation” for the rate of change in time of a probability distribution function P( I’/,t) :
“pY)l&t. =S
[-W(V,
For brevity *)
Uhlenbeck
we consider and Wang
V’) P(V, t) + W(V’,
V) P(V’,
the simple case where P(V, t) etc.
Chang2)
have
considered
the problem
in part
t)]dV.
(3)
are functions
by asking
whether
the
Kayleigh
piston model, under an external alternating field, in what they call “the strong-coupling i.e. the Lorentz limit (vi& infra), leads to the BR equation as a good approximation. approximation” They them.
conclude
that it is not an adequate
approximation
in the particular
problem
considered
by
SCATTERING
AND
THE
BOLTZMANN
RELAXATION
EQUATION
311
only of (a single scalar) velocity, V, and of time, t; the results of this section could however be readily generalized to include e.g. a three-dimensional velocity, V. There seem to be two principal assumptions in writing down an equation such as equation (3): (i) that the system can be adequately represented by a probability distribution, P(V, t), dependent only on the present state of the system (i.e. the Markoffian assumption) and (ii) that transition probabilities exist; (W(V, V’)dV’ is here the probability per unit time for a system in state I/ to make a transition into a range dV’ at V’) *). Our elders and betters tell us that these two assumptions should be valid for a wide range of scattering problems, so that we expect the master equation to be of rather general applicability. We shall now indicate the sort of “hand-waving” argument that is often presented to justify replacing the master equation (equation (3)) by the much simpler BR equation:
, aw, t)1 at
P(V, t) -
=-
scatt.
Pqv)
m
where Peq.( V) is the equilibrium probability distribution function and T(V) is a “relaxation time”. The argument might run typically as follows: We note that the first half of equation (3) that involving transitions out of state V) can be written immediately as -P(V, t)/T(V) if we define -
1 T(V)
=
It is then said that the “principle i.e.
W(V,
s
W(V,
V’)dV’.
of detailed
V’)Peq.(V)
enables the second half of equation
balance”,
= W(Vf,
Vp.(V’),
(3) to be written as
P”q.( V) 1 W( V, V’)
;Jqy;;f)
dV’.
The argument then runs that, at least for small deviations of P(V, t) from equilibrium, the factor P(V’, t)/Peq*(V’) in equation (6) will have negligible effect on the integral and so, using (4) again, we obtain the BR equation (1). This final step is appealing because it is often in fact a valid procedure (e.g. in the “drift terms” in transport theory - see below) to replace P(V) by Peg.(V) wherever possible, for small departures of P from Peq.. We shall now criticize the argument and analyze more precisely the conditions under
*) We should perhaps remark that if we are dealing metals and alloys, then in order to satisfy Fermi-Dirac include
specifically
a factor
(1 -
flP( V’)),
where
specifically with problems of electrons m statistics it is necessary for W(V, V’) to
l/,9 is the appropriate
density
of states.
312
A. M. GUdNAULT
which the master fined it.
equation
AND
D. K. C. MACDONALD
will reduce to the BR equation,
as we have de-
First let us note that we used the principle of detailed balance, equation (5). It is worth noticing that this principle is not of entirely general validity; however a general principle obeyed by mechanical systems is that of “microscopic reversibility”. On applying this to the present problem we obtain not equation (5), but: W(V,
V’)PqV)
= W(-VI’,
-V)P=(-V’).
(7)
It is only in symmetrical problems that this reduces further to “detailed balancing”, equation (5), (cf. e.g. Alkemade, van Kampen and MacDonalds) for study of an asymmetric problem in which (7) but not (5) is to be used). Hence we can only continue with the above argument for symmetrical systems, as we shall now do - this will not be a too serious restriction in practice, but must affect any claim of general validity for the BR equation. Secondly, we can rapidly expose the dubious nature of the argument which leads to replacement of the expression of equation (6) by Peq.( V)lT( V) in the master equation. Let us write: P(V, Using equations
+ Pl(V,
t).
(4) and (6), we may now write the master
ap(v, 2)
at
t) = P=(V)
Pl(V>
+,)
zz-
1 soatt.
t)
+ j-V’>
(8) equation:
V)Pl(V’, WV/‘.
(9)
The BR equation includes only the first term on the right hand side of argument, which in its conequation (9) - and the earlier “hand-waving” text seemed quite appealing, is now seen to amount to saying that “as Pl(V, t) is small, the integral in equation (9) can be neglected” *). This is palpably untrue in general, as clearly both terms on the right of equation (9) involve PI linearly, and consequently the smallness or otherwise of PI is irrelevant in this respect. We are therefore left with our original question - can the BR equation ever be a general expression for scattering? That is to say, we wish to ask what form W(V, V’) must take in order that the BR equation may follow from the master equation for an arbitrary perturbation Pl(V, t) from equilibrium. Restricting ourselves to symmetrical problems, we find equation (9) *) cf. e.g. Wilson ‘) in the first edition who states (p. 19): “Since the distribution function, f, never differs very much from the equilibrium distribution fo, the net number of electrons ejected per unit time (from dV) will be proportional to (f - fo), so that we can write
a [ at
1
call. =
The second edition (1953) is much more guarded
f - f, - --. T(V)
on the matter
(see p. 6).
SCATTERING
a convenient
AND
statement
THE
BOLTZMANN
RELAXATION
of the master equation,
EQUATION
313
and we can answer the
question as follows. (a) If equation (9) is to reduce to the BR form, we must have:
s
W(V’, V)P1(V’, t)dV’ = il
where 1 is a constant,
ply’,
t)
m
’
(10)
and hence
ap(v, t) = at 1 scatt.
(1 -
il) ‘:$;
t, .
(11)
(b) As the probability distribution functions are normalized functions, we have /P(V, t)dV = constant, so that in particular SPl(V, t)dV = 0. Furthermore, if we integrate equation (11) with respect to V, the left-hand side must vanish, and we have: O=
-(I
-A)
/p:;;;t)
dV.
(14
Remembering now that JPl(V’, t)dV = 0, but that otherwise we are assuming Pl(V, t) to be entirely arbitrary, the only possible conclusion from equation (12) is that T(V) is a constant, independent of V. (c) Equation (10) can now be written:
s
w(V’,
V) Pi(V’,
t) dV’ = ;Pl(V,
t).
The appropriate general solution of this equation for W(V’, V), remembering that PI(V) is to be an arbitrary function, satisfying onlyJP1(V, t)dV = 0, is readily shown to be W(V, V’) = f(V’)
+ +3(V
-
V’),
where 6(x) is the Dirac delta-function. However, we may reject the deltafunction solption ,as it does not contribute to aP/at],,,,,.; i.e. for our purposes, we set 1 = 0. Hence we see that W(V, V’) is to be a function of V’ only. (d) The principle of detailed balance, eqn. (5), tells us further that we must have: W(V, V’) = A Peq.(V’),
(13)
where A is a constant, and this completes our analysis. Our conclusions are thus that if the BR equation is to be valid for relaxatiom from an arbitrary initial distribution function, then: (i) the transition probabilities must necessarily have the somewhat re-
314
A. M. GUkNAULT
AND
D. K. C. MACDONALD
markable form W(V, V’) = A Pq.(V’), where A is independent of V and of v’. How closely the transition probabilities derived from any physical model might be made to approach to this we can only guess at present, but it is perhaps worth nothing that this form of W(V, V’) would evidently involve a strong component of “large-jump” scattering (i.e. [lr’ - VI “large” - or equivalently large-angle scattering in a three- dimensional mo-
del) . (ii) it also follows that the BR equation
aP(v, t) at
can only strictly
take the form:
P(V, t)-Peq*(V)
_-
7
1 scatt.
where T is a constant, independent of V. These general results seemed to us of sufficient interest to lead us to examine the analysis in some detail of a particular model subject to relaxation by scattering. In sections 4 and 5 below we shall discuss the one-dimensional “Rayleigh piston” model (or Rayleigh particle model) introduced by Lord RayleighJ) “in order to bring out fundamental statistical questions, unencumbered with other difficulties . . .“. Before doing so, however, let us consider briefly the solution of the steady-state transport problem. 3. Solution of transport problems. In the above section we discussed the analytical form of the change in time of P(V, t) due to scattering, i.e. In principle such an expression tells us the relaxation beap(v/‘, t)iati,,,,,.. haviour of a system described by P(V, t) towards its equilibrium distribution reached in the absence of any external forces as t + co. Very frequently (and particularly in metals theory), we are not interested in such general information, but rather we require a steady-state solution of an equation :
qv,
at
ap(v, 4
4 1 drift.
which we may call a (Boltzmann) written :
+z*vax
F ap(v/‘)ax MTF
-
ax
=
s
+at
=o 1scatt.
transport epation.
[-W(V,
Usually,
(14) this may be
V’)P(V) + W(V’, V)P(V’)]dV’
(15)
The two terms on the left represent drift terms in aP/at, arising from a force, F, and some gradient dX/dx. We require to solve equation (15) for P(V). Usually the forces and gradients are small, and the analysis shows that P(V) can be replaced by Peq.(V) in the drift term, although not of course in the scattering term (cf. previous section). If now the scattering
SCATTERING
term
AND
THE
BOLTZMANN
(the right hand side of equation
RELAXATION
315
EQUATION
by a BR ex-
(15)) can be replaced
pression, i.e. by -(P(V) - Peq.( V))/T( I’), then the solution for P(V) and hence for transport currents can be written down immediately. As we have seen, in general the collision term cannot be simplified in this dramatic way*); however, as we shall exemplify below, the actual steady-state solution of eqn. (15) often does have the same analytical appearance as the steady-state solution which would arise using a BR collision term with however an “effective relaxation time” a@ro$wiate to the specific kind of perturbation acting. We would suggest that this rather surprising situation has tended in the past to give a somewhat spurious air of general authority to the BR equation**). But let us now examine detailed models to try to understand this further. 4. Model I. Brownian movement limit of the Rayleigh piston model. As a first example, we consider the model discussed for instance by van Kampens), that of a Rayleigh piston (mass M, velocity V) which is bombarded by gas molecules (mass m < 1M). As the mass ratio m/M is small, this is a “small jump” model (i.e. W(V, V’) = 0 unless V= I”), and Alkemade, van Kampen and MacDonalds) have shown that the master equation (3) can be expanded consistently in powers of (m/M). The lowest order approximation, which is all we require here, gives the Fokker-Planck equation (or, briefly, the FP equation) :
qv/‘, 4 at
--a
[
1 scatt.
g
(VP(V)) +
$ g-J
(16)
where the parameter, a, depends on the details of the model. This is quite unlike the simple BR equation, and a general solution for equation (16) with arbitrary initial conditions is not straightforward; however in equilibrium (setting
1.h.s. = 0), equation
(16) yields directly
the gaussian
distribution,
i.e. Peq.(V) = 1/M/2nkT e- MV212kT.Let us look at the situation in more detail : (i) To deal with the relaxation of the piston, we only need to compute d/dt = f VP(V)dV. If we multiply (16) by V and integrate over V, the FP equation gives immediately
& =
-
a
-, 7’
say
(17)
where we have set r’ = l/u. *) The region
T <
8 where the phonon-electron
interaction
is to be treated
quantally,
scattering as is well known does not concern us here. **) Rroadly speaking this must arise from the fact that the perturbing forces and/or are supposed modest, and that the appropriate approximations to the master equation various models turn out to be linear in P(V) (and its derivatives). a rather drastic form of linearization of the collision integral.
The BR equation
and
the
gradients based on
then represents
316
A. M. GUkNAULT
At first sight this might the BR equation, namely
AND
D.
K. C. MACDONALD
seem to be identical
with the result given by
(18) However,
we recall that in equation -
whereas in the Fokker-Planck
1 7
=
s
(18) we have from equation
W(V, V’)dV’,
equation,
(4) : (4)
we have:
(19) (cf. e.g. Alkemade, leigh piston model
van
Kampen
and MacDonalds);
and for the Ray-
_4~GJ/5 (where v = number of gas molecules per unit length cf. Alkemade cu., appendix II). In the same notation, equation (4) gives in contrast 1 _N 7
of piston
cylinder;
kT --v 1/- m
(i.e. essentially the total no. of collisions per second), and hence we see that T’/T N M/m, > 1, i.e. r’ and 7 are of entirely different orders of magnitude. We may readily understand the reason for this difference in the following terms. Physically one would say that the BR equation assumes that each collision brings the system essentially into its equilibrium state (cf. also the form of W( I/, V’) derived in section 2 above). The FP equation, on the other hand, recognizes that we have a small-jump model and that -(M/m) collisions are required to dissipate its momentum. Thus although the BR equation would give a superficially correct answer for the decay of in this problem (namely an exponential decay in time), it would be most misleading in giving an entirely incorrect relaxation time. (ii) In (i) we discussed the approach of
SCATTERING
AND
THE
BOLTZMANN
well known (cf. e.g. MacDonaldc), P(V, t) =
1/
RELAXATION
317
EQUATION
namely
A4
2nkT( 1 -e-2at)
exp [
--M(V-Vo e-nt)2 2kT( 1 -e-2at)
1’ (20)
This is illustrated in fig. 1a. The solution of the BR equation has alreay been given in the introduction (eqn. (2)), and its behaviour in the present problem is illustrated in fig. lb. P(v,i P(v,t
Fig. la. Rough sketch of the solution of the Fokker-Planck equation, starting from a delta-function at Vo at time t = 0. Curve A (dotted) is for t N 0, B (dashed) for an intermediate time and curve C (full) fort + CO. Fig. lb. Rough sketch of the solution of the Boltzmann relaxation equation under conditions similar to those for fig. la.
For a system whose scattering is governed by the BR equation we see that the delta function component decays in size exponentially but remains fixed in position at Ve, while at the same time the component of the (equilibrium) gaussian distribution grows from nothing towards its full amplitude. In contrast, equation (20) (Fig. la) shows the actual mode of approach to equilibrium for our present model; namely the initial delta-function broadens and simultaneously shifts its mean position until it ultimately becomes =
the stationary
gaussian
equilibrium
distribution
centred
about
V
(Ei) Thirdly, I‘t IS ’ instructive to consider the solution of transport problems in this Brownian movement limit (cf. section 3 above). When a force F is applied to the piston, the transport equation will read (cf. eqns. (15) and (16))
(21) We can readily proceed to solve this equation Integrating eqn (21) we obtain VP+---
quite generally
kT
aP
nf
av > ’
as follows.
(22)
318
A. M. GUtiNAULT
(The constant of integration solution for F = 0).
AND
D. K. C. MACDONALD
is zero, as we know that P = Peq.(V)
is to be a
for P which is readily
Equation (22) is now a linear first order equation solved to give
For small fields, F, this can be written
Pb) (We note that this is equivalent to writing Peg.(V) instead of P(V) in the left-hand side of eqn. (21) or (22).) In contrast, the BR scattering expression leads to a transport equation: F ap ------_LM.4 av
p -
pea. 7
which then leads to a solution which is quite unlike the above, namely:
P(V)
s
eMV’/FT
=
e-
MV”=/ZkT
dj,T’. (244
--oo
However, for small fields, performing this expression simplifies to : P(V)
=
two successive
integrations
by parts,
W)
which is analytically identical in form to (23b) (and can also be obtained by direct approximation of P(V) by Pea.(V) in the “drift” term). Despite this analytic similarity for small fields, we note that exactly the same remarks are to be made as have been made under section (i) of the present discussion, where we analysed (V) ; namely for the Rayleigh piston problem, 1/CC( = T’) and 7 are of quite different orders of magnitude, so that the superficially correct result given by the BR approach is illusory when it comes to a calculation of the relaxation time (or equivalently a transport current) from first principles. We should note, however, that if (a) we are interested only in steady-state transport currents, and if (b) we had no method of calculating the relaxation time a priori, then the BR equation would suffice for these purposes despite its lack of general foundation. We repeat however that this is surely no guarantee of its validity, or even utility, in any other application.
SCATTERING
5. Model II.
AND
Lore&
section we discussed
THE
BOLTZMANN
RELAXATION
EQUATION
315
limit of the Rayleigh #&on model. In the preceding of a heavy Rayleigh
the behaviour
piston having mass
M much greater than that of the colliding gas molecules . The problem was then susceptible to analysis because expansion methods could be used with effectively m/M as a smallness parameter. In the opposite limit of M Q m, i.e. an extremely light piston bombarded by heavy molecules*), we shall show that expansion methods can again be used. This so-called Lorentz limit might be expected to bear some relation to that of the scattering of conduction electrons by very heavy ions in a metal*). To gain a little insight into the problem we first consider the simplest situation, that of a piston whose mass is effectively zero free to move in only the surrounding gas molecules have one dimension**) (or e q uivalently effectively infinite mass). The master equation for i3P(V)/i%],,,t, now takes a very simple form as the only transitions which can take place are from state V to -V or from -V to V, both with average frequency Y(V[, where v is the number of gas molecules per unit length of the cylinder, as before. Hence we have P(V)-P(-V)
WV _=-
at
(25)
r(V)
where 1 4V) It is particularly
instructive
= VII/‘].
to note that
WV) + P(-V)) =
o
at so that (P(V)
+ P(-
V)) remains constant
in time.
As an example of the behaviour of equation for P(V, t) for all t > 0, starting from an initial Vo, namely : P(V,t)
=0
(25), we give the solution delta-function P(Vo, 0) at
if V#*Vo
P(V0, t) = 4P(Vo, 0) [ 1 $ e-w Vo)1 1 q--o,
t) = iP(V,,
0) [I
_
,-t/dVO)
(2-W I
*) Actually, the strictly one-dimensional piston model in the ultimate limit of M/m=0 cannot be wholly satisfactory as a model for scattering in practice. The relaxation in time of the piston from some initially perturbed state (i.e. the return to momentum equilibrium) should be quite realistic, but the usually analogous model in the limit. **)
As a rather
vironment
bizarre
linear steady-state example
of “super-uranium”
transport
one might perhaps
gas molecules.
or “flow” envisage
solution
does not exist
for this
a piston of solid sHe in a dilute en-
320
A. M. GUkNAULT
AND
D. K. C. MACDONALD
>
Figure 2 illustrates this solution. Let us note in particular that: (i) momentz~~ equilibrium is established with a time constant T( VO); i.e. (v> = v. e-t/T(Vo), in a manner similar to that which one would obtain from a BR equation. P1v.t)
;‘i IlA :: ii II”
C C
Fig. 2. Rough piston
sketch
(equation
As in fig. C (full) fort
1, curves
of the solution
(25a)),
starting
A (dotted)
from
P( I/, t) for the an initial
Lorentz
delta-function
are for t = 0, B (dashed)
limit
of the
Rayleigh
at VO at time
for an intermediate
t = 0.
time
and
--f 00.
(ii) energetic equilibrium, strictly speaking, is never reached (in the limit of M/m + 0). Collisions are very effective in transferring piston momentum to the gas molecules, but contrariwise are most ineffective in altering the energy of the piston. In fact any symmetrical distribution for P(V) will satisfy equation (25), in direct contrast to the more usual situation (given for example by the BR equation, the FP equation or the Q equation discussed below) where, as t + co, Peq.(V) is the only solution. It is therefore clear from the solution of equation (25) that (in contrast to a BR scattering model) different properties may relax at different rates, and that we have here no single zLniversaZ relaxation time. These general features would still be maintained by a three-dimensional model in which only elastic collisions are possible. The momentum of a light “Rayleigh particle” would presumably have its direction randomized over one or two collisions, whereas its energy would remain essentially unchanged even after a large number of collisions. This behaviour is quite unlike that described by the BR equation, in which momentum and energy equilibrium are restored with the same relaxation time, indeed by the same collisions. We now study the light Rayleigh piston model in greater detail, considering expansion of the master equation to first order in ,u(= M/m), i.e. when we allow that M/m is small, but not now negligible. We cannot expand the transition probabilities in moments of the jump size (as is done for the heavy Rayleigh piston model) since we have now a “large-jump” model. However we recall that each collision almost reverses the velocity V, and we can use this to establish an expansion procedure by expressing the velocity V’ in the master equation as (I/‘*+ y) where V* = - V( 1 +,u) /(1 -,u) ;
SCATTERING
AND
THE
i.e., apart from a “recoil factor”
BOLTZMANN
(1 +p)/(
RELAXATION
1 -,u),
321
EQUATION
evidently
the velocity
V* is
just -V. Elementary analysis can now be performed, in which we start from the master equation and use transition probabilities given by ordinary collision mechanics (i.e. conservation of momentum and of energy in a collision), exactly as in the contrasting Brownian movement limit (M/m > 1). We do not reproduce the details here, but merely state the result that the master equation (equation (3)) can be written for ,u < 1:
wv,
t)
= -YIVj P(V,t) +
at f
1J&
(:‘)“f(2V
-
y) P (V*
+ -&-)
e--myzi8kT dy.
(27)
--oo The only approximation made in deriving this equation from the master equation is in neglect of terms in e -MVZ~2rkT. Unless we wish to deal with extremely low piston velocities (i.e. very much less than thermal velocities) this approximation will be entirely justified because so long as &IIVs 2 kT, e-MV=12pkT < -lip which can then be entirely neglected for expansions in We fioz~~ersof the small parameter ,u. Apart from this, we repeat, equation (27) is exact. The integro-differential equation (27) can be shown to be satisfied by the Maxwell-Boltzmann distribution (a pleasing result ) : i.e.
am.(v)
M
= 0, where Peg*(V) =
e-
MV212kT
2nkT
at
Next, if we expand the integral to the lowest order in p, we obtain equation (25). To the next highest order we have
wv,
___ at
4
= Y/V/l -p(v)+p(-v)+4pp(-v)--hi [
( + p
i+-gvy kT)
;&-
I”‘(-
VP’(-v)+ V)].
(28)
The form of the zeroth order equation (25) suggests that we should define two functions R(V) and Q(V) (which need only be defined for positive V) : R(V)
= P(V)
- P(-V)
Q(v) =P(V) +P(-VI
I
(29)
Broadly speaking, the approach of R(V) to zero will give the approach to momentum equilibrium. On the other hand, the way in which Q(V) tends to the Boltzmann distribution will now give the approach of the piston to
322
A. M. GUfiNAULT
AND
D. K.
C. MACDONALD
energetic equilibrium. We first consider R(V), and also for convenience write T(V) = l/vjVl (cf. eqn. (26)). To lowest order we have:
aqv, 4 at
= -2R(V,
t)/T(v)
i.e. R(V,
t) = R(V,
0) e-2t1T(v).
(30)
This tells us that momentum is equilibrated (i.e. R(V) = 0) after a time of order T(V) - i.e. after one or two collisions only. Inclusion of the next order terms will modify equation (30) a little, but not substantially. We turn now to Q(V). T o zeroth order, as already pointed out, we have @(V, t)/at = 0 so that energetic equilibrium is never reached. To first order in p we obtain:
T(V)
aQV'> 4
VQ’ +
at
2p j!
Q”
(31)
where T(V) is given still by eqn. (26). This equation is surprisingly*) similar in some respects to the familiar Fokker-Planck equation (cf. eqn. (16)) which if written in similar analytical form, would read: 7
wv,
4
= 2/_&P+ 2p VP’ + 2fA $
at
P”.
(32)
However it is analytically different in the essential respect of the form of T(V). On the one hand our equation (31) reduces to
w a -=at*
u2Q +
au
aQ
(33)
~4,x
where we have substituted t* = 2,uv(kT/M)tt, u = (M/kT)*V. Planck equation on the other hand reduces under similar (t* = 2,&/T, U = (M/kT)*V) to:
aP -=-
at*
a
The Fokkersubstitutions
ap1 1up+xJ
a24
(34
‘.
Equation (34) can be solved analytically for given initial conditions (cf. e.g. eqn. (20) and fig. la), and we know that after time t* > 1 we obtain the Boltzmann distribution, P cc e-*““, from arty initial conditions; i.e. the characteristic relaxation time is of order unity in units of t*. On the other hand the Q-equation (equation (33)) cannot unfortunately to our knowledge be solved analytically* *) but we expect the general look *) When we bear in nlind that we are involved here in a large-jump statistical to a smnll-jump model in the Brownian movement (Fokker-Planck) limit. **) We are grateful to Dr. V. Linis (University of Technology) for their advice on this question.
of Ottawa)
and Dr. A. Erdelyi
model, as opposed (California
Institute
SCATTERING
of the solution
AND
THE
to be similar
BOLTZMANN
RELAXATION
EQUATION
to that of eqn. (34) on physical
grounds.
323 The
Boltzmann distribution satisfies eqn. (33) as well as eqn. (34) in equilibrium. Hence we expect the Boltzmann distribution (Q cc eebU2) to be reached with In real time, the rerelaxation time T* - 1 from the initial distribution. laxation time is 7*/2v,u (M/kT)* - l/,~ T( Vtlth.), where T(V,,) is defined as usual from eqn. (26), with I’ = I’,, - (M/M)*, a typical thermal velocity consequently of the piston. energetic equilibrium is only reached after, roughly, (l/p) collisions of the piston with gas molecules whereas momentum equilibrium was reached after one or two collisions. Hence we see that for this light Rayleigh piston problem, the BR equation irt general would be quite inadequate to describe the relaxation of P(V) towards Pea.(V) for the following reasons. (i) The BR equation has a single relaxation time for momentum and energetic equilibrium. Here we have two different relaxation times, which differ by a large factor l/,u. (ii) The approach to energetic equilibrium, given essentially by Q(V), is qualitatively similar to that described by the Fokker-Planck equation; and we have already seen in the preceding section that the BR equation gives a form of approach to Peg.(V) which is quite foreign to the FP form of approach. Received
23-4-63
REFERENCES 1) Wilson, A. H., Theory of Metals (Cambridge University Press) (1936, 2nd ed. 1953). 2) Uhlenbeck, G. E. and Wang Chan g, C. S., Brussels Conference on “Transport Processes in Statistical Mechanics” (1956) (Interscience Publishers 1958) p. 611. See also Technical report 2457-3-T, Engineering Research Institute, Ann Arbor (1956). 3) Alkemade, C. T. J., Van Kampen, N. G. and MacDonald, D. K. C., Proc. Roy. Sot. A271 (1963) 449. 4) Rayleigh, Lord, Phil. Msg. 32 (1891) 424. 5) Van Kampen, N. G., Canad. J. Phys. 39 (1961) 551. 6) MacDonald, D. K. C., Noise and Fluctuations (John Wiley) (1962).
Physica
A. M.
MacDonald,
30
309-323
D. K. C.
1964
SCATTERING
AND THE BOLTZMANN EQUATION
by A. M. GUENAULT
RELAXATION
D. K. C. MACDONALD
and
*)
Division of Pure Physics, National Research Council Ottawa, Canada
Synopsis The “Boltzmann able
popularity,
change
possible
as specific previous
examples,
assumptions, equation
function
solving
statistical
we cannot relaxation
is valid
study
-(P
problems.
piston
expect
scattering
to be represented
Moreover
when,
assumption
rate
function, We
of
P,
consider
and also,
as we are not aware
of it. The results show that, without
consistent
the
here how far
model in both the “Brownian
in some detail
and
has consider-
We examine
equation.
the only
Peq.)/T,
describing
distribution
transport
this latter
-
for
for an arbitrary
of the Rayleigh
limits,
=
physicists,
due to scattering.
steady-state
the behaviour
and “Lorentz” detailed
Boltzmann
for
kP/at],,,,t. metal
can have general validity utility
movement”
strictly
equation”, amongst
in time of a distribution
this equation its
relaxation particularly
if, the
is that
rather
at all generally Boltzmann
7, the “relaxation
of a
unusual by the
relaxation time”,
is
constant.
1. Introduction. In non-equilibrium statistical mechanics, but particularly in the field of electron transport theory in metals and alloys, it is sometimes suggested that it is a good approximation to write
p _ pea.
--I 8P
at scatt. =-
(1)
7
for the I ate of change with time of the distribution function due to scattering. In this equation Peq. denotes specifically the distribution function in true thermal equilibrium, i.e. usually the Maxwell-Boltzmann for gas molecules, and the Fermi-Dirac distribution for electrons in metals; it is assumed that T, the so-called “relaxation time”, may itself be a function of the velocity, I’, or energy, E, of the particles (i.e. 7 = T(V) or T(E)) but is otherwise to be treated as a constant. Equation (1) as it stands may be solved quite generally for arty arbitrary initial distribution, however caused, to yield P(V, t) = P( V’, 0) ectit + Peq.( V) (1 -
e-t/r)
@a)
or
P(V, t) *j
Dr. MacDonald
Peq.( V) = [P(V, 0) -
died July
28, 1962. -
309
-
Peqv)]
e-t/t.
(2b)
310
A. M. GUl?NAULT
AND
D. K. C. MACDONALD
This particular form of equation (1) for the change in P due to scattering (of one kind or another) seems to have different terminology depending on the field of work involved. For the sake of definiteness let us in what follows here call an equation of the type indicated in equation (1) a “BR equation” (i.e. a Boltzmann relaxation equation). It is true that for certain specific types of perturbation (e.g. an electric field applied to a typical metal at “high” temperatures (T 2 0); cf. e.g. Wilsonl) a steady state is maintained in which the scattering to a first approximation can be considered to obey equation (l), but we find ourselves asking the following question. Is there, in fact, a model of scattering which leads to equation (1) for the return to equilibrium for entirely arbitrary initial disturbance of the distribution function? Starting from the so-called master equation we can give what we believe is a rather general answer to this question by showing that the BR equation will follow for an arbitrary initial disturbance if the transition probabilities can be assumed to follow an unusual and somewhat remarkable form. We have then proceeded further to examine in particular the relaxation due to scattering of the Rayleigh piston model in two limiting situations, namely the familiar Brownian movement approximation governed by the FokkerPlanck equation, and the somewhat less familiar Lorentz limit (this latter being examined in some detail). We conclude that a BR equation will not be found for the return to equilibrium due to scattering*) on either of these models. Consequently it is our opinion that it is improbable that any normal scattering mechanism would yield equation (1) for an airbtrary perturbation. It seems worthwhile to draw attention to this situation, and to invite possible rebuttal, because, in metal physics at least, it appears to be commonly argued that the BR equation should be a good first approximation in many scattering situations for all types of perturbation; indeed “hand-waving” arguments (which we shall outline and criticize below) are sometimes adduced to support this thesis. 2. The general validity of the BR equation. We take as our starting point in the argument the “master equation” for the rate of change in time of a probability distribution function P( I’/,t) :
“pY)l&t. =S
[-W(V,
For brevity *)
Uhlenbeck
we consider and Wang
V’) P(V, t) + W(V’,
V) P(V’,
the simple case where P(V, t) etc.
Chang2)
have
considered
the problem
in part
t)]dV.
(3)
are functions
by asking
whether
the
Kayleigh
piston model, under an external alternating field, in what they call “the strong-coupling i.e. the Lorentz limit (vi& infra), leads to the BR equation as a good approximation. approximation” They them.
conclude
that it is not an adequate
approximation
in the particular
problem
considered
by
SCATTERING
AND
THE
BOLTZMANN
RELAXATION
EQUATION
311
only of (a single scalar) velocity, V, and of time, t; the results of this section could however be readily generalized to include e.g. a three-dimensional velocity, V. There seem to be two principal assumptions in writing down an equation such as equation (3): (i) that the system can be adequately represented by a probability distribution, P(V, t), dependent only on the present state of the system (i.e. the Markoffian assumption) and (ii) that transition probabilities exist; (W(V, V’)dV’ is here the probability per unit time for a system in state I/ to make a transition into a range dV’ at V’) *). Our elders and betters tell us that these two assumptions should be valid for a wide range of scattering problems, so that we expect the master equation to be of rather general applicability. We shall now indicate the sort of “hand-waving” argument that is often presented to justify replacing the master equation (equation (3)) by the much simpler BR equation:
, aw, t)1 at
P(V, t) -
=-
scatt.
Pqv)
m
where Peq.( V) is the equilibrium probability distribution function and T(V) is a “relaxation time”. The argument might run typically as follows: We note that the first half of equation (3) that involving transitions out of state V) can be written immediately as -P(V, t)/T(V) if we define -
1 T(V)
=
It is then said that the “principle i.e.
W(V,
s
W(V,
V’)dV’.
of detailed
V’)Peq.(V)
enables the second half of equation
balance”,
= W(Vf,
Vp.(V’),
(3) to be written as
P”q.( V) 1 W( V, V’)
;Jqy;;f)
dV’.
The argument then runs that, at least for small deviations of P(V, t) from equilibrium, the factor P(V’, t)/Peq*(V’) in equation (6) will have negligible effect on the integral and so, using (4) again, we obtain the BR equation (1). This final step is appealing because it is often in fact a valid procedure (e.g. in the “drift terms” in transport theory - see below) to replace P(V) by Peg.(V) wherever possible, for small departures of P from Peq.. We shall now criticize the argument and analyze more precisely the conditions under
*) We should perhaps remark that if we are dealing metals and alloys, then in order to satisfy Fermi-Dirac include
specifically
a factor
(1 -
flP( V’)),
where
specifically with problems of electrons m statistics it is necessary for W(V, V’) to
l/,9 is the appropriate
density
of states.
312
A. M. GUdNAULT
which the master fined it.
equation
AND
D. K. C. MACDONALD
will reduce to the BR equation,
as we have de-
First let us note that we used the principle of detailed balance, equation (5). It is worth noticing that this principle is not of entirely general validity; however a general principle obeyed by mechanical systems is that of “microscopic reversibility”. On applying this to the present problem we obtain not equation (5), but: W(V,
V’)PqV)
= W(-VI’,
-V)P=(-V’).
(7)
It is only in symmetrical problems that this reduces further to “detailed balancing”, equation (5), (cf. e.g. Alkemade, van Kampen and MacDonalds) for study of an asymmetric problem in which (7) but not (5) is to be used). Hence we can only continue with the above argument for symmetrical systems, as we shall now do - this will not be a too serious restriction in practice, but must affect any claim of general validity for the BR equation. Secondly, we can rapidly expose the dubious nature of the argument which leads to replacement of the expression of equation (6) by Peq.( V)lT( V) in the master equation. Let us write: P(V, Using equations
+ Pl(V,
t).
(4) and (6), we may now write the master
ap(v, 2)
at
t) = P=(V)
Pl(V>
+,)
zz-
1 soatt.
t)
+ j-V’>
(8) equation:
V)Pl(V’, WV/‘.
(9)
The BR equation includes only the first term on the right hand side of argument, which in its conequation (9) - and the earlier “hand-waving” text seemed quite appealing, is now seen to amount to saying that “as Pl(V, t) is small, the integral in equation (9) can be neglected” *). This is palpably untrue in general, as clearly both terms on the right of equation (9) involve PI linearly, and consequently the smallness or otherwise of PI is irrelevant in this respect. We are therefore left with our original question - can the BR equation ever be a general expression for scattering? That is to say, we wish to ask what form W(V, V’) must take in order that the BR equation may follow from the master equation for an arbitrary perturbation Pl(V, t) from equilibrium. Restricting ourselves to symmetrical problems, we find equation (9) *) cf. e.g. Wilson ‘) in the first edition who states (p. 19): “Since the distribution function, f, never differs very much from the equilibrium distribution fo, the net number of electrons ejected per unit time (from dV) will be proportional to (f - fo), so that we can write
a [ at
1
call. =
The second edition (1953) is much more guarded
f - f, - --. T(V)
on the matter
(see p. 6).
SCATTERING
a convenient
AND
statement
THE
BOLTZMANN
RELAXATION
of the master equation,
EQUATION
313
and we can answer the
question as follows. (a) If equation (9) is to reduce to the BR form, we must have:
s
W(V’, V)P1(V’, t)dV’ = il
where 1 is a constant,
ply’,
t)
m
’
(10)
and hence
ap(v, t) = at 1 scatt.
(1 -
il) ‘:$;
t, .
(11)
(b) As the probability distribution functions are normalized functions, we have /P(V, t)dV = constant, so that in particular SPl(V, t)dV = 0. Furthermore, if we integrate equation (11) with respect to V, the left-hand side must vanish, and we have: O=
-(I
-A)
/p:;;;t)
dV.
(14
Remembering now that JPl(V’, t)dV = 0, but that otherwise we are assuming Pl(V, t) to be entirely arbitrary, the only possible conclusion from equation (12) is that T(V) is a constant, independent of V. (c) Equation (10) can now be written:
s
w(V’,
V) Pi(V’,
t) dV’ = ;Pl(V,
t).
The appropriate general solution of this equation for W(V’, V), remembering that PI(V) is to be an arbitrary function, satisfying onlyJP1(V, t)dV = 0, is readily shown to be W(V, V’) = f(V’)
+ +3(V
-
V’),
where 6(x) is the Dirac delta-function. However, we may reject the deltafunction solption ,as it does not contribute to aP/at],,,,,.; i.e. for our purposes, we set 1 = 0. Hence we see that W(V, V’) is to be a function of V’ only. (d) The principle of detailed balance, eqn. (5), tells us further that we must have: W(V, V’) = A Peq.(V’),
(13)
where A is a constant, and this completes our analysis. Our conclusions are thus that if the BR equation is to be valid for relaxatiom from an arbitrary initial distribution function, then: (i) the transition probabilities must necessarily have the somewhat re-
314
A. M. GUkNAULT
AND
D. K. C. MACDONALD
markable form W(V, V’) = A Pq.(V’), where A is independent of V and of v’. How closely the transition probabilities derived from any physical model might be made to approach to this we can only guess at present, but it is perhaps worth nothing that this form of W(V, V’) would evidently involve a strong component of “large-jump” scattering (i.e. [lr’ - VI “large” - or equivalently large-angle scattering in a three- dimensional mo-
del) . (ii) it also follows that the BR equation
aP(v, t) at
can only strictly
take the form:
P(V, t)-Peq*(V)
_-
7
1 scatt.
where T is a constant, independent of V. These general results seemed to us of sufficient interest to lead us to examine the analysis in some detail of a particular model subject to relaxation by scattering. In sections 4 and 5 below we shall discuss the one-dimensional “Rayleigh piston” model (or Rayleigh particle model) introduced by Lord RayleighJ) “in order to bring out fundamental statistical questions, unencumbered with other difficulties . . .“. Before doing so, however, let us consider briefly the solution of the steady-state transport problem. 3. Solution of transport problems. In the above section we discussed the analytical form of the change in time of P(V, t) due to scattering, i.e. In principle such an expression tells us the relaxation beap(v/‘, t)iati,,,,,.. haviour of a system described by P(V, t) towards its equilibrium distribution reached in the absence of any external forces as t + co. Very frequently (and particularly in metals theory), we are not interested in such general information, but rather we require a steady-state solution of an equation :
qv,
at
ap(v, 4
4 1 drift.
which we may call a (Boltzmann) written :
+z*vax
F ap(v/‘)ax MTF
-
ax
=
s
+at
=o 1scatt.
transport epation.
[-W(V,
Usually,
(14) this may be
V’)P(V) + W(V’, V)P(V’)]dV’
(15)
The two terms on the left represent drift terms in aP/at, arising from a force, F, and some gradient dX/dx. We require to solve equation (15) for P(V). Usually the forces and gradients are small, and the analysis shows that P(V) can be replaced by Peq.(V) in the drift term, although not of course in the scattering term (cf. previous section). If now the scattering
SCATTERING
term
AND
THE
BOLTZMANN
(the right hand side of equation
RELAXATION
315
EQUATION
by a BR ex-
(15)) can be replaced
pression, i.e. by -(P(V) - Peq.( V))/T( I’), then the solution for P(V) and hence for transport currents can be written down immediately. As we have seen, in general the collision term cannot be simplified in this dramatic way*); however, as we shall exemplify below, the actual steady-state solution of eqn. (15) often does have the same analytical appearance as the steady-state solution which would arise using a BR collision term with however an “effective relaxation time” a@ro$wiate to the specific kind of perturbation acting. We would suggest that this rather surprising situation has tended in the past to give a somewhat spurious air of general authority to the BR equation**). But let us now examine detailed models to try to understand this further. 4. Model I. Brownian movement limit of the Rayleigh piston model. As a first example, we consider the model discussed for instance by van Kampens), that of a Rayleigh piston (mass M, velocity V) which is bombarded by gas molecules (mass m < 1M). As the mass ratio m/M is small, this is a “small jump” model (i.e. W(V, V’) = 0 unless V= I”), and Alkemade, van Kampen and MacDonalds) have shown that the master equation (3) can be expanded consistently in powers of (m/M). The lowest order approximation, which is all we require here, gives the Fokker-Planck equation (or, briefly, the FP equation) :
qv/‘, 4 at
--a
[
1 scatt.
g
(VP(V)) +
$ g-J
(16)
where the parameter, a, depends on the details of the model. This is quite unlike the simple BR equation, and a general solution for equation (16) with arbitrary initial conditions is not straightforward; however in equilibrium (setting
1.h.s. = 0), equation
(16) yields directly
the gaussian
distribution,
i.e. Peq.(V) = 1/M/2nkT e- MV212kT.Let us look at the situation in more detail : (i) To deal with the relaxation of the piston, we only need to compute d/dt
&
-
a
say
(17)
where we have set r’ = l/u. *) The region
T <
8 where the phonon-electron
interaction
is to be treated
quantally,
scattering as is well known does not concern us here. **) Rroadly speaking this must arise from the fact that the perturbing forces and/or are supposed modest, and that the appropriate approximations to the master equation various models turn out to be linear in P(V) (and its derivatives). a rather drastic form of linearization of the collision integral.
The BR equation
and
the
gradients based on
then represents
316
A. M. GUkNAULT
At first sight this might the BR equation, namely
AND
D.
K. C. MACDONALD
seem to be identical
with the result given by
(18) However,
we recall that in equation -
whereas in the Fokker-Planck
1 7
=
s
(18) we have from equation
W(V, V’)dV’,
equation,
(4) : (4)
we have:
(19) (cf. e.g. Alkemade, leigh piston model
van
Kampen
and MacDonalds);
and for the Ray-
_4~GJ/5 (where v = number of gas molecules per unit length cf. Alkemade cu., appendix II). In the same notation, equation (4) gives in contrast 1 _N 7
of piston
cylinder;
kT --v 1/- m
(i.e. essentially the total no. of collisions per second), and hence we see that T’/T N M/m, > 1, i.e. r’ and 7 are of entirely different orders of magnitude. We may readily understand the reason for this difference in the following terms. Physically one would say that the BR equation assumes that each collision brings the system essentially into its equilibrium state (cf. also the form of W( I/, V’) derived in section 2 above). The FP equation, on the other hand, recognizes that we have a small-jump model and that -(M/m) collisions are required to dissipate its momentum. Thus although the BR equation would give a superficially correct answer for the decay of
SCATTERING
AND
THE
BOLTZMANN
well known (cf. e.g. MacDonaldc), P(V, t) =
1/
RELAXATION
317
EQUATION
namely
A4
2nkT( 1 -e-2at)
exp [
--M(V-Vo e-nt)2 2kT( 1 -e-2at)
1’ (20)
This is illustrated in fig. 1a. The solution of the BR equation has alreay been given in the introduction (eqn. (2)), and its behaviour in the present problem is illustrated in fig. lb. P(v,i P(v,t
Fig. la. Rough sketch of the solution of the Fokker-Planck equation, starting from a delta-function at Vo at time t = 0. Curve A (dotted) is for t N 0, B (dashed) for an intermediate time and curve C (full) fort + CO. Fig. lb. Rough sketch of the solution of the Boltzmann relaxation equation under conditions similar to those for fig. la.
For a system whose scattering is governed by the BR equation we see that the delta function component decays in size exponentially but remains fixed in position at Ve, while at the same time the component of the (equilibrium) gaussian distribution grows from nothing towards its full amplitude. In contrast, equation (20) (Fig. la) shows the actual mode of approach to equilibrium for our present model; namely the initial delta-function broadens and simultaneously shifts its mean position until it ultimately becomes =
the stationary
gaussian
equilibrium
distribution
centred
about
V
(Ei) Thirdly, I‘t IS ’ instructive to consider the solution of transport problems in this Brownian movement limit (cf. section 3 above). When a force F is applied to the piston, the transport equation will read (cf. eqns. (15) and (16))
(21) We can readily proceed to solve this equation Integrating eqn (21) we obtain VP+---
quite generally
kT
aP
nf
av > ’
as follows.
(22)
318
A. M. GUtiNAULT
(The constant of integration solution for F = 0).
AND
D. K. C. MACDONALD
is zero, as we know that P = Peq.(V)
is to be a
for P which is readily
Equation (22) is now a linear first order equation solved to give
For small fields, F, this can be written
Pb) (We note that this is equivalent to writing Peg.(V) instead of P(V) in the left-hand side of eqn. (21) or (22).) In contrast, the BR scattering expression leads to a transport equation: F ap ------_LM.4 av
p -
pea. 7
which then leads to a solution which is quite unlike the above, namely:
P(V)
s
eMV’/FT
=
e-
MV”=/ZkT
dj,T’. (244
--oo
However, for small fields, performing this expression simplifies to : P(V)
=
two successive
integrations
by parts,
W)
which is analytically identical in form to (23b) (and can also be obtained by direct approximation of P(V) by Pea.(V) in the “drift” term). Despite this analytic similarity for small fields, we note that exactly the same remarks are to be made as have been made under section (i) of the present discussion, where we analysed (V) ; namely for the Rayleigh piston problem, 1/CC( = T’) and 7 are of quite different orders of magnitude, so that the superficially correct result given by the BR approach is illusory when it comes to a calculation of the relaxation time (or equivalently a transport current) from first principles. We should note, however, that if (a) we are interested only in steady-state transport currents, and if (b) we had no method of calculating the relaxation time a priori, then the BR equation would suffice for these purposes despite its lack of general foundation. We repeat however that this is surely no guarantee of its validity, or even utility, in any other application.
SCATTERING
5. Model II.
AND
Lore&
section we discussed
THE
BOLTZMANN
RELAXATION
EQUATION
315
limit of the Rayleigh #&on model. In the preceding of a heavy Rayleigh
the behaviour
piston having mass
M much greater than that of the colliding gas molecules . The problem was then susceptible to analysis because expansion methods could be used with effectively m/M as a smallness parameter. In the opposite limit of M Q m, i.e. an extremely light piston bombarded by heavy molecules*), we shall show that expansion methods can again be used. This so-called Lorentz limit might be expected to bear some relation to that of the scattering of conduction electrons by very heavy ions in a metal*). To gain a little insight into the problem we first consider the simplest situation, that of a piston whose mass is effectively zero free to move in only the surrounding gas molecules have one dimension**) (or e q uivalently effectively infinite mass). The master equation for i3P(V)/i%],,,t, now takes a very simple form as the only transitions which can take place are from state V to -V or from -V to V, both with average frequency Y(V[, where v is the number of gas molecules per unit length of the cylinder, as before. Hence we have P(V)-P(-V)
WV _=-
at
(25)
r(V)
where 1 4V) It is particularly
instructive
= VII/‘].
to note that
WV) + P(-V)) =
o
at so that (P(V)
+ P(-
V)) remains constant
in time.
As an example of the behaviour of equation for P(V, t) for all t > 0, starting from an initial Vo, namely : P(V,t)
=0
(25), we give the solution delta-function P(Vo, 0) at
if V#*Vo
P(V0, t) = 4P(Vo, 0) [ 1 $ e-w Vo)1 1 q--o,
t) = iP(V,,
0) [I
_
,-t/dVO)
(2-W I
*) Actually, the strictly one-dimensional piston model in the ultimate limit of M/m=0 cannot be wholly satisfactory as a model for scattering in practice. The relaxation in time of the piston from some initially perturbed state (i.e. the return to momentum equilibrium) should be quite realistic, but the usually analogous model in the limit. **)
As a rather
vironment
bizarre
linear steady-state example
of “super-uranium”
transport
one might perhaps
gas molecules.
or “flow” envisage
solution
does not exist
for this
a piston of solid sHe in a dilute en-
320
A. M. GUkNAULT
AND
D. K. C. MACDONALD
>
Figure 2 illustrates this solution. Let us note in particular that: (i) momentz~~ equilibrium is established with a time constant T( VO); i.e. (v> = v. e-t/T(Vo), in a manner similar to that which one would obtain from a BR equation. P1v.t)
;‘i IlA :: ii II”
C C
Fig. 2. Rough piston
sketch
(equation
As in fig. C (full) fort
1, curves
of the solution
(25a)),
starting
A (dotted)
from
P( I/, t) for the an initial
Lorentz
delta-function
are for t = 0, B (dashed)
limit
of the
Rayleigh
at VO at time
for an intermediate
t = 0.
time
and
--f 00.
(ii) energetic equilibrium, strictly speaking, is never reached (in the limit of M/m + 0). Collisions are very effective in transferring piston momentum to the gas molecules, but contrariwise are most ineffective in altering the energy of the piston. In fact any symmetrical distribution for P(V) will satisfy equation (25), in direct contrast to the more usual situation (given for example by the BR equation, the FP equation or the Q equation discussed below) where, as t + co, Peq.(V) is the only solution. It is therefore clear from the solution of equation (25) that (in contrast to a BR scattering model) different properties may relax at different rates, and that we have here no single zLniversaZ relaxation time. These general features would still be maintained by a three-dimensional model in which only elastic collisions are possible. The momentum of a light “Rayleigh particle” would presumably have its direction randomized over one or two collisions, whereas its energy would remain essentially unchanged even after a large number of collisions. This behaviour is quite unlike that described by the BR equation, in which momentum and energy equilibrium are restored with the same relaxation time, indeed by the same collisions. We now study the light Rayleigh piston model in greater detail, considering expansion of the master equation to first order in ,u(= M/m), i.e. when we allow that M/m is small, but not now negligible. We cannot expand the transition probabilities in moments of the jump size (as is done for the heavy Rayleigh piston model) since we have now a “large-jump” model. However we recall that each collision almost reverses the velocity V, and we can use this to establish an expansion procedure by expressing the velocity V’ in the master equation as (I/‘*+ y) where V* = - V( 1 +,u) /(1 -,u) ;
SCATTERING
AND
THE
i.e., apart from a “recoil factor”
BOLTZMANN
(1 +p)/(
RELAXATION
1 -,u),
321
EQUATION
evidently
the velocity
V* is
just -V. Elementary analysis can now be performed, in which we start from the master equation and use transition probabilities given by ordinary collision mechanics (i.e. conservation of momentum and of energy in a collision), exactly as in the contrasting Brownian movement limit (M/m > 1). We do not reproduce the details here, but merely state the result that the master equation (equation (3)) can be written for ,u < 1:
wv,
t)
= -YIVj P(V,t) +
at f
1J&
(:‘)“f(2V
-
y) P (V*
+ -&-)
e--myzi8kT dy.
(27)
--oo The only approximation made in deriving this equation from the master equation is in neglect of terms in e -MVZ~2rkT. Unless we wish to deal with extremely low piston velocities (i.e. very much less than thermal velocities) this approximation will be entirely justified because so long as &IIVs 2 kT, e-MV=12pkT < -lip which can then be entirely neglected for expansions in We fioz~~ersof the small parameter ,u. Apart from this, we repeat, equation (27) is exact. The integro-differential equation (27) can be shown to be satisfied by the Maxwell-Boltzmann distribution (a pleasing result ) : i.e.
am.(v)
M
= 0, where Peg*(V) =
e-
MV212kT
2nkT
at
Next, if we expand the integral to the lowest order in p, we obtain equation (25). To the next highest order we have
wv,
___ at
4
= Y/V/l -p(v)+p(-v)+4pp(-v)--hi [
( + p
i+-gvy kT)
;&-
I”‘(-
VP’(-v)+ V)].
(28)
The form of the zeroth order equation (25) suggests that we should define two functions R(V) and Q(V) (which need only be defined for positive V) : R(V)
= P(V)
- P(-V)
Q(v) =P(V) +P(-VI
I
(29)
Broadly speaking, the approach of R(V) to zero will give the approach to momentum equilibrium. On the other hand, the way in which Q(V) tends to the Boltzmann distribution will now give the approach of the piston to
322
A. M. GUfiNAULT
AND
D. K.
C. MACDONALD
energetic equilibrium. We first consider R(V), and also for convenience write T(V) = l/vjVl (cf. eqn. (26)). To lowest order we have:
aqv, 4 at
= -2R(V,
t)/T(v)
i.e. R(V,
t) = R(V,
0) e-2t1T(v).
(30)
This tells us that momentum is equilibrated (i.e. R(V) = 0) after a time of order T(V) - i.e. after one or two collisions only. Inclusion of the next order terms will modify equation (30) a little, but not substantially. We turn now to Q(V). T o zeroth order, as already pointed out, we have @(V, t)/at = 0 so that energetic equilibrium is never reached. To first order in p we obtain:
T(V)
aQV'> 4
VQ’ +
at
2p j!
Q”
(31)
where T(V) is given still by eqn. (26). This equation is surprisingly*) similar in some respects to the familiar Fokker-Planck equation (cf. eqn. (16)) which if written in similar analytical form, would read: 7
wv,
4
= 2/_&P+ 2p VP’ + 2fA $
at
P”.
(32)
However it is analytically different in the essential respect of the form of T(V). On the one hand our equation (31) reduces to
w a -=at*
u2Q +
au
aQ
(33)
~4,x
where we have substituted t* = 2,uv(kT/M)tt, u = (M/kT)*V. Planck equation on the other hand reduces under similar (t* = 2,&/T, U = (M/kT)*V) to:
aP -=-
at*
a
The Fokkersubstitutions
ap1 1up+xJ
a24
(34
‘.
Equation (34) can be solved analytically for given initial conditions (cf. e.g. eqn. (20) and fig. la), and we know that after time t* > 1 we obtain the Boltzmann distribution, P cc e-*““, from arty initial conditions; i.e. the characteristic relaxation time is of order unity in units of t*. On the other hand the Q-equation (equation (33)) cannot unfortunately to our knowledge be solved analytically* *) but we expect the general look *) When we bear in nlind that we are involved here in a large-jump statistical to a smnll-jump model in the Brownian movement (Fokker-Planck) limit. **) We are grateful to Dr. V. Linis (University of Technology) for their advice on this question.
of Ottawa)
and Dr. A. Erdelyi
model, as opposed (California
Institute
SCATTERING
of the solution
AND
THE
to be similar
BOLTZMANN
RELAXATION
EQUATION
to that of eqn. (34) on physical
grounds.
323 The
Boltzmann distribution satisfies eqn. (33) as well as eqn. (34) in equilibrium. Hence we expect the Boltzmann distribution (Q cc eebU2) to be reached with In real time, the rerelaxation time T* - 1 from the initial distribution. laxation time is 7*/2v,u (M/kT)* - l/,~ T( Vtlth.), where T(V,,) is defined as usual from eqn. (26), with I’ = I’,, - (M/M)*, a typical thermal velocity consequently of the piston. energetic equilibrium is only reached after, roughly, (l/p) collisions of the piston with gas molecules whereas momentum equilibrium was reached after one or two collisions. Hence we see that for this light Rayleigh piston problem, the BR equation irt general would be quite inadequate to describe the relaxation of P(V) towards Pea.(V) for the following reasons. (i) The BR equation has a single relaxation time for momentum and energetic equilibrium. Here we have two different relaxation times, which differ by a large factor l/,u. (ii) The approach to energetic equilibrium, given essentially by Q(V), is qualitatively similar to that described by the Fokker-Planck equation; and we have already seen in the preceding section that the BR equation gives a form of approach to Peg.(V) which is quite foreign to the FP form of approach. Received
23-4-63
REFERENCES 1) Wilson, A. H., Theory of Metals (Cambridge University Press) (1936, 2nd ed. 1953). 2) Uhlenbeck, G. E. and Wang Chan g, C. S., Brussels Conference on “Transport Processes in Statistical Mechanics” (1956) (Interscience Publishers 1958) p. 611. See also Technical report 2457-3-T, Engineering Research Institute, Ann Arbor (1956). 3) Alkemade, C. T. J., Van Kampen, N. G. and MacDonald, D. K. C., Proc. Roy. Sot. A271 (1963) 449. 4) Rayleigh, Lord, Phil. Msg. 32 (1891) 424. 5) Van Kampen, N. G., Canad. J. Phys. 39 (1961) 551. 6) MacDonald, D. K. C., Noise and Fluctuations (John Wiley) (1962).