Statistical Mechancis

227 Pages·2011·8.368 MB·English

by  Carlo Cercignani (auth.), Prof. Giovanni Gallavotti (eds.)

Giovanni Gallavotti (Ed.)

Statistical Mechanics

Lectures given at a S ummer School o f the
Centro Internazionale Matematico Estivo (C.I.M.E.),
held in Bressanone (Bolzano), Italy,
June 21-27, 1976
C.I.M.E. Foundation
c/o Dipartimento di Matematica “U. Dini”
Viale Morgagni n. 67/a
50134 Firenze
Italy
[email protected]

ISBN 978-3-642-11107-5 e-ISBN: 978-3-642-11108-2
DOI:10.1007/978-3-642-11108-2
Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010
Reprint of the 1sted. C.I.M.E., Ed. Liguori, Napoli 1 976
With kind permission of C.I.M.E.

Printed on acid-free paper

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C mTRO INTERNAZIONALE MATEMATICO ESTIVO
(c.I.M.E.)
-
I Ciclo Bressanone dal 21 giugno a1 24 giugno 1976
STATISTICAL MECHANCIS
Coordinatore: Prof. Giovanni Gallavotti
P. Cartier: Theorie de la mesure. Introduction B la mecanique
statistique classique (Testo non pervenuto)
C. Cercignani: A sketch of the theory of the Boltzmann equation.-
O.E. Lanford: Qualitative and statistical theory of dissipative systems.-
E.H. Lieb: many particle Coulomb systems.-
B. Tirozzi: Report on renormalizationg roup.-
A. Wehrl: Basic properties of entropy in quantum mechanics.
P. CARTIER
Theorie de la misure
Introduction a la me canique statistique classique.
(Testo non ~ervenuto)
2 ENTRO INTERN AZIONALE MATEMATICO ESTIVO
(c.I.M.E.)
A SKETCH OF THE THEORY
OF THE BOLTZMANN EQUATION
C. CERCIGNANI
Istituto di Matematica, Politecnico d i Milano
Corso tenuto a Bressanone dal 21 giugno a1 24 giugno 1976
A Sketch of the Theory of the
Bolt zmann equation
Carlo Cercignani
Istituto di Matemeticz
Politecnico di Tfiilano
Milano, Italy
In this seminar, I shall briefly review the theop of the
Boltzmann equation. How the latter arises from the 1,iouville
equation has been discussed in 0. Lanf ord 's lectures.
We shall write the Boltzmann equztion in thin form
5
where t, 2, denote the time, space and velocjty vn-
4
riables, while is the distribution function, normalized
in such a way that
where M is the mass contained in the re~iono ver which the
intearation with respect to 2 extends.
Q({,{)
is the so called collision term, explicitly obtai-
nable from the following definition
$
where is an ausiliary velocity vector, V is the re-
y -I*,
lative speed, i.e.. the mamitude of the vector = f
-
#'=Q&~),$= ' 5;
etc., where and are releted
to -f and f, through the relations expressing conservation
of momentum and energy in a collision
where 2 is a unit vector, whose polar angles are 4 ma
Y
2 in R polar coordinate system with as polar axis.
Tnt~rr~tioenxt ends to ~ lvlal ues of and between 0
and rr/2 with reroect to 6, from 6 to 21 with respect
B(~v)
to 6 . Finally is related to the differential cross
q(qv
section by the relation
and m is the mass of a gas molecule. For further details
one should consult one of my books [I ,2].
Eq. (1 ) is valid for monatomic molecules and is more Ke-
-
nernl than the Boltzrnann equation considered by Lanford in
his lectures, because it is not restricted to rigid spheres,
but allows molecules with any differential cross section.
B(~v)
The case of rigid spheres is obtained by specializing
as follows
d
where is the sphere diameter. Another importa?t cF.se i s
offered by the so called 1~;axwellm olecules.
The latter are clzssical point masses interactinr with s
central force inversely proportional to the fifth Dower of
their mutual distence; as a consequence, it turns out that
B(6,V) V.
is independent of
It is clear that initial and boundary conditions are requi-
red in order to solve the Boltzmann equation, since the lattrr
f.
contain3 the time and space derivatives of The bound:- 4CO~?I-
ditions are particularly important since they describe the in-
teraction of the gas molecules with solid walls, but part) cu-
lar difficult to establish; the difficulties are due, mainly,
to our lack of bowledge of the structure of the surface lcy~r-.
of solid bodies and hence of the interaction potential of tlir
gas molecules with molecules of the solid. \'!hen a molecule i::-
pinces upon s surface, it is adsorbed and may form chemi cn.1
bonds, dissociate, become ionized or displace surface atoms.
The simplest possible model of the pas-surface interaction
is to assume that the molecules are specularly reflected at
the solid boundary. This assumption is extremely unrenli stlc
in eeneral and can be used only in particular caws. Tn pene-
9
/
ral, a molecule striking a surface at a velocity reflects
-
from it at a velocity which is strictly determined only
if the path of the molecule within a wbll can be computed exac-
tly. This computation is impossible because it depends upon a
great number of details, such as the locations and velocities
of all the molecules of the wall. Hence vze m&y only hope to
&' -45)
compute the probability density R thrt 2 no1 cc:r? e
-e ' -
-fi +dY
striking the surface with velocity between and re-
t .
emerces with velocity between - and f. +df If R is hown,
@
it is easy to write the boundary condition for
where g is the unit vector nonnal to the wall and we assumed
2,
the wall to be at rest (otherwise must be replaced
f-%,tL,% de not ins the wall ' s velocity. )
by
In ~eneral,R will be different at different points of the
w?11 and different times; the dependence on 5 and t is not shown
exnlicitly to make the equations shorter.
If the wall restitutes all the eas molecules (i.e. it is non-
porous md nonadsorbing 1, the tota 1 probability for an impinginp
aolecule to be re-emitted, with no matter what velocity is
-I
mity:
Kt!?!)
An obvious property of the kernel is that it cannot
assume negative values
,
R
Another basic property of the kernel which can be cal-
is
led the "reciprocity laww or the "detailed balance", written
as follows [I, 21 :
$&)
where is pmportlond to u*p[-.!!y(2KQJ,where To is
{(S)
the temperature of the well (in other wonls, j 5 E ;:F.Y-~?-
lian distribution for a pas at rest at the tennerature of tho
wall 1.
.-
-
We note a simple consequence of reciprocity; if tl-e Y --rln uj
4
distribution is the wall ilaxwellian md aess IS ror.t.rne6
at the wall according to Eq.(ll), then the distribution fimctior,
4
of the emerging molecules is again or, in other words, the
wall >Iaxwellian satisfies the boundary conditions. In fact, 1f
-8'
we inteprate Eq. (13) with respect t o and use Eq. (1 1 ) we
obtain
and this equation proves our statement, according to Eq. (10).
It is to be remarked that Eq. (14 1, although a consenuence of
Eq. (13) (when Eq. (11) holds) is less restrictive t h m Xq. (1')
and could be satisfied even if Eq. (13) failed.
As a consequence of the above properties, one can pmve [2]
the following remarkable theorem:
C(I)
Let be a strictly convex continuous function of its
2. R(k'd3)
arpment Then for any scatterinp kernel sati-
sfyine Eqs. (11 ), (121, (14), the follow in^ inequality holds
g
where is the wall Idaxwellian, 3 = $/fo and inte ~retion
extends to the full ranges of values of the components of'
4 Zq9J
the values of for being related to those for ).9C~
thero=ug h Eq. (1.6). Equality in Eq. (15) holds if and only if
R (EL5 )
almost everywhere, unless is proportionzl