Lecture 7 : Refs : D . Arovas notes ch

.

5 z . ,

kinetic theory& the Boltzmann equation

Reit ch . 12-14Kardar

,Stat . phys of particles .

Kinetic theory : Study of macroscopic properties of many- particle

Systems using microscopic equation , of motion.

Imagine we have a systemof N particles described by

the Hamiltonian

H = EYL'¥m + ukil ) + ,§ ucxix ;)

The probability of finding particlei at x ; within dei and

with momentum p ; within dpi is

g( × , , ... ,×n ; P , , ... ,pn ) dxidp , . . . . dxwdpnat

Remebr from the first lecture that if (there denoted by f )

satisfies the Lionville 's theorem ,which States

that g

behaves like an incompressible fluid , that is

dg dg- =o or a =

- { 9. H }dt

where the Poisson bracket { A ,B } isdefined by

{ A B}=§(°Fga9±pa - ftp.ofa )= . { Bint

This results in the continuity equation

0g with x=( 7. P )

of+ 17×194=0

v = 1 if ,p.

)

7-2

For macroscopic observables , g contains wayto much

information. often it will be sufficient to know only

one or few body densities . we are now particularlyinterested in the one . body

distribution

f trip ;t ) = § ( 8( Iiltl-FISIF ,.lt ' - F ) )

= N f,

.tt#ddxiddPiglr,xr....xn;F.Fu...Fn;t )

The one - body distributionfunction satisfies

) d3p fliip , E) = n( t.tl densityof particles

fd3r nc Fit ) = Ntotal number of particles

and

f ( F , F ; t ) dridp

is the number of particles in a volumeDF around

F and with momentum range d§ around I

If there is no potential v ( 51 , suchthat

translation symmetry is preserved ,the equilibrium

distribution is

f°lF .FI = pthfmksphe- %mkBT

with n=F

7-3

If we know f we can compute expectation values ofobservables that only depend

on F and F,

such as

particle current

%it) = Sdf

fliip;t )

IM

and energycurrentjdritl

= fdf flip ;tl Elp )Em

etc . we now want to obtain an equation that

allows us to calculate f systematically .

The Boltzmann equation

Consider the flow of phase space volume in the

absence of collisions between particles

- ttdtt ^

+ n Q"% drtdp

'

DFDF>

,

Lionville 's theorem States the flow in phase spaceis

Incompressible , i.e. , that dFdF=dF' df'

,

or

f ( Ftidt , ftp.dt , ttdt )dF' df'

= f( i. Fit ) didp

Taylor expandingwe then obtain

¥+o±÷+ :÷i=o

7-4

Keeping in mind that we are assuming there are no

collisions,

that is H = EIP ) + Ue×+( F ),

we have

E= dat = 9¥ = off

=icp)

is = -9¥ = . off = text

Therefore

I+

t.SI+ E. ftp. = 0

at

This onlydescribes the eftet of the motion without

ay

collision . To descnipe themotion realistically we need

to take into account theeffects of collisions .

This Is written in terms of acollision integral , which

at this point we simply write as

÷+o . :÷+÷o¥=l¥t . "Note : sometimes people write

tooth.

= - t.IE . E. of

for a streaming term ,and then

o¥=l¥hwH¥h. "

making explicit that there aretwo distinct contributions

to

the time evolution of f , streamingmotion and collisions .

7-5

Collisions

We assume collisions are elastic Naturally , there will be-

Some observables that are invariants ofthe motion and

need to be conserved by the collisions( such as energy ,

Momentum,

particle number ) .Consider an observable

given by the functionAir , F)

,

such that its expectation

value is

AH ) = fdidp Ali , F) f IF , F)

Taking thetime derivative

date = fdidf Air.FI#=sa=aeairiFlfiooota-EfIttEh

. ")=saiap[o÷. # f+F÷IIf +

Aloft. " ]

where we use thatit is independent of F

and E= jo .

Now

oo÷oE+o÷¥t=f÷9¥io÷. :# =Landwhere Ho is the Hamiltonian

without collisions . Since

{ A. Ho } = It gires the timeevolution between

collisions,

It

and we assumeA to be conserved by

that motion,

the

first two terms above must give zero

.

That rs

day = Sdidpaeriplfott). . "

7-6

Quantities that are conserved by the collisions have £¥←=o .

Examples : A=1 particle number

A =pmomentum

A= Elf ) energy.

Scatteringprocesses

What processescontribute to the collison integral ?

i ) single particle scattering #}Define the rate w(P→P

'

) such that

w( Pip'

)f( F,

F. E) df 'dF

is the rate at which particles within df of ( F.F) scatter

into within df'

of IF'ir ) at time t .

The differential cross section is then

do = w(Pi'P# delnivl

- 1

ii ) Two . particle scattering PIXXP'

x ×/ F

'

PT i

In similar way asin

the single particle scattering we definethe rate

WCPTI -7 PYFZ'

)fz( F. F, ;i ,Fa ;t1dEdF2

at which two particles within df , of P ,and dpzot Fz

scatter within DF ,'

ot F,' and dfz ' of Fz

'

at time t .

7-7

Here fz is the two . particle distribution function

fzlip ; F 'F'

;t)=§ <81 Iiltl - F) SIPIHI - F) SIXTH - FYJIPTHI - F

'

) >

and the differential cross section

do = ¥521152'

) IF 'dpi'

IT , - Jzl

Note that we have assumed the scattering Occurs locally .

This essentially amountsto treating each scattering process

independently and that the scatteringtakes place on length

scale small compared with meandistance between particles .

Equivalently , the scatteringtime to is the smallest time

scale in the problem andin particular is

much smaller

Mean free time T , i.e. tea< < I

.

This assumption

requires low density ,such as is

obtained in gases

but not in dense liquids.

If 5 is the totalcross section

,the mean free time is

defined such thatin the Volume

orx ✓ E there

#-) trotmeanehr,

• has units

c- ve -

should be on average one particle ,i. a.

,

or t =

nvooven= 1

E

For single particle scattering v is the average velocity ,while

7-8

for two particle scattering it 's the average relative velocity .

V = < Iv , - ✓ 217.

Let's focus on the two particle scattering from now on .

N±oe: The one particle scattering requiresa potential that

breaks translation invariance . This can for example happen

if there is disorderin the system . In a gas this might

be dust particles . In a relatively clean environmentthe

two particle processes maydominate . This is not

necessarily always true .

With the definition of the rates , weknow how to

wrote

down the collision integral :

#t )a , ,

= |dFsdIidPi WCP , 'Pz'

-7 Fizlfzlioir; Pir ;t )[

e. , ; # it ')- w( Fit → Bi Fi )fzlPNote that this depends on fz and not only f .

The differential

equation for f therefore doesnot close

.

we wouldin

Principle nowneed to write

down a differential equation

for fz but wewould find that it depends

on

fz the three - bodydistribution

,which In turn depends

on

f , etc . ( seeKardw for details ) . To close the set of

equation we new to makean approximation .

7- 9We therefore make the assumption of Molecular

chaos

( or Stoss Zal - Ausatz ) that

fz( F. R,

;F. Fz ) = f ( i ,E) FIF , PI ) .

Note that when we use this approximation in the

expression for ( ft ). , ,

it is used for the two particles

just betorc the collisiontakes place . If the motion

of

all the particles in the long finebetween the

collision

is chaotic ( hence the name ) we can safely

assume

that the particles are not correlatedbefore the

collision.

To make further progress , we usetime - reversal

invariance and parity symmetry ofthe microscopic

motion which together imply

wlpi ,I→ PYE'

) Iwl - Fi , - Fi - ' - Fi, -E) I wtpipz'

-7 I. E)

Using this weobtain ( surpassing

F in the notation )

( 0¥ ). , ,

=|dFzdpYdPi' WIFPI 'PYFil[ftp.lflpi ) - ftp.lflp . ) ]

7-10

The transition probability wlpipz- s PYPI' ) is highly constrained

by conservationlaws :

B. tpz = F.'

+ Ez'

conservation of momentum

¥L+ Man = HE + If'

conservation at energy

This means that the length of the relativemomentum

191=1152-In = pit P ,'

- zi , .E = (Pip + IPi )'

- 2,5,

!Fi= lpi'

- Fil

is conserved . It can however be rotated by a solid angle£

i. c .

Pz'

- PY = II. . Fire where bil=1 .

Thus, Knowing F, ,F, and a gives F,

'and PI via

15,'

=I Iftp.lpipili )Fi = ±( PT + In + lpi - Pilr )

Taken together this meansthat

Sdi, 'fdF .' wlpipi → F.

' PI ) ( ftp.ilflpil-ftplffpz ) ]

2 O_0 oo

= fdr or w -KI[ ftp.llflpil - ftp.lflpz ) ]where we used the definition of the differential Cross section .Altogether we get the Boltzmann equation ;

Get + ii. ftp.E.fpf-fdpzdpidpi' WIFPI ' ' PYPI 'l[ftp.lflpi ) - ftp.lflp . ) ]do

= fdfzfd's

Jr 15 -521 [ ftp.llflpil - ftp.lflpz ) ]

7-11

Boltzmann 's H - theorem

DefineHIT ) = fdidpflr.pt ) ln f C F. F. t )

If f satisfies the Boltzmann equation , then

¥I 0

dt

-Proof :

daft = fdidp ¥ ( lnfti ) = fdrtdp ¥lnf[

since fdoidpf = N independentof time .

f F, =f( By)

= Said ,i{[ i. of' - E. 0¥' ]enfI , I+

ftp.dhdfelfuillfifziff

, ] lnf,

}£

fz=f( Fz ) etc .

The first part with the streaming terms is Zero , ascan be

Checked by repeated partial integration( check ! )

since we are integrating over F , and iz we can change thelabels without changing the integral . Using ( schematically )

I = t(fdPTdPz( IIP , ,Pr ) + IIPZ , P , ) ) and lufitlnfi = lnlf.fr )

we then get

dzft =

tzfdr,dF ,

dpidrffrhi

- oil [ fifzi - f. fz ] lufifz )

Finally , using the symmetryof the cross section under time reversal

and parity we cantale I, pit ) I,

' PI without changing

7-12

the value of the integral . Again averaging overthe two we get

dk I-

= gfdridp , dpidrffrlv- oil [ fifz ' - fifz ][lufifz ) .hn/f,ify )]

dt

I 0

since ( y- × ) ( lug

- lnx ) 20

-

In equilibrium S = - KBH gives thermal entropy . H is a generalized -

to non equilibrium and reflects irreversibility .

Q : At which point in our derivation did irreversibility sneak in?

A : when we tool fz ( F,§, ;I. Fz ) . - f ( F. F.) f ( FiP2 )

Note that dtt= o if f( F.) fc Fz ) = ftp.llflpz

'

) which is a

dt

version of detailed balance which is satisfied in equilibrium .

One can in fact use this necessaryconation to derive a

functional form for the equilibriumdistribution feq .