principle of equal a priori probability principle of equal a priori probabilities: an isolated...
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Principle of equal a priori probability
Principle of equal a priori probabilities:
An isolated system, with N,V,E, has equal probability to be in any of the (N,V,E) quantum states or
Each and every one of the (N,V,E) quantum states is represented with equally probability
The probability of finding a state 1
P( ,
is ,
)N V E
nd S(N,V,E;int.constr)<S(N 2 Law ,V,E)
(1) (1) (1) (2) (2) (2)
states states
int. constr. N,V,E ,V ,E & ,V ,E
# obeying the int. c
(N,V,E; int.co
onstr.<# with n
nstr.) (N,V,E; int.constr.)
o int. constr.
N N
Maximum
Maxim
um S
,
ln ( , , ) We define
1
B
N V
ST
E
N V E
T
S k
, ,
1 ln ( , )
T
Since 0 ( , , ) when E
BN V N V
S N Vk
E E
T N V E
Canonical ensembles
Canonical N,V,T
I II III AThere are A identical replicas
with same N,V, and T
V=AxV N=AxN EEach microsystem has an energy value Ej(N,V).Each Ej is (Ej) times degenerate
al number of systems in state l , occupation number
heat bath(T)
Set of {al } = distribution = a describes the state of the Ensemble
A=j aj
E=j aj Ej
One possible state of ensemble{ 1 2 3 lEl E1 E2 … El
al a1 a2…al
Canonical as subsystem of microcanocal
heat bath(T)EB
Ej(N,V)
N,V,E E=Ej+EB
Distributions
Principle of equal priori probability every {a} is equally probable
Since there are A systems (each with energy Ej)
there are A number distinguishable particles that can be
distributed according to their al value
a1 in group 1, a2 in group 2, al in group l…
How many times a particular distribution can be found in the ENSEMBLE?
We know the answer to this one, right?
! ( )
!kk
W {a}Aa
fraction systems with in any particular microsate is jE ja
A
Consider 3 oscillators, with 3TE
0 ,1 ,2 ,3
For this ,
there are 10 states of the en
canonic
emble w
al en
ith
semble
=3
jE
Esame distributa group of states can have the with
What is the # of states in each distributi
ion
on?
IIII II
distribution IThere are 3 2 1 6 possible states for
of those, some are equivalent example
3!
# of nonequivalent states in distribution I 32!1!
3 systems
Example (Nash)II
0
1
2
3
2 oscillators in E 0
0 oscillators in E 1distribution with 3 times
0 oscillators in{2,
E 2
1 os
0,0,1};
cillators in E 3
the other two distributions are
3!=6 times
1!1!1!
3! =1 time
3!
{1,1,1,0}
{0,3,0,0}
10 states in 3 different distribution
( ) 3; ( ) 6; ( ) 1
3 3
s
W I W II W III
Eii i
i ia EaA
If we increase to 5 8
there are 7 different distributions
E A=
8 ! 8 ! 8 ! ( ) 8 ( ) 56 ( ) 56
1! 7 ! 6 ! 1! 1! 6 ! 1! 1!
8 ! 8 ! 8 !( ) 168 ( ) 168 ( ) 280
5 ! 2 ! 1! 5 ! 2 ! 4 ! 3 !
8 !( )
5 ! 3 !
792 states
W I W II W III
W IV W V W VI
W VII W0 2 4 6
W
B
systems
For E=1000 and A=1000 W=10600
there are 1070 atoms in the galaxyFor larger and larger A values, the ratioWn/ Wmax=An 0 20 40 60 80 100
An
B
( ) is a function which sharply peaks at * ( )W W{a} {a}
Probability of finding a system in Ej is obtained by
j
WP
Wa
a
j {a}{a}
{a}
( )×A
( )
a ( )aA
k
Since W( )= is a multinomial coeff., the
which maximizes ( ) is = which maximizes ln ( )
!
W W
kk
k
{a}
{a} {a}
aa
a
A!
knd
If there is a single constraint , the max of W( )will
happen for { } .
But with a 2 constraint
( )
=
E
will maximize for { }W
a
k
k
k
1 2
k
k
k
k
{a}
a
Aa a a
=A
= =...=
E
ak
a
a
=
Distribution for max W(a)
j(to find P we find which maximizes W( ) j {a } {a}a
( ) is a function which sharply peaks at * ( )
( ) ( )
all ( ) ( ) can be neglected in the sum
W W
W W
W W
{a} {a}
{a} {a }
{a} {a }
1
=
where = in the
j
WW
PW W
a
a
j
j j
j j
{a}{a}
{a }
{a} {a }
{a }
( )× ( )A( )
a ( )a
a a
A A
a
( )
Having 2 constraints, we need to use Lagrange multipliers
Maximazing lnW(a)
n f ln ( ) undetermined multipliers: ,
constraints:
W
Ek k kk k
{a}
a =A a =E
for each given 1,2,...
ln 0
j k
W Ek k kk kj
a a aa
0 ln ! ln ! Ek k k kk k kj
A a a aa
0 lnj
A A- Aa
lnk k kk
a a a
Ek k kk k k
a a
10 0 ln Ej j j
j
a aa
1
ln 1 0
for every j
= 1,2,...k
jEe
E
e
j
*j
ja +
a
Evaluating the multiplier
1constraint #1 jEe e*j
j j
a =A=
1
jEe
e j
A
( 1)
jE
j
e eP
*ja
A A
1
j
j
j
j
E
j
E
E
E
e
e
e
P
ej
*j
j
Aa
A A
From here we can calculate all othermechanical thermodynamic properties
Probability of finding the quantum state with Ej at a given N,V
j
j
E
j j jEj
eP M P M
ej