Transcript Document

Lecture 5
•The grand canonical ensemble.
•Density and energy fluctuations in the grand canonical
ensemble: correspondence with other ensembles.
•Fermi-Dirac statistics.
•Classical limit.
•Bose-Einstein statistics.
1
The grand canonical ensemble.
We now consider a subsystem s which can exchange particles and
energy with the heat reservoir r, the total system t being represented
by a microcanonical ensemble with constant energy and constant
number of particles.
We want the probability dws(Ns) of a state of the subsystem in which
the subsystem contains Ns particles and is found in the element
ds(Ns) of its phase space. The notation ds(Ns) reminds us that the
nature of phase space s changes with Ns: the number of dimensions
will change.
Rest ( r ) or heat
reservoir
(s)
Total system (t)
Subsystem
2
We do not care about the state of the remainder of the system provided
only that
(5.1)
E E E , N N  N
s
r
t
s
r
t
Then, by analogy with the treatment of the canonical ensemble,
dws (Ns )  C ds (Ns )r (Nt - Ns )
or
dws  Cer ( Et  Es ;Nt  Ns )ds ( N s )
(5.2)
(5.3)
We expend r in a power series:
 r ( Et , N t )
 ( E , N )
Es  r t t N s  ...
Et
 `N t
E N
  r ( Et , N t )  s  s s


 r ( Et  Es ; N t  N s )   r ( Et , N t ) 
recalling that
  
 
;



E V , N
1
   
- 



N  E ,V
(5.4)
(5.5)
3
Dropping the subscript s, we have
dw( N )  Ae( N  E ) / kT d ( N )
(5.5)
where A is normalization constant. Writing by convention
A  e  / kT
(5.6)
we have
dw( N )  e(   N  E ) / kT d ( N )  ( N )d ( N )
(5.7)
where
( N )  e(   N  E ) / kT
(5.8)
is the grand canonical ensemble.
If several molecular species are present, N is replaced by  Nii. The
quantity  is called the grand potential.
4
Grand partition function
The normalization is
(   N  E ) / kT

(
N
)
d

(
N
)

e
d( N ) 1


N
(5.9)
N
We define the grand partition function
Z  e / kT   eN / kT  e E / kT d ( N )
(classical)
(5.10)
N
Z   e
N
( N  EN ,i ) / kT
(quantum) (5.11)
i
5
Connection with thermodynamic
functions
Proceeding at the same way that in the case of the canonical
ensemble, we get for the entropy
  ln    ln ( N , E )  (  N  E ) / kT
or
E      N  F
(5.12)
(5.13)
Let us prove now that = N G, where G is the Gibbs free energy
Now by
whence
G  E    pV
(5.14)
dG  dE - d  d  pdV  Vdp
(5.15)
dE  d  pdV  dN
(5.16)
dG  -d  Vdp  dN
(5.17)
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 G 



 N  p ,
(5.18)
Now G may be written as N times a function of p and  along. Both p
and  are intrinsic variables and do not change value when two
identical systems are combined in one. For fixed p and , G is
proportional to N and consequently
G  Ng( p , )
(5.19)
where g is the Gibbs free energy per particle. In this case, we have
 G 

  g(p, )  
 N  p ,
whence
G  N( p , )
Then from (5.13)
G  F  pV
E      N  F
(5.20)
(5.21)
(5.22)
(5.13)
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and by comparing with (5.13) we see that
   pV
(5.23)
Other thermodynamic quantities may be calculated from . We can
easily get
  E    N
d  dE  d  d  dN  Nd   pdV  d  Nd
(5.24)
(5.25)
  
p  

 V   ,
(5.26)
  
   
   V ,
(5.27)
  
N  

   V ,
(5.28)
8
Fermi-statistics and Bose Statistics
The occupation numbers, or number of particles in each one-particle
state are strongly restricted by a general principle of quantum
mechanics. The wave function of a system of identical particles must
be either symmetrical (Bose) or antisymmetrical (Fermi) in
permutation of a particle of the particle coordinates (including spin).
It means that there can be only the following two cases:
for Fermi-Dirac Distribution (Fermi-statistics) n=0 or 1
for Bose-Einstein Distribution (Bose-statistics) n=0,1,2,3......
The differences between the two cases are determined by the nature
of particle. Particles which follow Fermi-statistics are called Fermiparticles (Fermions) and those which follow Bose-statistics are
called Bose- particles (Bosones).
Electrons, positrons, protons and neutrons are Fermi-particles,
whereas photons are Bosons. Fermion has a spin 1/2 and boson has
integral spin. Let us consider this two types of statistics consequently.
9
10
Fermi-Dirac Distribution
Enrico Fermi
Physicist
1901 - 1954
There are two possible outcomes:
If the result confirms the hypothesis, then
you've made a measurement.
If the result is contrary to the hypothesis,
then you've made a discovery.
Born: 8 Aug 1902 in
Bristol, England
Died: 20 Oct 1984 in
Tallahassee, Florida, USA
11
Fermi-Dirac Distribution
We consider a system of identical independent non-interacting particles
sharing a common volume and obeying antisymmetrical statistics: that
is, the spin 1/2 and therefore, according to the Pauli principle, the total
wave function is antisymmetrical on interchange of any two particles.
As the particles are assumed to be non-interacting it is convenient to
discuss the system in terms of the energy states i of one particle in a
volume V. We specify the system by specifying the number of particles
ni , occupying the eigenstate i . We classify i in such way that i
denotes a single state, not the set of degenerate states which may
have the same energy.
On the above model the Pauli principle allows only the values ni=1,0.
This is, of course, just the elementary statement of the Pauli principle:
a given state may not be occupied by more than one identical particle.
The partition function of the system is
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ni i

Z  e

(5.29)
{ ni }
subject to  ni  N. We note that the  in the exponent runs over all
i
one-particle states of the system; {ni} represents n allowed set of
values of the ni ; and
runs over all such sets. Each ni may be 0 or
{ ni }
1.
Let us consider as an example a system with two states 1 and 2. The
upper sum reads

e
  ( n11  n2  2 )
the other sum reads
  ( 01  0 2 )
Z e
e
(5.30)
 e  ( 11 0 2 )  e  ( 01 1 2 )
  ( 11 1 2 )
(5.31)
but we have not included the requirement n1+n2=N. If we take N=1,
we have
 1
  2
(5.32)
Ze
e
13
For a system with many states and many particles it is difficult
analytically to take care of the condition ni=N. It is more convenient
to work with grand canonical ensemble. We have for the grand
partition function
Z  e
so that
( 
{ ni }
 ni  nii )  e   (  i )ni

{ ni }
Z    e  (  i )ni
{ ni }
(5.33)
(5.34)
i
A simple consideration shows that we may reverse the order of the 
and  in (5.34). We note that the significance of the  changes
entirely, from {ni}=0,1. Every term, which occurs, for one order will
occur for the other order
Z    e  (  i )ni    x ni
i
where
ni 0 ,1
xi  e
 (   i )
i
(5.35)
ni 0,1
(5.36)
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Now from the definition of the grand partition function
Z  e     e  ( N  E )
(5.37)
N
we have


ni
   kT ln Z   kT  ln  x i    i
 ni

i
i
where
i  kT ln  x
ni
ni
i
(5.38)
(5.39)
For ni restricted to 0,1, we have
 i  kT ln(1  xi )
(5.40)
Now
  
 i 
N  
   

    ,V
i   
(5.41)
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with N   ni it appears reasonable to set
i
 j
1
ni  
 (   )
 e
1
(5.42)
i
The same result can be provided by direct use of averaging in the
grand canonical ensemble

  (  i )ni 
 n j e

{ ni }

nj 

  (  i )ni 
 e

{ ni }

This may be simplified using the form (5.36):
x j  ( 1  xi )
nj 
i j
 (1  x
i
i
)

xj
xi  e
(5.43)
 (   i )
(5.44)
1 xj
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Fermi-Dirac distribution law.
or
ni 
1
e
 ( i   )
(5.45)
1
in agreement with (5.42).
This is the Fermi-Dirac distribution law. It is often written in terms
of f(), where f is the probability that a state of energy is occupied:
f ( ) 
1
e (   )  1

is called the Fermi level, or, for free electron gas, the Fermi energy EF.
It is implicit in the derivation that

(5.46)
is the chemical potential. Often
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Classical limit
For sufficiently large  we will have (-)/kT>>1, and in
this limit
(    ) / kT
f(  )  e
(5.47)
This is just the Boltzmann distribution. The high-energy tail
of the Fermi-Dirac distribution is similar to the Boltzmann
distribution. The condition for the approximate validity of
the Boltzmann distribution for all energies  0 is that
e
  / kT
 1
(5.48)
18
Bose-Einstein Distribution
19
Bose-Einstein Distribution
Particles of integral spin (bosons) must have symmetrical wave
functions. There is no limit on the number of particles in a state, but
states of the whole system differing only by the interchange of two
particles are of course identical and must not be counted as distinct.
For bosons we can use the results (5.38) and (5.39), but with
ni=0,1,2,3,...., so that
1 ni 
i kT kT
ln 
xi  kT
kT
lnln  xi kT ln(
1i xi )
ln Z

 nxi i 
i 1
i
ni

ni
where
Thus
(5.38)
(5.49)
xi  e  (  i )
 j
xj
1
nj  

 (   )
 1  x j e
1
(5.50)
j
or
n(  ) 
1
e
(   )
1
This is the Bose-Einstein distribution
(5.51)
20
We can confirm (5.50) by a direct calculation on nj. Using the previous
result
 (  i )ni
ni
Z   e
i
we have
nj
n x


x
j
nj
j
nj
j
  x
ni 0 ,1
i
ni 0,1


1
n
 xj
ln  x j x j
ln
 xj
 x j 1 x j
j
or
nj 
(5.52)
1
e
(  j  )
1
(5.53)
in agreement with (5.50).
21