Lecture 22. Ideal Bose and Fermi gas (Ch.
7)
N ( ) − E ( ) N ( ) − E ( )
Gibbs factor exp Z = exp
k BT k BT
the grand partition function Z = Z i Zi exp − ni ( i − )
of ideal quantum gas: i ni kBT
fermions: ni = 0 or 1 bosons: ni = 0, 1, 2, .....
Outline
1. Fermi-Dirac statistics (of fermions)
2. Bose-Einstein statistics (of bosons)
3. Maxwell-Boltzmann statistics
4. Comparison of FD, BE and MB.
The Partition Function of an Ideal Fermi Gas
The grand partition function for all particles in the ith
single-particle state (the sum is taken over all possible
values of ni) :
ni ( − i )
Z i = exp
ni k BT
If the particles are fermions, − i
n can only be 0 or 1: Z i
FD
= 1 + exp
k BT
Putting all the levels − i
together, the full partition Z FD = 1 + exp
function is given by: i k BT
Fermi-Dirac Distribution
The probability of a state to be occupied by a fermion:
ni ( − i )
P( i , ni ) = exp
1
ni = 0, 1
Zi k BT
The mean number of fermions in a particular state:
1
ni =
Zi
Zi =
1
1 + exp ( − i )
1 + exp ( − i )
exp ( − i ) 1
= =
1 + exp ( − i ) exp ( i − ) + 1
Fermi-Dirac distribution
nFD ( ) =
1
( is determined by T −
and the particle density)
exp + 1
k BT
Fermi-Dirac Distribution
At T = 0, all the states with < 1
have the occupancy = 1, all the ~ kBT
states with > have the
occupancy = 0 (i.e., they are T=0
0
unoccupied). With increasing T, the
step-like function is “smeared” over (with respect to ) =
the energy range ~ kBT.
The macrostate of such system is completely defined if we
know the mean occupancy for all energy levels, which is
f (E ) n (E )
often called the distribution function:
While f(E) is often less than unity, it is not a probability:
f (E ) = n n=N/V – the average
i density of particles
The Partition Function of an Ideal Bose Gas
The grand partition function for all particles in the ith
single-particle state (the sum is taken over all possible
values of ni) : ni ( − i )
Zi = exp
ni k BT
If the particles are Bosons, n can be any #, i.e. 0, 1, 2, …
ni ( − i ) ( − i ) 2 ( − i )
Zi = exp = 1 + exp + exp +
ni = 0 k BT k BT k BT
−1
+ + 2
+ =
1 − i
If x 1, 1 x x
1 − x Zi BE
= 1 − exp
k BT
−1
Putting all the levels − i
together, the full partition Z BE = 1 − exp min ( i )
function is given by: i k BT
Bose-Einstein Distribution
The probability of a state to be occupied by a Boson:
1 ni ( − i )
P ( i , ni ) = exp ni = 0,1,2,
Zi k BT
The mean number of Bosons in a particular state:
− i
1 − exp ( − i )
1 −1
ni = Z i = 1 − exp
Z i ( ) k BT ( )
− i exp ( − i ) 1
= 1 − exp =
k BT 1 − exp ( − i )
2
exp ( i − ) − 1
min ( )
Bose-Einstein distribution
1 The mean number of particles in a
nBE ( ) =
− given state for the BEG can exceed
exp − 1 unity, it diverges as → min().
k BT
Comparison of FD and BE Distributions
2
nFD ( ) =
1
BE −
exp + 1
<n>
1
k BT
FD
1
n n nBE ( ) =
−
exp −1
0
-6 -4 -2 0 2 4 6 k BT
(−)/kBT
1
− −
1, exp nFD ( ) nBE ( )
when 1
−
k BT B
k T exp
B
k T
1
Maxwell-Boltzmann nMB ( ) =
distribution: −
exp
B
k T
Maxwell-Boltzmann Distribution (ideal gas model)
Recall the Boltzmann distribution (ch.6) derived from canonical ensemble:
2 mk BT
3/ 2
V 1 N V
Z1 = V = Z= Z1 F = −k BT ln Z = − Nk BT ln + 1
h
2
VQ N!
NVQ
F V 1 N N
= = −k BT ln = ln = exp ( )
N T ,V NVQ Z1 Z1
The mean number of particles in a particular state of N particles in
volume V:
N
nMB ( ) = N P ( ) = exp ( − ) = exp ( ) exp ( − ) = exp − ( − )
Z1
Maxwell-Boltzmann distribution −
MB is the low density limit where the nMB ( ) = exp −
difference between FD and BE disappears. k B T
nVQ 1 ( i.e. N Z1 1 and 0 ) nVQ 1
Comparison of FD, BE and MB Distribution
2
nFD ( ) =
1
MB BE −
exp + 1
k BT
<n>
1 1
FD nBE ( ) =
−
n n exp −1
k BT
0 −
-6 -4 -2 0 2 4 6 nMB ( ) = exp −
(−)/kBT k B T
what are the possible values of MB , FD , and BE ? ( assume 0 )
MB 0 FD F ( 0 ) BE min ( ) = 0
Comparison of FD, BE and MB Distribution
(at low density limit)
1.0
= - kBT The difference between FD, BE
and MB gets smaller when gets
MB
<n>
FD
0.5 BE more negative.
i.e., when 0, nFD nBE nMB
0.0
0 1 2 3
/kBT
0.2 MB is the low density limit where
= - 2kBT MB the difference between FD and BE
FD
BE
disappears.
<n>
0.1
nVQ 1 ( i.e. N Z1 1)
0.0
0 1 2 3
/kBT
Comparison between Distributions
Bose Fermi
Boltzmann
Einstein Dirac
1 1 1
nk = nk = nk =
− − −
exp exp − 1 exp + 1
k BT k BT k BT
indistinguishable indistinguishable indistinguishable
Z=(Z1)N/N! integer spin 0,1,2 … half-integer spin 1/2,3/2,5/2 …
nK<<1
spin doesn’t matter bosons fermions
localized particles wavefunctions overlap wavefunctions overlap
don’t overlap total symmetric total anti-symmetric
gas molecules photons free electrons in metals
at low densities 4Heatoms electrons in white dwarfs
“unlimited” number of unlimited number of never more than 1
particles per state particles per state particle per state
nK<<1
“The Course Summary”
Ensemble Macrostate Probability Thermodynamics
micro- U, V, N
Pn =
1 S (U , V , N ) = k B ln
canonical (T fluctuates)
En
canonical T, V, N 1 − kB T
Pn = e F (T , V , N ) = −k B T ln Z
(U fluctuates) Z
( En − N n )
T, V, 1 −
grand Pn = e (T , V , ) = −k B T ln Z
kB T
canonical (N, U fluctuate) Z
The grand potential − k BT ln Z (the Landau free energy) is a generalization
of F=-kBT lnZ
- the appearance of μ as a variable, while
d = − SdT − PdV − Nd computationally very convenient for the grand canonical
ensemble, is not natural. Thermodynamic properties of
systems are eventually measured with a given density of particles. However, in the
grand canonical ensemble, quantities like pressure or N are given as functions of the
“natural” variables T,V and μ. Thus, we need to use ( / )T ,V = − N to eliminate μ
in terms of T and n=N/V.
