Download Lecture 23. Statistics of Ideal Quantum Systems

Lecture 23. Systems with a Variable Number of
Particles.
Ideal Gases of Bosons and Fermions (Ch. 7)
In L22, we considered systems with a fixed number of particles at low particle
densities, n<<nQ. We allowed these systems to exchange only energy with the
environment. Today we’ll remove both constraints: (a) we’ll extend our analysis to the
case where both energy and matter can be exchanged (grand canonical ensemble),
and (b) we’ll consider arbitrary n (quantum statistics).
When we consider systems that can exchange particles and energy with a large
reservoir, both  and T are dictated by the reservoir (they are the reservoir’s
properties). In particular, the equilibrium is reached when the chemical potentials of a
system and its environment become equal to one another. In equilibrium, there is no
net mass transfer, though the number of particles in a system can fluctuate around its
mean value (diffusive equilibrium).
For a system with a fixed number of particles, we found that the probability P(i) of
finding the system in the state with a particular energy i is given by the canonical
distribution:
P i  
1
exp  i 
Z T ,V ,...
We want to generalize this result to the case where both energy and particles can be
exchanged with the environment.
The reservoir is now both a heat reservoir with
The Gibbs Factor
the temperature T and a particle reservoir with
chemical potential . Because each single-particle energy level is populated from a particle
reservoir independently of the other single particle levels, the role of the particle reservoir
is to fix the mean number of particles.
Reservoir
UR, NR, T, 
System
E, N
1 and 2 - two microstates of
the system (characterized by the
spectrum and the number of
particles in each energy level)
R
2
1
S
According to the fundamental assumption of thermodynamics, all the states of the
combined (isolated) system “R+S” are equally probable. By specifying the microstate of
the system i, we have reduced S to 1 and SS to 0. Thus, the probability of occurrence of a
situation where the system is in state i is proportional to the number of states accessible to
the reservoir R . The total multiplicity:
 i   S  i   R  i   1  R  i    R  i 
 S    S R 1 
P 2   R  2  exp S R  2  / k B 


 exp  R 2

P1   R 1  exp S R 1  / k B 
kB


dS R 
1
dU R  PdVR  dN R 
T
neglect
The changes U and N for the reservoir = -(the corresponding changes for the system).
S R  2   S R 1   
1
ES  2   ES 1   N S  2   N S 1 
T
P 2  exp N S  2   ES  2  / k BT 

P1  exp N S 1   ES 1  / k BT 
N S   ni
Gibbs factor =
i
ES   ni Ei
i
 N    E  
exp 

k
T
B


Gibbs factor =
The Grand Partition Function
- proportional to the probability that the system in
 N    E  
exp


k BT


the state  contains N particles and has energy E
the probability that the system is in state 
with energy E and N particles:
the grand partition function or the Gibbs sum
 T ,V , N   n  N / V    T , n 
P  
 N    E  
1
exp 

Z
k
T
B


 N    E  
Z   exp 

k
T

B


 is the index that refers to a specific microstate of the system, which is specified by the
occupation numbers ni: s  {n1, n2,.....}. The summation consists of two parts: a sum
over the particle number N and for each N, over all microscopic states i of a system
with that number of particles.
The systems in equilibrium with the reservoir that supplies both energy and particles
constitute the grand canonical ensemble.
In the absence of interactions between the particles, the energy levels Es of the system
as a whole are determined by the energy levels of a single particle, i: i - the index that
refers to a particular single-particle state.
As with the canonical ensemble, it would be convenient to represent this sum as a
product of independent terms, each term corresponds to the partition function of a
single particle. However, this can be done only for ni<<1 (classical limit). In a more
general case, this trick does not work: because of the quantum statistics, the values of
the occupation numbers for different particles are not independent of each other.
From Particle States to Occupation Numbers
Systems with a fixed number of
particles in contact with the reservoir,
occupancy ni<<1
4
3
2
1
Systems which can exchange both
energy and particles with a reservoir,
arbitrary occupancy ni
1 N
Z1
N!

U1  
ln Z1

3
U  n U1
1
Z total 
The energy was fluctuating, but the total
number of particles was fixed. The role of
the thermal reservoir was to fix the
mean energy of each particle (i.e., each
system). The identical systems in contact
with the reservoir constitute the canonical
ensemble. This approach works well for
the high-temperature (classical) case,
which corresponds to the occupation
numbers <<1.
4
2
nE4 
N   ni
i
E   ni Ei
i
When the occupation numbers are ~ 1, it
is to our advantage to choose, instead of
particles, a single quantum level as the
system, with all particles that might
occupy this state. Each energy level is
considered as a sub-system in equilibrium
with the reservoir, and each level is
populated from a particle reservoir
independently of the other levels.
From Particle States to Occupation Numbers (cont.)
We will consider a system of identical non-interacting particles at the temperature T, i
is the energy of a single particle in the i state, ni is the occupation number (the
occupancy) for this state:
N n

i
i
The energy of the system in the state s  {n1, n2, n3,.....} is:
E s   n11  n2 2  n3 3  ...   ni i
i
The grand partition function:
 n   ni 

Z   exp   i i
k
T
i ,n
B


The sum is taken over all possible occupancies and all states for each occupancy.
The Gibbs sum depends on the single-particle spectrum (i), the chemical potential, the
temperature, and the occupancy. The latter, in its tern, depends on the nature of
particles that compose a system (fermions or bosons). Thus, in order to treat the ideall
gas of quantum particles at not-so-small ni, we need the explicit formulae for ’s and ni
for bosons and fermions.
“The Course Summary”
Ensemble
Macrostate
microcanonical
U, V, N
(T fluctuates)
canonical
T, V, N
(U fluctuates)
grand
canonical
Probability
Pn 
The grand potential
  kBT ln Z
S U ,V , N   kB ln 
1

En
1  kB T
Pn  e
Z
1 
T, V, 
Pn  e
(N, U fluctuate)
Z
Thermodynamics
 En   N n 
kB T
F T ,V , N   kB T ln Z
 T ,V ,    kB T 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
in terms of T and n=N/V.
 /  T ,V
  N to eliminate μ
Bosons and Fermions
One of the fundamental results of quantum mechanics is that all particles can be
classified into two groups.
Bosons: particles with zero or integer spin (in units of ħ). Examples: photons, all
nuclei with even mass numbers. The wavefunction of a system of bosons is
symmetric under the exchange of any pair of particles: (...,Qj,...Qi,..)=
(...,Qi,...Qj,..). The number of bosons in a given state is unlimited.
Fermions: particles with half-integer spin (e.g., electrons, all nuclei with odd mass
numbers); the wavefunction of a system of fermions is anti-symmetric under the
exchange of any pair of particles: (...,Qj,...Qi,..)= -(...,Qi,...Qj,..). The number of
fermions in a given state is zero or one (the Pauli exclusion principle).
The Bose or Fermi character of composite objects: the composite objects that
have even number of fermions are bosons and those containing an odd number of
fermions are themselves fermions.
(an atom of 3He = 2 electrons + 2 protons + 1 neutron  hence 3He atom is a fermion)
In general, if a neutral atom contains an odd # of neutrons then it is a fermion, and if it
contains en even # of neutrons then it is a boson.
The difference between fermions and bosons is specified by the possible values of ni:
fermions: ni = 0 or 1
bosons: ni = 0, 1, 2, .....
Bosons & Fermions (cont.)
distinguish.
n1
particles
n2
Bose
n1
statistics
n2
Fermi
n1
statistics
n2
1
2
1
2
3
1
3
2
3
4
1
4
2
4
3
1
1
2
2
1
3
2
3
3
1
4
2
4
3
4
1
2
1
1
2
1
2
3
2
1
3
1
3
2
3
2
3
4
3
1
4
1
4
2
4
2
Consider two noninteracting particles
in a 1D box of
length L. The total
energy is given by
En1 ,n2

h2
2
2

n

n
1
2
8mL2
The Table shows all
possible states for
the system with the
total energy
n1  n2  25
2
4
3
4
3
2

The Partition Function of an Ideal Fermi Gas
Let’s consider a system that consists of just one single
state of energy i. The total energy of this state: ni i. The
probability of this state to be occupied by ni particles:
The grand partition function for all particles in the ith singleparticle state (the sum is taken over all possible values of ni) :
If the particles are fermions, n can only be 0 or 1:
Putting all the levels together, the
full partition function is given by:
P i , ni  
 n    i 
1
exp  i

Z
 k BT 
 n    i 
Z i   exp  i

k
T
ni
B


   i 

Z iFD  1  exp 
k
T
 B 

    i 

Z FD   1  exp 
i 
 k BT  
The partition functions of different levels are multiplied because they are independent of
one another (each level is an independent thermal system, it is filled by the reservoir
independently of all other levels).
kBT  , 1
1

   1 

Z  1  exp 
 k BT 
1
P 1 ,0  
1  exp     i 
FD
1
1   1
exp   1   1   1
P 1 ,1 
exp     i 
1  exp     i 
Problem (partition function, fermions)
Calculate the partition function of an ideal gas of N=3 identical fermions in equilibrium
with a thermal reservoir at temperature T. Assume that each particle can be in one of
four possible states with energies 1, 2, 3, and 4. (Note that N is fixed).
1
1
1
1
0
2
1
0
1
1
The Pauli exclusion principle leaves only four
accessible states for such system. (The spin
degeneracy is neglected).
3
1
1
0
1
the number of particles in the single-particle state
4
0
1
1
1
the system is in a state with Ei
The partition function:
Z 3   exp  Ei  
Ei
exp    1   2   3  exp    1   3   4  exp    1   2   4  exp    2   3   4 
4
3
Calculate the grand partition function of an ideal gas of fermions in equilibrium
with a thermal and particle reservoir (T, ). Fermions can be in one of four
possible states with energies 1, 2, 3, and 4. (Note that N is not fixed).
2
1
each level I is a sub-system independently “filled” by the reservoir
Z  1  exp     i   1  exp     1  exp     2  exp     3 
i
exp     4  exp  2   1   2  exp  2   2   3  ...
Fermi-Dirac Distribution
The probability of a state to be occupied by a fermion:
P i , ni  
 n    i 
1
exp  i
 ni  0,1
Z
 k BT 
The mean number of fermions in a particular state:
nFD   
   

exp 
k
T
1
 B  
ni   ni Pni   0  P0  1 P1 
   
  
ni
 exp 
  1
1  exp 
 k BT 
 k BT 
1
  
- the Fermi-Dirac distribution
  1
exp 
( is determined by T and the particle density)
 k BT 
At T = 0, all the states with  <  have the occupancy
= 1, all the states with  >  have the occupancy = 0
(i.e., they are unoccupied). With increasing T, the
step-like function is “smeared” over the energy range
~ kBT.
1
~ kBT
The macrostate of such system is completely defined
if we know the mean occupancy for all energy
levels,
which is often called the distribution
function:
0
f E   n E 
While f(E) is often less than unity, it is not a probability:
T=0
(with respect to )
 f E   n
i
=
n=N/V – the average
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 the possible values of ni)
If the particles are bosons,
n can any integer  0:
Z iBE
  
 1  exp 
 k BT
 n    i 
Z i   exp  i

ni
 k BT 
  
 2   
 3   
Z i  1  exp 

exp

exp




  ....
 k BT 
 k BT 
 k BT 
    
  exp 
   k BT
2
     
  exp 
    k BT
3

1
  .... 
  

1  exp 
 k BT
- the partition function for the Bose-Einstein gas



Bose-Einstein Distribution
 n     
exp  i
k BT 

ni   ni P ni   0  P0   1 P 1  2  P2   ...   ni
Z
ni
ni
The mean
number of
bosons in a
given state:

   1
 x 

k
T
B

 Z

1 Z


exp
n
x


i
Z x
ni x
1 Z
ex
1
x   1 
ni 
 1 e




Z x
x  1  e x  1  e x ex  1


nBE   
1
The Bose-Einstein
  
distribution
  1
exp 
k
T
 B 
The mean number of particles in a given state for the BEG can exceed unity, it
diverges as   .
2
1
Comparison of the FD and BE distributions
plotted for the same value of .
FD
=
0
BE
The Classical Regime Revisited
2
MB
1
BE
FD
The FD and BE distributions are reduced to the
Boltzmann distribution in the classical limit:
=
0
nBE  nFD 
Comparison of the FD and BE distributions plotted
for the same value of . Note that the MB
distribution makes no sense when the average #
of particle in a given state becomes comparable to
1 (violation of the classical limit).
  
  
  
 exp 
  1 exp 
  1
ni  1 exp 
k
T
k
T
k
T
 B 
 B 
 B 
     

 exp  
  
  k BT 

exp 
 k BT 
1
- this is still not the Boltzmann factor: we deal with
the -fixed formalism whereas the Boltzmann factor
is the distribution function in the N-fixed formalism.
To get to the N-fixed formalism, let’s add all nk for all single-particle states and demand
that  be such that the total number of occupancies is equal to N:  ni  N
i
  
  
  
 exp  i   exp 
 Z1  N
exp 
 k BT  i
 k BT 
 k BT 
This is consistent
with
our
initial
assumption that
  
  1
exp 
k
T
 B 
   N 
V  n
 
exp 
  Z1   
 1
k
T
Z
V
n

1
Q
Q
 B 

The resulting chemical potential
is the same as what we
obtained in the classical regime:
 nQ 
  k BT ln  
 n 
The Classical Regime Revisited (cont.)
The free energy in the classical regime:
  V  
  1
F T ,V , N   k B T ln Z   NkB T ln 
  NVQ  
The chemical potential of Boltzmann gas
(the classical regime):
 nQ 
 F 
 Boltzmann  
  k BT ln  
 N T ,V
 n 
μ for an ideal gas is negative: when you add a particle to a system and want to
keep S fixed, you typically have to remove some energy from the system.
In terms of the density, the classical limit
corresponds to n << the quantum density:
 2 mkBT 

n  nQ  
2
h


3/ 2
We can also rewrite this condition as T>>TC where TC is the so-called degeneracy
temperature of the gas, which corresponds to the condition n~ nQ. More accurately:
h2
TC 
2 mkB
 n 


 2 .6 
2/3
For the FD gas, TC ~ EF/kB where EF is the Fermi energy (Lect. 24) , for the BE gas
TC is the temperature of BE condensation (Lect. 26).
nFD    f FD   
1
  
  1
exp 
 k BT 
 T ,V , N   n  N / V    T , n
 for Fermi Gases
g  
d
    T , n  
0
  1
exp 
k
T
B




n   g   f   d  
0
When the average number of fermions in a system (their density) is known, this equation
can be considered as an implicit integral equation for (T,n). It also shows that 
determines the mean number of particles in the system just as T determines the mean
energy. However, solving the eq. is a non-trivial task.
 /EF
n ~ nQ
2
2
depending on n and T,  for 1

EF
 1
  k BT 

  ....
12  EF 
fermions may be either
positive or negative.
The limit T0: adding one fermion to the system at T=0 increases
its energy U by EF. At the same time, S remains 0: all the fermions
are packed into the lowest-energy states.
1
dS 
T
kBT/EF
1
dU  dN 
 nQ 

 n 
 Boltzmann  k BT ln 
 T  0  EF
 F 
The same conclusion you’ll reach by considering F =U-TS=UT=0 and recalling that
 

the chemical potential is the change in F produced by the addition of one particle:

N

T ,V
The change of sign of (n,T) indicates the crossover from the
degenerate Fermi system (low T, high n) to the Boltzmann statistics.
The condition kBT << EF is equivalent to n >> nQ:
n
4

nQ 3 
 EF

 k BT



3/ 2
 for Bose Gases
Bose
Gas
nBE
1

  
  1
exp 
 k BT 

g  
d
  
0
  1
exp 
k
T
 B 

n   g   f   d  
0
The occupancy cannot be negative for any , thus, for bosons,
  0 ( varies from 0 to ). Also, as T0,   0
nBE T 0 
1

exp 0 / 0  1

T
0,   0
1,   0
nBE T 0  
For bosons, the chemical potential is a non-trivial function of the density and temperature
(for details, see the lecture on BE condensation).
Comparison between Distributions
S
Nk B
Fermi-Dirac
Boltzmann
3
Bose-Einstein
U
k BTC
2
3
1
2
1
zero-point
energy,
Pauli
principle
2
T
3 TC
1
1
2
3
T
TC
h2
TC 
2 mkB
 n 


 2 .6 
2/3
Comparison between Distributions
CV /NkB
Fermi-Dirac
Boltzmann
Bose-Einstein
2
1.5
0
1
T/TC
Comparison between Distributions
Boltzmann
nk 
1
  

exp 
 k BT 
Bose
Einstein
nk 
1
  
  1
exp 
 k BT 
Fermi
Dirac
nk 
1
  
  1
exp 
 k BT 
indistinguishable
Z=(Z1)N/N!
nK<<1
indistinguishable
integer spin 0,1,2 …
indistinguishable
half-integer spin 1/2,3/2,5/2 …
spin doesn’t matter
bosons
fermions
localized particles
 don’t overlap
wavefunctions overlap
total  symmetric
wavefunctions overlap
total  anti-symmetric
photons
atoms
free electrons in metals
electrons in white dwarfs
unlimited number of
particles per state
never more than 1
particle per state
gas molecules
at low densities
“unlimited” number of
particles per state
nK<<1
4He