1. Lecture 21. Grand canonical ensemble (Ch. 7)
Boltzmann statistics (T, V, N) :
Reservoir R
U0 - 
System S 
Plan: we want to generalize this result to the case where both energy and particles can be exchanged with the environment.
Reservoir
UR, NR, T, 
System
E, N
Quantum statistics (T, V, ):
In L18, 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).
Gibbs factor
Grand part. Func.
Grand free energy
(or grand potential)

2. The Gibbs Factor R S 1 2
1 and 2 - two microstates of the system (characterized by the spectrum and the number of particles in each energy level)
Reservoir
UR, NR, T, 
System
E, N
neglect
The reservoir is now both a heat reservoir with 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.
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 changes U and N for the reservoir = -(U and N for the system).
Gibbs factor

3. The Grand Partition Function
- proportional to the probability that the system in the state  contains N particles and has energy E
Gibbs factor
the probability that the system is in state  with energy E and N particles:
the grand partition function or the Gibbs sum:
 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.
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.
The systems in equilibrium with the reservoir that supplies both energy and particles constitute the grand canonical ensemble.

4. From Particle States to Occupation Numbers
Systems with a fixed number of particles in contact with the reservoir, occupancy ni<<1
Systems which can exchange both energy and particles with a reservoir,
arbitrary occupancy ni
 4
 4
 3
 3
 2
 2 1 1
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.
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.

5. 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:
The energy of the system in the state s  {n1, n2, n3,.....} is:
The grand partition function:
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 ideal gas of quantum particles at not-so-small ni, we need the explicit formulae for ’s and ni for bosons and fermions.
The sum is taken over all possible occupancies and all states for each occupancy.

6. Grand free energy (Landau free energy)
The grand partition function:
Grand free energy
Pr. 7.7 (Pg. 262)
In thermodynamics (Pr. 5.23, Pg. 166)

7. The Grand Partition Function of an Ideal Quantum Gas
a microstate s  {n1, n2, n3,.....}
The sum is taken over all possible values of ni
depending on the quantum nature of particles

8. The Grand Partition Function of an Ideal Quantum Gas
What is the meaning of the grand partition function formula:
The partition function of each quantum level is independent of other levels.
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.
Page 266: The “system” and the “reservoir” therefore occupy the same physical space.

9. The mean occupancy at ith level
The probability of a state (ith level) to be occupied by ni quantum particles:

10. 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).

11. Bosons and Fermions (cont.)
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, .....

12. Bosons and Fermions (cont.)
fermions: ni = 0 or 1 bosons: ni = 0, 1, 2, .....
Consider two non-interacting particles in a 1D box of length L. The total energy is given by
distinguish.
particles
Bose
statistics
Fermi n1 n2 1 2 3 4
The Table shows all possible states for the system with the total energy

13. 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 2 3 4
The Pauli exclusion principle leaves only four accessible states for such system. (The spin degeneracy is neglected).
the number of particles in the single-particle state
a state with Ei
The partition function (canonical ensemble):

14. Problem (partition function, fermions)
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).
 4
 3
each level I is a sub-system independently “filled” by the reservoir
 2 1