Lecture 21. Grand canonical ensemble (Ch. 7)

Plan: we want to generalize this result to the case where both energy and particles can be exchanged with the environment.

Reservoir RU0 -

System S

ii VTZP exp

,...,

1Boltzmann statistics (T, V, N) :

ReservoirUR, NR, T,

SystemE, NQuantum statistics (T, V, ):

Gibbs factor

: a (quantum) microstate

exp

B

N E

k T

lnBk T Z Grand free energy

(or grand potential)Z

Grand part. Func.

1For reservoir: (thermodynamic identity)R R R RdS dU PdV dN

T

The Gibbs Factor

ReservoirUR, NR, T,

SystemE, N

1 and 2 - two microstates of the system (characterized by the spectrum and the number of particles in each energy level)

121212

1 SSSSRR NNEET

SS

The changes U and N for the reservoir = -(U and N for the system).

TkEN

TkEN

P

P

BSS

BSS

/exp

/exp

11

22

1

2

Gibbs factor

Tk

EN

B

exp

neglect

RS

1

2

i

iiS EnEi

iS nN

the total multiplicity: 1Total i S i R i R i R i

22 2 2 1

1 1 1

exp /exp

exp /R BR R R

R BR B

S kP S S

P kS k

The Grand Partition Function

the probability that the system is in state with energy E and N particles:

Tk

EN

ZP

B

exp1

Tk

ENZ

B

exp

the grand partition function or the Gibbs sum:

nTVNnNVT ,/,,

The systems in equilibrium with the reservoir that supplies both energy and particles constitute the grand canonical ensemble.

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.

- proportional to the probability that the system in the state contains N particles and has energy E

Gibbs factor

Tk

EN

B

exp

From Particle States to Occupation Numbers

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.

1

2

3

4

Systems with a fixed number of particles Systems with a fixed number of particles in contact with the reservoir, occupancy in contact with the reservoir, occupancy nnii<<1<<1

Systems which can exchange both Systems which can exchange both energy and particles with a reservoir,energy and particles with a reservoir,arbitrary occupancy narbitrary occupancy ni 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.

11 ln ZU

1UnU

i

inN

4En

i

iiEnE

Ntotal Z

NZ 1!

1

1

2

3

4

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:

i

iinnnnsE ...332211

The energy of the system in the state s {n1, n2, n3,.....} is:

i

insN

The grand partition function:

The sum is taken over all possible occupancies and all states for each occupancy.

Z

exp exp exp

i i

i i ii i i i i

s n n iB B B

n nN s E s n n

k T k T k T

Z

exp

i

i i

n i B

n

k T

Grand free energy (Landau free energy)

The grand partition function: Z

exp

i

i i

n i B

n

k T

Pr. 7.7 (Pg. 262) , , lnBT V k T Z Grand free energy

U TS N

In thermodynamics (Pr. 5.23, Pg. 166)

Similar to partition function which is related to

Helmholtz free energy , grand partition function Z

Z

F

is related to another thermodynamic potential:

PV

d SdT PdV Nd

The Grand Partition Function of an Ideal Quantum Gas

i

inN i ii

E s n a microstate s {n1, n2, n3,.....}

Z

exp

i

i i

n i B

n

k T

0

exp expi i

i i i ii

n i i n iB B

n nZ

k T k T

1: a, be.g.

2: c, d

The sum is taken over all possible values of ni

Z Z i

i

expi

i ii

n B

nZ

k T

depending on the quantum nature of particles

ac ad bc bd a b c d

The partition function of each quantum level is independent of other levels.

What is the meaning of the grand partition function formula:

The Grand Partition Function of an Ideal Quantum Gas

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.

Z Z i

i

The mean occupancy at ith level

The probability of a state (ith level) to be occupied by ni quantum particles:

Gibbs factor,

grand partition functioni iP n 1

exp i i

i B

n

Z k T

, , sum over all possible . i

i i i i in

n n P n n

1exp

1 1exp

1

ln1

i

i

i ii i

n i B

i ini

ii

i

nn n

Z k T

nZ

ZZ

Z

1

Bk T

2 22

2

Pr. 7.6 (Pg. 261)

B

B

N B

k T ZN

Z

k T ZN

Z

Nk T

exp

i

i ii

n B

nZ

k T

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).

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, .....

distinguish. particles Bose statistics Fermi statistics

n1 n2 n1 n2 n1 n2

1 1 1 1

2 1 2 1 2 1

1 2

2 2 2 2

3 1 3 1 3 1

1 3

3 2 3 2 3 2

2 3

3 3 3 3

4 1 4 1 4 1

1 4

4 2 4 2 4 2

2 4

4 3 4 3 4 3

3 4

Consider two non-interacting particles in a 1D box of length L. The total energy is given by

The Table shows all possible states for the system with the total energy

22

212

2

, 821nn

mL

hE nn

2522

21 nn

fermions: ni = 0 or 1 bosons: ni = 0, 1, 2, .....

Bosons and Fermions (cont.)

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

3 1 1 0 1

4 0 1 1 1

the number of particles in the single-particle state

The Pauli exclusion principle leaves only four accessible states for such system. (The spin degeneracy is neglected).

The partition function (canonical ensemble):

a state with Ei

3

1 2 3 1 3 4

1 2 4 2 3 4

exp

exp exp

exp exp

i

iE

Z E

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).

1

2

3

4

each level I is a sub-system independently “filled” by the reservoir

1 2 3 4

1 2 3 4

1 2 2 3

1 exp

1 exp 1 exp 1 exp 1 exp

1 exp exp exp exp

exp 2 exp 2 ...

ii

Z