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<

When we consider systems that can exchange particles and energy with a large reservoir, both μ 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(εi) of finding the system in the state with a particular energy εi is given by the canonical distribution: 1 P()ε = exp()− βε i ()VTZ ,..., i 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 T and a particle reservoir with μ. 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. α and α - two microstates of R 1 2 S Reservoir System the system (characterized by the ε2 spectrum and the number of ε1 UR, NR, T, μ E, N particles in each energy level)

According to the fundamental assumption of , all the states of the combined (isolated) system “R+S”are equally probable. By specifying the microstate of the system i, we have reduced ΩS 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: Ω(α )()= Ω α ×Ω (αiRiSi )=1×Ω (α )= Ω (αiRiR )

⎡ ⎤ 1 P()α 2 ΩR ()α 2 []R (α 2 )/exp kS B R ( 2 )− SS R (αα1 ) dS ()−+= μdNPdVdU = = = exp⎢ ⎥ R R R R P()α Ω ()α []()α /exp kS k T 1 R 1 R 1 B ⎣ B ⎦ neglect The changes ΔU and ΔN for the reservoir = -(the corresponding changes for the system). 1 = nN = EnE R ()2 SS R αα1 ()−=− []S ()2 S 1 ()S 2 +−− ()NNEE S αμαμαα1 ()S ∑ i S ∑ ii T i i

P()α 2 exp {}[]μ S ()α 2 − S (α 2 ) / BTkEN ⎧ ( )− EN (ααμ)⎫ = exp P()α exp {}[]()− ααμ ()/ TkEN Gibbs factor = ⎨ ⎬ 1 S 1 S 1 B ⎩ BTk ⎭ The Grand Partition Function ⎧ ( )− EN (ααμ)⎫ - proportional to the probability that the system in Gibbs factor = exp⎨ ⎬ the state α contains N particles and has energy E ⎩ BTk ⎭ 1 ⎧ ( )− EN (ααμ)⎫ the probability that the system is in state α P()α = exp⎨ ⎬ with energy E and N particles: Z ⎩ BTk ⎭

the grand partition function or the Gibbs sum ⎧ ( )− EN (ααμ)⎫ Z = ∑exp⎨ ⎬ μ(),, []=== μ( ,/ nTVNnNVT ) α ⎩ BTk ⎭ α 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 .

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, εi: 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 Systems which can exchange both particles in contact with the reservoir, energy and particles with a reservoir,

occupancy ni<<1 arbitrary occupancy ni 1 ε N 4 Ztotal = Z1 ε (En ) N ! 4 4 ε 3 ε ∂ 3 = ∑ nN i ε U1 −= ln Z1 2 ∂β ε 2 i

ε = EnE ii 1 = UnU 1 ε1 ∑ i The energy was fluctuating, but the total When the occupation numbers are ~ 1, it number of particles was fixed. The role of is to our advantage to choose, instead of the thermal reservoir was to fix the particles, a single quantum level as the mean energy of each particle (i.e., each system, with all particles that might system). The identical systems in contact occupy this state. Each energy level is with the reservoir constitute the canonical considered as a sub-system in equilibrium ensemble. This approach works well for with the reservoir, and each level is the high-temperature (classical) case, populated from a particle reservoir which corresponds to the occupation independently of the other levels. numbers <<1. From Particle States to Occupation Numbers (cont.)

We will consider a system of identical non-interacting particles at the temperature T, εi is the energy of a single particle in the i state, ni is the occupation number (the occupancy) for this state: = ∑ nN i i

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

( ) nnnsE εεε332211 ... =+++= ∑ n ε ii i

The grand partition function: ⎛ − μεnn iii ⎞ Z ∑exp⎜−= ⎟ ,ni ⎝ BTk ⎠

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

The Gibbs sum depends on the single-particle spectrum (εi), 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 μ’s and ni for bosons and fermions. “The Course Summary”

Ensemble Macrostate Probability Thermodynamics

micro- U, V, N 1 P = ( )= kNVUS B ln,, Ω canonical (T fluctuates) n Ω E T, V, N 1 − n canonical B Tk P = e ( ,, )= − B ln ZTkNVTF (U fluctuates) n Z ( −μ NE ) 1 − nn grand T, V, μ B Tk Pn = e Φ ( ,, μ)= − ln ZTkVT canonical (N, U fluctuate) Z B

(the Landau free energy) is a generalization The grand potential Φ ≡ − B ln ZTk of F=-kBTlnZ - the appearance of μ as a variable, while −−−=Φ NdPdVSdTd μ 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 or N are given as functions of the

“natural” variables T,V and μ. Thus, we need to use () ∂ Φ / ∂ μ , VT = − N to eliminate μ in terms of T and n=N/V. 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: Ψ(...,Qj,...Qi,..)= Ψ(...,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: Ψ(...,Qj,...Qi,..)= -Ψ(...,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. particles Bose statistics Fermi statistics Consider two non- n n n n n n 1 2 1 2 1 2 interacting particles 1111 in a 1D box of 212121length L. The total 12 energy is given by 2222 313131 h2 E = ( 2 + nn 2 ) 13 ,nn 21 8mL2 1 2 323232 23 The Table shows 3333 all possible states 414141for the system with 14 the total energy 424242 24 2 nn 2 ≤+ 25 434343 1 2 34 The Partition Function of an Ideal

Let’s consider a system that consists of just one single 1 ⎡n ( − εμii )⎤ state of energy εi. The total energy of this state: ni εi. The ()ε ,nP ii = exp⎢ ⎥ Z ⎣ BTk ⎦ probability of this state to be occupied by ni particles:

The grand partition function for all particles in the ith single- ⎡n ( −εμii )⎤ Zi = exp⎢ ⎥ particle state (the sum is taken over all possible values of n ) : ∑ Tk i ni ⎣ B ⎦

⎛ −εμ⎞ If the particles are fermions, n can only be 0 or 1: FD ⎜ i ⎟ Zi += exp1 ⎜ ⎟ ⎝ BTk ⎠

Putting all the levels together, the ⎡ ⎛ − εμ⎞⎤ Z +Π= exp1 ⎜ i ⎟ full partition function is given by: FD ⎢ ⎜ ⎟⎥ ε i ⎣ ⎝ BTk ⎠⎦

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).

1 μ < ε1 BTk << μ,ε1 ⎛ − εμ⎞ FD ⎜ 1 ⎟ Z1 += exp1 ⎜ ⎟ exp[β (μ − ε1 )] 1 μ >>> ε1 ε1 ⎝ BTk ⎠ 1 μ P()ε 0, = exp[β (μ − ε )] 1 P()ε 1, = i exp1 []()−+ εμβi 1 exp1 []()−+ εμβi 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 ε1, ε2, ε3, and ε4. (Note that N is fixed). The Pauli exclusion principle leaves only four ε1 1110 accessible states for such system. (The spin degeneracy is neglected). ε2 1011

ε3 1101 the number of particles in the single-particle state ε 0111 4 The partition function: Z3 ∑exp{ βEi }=−= the system is in a state with Ei Ei

exp{}[]εεεβ321 exp{}[]εεεβ431 exp{}[]εεεβ421 exp{}[]++−+++−+++−+++− εεεβ432

Calculate the grand partition function of an ideal gas of fermions in equilibrium with a thermal and particle reservoir (T, μ). Fermions can be in one of four ε 4 possible states with energies ε1, ε2, ε3, and ε4. (Note that N is not fixed). ε 3

ε 2 each level εI is a sub-system independently “filled” by the reservoir

ε 1 Z []+Π= exp1 {β(μ −εi )}= +exp1 {β(μ −ε1)}+exp{β(μ −ε2 )}+exp{β(μ −ε3 )}+ i

exp{}(()4 +− {}()2exp 21 +−− {}2exp )εεμβ32 +−− ... εεμβ εμβ Fermi-Dirac Distribution

1 ⎡n ( − εμii )⎤ The probability of a state to be occupied by a fermion: ()ε , nP ii = exp⎢ ⎥ ni = 1,0 Z ⎣ BTk ⎦ The mean number of fermions in a particular state: ⎛ −εμ⎞ exp⎜ ⎟ ⎝ BTk ⎠ 1 i = ∑ ()ii ()PPnPnn 1100 =×+×= () = ni ⎛ − ⎞ ⎛ − μεεμ⎞ + exp1 ⎜ ⎟ exp⎜ ⎟ +1 1 ⎝ BTk ⎠ ⎝ BTk ⎠ nFD ()ε = ⎛ − με⎞ - the Fermi-Dirac distribution exp⎜ ⎟ +1 ⎜ ⎟ (μ is determined by T and the particle density) ⎝ BTk ⎠

At T = 0, all the states with ε < μ have the occupancy 1 = 1, all the states with ε > μ have the occupancy = 0 (i.e., they are unoccupied). With increasing T, the ~ kBT step-like function is “smeared” over the energy range

~ kBT. The macrostate of such system is completely defined T=0 if we know the mean occupancy for all energy 0 levels, which is often called the distribution function: ε = μ ()≡ ()EnEf (with respect to μ)

While f(E) is often less than unity, it is not a probability: ∑ ( )= nEf n=N/V – the average i density of particles The Partition Function of an Ideal The grand partition function for all particles in the ⎡n ( − εμii )⎤ th i single-particle state: Zi = exp⎢ ⎥ ∑ Tk ni ⎣ B ⎦ (the sum is taken over the possible values of ni)

If the particles are bosons, ⎡ − εμ⎤ ⎡2( − εμ)⎤ ⎡3( − εμ)⎤ Z += exp1 + exp + exp + .... n can any integer ≥ 0: i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ BTk ⎦ ⎣ BTk ⎦ ⎣ BTk ⎦

2 3 ⎛ − εμ⎞ ⎡ ⎛ − εμ⎞⎤ ⎡ ⎛ −εμ⎞⎤ 1 BE ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Zi += exp1 ⎜ ⎟ + ⎢exp⎜ ⎟⎥ + ⎢exp⎜ ⎟⎥ .... =+ ⎝ BTk ⎠ ⎣ ⎝ BTk ⎠⎦ ⎣ ⎝ BTk ⎠⎦ ⎛ − εμ⎞ − exp1 ⎜ ⎟ ⎝ BTk ⎠ - the partition function for the Bose-Einstein gas Bose-Einstein Distribution ⎡ n ( − εμ)⎤ The mean exp i ⎢ Tk ⎥ number of = ()PnPnn ()P ()P ()...221100 =+×+×+×= n ⎣ B ⎦ i ∑ ii ∑ i Z bosons in a n i n i given state: ⎡ − εμ⎤ 1 ∂ 1 ∂ Z ⎢ x ≡= ⎥ = exp ()i xn = ∑ ∂xZTk Z ∂x ⎣ B ⎦ n i

x 1 1 ∂Z ∂ 1 e 1 x ⎛ ⎞ nBE ()ε = The Bose-Einstein ni = ()1−= e ⎜ x ⎟ = = −xx ⎛ ⎞ Z ∂x −∂ ex 11 − ee −1 − με ⎝ ⎠ exp⎜ ⎟ −1 distribution ⎝ BTk ⎠

The mean number of particles in a given state for the BEG can exceed unity, it diverges as μ→ε.

2 Comparison of the FD and BE distributions plotted for the same value of μ. BE 1 FD

ε = μ 0 The Classical Regime Revisited 2 Comparison of the FD and BE distributions plotted MB for the same value of μ. Note that the MB distribution makes no sense when the average # BE of particle in a given state becomes comparable to 1 FD 1 (violation of the classical limit).

ε = μ The FD and BE distributions are reduced to the Boltzmann distribution in the classical limit: 0 ⎛ ε ⎞ ⎛ − μ ⎞ ⎛ − μ ⎞ exp⎜ ⎟exp⎜ ⎟ >> 1 exp⎜ ⎟ >> 1 ni << 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ B ⎠ ⎝ BTkTk ⎠ ⎝ BTk ⎠ 1 ⎡ ⎛ − με⎞⎤ - this is still not the Boltzmann factor: we deal with ⎜ ⎟ BE nn FD ≈≈ exp⎢−= ⎜ ⎟⎥ the μ-fixed formalism whereas the Boltzmann factor ⎛ − με⎞ ⎣ ⎝ BTk ⎠⎦ exp⎜ ⎟ is the distribution function in the N-fixed formalism. ⎝ BTk ⎠

To get to the N-fixed formalism, let’s add all nk for all single-particle states and demand that μ be such that the total number of occupancies is equal to N: ∑ i = Nn i ⎛ μ ⎞ ⎛ − ε ⎞ ⎛ μ ⎞ ⎛ μ ⎞ N ⎡ V ⎤ n ⎜ ⎟ ⎜ i ⎟ ⎜ ⎟ exp⎜ ⎟ Z === <<= 1 exp⎜ ⎟∑ exp⎜ ⎟ = exp⎜ ⎟ 1 = NZ ⎜ ⎟ ⎢ 1 ⎥ ⎝ B ⎠ i ⎝ B ⎠ ⎝ BTkTkTk ⎠ ⎝ BTk ⎠ Z1 ⎣⎢ VQ ⎦⎥ nQ This is consistent ⎛ − μ ⎞ The resulting chemical potential ⎛ nQ ⎞ with our initial exp⎜ ⎟ >> 1 μ −= BTk ln⎜ ⎟ ⎜ Tk ⎟ is the same as what we ⎜ n ⎟ assumption that ⎝ B ⎠ obtained in the classical regime: ⎝ ⎠ The Classical Regime Revisited (cont.) ⎡ ⎛ V ⎞ ⎤ The free energy in the classical regime: (),, ln −=−= TNkZTkNVTF ⎢ln⎜ ⎟ +1⎥ B B ⎜ NV ⎟ ⎣⎢ ⎝ Q ⎠ ⎦⎥

The chemical potential of Boltzmann gas ⎛ ∂F ⎞ ⎛ nQ ⎞ ⎜ ⎟ (the classical regime): μBoltzmann = ⎜ ⎟ −= BTk ln⎜ ⎟ ⎝ ∂N ⎠ ,VT ⎝ 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.

2/3 ⎛ 2π Tmk ⎞ In terms of the density, the classical limit nn =<< ⎜ B ⎟ Q ⎜ 2 ⎟ corresponds to n << the quantum density: ⎝ h ⎠

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: 3/2 h2 ⎛ n ⎞ ⎜ ⎟ TC ≅ ⎜ ⎟ π mkB ⎝ 6.22 ⎠

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). 1 FD ()fn FD εε ()== μ for Fermi Gases ⎛ − με⎞ exp⎜ ⎟ +1 ∞ ∞ ⎜ Tk ⎟ g(ε ) ⎝ B ⎠ = ()()dfgn εεε= ∫∫ dε μ(),, []== = μ( ,/ nTVNnNVT ) 0 0 ⎛ − με(),nT ⎞ exp⎜ ⎟ +1 ⎝ BTk ⎠ When the average number of fermions in a system (their density) is known, this equation can be considered as an implicit integral equation for μ(T,n). It also shows that μ 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. μ/E F n ~ n 2 1 Q πμ2 ⎛ Tk ⎞ depending on n and T, μ for 1−= ⎜ B ⎟ +.... fermions may be either ⎜ ⎟ k T/E EF 12 ⎝ EF ⎠ positive or negative. 1 B F ⎛ n ⎞ The limit T→0: adding one fermion to the system at T=0 increases μ −= Tk ln⎜ Q ⎟ Boltzmann B ⎜ n ⎟ its energy U by EF. At the same time, S remains 0: all the fermions ⎝ ⎠ are packed into the lowest-energy states. 1 dS ()−= μdNdU μ( = 0)= ET T F The same conclusion you’ll reach by considering F=U-TS=U and recalling that ⎛ ∂F ⎞ T=0 μ = ⎜ ⎟ the chemical potential is the change in F produced by the addition of one particle: ⎝ ∂N ⎠ ,VT

The change of sign of μ(n,T) indicates the crossover from the 2/3 n 4 ⎛ EF ⎞ degenerate Fermi system (low T, high n) to the Boltzmann statistics. = ⎜ ⎟ nQ 3 π ⎝ BTk ⎠ The condition kBT<< EF is equivalent to n >> nQ: μ for Bose Gases

1 ∞ ∞ g(ε ) Bose n = = ()()dfgn εεε= dε Gas BE ⎛ − με⎞ ∫∫ ⎛ − με⎞ exp⎜ ⎟ −1 0 0 exp⎜ ⎟ −1 ⎜ Tk ⎟ ⎜ ⎟ ⎝ B ⎠ ⎝ BTk ⎠ The occupancy cannot be negative for any ε, thus, for bosons, μ μ≤0 (ε varies from 0 to ∞). Also, as T→0, μ → 0 T

1 ⎧ ε > 0,0 ()nBE T =0 = → ()nBE T =0 = ⎨ ()−10/0exp ⎩ ε = 0,1

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 Fermi-Dirac NkB Boltzmann 3 Bose-Einstein

U

Tk CB 2 3

1

2 T 1 23 TC zero-point 1 energy, Pauli 3/2 h2 ⎛ n ⎞ principle ⎜ ⎟ TC ≅ ⎜ ⎟ T π mkB ⎝ 6.22 ⎠ 1 23 TC Comparison between Distributions

C /Nk V B Fermi-Dirac Boltzmann Bose-Einstein 2

1.5

0 1 T/TC Comparison between Distributions Bose Fermi Boltzmann Einstein Dirac 1 1 1 n = n = n = k ⎛ − με⎞ k ⎛ − με⎞ k ⎛ − με⎞ exp⎜ ⎟ exp⎜ ⎟ −1 exp⎜ ⎟ +1 ⎝ BTk ⎠ ⎝ BTk ⎠ ⎝ BTk ⎠ indistinguishable indistinguishable indistinguishable N Z=(Z1) /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 Ψ don’t overlap total Ψ symmetric total Ψ anti-symmetric

gas molecules photons free electrons in metals at low densities 4He atoms 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