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 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 temperature T and a particle reservoir with chemical potential 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 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 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: Ωi(α )()= Ω Sα i×Ω R(α ) = i 1×ΩR(α i) = Ω R(α ) i ⎡ ⎤ 1 P()α 2 ΩR ()α 2 expS[]R (α 2 ) B k / SSRα( 2 )− R ( α1 ) dS=dU + PdVμ − dN = = = exp⎢ ⎥ R ()R R R P α Ω α expSα k / 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 N= n E= n E SSRα()−2 R () α1 =EENNα −[]S ()−2 αS ()1 − μS () α2 + S () μ1 α S ∑ i S ∑ i i T i i P()α 2 expN{}[]μ S ()α 2 − ES (α 2 ) / B k T μ⎧NE( α)− ( )⎫ α = exp P()α expμN{}[]() α− E () α/ k T Gibbs factor = ⎨ ⎬ 1 S 1 S 1 B ⎩ kB T ⎭ The Grand Partition Function μ⎧NE( α)− ( )⎫ α - proportional to the probability that the system in Gibbs factor = exp⎨ ⎬ the state α contains N particles and has energy E ⎩ kB T ⎭ 1 μ⎧NE( α)− ( )⎫ α the probability that the system is in state α P()α = exp⎨ ⎬ with energy E and N particles: Z ⎩ kB T ⎭ the grand partition function or the Gibbs sum μ⎧NE( α)− ( )⎫ α Z = ∑exp⎨ ⎬ T Vμ(),, N= n[] = N/, Vμ =( ) T n α ⎩ kB T ⎭ 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 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, 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 n E N ! 4 ( 4 ) ε 3 ε ∂ 3 N= ∑ i n ε U1= − ln Z1 2 ∂β ε 2 i ε E= ni i E 1 U= n1 U 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: N= ∑ i n i The energy of the system in the state s → {n1, n2, n3,.....} is: E= s( )1ε n 1 + 2 ε n 2 + 3 ε n... 3 +∑ n =iε i i The grand partition function: ⎛ nεi i− μ n i⎞ Z=∑exp⎜ − ⎟ i, n ⎝ kB T ⎠ 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 = S U(, V ,) = NB lnΩ k canonical (T fluctuates) n Ω E T, V, N 1 − n canonical kB T P = eF T( V,, N) = − B kln T Z (U fluctuates) n Z EN( −μ ) 1 − n n grand T, V, μ kB T Pn = e TΦ ( V,,μ) = − kln T Z canonical (N, U fluctuate) Z B (u free energythe Landa) is a generalization The grand potential Φ ≡k− B Tln Z of F=-kZB n Tl - the appearance of μ elle, whias a variab dΦ SdT = − − PdVμ − computationally Nd 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 () ∂ Φ / ∂ μ TV , =t −eliminateN o 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 2 313131 h 2 2 En, n = n( 1 + 2 n ) 13 1 2 8mL2 323232 23 The Table shows 3333 all possible states 414141for the system with 14 the total energy 424242 24 n2+ n2 ≤25 434343 1 2 34 The Partition Function of an Ideal Fermi Gas just of sts consier a system that Let’s considone single 1 ⎡niμ( − ε i)⎤ state of energy εi.