8.333: Statistical Mechanics I Problem Set # 4 Due: 11/13/13 Non-interacting particles

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1 8.333: Statistial Mehanis I Problem Set # 4 Due: 11/13/13 Non-interating partiles 1. Rotating gas: Consider a gas of N idential atoms onfined to a spherial harmoni trap in three dimensions, i.e. the partiles are subjet to the Hamiltonian N [ ] 2 pi K 2 H = + r i. 2m 2 (a) Show that the angular momentum of eah partile L i i = ir i ip i is a onserved quantity, { i } i.e. L i,h = 0. (Note that one interations between partiles are inluded, individ- ual angular momenta are no longer onserved, while their sum L i = onserved quantity.) L N L i i remains a (b) Sine angular momentum is onserved, a generalized anonial distribution an be defined with probability 1 p[µ {ip i,ir i }] = exp βh(µ) βω i L i (µ). Z(β, Ω) i Compute the lassial partition funtion for this gas of idential partiles, assuming Ω i = J Ωẑ, with Ω < K/m. () Find the expetation value of angular momentum (L z ) in the above ensemble. (d) Write down the probability density of finding a partile at loation (x,y,z), and hene obtain the expetation values x, y, and z. 2. Moleular adsorption: N diatomi moleules are stuk on a metal surfae of square symmetry. Eah moleule an either lie flat on the surfae in whih ase it must be aligned to one of two diretions, x and y, or it an stand up along the z diretion. There is an energy ost of ε > 0 assoiated with a moleule standing up, and zero energy for moleules lying flat along x or y diretions. (a)how manymirostateshavethesmallestvalueof energy? Whatisthelargestmirostate energy? (b) For miroanonial marostates of energy E, alulate the number of states Ω(E,N), and the entropy S(E,N). 1

2 () Calulate the heat apaity C(T) and sketh it. (d) What is the probability that a speifi moleule is standing up? (e) What is the largest possible value of the internal energy at any positive temperature? 3. Curie suseptibility: Consider N non-interating quantized spins in a magneti field ib = Bẑ, and at a temperature T. The work done by the field is given by BM z, with a L N magnetization M z = µ m i. For eah spin, m i takes only the 2s +1 values s, s + 1,,s 1,s. (a)calulatethegibbs partitionfuntion Z(T,B). (Notethatthe ensemble orresponding to the marostate (T,B) inludes magneti work.) (b) Calulate the Gibbs free energy G(T,B), and show that for small B, Nµ 2 s(s +1)B 2 G(B) = G(0) + O(B 4 ). 6k B T () Calulate the zero field suseptibility χ = M z / B B=0, and show that is satisfies Curie s law χ = /T. (d) Show that C B C M = B 2 /T 2 where C B and C M are heat apaities at onstant B and M respetively. 4. Langmuir isotherms: An ideal gas of partiles is in ontat with the surfae of a atalyst. (a) Show that the hemial potential of the gas partiles is related to their temperature [ J and pressure via µ = k B T ln P/T 5/2 + A 0, where A 0 is a onstant. (b) If there are N distint adsorption sites on the surfae, and eah adsorbed partile gains an energy ǫ upon adsorption, alulate thegrandpartitionfuntion for the two dimensional gas with a hemial potential µ. () In equilibrium, the gas and surfae partiles are at the same temperature and hemial potential. Show that the fration of oupied surfae sites is then given by f(t,p) = P/ P + P 0 (T). Find P 0 (T). 2

3 (d)inthegrandanonial ensemble, thepartilenumber N is a random variable. Calulate its harateristi funtion (exp( ikn)) in terms of Q(βµ), and hene show that m G (N m ) = (k B T) m 1, µ m T where G is the grand potential. (e) Using the harateristi funtion, show that (N) N 2 = k B T. µ T (f) Show that flutuations in the number of adsorbed partiles satisfy N 2 1 f =. (N) 2 Nf 5. Moleular oxygen has a net magneti spin, S i, of unity, i.e. S z is quantized to -1, 0, or +1. The Hamiltonian for an ideal gas of N suh moleules in a magneti field B i I ẑ is N [ 2 pi ] i H = µbsi z, 2m where {pi i } are the enter of mass momenta of the moleules. The orresponding oordinates {iq i } are onfined to a volume V. (Ignore all other degrees of freedom.) (a) Treating {pi i,iq i } lassially, but the spin degrees of freedom as quantized, alulate the Gibbs partition funtion, Z(T,N,V,B). (b) What are the probabilities for S i z of a speifi moleule to take on values of -1, 0, +1 at a temperature T? L N () Find the average magneti dipole moment, (M)/V, where M = µ Si z. (d) Calulate the zero field suseptibility χ = (M)/ B B=0. 6. One dimensional polymer: Consider a polymer formed by onneting N dis shaped moleules into a one dimensional hain. Eah moleule an align either along its long axis 3

4 (of length2a) or short axis(length a). The energy of the monomer aligned along itsshorter axis is higher by ε, i.e. the total energy is H = εu, where U is the number of monomers standing up. (a) Calulate the partition funtion, Z(T,N), of the polymer. (b) Find the relative probabilities for a monomer to be aligned along its short or long axis. () Calulate the average length, (L(T,N)), of the polymer. (d) Obtain the variane, L(T,N) 2. (e) What does the entral limit theorem say about the probability distribution for the length L(T,N)? 7. Polar rods: Consider rod shaped moleules with moment of inertia I, and a dipole moment µ. The ontribution of the rotational degrees of freedom to the Hamiltonian is given by p φ H rot. = p θ + µeosθ, 2I sin 2 θ where E is an external eletri field. (φ [0,2π], θ [0,π] are the azimuthal and polar angles, and p φ, p θ are their onjugate momenta.) (a) Calulate the ontribution of the rotational degrees of freedom of eah dipole to the lassial partition funtion. (b) Obtain the mean polarization P = (µosθ), of eah dipole. () Find the zero field polarizability P χ T =. E E=0 (d) Calulate the rotational energy per partile (at finite E), and omment on its high and low temperature limits. (e) Sketh the rotational heat apaity per dipole. 8. (Optional) Disordered glass: The heatapaity of many disordered materialsvanishes linearly at low temperatures. A ommonly used model suh glassy materials materials is 4

5 a olletion of N non-interating defets in thermal equilibrium at temperature T. Eah defet is assumed to have two possible energies ǫ i and ǫ i + δ i, with different values of ǫ i and δ i for eah defet. (a) Compute the partition funtion Z(T), the average energy E(T), and the ontribution to heat apaity C(T) from these independent defets. (b) The number of defets with exitation energies between δ and δ+dδ is given by ρ(δ)dδ, where ρ(δ) is the density of states of defets as a funtion of exitation energy. Assuming that ρ(δ)isuniformly distributedbetween energiesof0andδ, find thedefet heat apaity C(T), and omment on its behavior at low and high temperatures. () A uniform density of states may not be realisti. What feature of ρ(δ) will ensure C T at low temperatures? 9. (Optional) Classial virial theorem: Let X = {qi i,ip i }denote anyofthe6n oordinates in phase spae, and onsider any funtion f(x). (a) Show that in a anonial ensemble governed by a Hamiltonian H(X) where β = 1/(k B T). f H = β f, X i X i (b) Find the forms of the virial theorem obtained by substituting f = q i and f = p i in the general expression. 5

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