Physics 562: Statistical MechanicsSpring 2002, James P. SethnaHomework 4, due Wednesday, April 3Latest revision: April 12, 2002, 11:23ReadingDavid Chandler, “Introduction to Modern Statistical Mechanics”, chapter 8.Pathria, chapters 6 & 7.Feynman, “Lectures on Physics” volume III, chapter 4Problems(4.1) Solving the Diffusion Equation. (Optional: for those for whom Fourier andGreens function methods are unfamiliar.)If needed: Matthews and Walker, Mathematical Methods of Physics, Chapter 8.4 p. 242-245 (Diffusion Equation).Consider a one-dimensional diffusion equation ∂ρ/∂t = D∂2ρ∂x2, with initial conditionperiodic in space with period L, consisting of a δ function at every xn= nL: ρ(x, 0) =∞n=−∞δ(x − nL).(a) Using the Greens function method, give an approximate expression for the the density,valid at short times and for −L/2 <x<L/2, involving only one term (not an infinitesum). (Hint: how many of the Gaussians are important in this region at early times?)(b) Using the Fourier method, give an approximate expression for the density, valid atlong times, involving only two terms (not an infinite sum). (Hint: how many of thewavelengths are important at late times?)(c) Give a characteristic time τ in terms of L and D, such that your answer in (a) is validfor t  τ and your answer in (b) is valid for t  τ.1(4.2) Coarse-Grained Magnetic Dynamics.If needed: Matthews and Walker, Mathematical Metho ds of Physics, Chapter 12 (Calculusof Variations).A one-dimensional magnet above its critical point is described by a free energy densityF[M]=(K/2)(∇M)2+(B/2)M2where M(x) is the variation of the magnetization with position along the single coordinatex. The average magnetization is zero, and the total free energy of the configuration M(x)is F [M ]=F[M]dx.The methods we developed in class to find the correlation functions and susceptibilitiesfor the diffusion equation can be applied with small modifications to this (mathematicallymore challenging) magnetic system.(a) Calculate the equilibrium equal-time correlation function for the magnetization,C(r, 0) = M (r, 0)M(0, 0).You’ll want to know that the Fourier transform ∞∞eikx/(1 + a2k2) dk =(π/a)exp(−|x|/a).(b) Assume the magnetic order parameter is not conserved, and is overdamped, so the timederivative of M is given by the inverse viscosity η times the variational derivativeof the free energy: ∂M/∂t = −ηδF/δM. M evolves in the direction of the totalforce on it. Here the average is over all future evolutions given the initial condition.Calculate the Greens function for M, G(r, t) giving the time evolution of an initialcondition M (r, 0) = G(r, 0) = δ(r). (Hint: You can solve this with Fourier transformsas in class.)(c) Using the Onsager regression hypothesis and your answer to part (a), calculate thespace-time correlation function C(r, t)=M(r, t)M(0, 0). (This part is a challenge:your answer will involve the error function.) If it’s convenient, plot it for short timesand for long times: does it look like exp(−|y|) in one limit and exp(−y2)inanother?Calculate the susceptibility χ(r, t)fromC(r, t) (Chandler p. 257).2(4.3) Phonons and Photons are Bosons.Phonons and photons are the elementary, harmonic excitations of the elastic and electro-magnetic fields. We’ve seen in an earlier problem that phonons are decoupled harmonicoscillators, with a distribution of frequencies ω. A similar analysis shows that the Hamil-tonian of the electromagnetic field can be decomposed into harmonic normal modes calledphotons.This problem will explain why we think of phonons and photons as particles, instead ofexcitations of harmonic modes.(a) Calculate the partition function for a quantum harmonic oscillator of frequency ω.(b) Calculate the grand canonical partition function for bosons multiply filling a singlestate with energy ¯hω.Forµ = 0, show that the grand canonical partition functionfor bosons is the same as that for a quantum harmonic oscillator up to a shift in thearbitrary zero of the total energy of the system.The Boltzmann filling of a harmonic oscillator is therefore the same as the Bose-Einsteinfilling of bosons into a quantum state, except for an extra shift in the energy of ¯hω/2. Thisextra shift is called the zero point energy.(c) A system has N harmonic modes with frequencies ωk. Show that the partition functionis the same as that of a grand-canonical system with zero point energyNk=1¯hωk/2and N distinguishable, non-interacting boson states with energy ¯hωkand chemicalpotential µk=0.Often, the label for the N states or modes in part (c) is the wavevector k (e.g., for phononsin crystals and photons in vacuum or crystals). In this case, instead of thinking of thesystem as N bosons each with one state, we can think of it as one boson with N possiblestates, with a dispersion relation ω(k). Real, massive bose particles like He4in free spacehave a similar dispersion relation Ek=¯h2k2/2m: phonons and photons are thus Boseparticles with dispersion relation Ek=¯hωk.(d) Can phonons or photons Bose condense at low temperatures? Why not?Be careful not to get confused: Bose atoms in a harmonic potential can Bose condense(next problem), but a harmonic vibrational excitation of a collection of atoms cannot. Theformer has a fixed number of bosons with a variety of (equally spaced) available energystates: the latter is viewed as a variable number of bosons occupying a single energy state.3(4.4) Bose Condensation in a Parabolic Potential.“Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor”, M.H. Anderson,J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science 269, 198 (1995).http://jilawww.colorado.edu/bec/Wieman and Cornell in 1995 were able to get a dilute gas of rubidium-87 atoms to Bosecondense. (The first author was the post-doc Anderson: Ensher and Matthews were thegraduate students.)(a) Is rubidium-87 a boson or a fermion?(b) At their quoted maximum number density of 2.5 × 1012/cm3,atwhattemperatureTpredictcdo you expect the onset of Bose condensation in free space? They claim thatthey found Bose condensation starting at a temperature of Tmeasuredc= 170nK. Is thatabove or below your estimate?Bose-Einstein Condensation at 400, 200, and 50 nano-Kelvin, from Jila (above reference).The pictures are spatial distributions 60ms after the potential is removed; the field of viewof each image is 200µm