Exam TFY4230 Statistical Physics kl Wednesday 01. June 2016

Transcription

1 TFY June 216 Side 1 av 5 Exam TFY423 Statistical Physics l Wednesday 1. June 216 Problem 1. Ising ring (Points: = 3) A system of Ising spins σ i = ±1 on a ring with periodic boundary conditions is defined by the Hamiltonian H = J σ i σ i+1 h σ i where i denotes a lattice site, and σ N+1 = σ 1. J is the strength of the nearest neighbor interaction between spins, and h is a uniform external magnetic field. The partition function for this system is given by Z = {σ i } e βh = e βg, where G is the Gibbs energy of the system. An explicit calculation yields Z = λ N + + λ N, where ] λ ± = e [cosh(ω) K ± sinh 2 (ω) + e 4K, where K βj and ω βh. Here, β 1/ B T, B is Boltzmann s constant, and T is temperature. Write out the sum in the partition function explicitly for N = 3, collecting all terms of equal Boltzmann weight e βh. Show that the magnetization m lim N (M/N) = lim N (1/N) N σ i of this system is given by the expression m = sinh(ω) sinh 2 (ω) + e 4K. Explain on physical grounds the difference in the results for J > and J <.

2 TFY June 216 Side 2 av 5 Consider now a slightly different model of Ising-spins on a ring with the following Hamiltonian H = [J 1 σ i σ i+1 + J 2 σ i σ i+2 ] where J 1 > is the interaction strength between nearest neighbor spins, and J 2 > is the interaction strength between next-nearest neighbor spins. There is no external magnetic field. Compute the Gibbs energy G for this system for N. From this, find the limiting values of G for low temperatures βj 1 1, βj 2 1. Give a physical explanation of the result. (Hint: Introduce the new spin variable τ i = σ i σ i+1 and use periodic boundary conditions on the τ i.)

3 TFY June 216 Side 3 av 5 Problem 2. Ideal gas in a 3D anharmonic trap (Points: 1+1+1=3) The canonical partition function Z for a system of N classical non-relativistic particles of equal mass m which are in thermal equilibrium with their surroundings and moving in three spatial dimension 3D in an anharmonic trap potential, is given by 1 Z = N!h 3N dr 1..dr N dp 1..dp N e βh = e βf where β = 1/ B T, B is Boltzmann s constant, T is temperature, h is Plan s constant, F = U T S is the Helmholz free energy, and the Hamiltonian H of the system is given by H = H i H i = p2 i 2m + α r i 3. Here, α is a dimensionful constant which gives the strength of the anharmonic trap-potential α r i 3. The 3D volume of the system to which the particles are confined is defined by a sphere of radius R, with volume V = 4πR 3 /3. The coordinates {r i } are all measured from the center of this sphere. Show that the partition function of the system is given by Z = 1 N! V N Λ 3N x 3βαV 4π h Λ 2πmB T. [ 1 e x x ] N Compute the internal energy U = H of the system for the limits 3βαV/4π 1 and 3βαV/4π 1. Explain your results on physical grounds in each case. Compute the pressure of the system for general values of 3βαV/4π. Consider then the limits 3βαV/4π 1 and 3βαV/4π 1 and compute the equation of state in these limits. Explain your results on physical grounds in each case. (Hint: Use F = U T S and T ds = du + pdv to express pressure p in terms of Z.) Useful formulae: U = 1 1 Z N!h 3N dr 1..dr N dp 1..dp N H e βh a d ν r F ( r ) = Ω ν dx x ν 1 e xν = 1 ν a ν dr r ν 1 F (r); du e u Ω ν = 2πd/2 Γ(d/2) ; Γ(z + 1) = z Γ(z)

4 TFY June 216 Side 4 av 5 Problem 3. d-dimensional Bose system (Points: 1+1+1=3) Ultra-relativistic non-interacting bosons moving in d dimensions have a Hamiltonian given by H = ε n ; n =, 1, 2,... ε = hc where c is the speed of light, h = h/2π with h Planc s constant, and is a wavenumber uniquely determining the single-particle bosonic states. The grand canonical partition function Z g for a system of non-interacting bosons is given by ln Z g = [ ] ln 1 e β(ε µ) = βpv Here, β = 1/ B T, B is Boltzmann s constant, T is temperature, p is pressure and V is the volume of the system. It can be shown (need not be shown!) that the density of states g(e) for this system is given by g(e) = where Θ(e) = 1, e ; Θ(e) =, e <. V 2π d/2 (2π hc) d Γ(d/2) ed 1 Θ(e) The average number of particles in the system is given by N = ln Z g / (βµ) = n where n is the average number of particles with wavenumber. The internal energy U is given by U = ε n. Show that N = 1 e β(ε µ) 1 U = ε e β(ε µ) 1 Introduce the density of states g(e) and the fugacity z = e βµ. Show that N = V βu d = V and thereby determine b l. Compute the ratio U/pV. l=1 l=1 l b l z l b l z l Compute the first and second virial coefficients E 1 (T ) and E 2 (T ) in the virial expansion for the internal energy, U = E 1 (T )ρ + E 2 (T )ρ Show that E 2 (T ) is a quantum effect, and give a physical explanation for the sign of E 2 (T ).

5 TFY June 216 Side 5 av 5 Useful formulae: F (ε ) = g(e) de g(e) F (e) δ(e ε ) Γ(z) = Γ(z + 1) = z Γ(z) dx x z 1 e x