California State University Bakersfield Physics Questions Exam Practice I have a 3 question assignment for statistical physics. the book is Pathria statist

California State University Bakersfield Physics Questions Exam Practice I have a 3 question assignment for statistical physics. the book is Pathria statistical mechanics. 5. Let the H = Ho + V be the Hamiltonian of a classical macrosopic system.
(A)(10 pts) Show that the following inequality holds
F < Fo + (V), (9) where F is the free energy of the system with Hamiltonian H and Fo is the free energy with Hamil- tonian Ho. The average (...)o denotes the average over the canonical distribution of the system with Ho. This inequality is called the Gibbs-Bogoliubov inequality. (B)[10 pts) We can use the inequality above to estimate F: Suppose that we can compute Fo and (V)o, then we can estimate the upper-bound of F. Thus, the minimum of Fo + (V)o provides a good approximation to F. This is the idea of the variational approximation. Let us consider N non-interacting anharmonic oscillators, whose single particle Hamiltonian is 2 1 H= 1 + kx² + 2m 2 (10) where m, k, and a are positive constants. Let us define Ho pa 2m +-Ka?, (11) Choose K to obtain the best estimate of F. 6. Consider the Ising model on a 3-dimensional cubic lattice with N sites, -BH[{0;}] :£ ??, ?? ?; + ????. (12) i=1 Here Kij BJ when i and j are nearest neighbor sites on the cubic lattice, and zero otherwise. Also, h = BH, where H is the applied magnetic field, and J is the Ising exchange coupling. We want to apply the results of problem 1 of this exam, to show that mean-field theory can be viewed as the best "one-body" approximation to the exact partition function, i.e. we use the "non-interacting" Hamiltonian - BHo[{0;}] = x;di. (13) N i=1 The Gibbs-Bogoliubov inequality can be written in this case as (14) F[Kij,h] = -kBT In Z[Kij,h] < Fo[{X;}] + (H - Ho), where Fo[{X;}] is the free energy associated with Ho. (A) [7 pts) Work out explicitly the partition function ??[{X;}] = ?... ???Ho, (15) 01=+1 on=+1 and (H – Holo. (B) [3 pts) Find an equation for the set of parameters {X;}, in terms of Kij and h, that best approximate F[Kij, h], according to the Gibbs-Bogoliubov inequality above. (C) [7 pts) Let us denote by FMFT[Kij, h] the value of the right-hand-side of Eq. (14) for the choice of the parameters {X;} in part (B). Derive a self-consistent equation for the magnetization a m =-B ah - FMFt[Kij, h], (16) in terms of Kij and h. (D) [3 pts ] Consider the case when h = 0. Compute the mean-field transition temperature TMFT. 2 +3mw?s?, 4. Consider an ideal bose gas consisting of N particles confined in a 2D harmonic potential. The single particle Hamiltonian is pa 1 Wii) (6) 2m 2 and the total Hamiltonian is ?n = L; (i) (A) (5 pts) Show that the density of states as a function of energy e can be written as ?? D(E)= (7) (kbT)2' where kb is the Boltzmann's constant, and T is the temperature. What is y? (Note: Measure the energy from the ground state and ignore the zero-point energy, i.e. the single particle energy eigenvalues are e(nz, ny) = ?w(nr + ny), where ne and ny are non-negative integers.] (B) (7 pts) Find the number of particle N as a function of the chemical potential, (T). [You may leave your answer in terms of a dimensionaless integral, which you don't have to evaluate.] Show that the integral is bounded from above, implying that the system undergoes a Bose-Einstein condensation. Find To as a function of N (C) (8 pts] Show that the number of particles occupying the one-particle ground state approaches zero as No(T) = N [1 – (T/T.)"], when T + Tc from below. Find a. Purchase answer to see full attachment