Fall 2016 Homework Assignments

**Homework 1: (Due Thursday, October 20)**1. A polymer is composed of a chain of N chemical pieces. Each piece can be in one of two states: "Short state" with length L

_{1}and energy E_{1}"Long state" with length L_{2}and energy E_{2}with L_{1}< L_{2}and E_{1}< E_{2}If n_{1}of the pieces are in the short state, and n_{2}in the long state (obviously with n_{1}+n_{2}=N), the energy of the system will be E = n_{1}E_{1}+ n_{2}E_{2}with total length L = n_{1}L_{1}+ n_{2}L_{2}. (a) The total length L of the polymer is then associated with its energy E. What is this relationship? (b) Find the entropy S(L) of the system when the total length of the polymer is L. (Hint: First find out in how many states can the system be in at that length.) (c) Differentiate S with respect to E to associate the temperature T with E. (i.e., find E(T) ) (d) In parts (a), (b) and (c) you have calculated S for a fixed E (the micro cannonical ensemble). Now, use the canonical ensemble, i.e. sum exp(-H_{states}/k_{B}T) over all states of the system to obtain the partition function, and obtain the energy from that. (e) Check the limits for small and large temperatures and see that you get expected results. 2. Consider a collection of N fixed, independent (i.e. not interacting with one another) spin-1/2 particles under the effect of an external magnetic field B in the z-direction. The Hamiltonian is then given by H = - B Σ_{i}S_{z}^{i}where S_{z}^{i}is the z-component of the spin operator of the i'th spin. Find the entropy and the energy of the system as a function of temperature and B.**Homework 2: (Due Monday, October 24)**Consider a one dimensional Ising spin system in which nearest as well as next-nearest neighbor spin interactions exist. The Hamiltonian of the system is then given by -β H = Σ

_{i}K S_{i}S_{i+1}+ Q S_{i}S_{i+2}where K and Q are the coupling constants for the nearest and the next-nearest interactions respectively. Find the transfer matrix associated with the partition function of this system.**Homework 3: (Due Monday, November 21)**Problem 3.8.6 in the texbook.

**Homework 4: (Due Thursday, December 29)**Using the Migdal bond-moving approximation, construct a Renormalization Group transformation for the two-state Ising model on a hexagonal lattice. Find the critical value of the nearest neighbor interaction and the critical exponents α, β, δ and γ predicted by your transformation. Compare these values with "well established" values.

Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.