• 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 L1 and energy E1
         "Long  state" with length L2 and energy E2  with L1 < L2  and E1 < E2
  
       If n1 of the pieces are in the short state, and n2 in the long state (obviously with n1+n2=N), the 
       energy of the system will be E = n1 E1 + n2 E2 with total length L = n1 L1 + n2 L2.
  
       (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(-Hstates/kBT) 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 Σi Szi
  
       where Szi 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 = Σi K SiSi+1 + Q SiSi+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.
  

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