Using the uniform random number generator provided by the computer system,

generate random numbers y distibuted as PY(y) = A y for 0 < y < 1 .

Generate sufficient number of these numbers to construct a histogram consistent with 

this distribution.

Two classical particles A and B are moving in a one-dimensional potential energy well 

   U(x) = q x2 / 2

where q is a constant. The particles do not interact with one another, apart from the 

limitation that they cannot occupy the same position so that xA < xB 

at all times. The particles are in contact with a heat bath at temperature T.



Construct a Monte Carlo simulation program using the Metropolis algorithm to obtain

the following distribution functions and the corresponding expectation values at 

equilibrium at several temperatures:

  P(xA)   ,  < xA >

  < xB >

  < xB - xA >

  P(UA)  ,  < UA + UB >

(Assume that the particles may move a small distance ±x0 at each step.)

1) Consider random numbers ri uniformly distributed within the interval  0 < r < 1 .
   Now, we generate a set of new random numbers x which are the sums of 10 random numbers r.
   We generate xk such that it has n values of r in common with xk-1 and n in common with xk+1.

   For example for n=2, some values of x would look like:

     x2 = r9  + r10 + r11 + r12 + r13 + r14 + r15 + r16 + r17 + r18

     x3 = r17 + r18 + r19 + r20 + r21 + r22 + r23 + r24 + r25 + r26

     x4 = r25 + r26 + r27 + r28 + r29 + r30 + r31 + r32 + r33 + r34 .


   Calculate the theoretical values for the correlations <xk xk+1> for 0 ≤ n ≤ 10.

   Construct a computer simulation of this process for n=2 and estimate the above correlation 
   numerically from N samples of x. Take N to be 102, 103, 104, 105, and 106.
   For each case, estimate the probable error in your result.

2) A random variable s can take only the values ±1. It is known that if k is even, sk+1 is equal to the
   value of sk with probability 0.8 and is equal to -sk with probability 0.2. If k is odd, then
   sk+1 is equal to the value of sk with probability 0.2 and is equal to -sk with probability 0.8.
   (Such distributions are relevant to physical systems with "staggered" magnetizations.)

   Construct a computer simulation to calculate the correlation <sk sk+m> with 0 ≤ m ≤ 16.
   Plot this correlation as a function of m. (Save these 17 values for your next homework.)

   Consider the 17 values obtained in the second part of Assignment 3. Suppose that the sk were random variables
   from a one-dimensional lattice of size N=32, with periodic boundary conditions. The correlations 
   Cm = <sk sk+m> then obey CN-m = Cm .

   Extend the correlation values thus to the range C0 to C31.

   Fourier transform these values to obtain the structure function corresponding to this correlation. 
   (It may be necessary to plot log |S(k)|2 versus k to see a better picture.) Interpret your result.

 Construct a program which determines the elements of a transfer matrix for a one dimensional
 Ising strip, composed of spins on a lattice of width m and length n (for a specific value
 of the coupling K, and for h=0). 

 The transfer matrix will be of size 2m by 2m.

 Then, determine the largest eigenvalue of this matrix to obtain the partition function Q(K).


 Numerically differentiate ln Q to obtain the unitless specific heat cv = K2 ∂2/∂K2 ln Q .
 (use ∂2/∂K2 f(k) ≈ [f(K+ΔK) + f(K-ΔK) - 2 f(K)]/(ΔK)2 )

 Plot the specific heat as a function of K for strips of witdh m=2 and m=3.

 If you do not have a routine for calculating eigenvalues, a relatively simple procedure is to start with a 
 random vector and multiply it with the matrix repeatedly a large number of times. The largest eigenvalue
 (and the corresponding eigenvector) will dominate:
 The recursion V' = M V will yield a better and better approximation to V ≈ Ψ>  and V' ≈ λ>Ψ>.
 It is a good idea to normalize V' before replacing it with V. The normalization factor will be the
 approximate value of λ>.

 Construct a program which carries out an RG transformation on a two dimensional Ising lattice, in such
 a way that the ferromagnetic as well as the anti-ferromagnetic ground states are conserved. This should
 enable you to analyze the model with negative, as well as positive values of the nearest coupling constant K,
 in the presence of a magnetic field h.

 I suggest that you use a Migdal bond-moving approximation with a scaling factor of b=3, but you may use any
 approximation, perhaps one of your own.

 Using this transformation, construct the phase diagram of this system in the K - h plane, and determine the
 critical exponents ν and β for the transitions.