Page 376 - Numerical Methods for Chemical Engineering

Problems 365

Problems

7.A.1. Consider a game involving two six-sided dice. Plot the probability of observing each
total value 2–12 if the dice are fair. How many times, on average, must one roll the dice
beforea7is observed? How many times, on average, can one roll the dice before a 2, “snake
eyes,” is observed?

7.A.2. Consider again the log normal random walk model of the spot (market) price S(t)of
an asset,

(7.269)
dS = µSdt + σ SdW t
Consider a stock with its price sampled each business day (assume 252 business days per
year), with time measured in years. Let the (annual) drift rate be 8%, and let the volatility of
the stock be 20%. Write a program that generates typical characteristic random trajectories
over a year, starting from a price of 100. Make a histogram of the final stock price after one
year.
7.A.3. The central limit theorem of statistics states that the sum of many independent random
variables tends towards a normal distribution without regard to the shape of the underlying
distribution. Consider the statistic X that is a sum of N random variables u j , each uniformly
distributed on [0, 1],
N

X = u j (7.270)
j=1
For large N, compute the distribution of X and fit it to a normal distribution. As N increases,
does P(X) indeed approach a normal distribution? Repeat the calculations for Y defined
as
N

Y = w j u j (7.271)
j=1
where the w j are random weights determined at the beginning of the calculation from a
uniform distribution on [0, 1]. Does P(Y) also approach a normal distribution?

7.A.4. Consider again the 2-D Ising lattice model, for which a Monte Carlo simulation
program has been provided. Using this program, you are asked here to investigate the
nature of the order–disorder transition as a function of the external field H and thermal
energy k b T. Simulate a 100 × 100 lattice, with µ = 1, J = 1. In the absence of an external
field, H = 0, the order–disorder transition occurs at the Curie point k b T c ≈ 2.269 J (for an
infinite lattice). Using the provided Monte Carlo program, compute the average magnetic
order parameter as a function of H at various temperatures above and below the Curie point.
Plot the order parameter for a very small field H > 0 as a function of temperature and
describe the qualitative nature of the transition. Then, plot the order parameter vs. H at
several temperatures. How does the transition from positive to negative H change above and
below the Curie point?