Physics 120/240

Homework 1

Due 4/19/06

This homework has a lot of overlap with introductory statistical mechanics that you've probably already seen, but it may very well bring a new perspective to it.

One important thing to be learned from this is an understanding of the applicability of the "Central Limit Theorem". Where it works and where it doesn't. Another thing is the relationship between what you already know and polymer statistics.

The last part of this assignment is to implement the ideas in this problem numerically. I encourage you to team up with others for some or all of this homework.

1. Consider $N$ noninteracting Ising spins $\sigma_1,\dots,\sigma_N$ where each spin is chosen with equal probabilities to have values $\pm 1$. The Hamiltonian (or energy) of this system is

$$E = h \sum_{i=1}^N \sigma_i$$

where $h$ is some parameter that can be thought of as a (negative) magnetic field. Using combinatorics (or the binomial distribution), calculate the number states that the system can have at energy $E$.

2. Define the entropy per spin $s \equiv S_{total}/N$ and the energy per spin as $e \equiv E/N$. Use the simplified Sterling's formula

$$n! \approx n^n e^{-n}$$

to find an expression form $s(e)$ in the limit of large $N$.

3. The result of the last problem is the entropy in the "microcanonical", or constant energy, ensemble. Let's do the calculation in the "canonical", or constant temperature, ensemble. Consider the same system but now held at constant temperature.

a

The partition function $Z$ for this system can be easily calculated. Do this by realizing that the spins are noninteracting so that the system breaks up into $N$ subsystems.

b

Now calculate the free energy $F$.

c

By the appropriate differentiation, calculate the energy $E$.

d

Recall that $F=E-TS$. From this and the answer to the last part, calculate $S$ as a function of $T$ (or $\beta$). Then find $s(e)$ as before. To do this, you'll have to invert a $\tanh$ which can be expressed in terms of a logarithm. Your result should be identical to the one found in the previous problem.

4.

a

Calculate $<E^2>$. The average is over all possible spin configurations all of which are weighted equally. This can be thought of several different ways. One is an average weighted by the probability distribution in problem 1. This isn't the easiest way to calculate this. Instead write $E$ in terms of the $\sigma_i$'s which leads to a double sum. Using independence of the spins the cross terms can be eliminated.

b

Calculate $<exp(cE)>$ where $c$ is an arbitrary constant. The easiest way to do this is similar to part a; if $E$ is expressed as a sum over spins, this can then be written as a product (similar to the calculation of the partition function). Using independence allows you to decouple all terms.

5. Find an expression for the probability distribution $P(E)$ near $E=0$ for large $N$. Do this by expanding out the logarithms found in problem 2. Hint: Make sure to go to 2nd order in the expansion of the logs! Your result should be a Gaussian.

6. Using the Gaussian approximation of problem 5, recalculate the averages of problem 4. The easiest way to do these is directly, for example

$$<E^2> = \int_{-\infty}^{\infty} E^2 P(E) dE.$$

Compare your results to the exact ones found in problem 4.

7. Do a rough sketch of

a

the exact $P(E)$ versus $E$, and also $\ln P(E)$ versus $E$,

b

the Gaussian approximation for both $P(E)$ versus $E$, and $\ln P(E)$ versus $E$.

Display the essential features, such as the width of the distribution and how the two cases differ.

8. Now explain qualitatively from the sketches in the last problem why the approximate results found in problem 6 either work well, or don't.

9. How do the last eight problems relate to the problem of a random walk? Find a mapping between a random walk and the spin system discussed above. How is the force related to the "spin" $\beta$ found before?

10. Calculate the probability distribution of a random walk this numerically as follows.

a

Write a program that generates a random sequences of $-1$'s and $1$'s and sums $N$ of them up at a time. You don't have to use the best random number generator in the assignment, for example, you can use the unix "rand" function. Be careful however not to do something like rand()%2. That is a big no no. Instead you want something that divides by the right big number to first create a random number that goes between $0$ and $1$. Use that to decide if the answer should be $1$ or $-1$. This (ran()) and many useful macros are defined in Josh's defs.h

b

Take the resulting sum and assign it to a "bucket" which takes the form of an array. You can allocate all the memory it uses in advance, or do so dynamically using a more general scheme such as Josh's hist.c and hist.h

c

After enough averages, you should be able to plot your histogram by writing it to a file and using gnuplot or some other graphing tool.