Physics 562: Statistical Mechanics Spring 2002, James P. Sethna Prelim, due Wednesday, March 13 Latest revision: March 22, 2002, 10:9 Open Book Exam Work on your own for this exam. You may consult your notes, homeworks and answer keys, books and published work, or Web pages as you find useful. The problems have been designed to be doable given only material already presented in the course. If you find something in the literature or on the Web that is particularly helpful (e.g., solvesthe problem), feel free to use it: but, just as in a publication, give a citation. (P.1) Pollen and Hard Squares. Problems Q B b Square pollen grain in fluid of oriented square molecules, next to a wall. The red lines represents the exclusion region around the pollen grain and away from the wall. We model the entropic attraction between a pollen grain and a wall using a two-dimensional ideal gas of classical indistinguishable particles as the fluid. For convenience, we imagine that the pollen grain and the fluid are formed from square particles lined up with the axes of the box, of length B and b, respectively. We assume no interaction between the ideal gas molecules (unlike in the problem set); the potential energy is infinite, though, if the gas molecules overlap with the pollen grain or with the wall. The container as a whole has one pollen grain, N gas molecules, and total area L L. 1
Assume the pollen grain is close to one wall. Let the distance from the surface of the wall to the closest face of the pollen grain be Q. (A similar square-particle problem with interacting small molecules was studied by D. Frenkel and A. A. Louis, Phase separation in a binary hard-core mixture. An exact result, Phys. Rev. Lett. 68, 3363 (1992).) (a) What is the area A(Q 0) available for the gas molecules, in units of (length) 2,when the pollen grain is far from the wall? What is the overlap of the excluded regions, A( ) A(Q), when the pollen grain touches the wall, Q =0? Giveaformulafor A(Q) as a function of Q. (b) What is the configuration-space volume Ω(Q) for the gas, in units of (length) 2N? What is the configurational entropy of the ideal gas, S(Q)? (c) Your answers to part (b) can be viewed as giving a free energy for the pollen grain after integrating over the gas degrees of freedom (also known as a partial trace, or coarse-grained free energy). What is the resulting free energy of the pollen grain, F(Q)? Is the force on the pollen grain positive (away from the wall) or negative? How does the force compare to the ideal-gas pressure on the far side of the pollen grain? 2
(P.2) Entropy Loss on Cycling through the Glass Transition. Specific heat of B 2 O 3 glass measured while heating and cooling. The glass was first rapidly cooled from the melt (500 C 50 C in a half hour), then heated from 33 C 345 C in 14 hours (solid curve with squares), cooled from 345 Cto room temperature in 18 hours (dotted curve with diamonds), and finally heated from 35 C 325 C (solid curve with crosses). Data from S. B. Thomas and G. S. Parks, J. Phys. Chem. 35, 2091 (1931); revisited in Nonequilibrium Entropy and Entropy Distributions, S. A. Langer, E. R. Grannan, and J. P. Sethna, Phys. Rev. B 41, 2261 (1990). The specific heat of glasses is not only time-dependent at cryogenic temperatures; it is also time dependent near the glass transition temperature. The experimentalist above is making the natural assumption (technically untrue) that the specific heat of the glass is 3
the measured heat flow out of the glass divided by the temperature change of the heat bath. (a) Consider the second cooling curve and the final heating curve, from 325 Ctoroom temperature and back. Assume that the liquid at 325 C is in equilibrium, and the glass configuration is frozen (does not leak heat) during the time at room temperature. Must the integral under these two curves be equal to one another? Why or why not? (b) Under the same conditions, must the net entropy flow from the heat bath be the same on heating and cooling? If not, which one must be larger on thermodynamic grounds, and why? Argue from the shape of the curves that thermodynamics is valid. The fact that the energy lags the temperature near the glass transition, in linear response, leads to the study of specific heat spectroscopy, (N. O. Birge and S. R. Nagel, Phys. Rev. Lett. 54, 2674 (1985).) (P.3) Shannon Entropy. An obscure language A bç! for long-distance communication has only three sounds: a hoot represented by A, a slap represented by B, and a click represented by C. In a typical message, hoots and slaps occur equally often (p =1/4), but clicks are twice as common (p = 1/2). Assume the messages are otherwise random. (a) What is the Shannon entropy in this language? More specifically, what is the Shannon entropy rate (entropy per sound, or letter, transmitted)? (b) Give a bound on how many bits per letter on average would need to be transmitted across a A bç! communication channel. (c) Find a compression scheme (a rule that converts a A bç! message to zeros and ones, that can be inverted to give back the original message) that is optimal, in the sense that it satisfies the bound you derived in part (b). 4
(P.4) Polymers and Random Walks. Polymers are long molecules, typically made of identical small molecules called monomers that are bonded together in a long, one-dimensional chain. When dissolved in a solvent, the polymer chain configuration often forms a good approximation to a random walk. Typically, neighboring monomers will align at relatively small angles: several monomers are needed to lose memory of the original angle. Instead of modeling all these small angles, we can produce an equivalent problem focusing all the bending in a few hinges: we approximate the polymer by an uncorrelated random walk of straight segments several monomers in length. The equivalent segment size is called the persistence length. (a) If the persistence length to bending of DNA is 50nm, with 3.4Åper nucleotide base pair, what will the root-mean-square distance R 2 be between the ends of a gene in solution with 100,000 base pairs, if the DNA is accurately represented as a random walk? Polymers are not accurately represented as random walks, however. Random walks, particularly in low dimensions, often intersect themselves. Polymers are best represented as self-avoiding random walks: the polymer samples all possible configurations that does not cross itself. (Greg Lawler, in the math department here, is an expert on self-avoiding random walks.) Let s investigate whether avoiding itself will change the basic nature of the polymer configuration. In particular, does the end-to-end typical distance continue to scale with the square root of the length L of the polymer, R L? (b) Two dimensional self-avoiding random walk. Give a convincing, short argument explaining whether or not a typical, non-self-avoiding random walk in two dimensions will come back after large numbers of monomers and cross itself. BU java applet. Run the Java applet linked to at http://polymer.bu.edu/java/java/saw/saw.html. They model a 2-dimensional random walk as a connected line between nearest-neighbor neighboring lattice points on the square lattice of integers. They start random walks at the origin, and discard them when they hit the same lattice point twice. As long as they survive, they average the squared length as a function of number of steps. (c) Measure for a reasonable length of time, print out the current state, and enclose it with your exam. Did the simulation give R L? If not, what s the estimate that your simulation gives for the exponent relating R to L? How does it compare with the two-dimensional exponent given at the Web site? The B3 machines apparently do not have Java enabled on their browsers, but CIT runs several computer facilities, see http://www.cit.cornell.edu/labs/labs.html. If you have trouble getting the Web site to work, let me know Monday two days before the exam is due. 5
(P.5) Microcanonical and Canonical Distributions. A defect in a crystal has one on-center configuration with energy zero, and M off-center configurations with energy ɛ, with no significant quantum tunneling between the states. The Hamiltonian can be approximated by the (M +1) (M +1)matrix H = 0 0 0 0 ɛ 0 0 0 ɛ There are N defects in the crystal, which can be assumed stuck in position (and hence distinguishable) and assumed not to interact with one another. (a) Write the canonical partition function Z(T ), the mean energy E(T ), the fluctuations in the energy, the entropy S(T ), and the specific heat C(T ) as a function of temperature. Plot the specific heat per defect C(T )/N for M = 6; set the unit of energy equal to ɛ and k B =1foryourplot. (b) Derive a simple relation between M and the change in entropy between zero and infinite temperature. Check this relation using your formula for S(T ). (P.6) Entropy Increases. We saw that entropy technically doesn t increase for a closed system, for any Hamiltonian, either classical or quantum. However, we can show that entropy increases for most of the coarse-grained effective theories that we use in practice: when we integrate out degrees of freedom, we provide a means for the information about the initial condition to be destroyed. Here you ll show that entropy increases for the diffusion equation. Diffusion Equation Entropy. Let ρ(x, t) obey the one-dimensional diffusion equation ρ/ t = D 2 ρ/ x 2. Assume that the density ρ and all its gradients die away rapidly at x = ±. Derive a formula for the time derivative of the entropy S = k B ρ(x)logρ(x) dx and show that it strictly increases in time. (P.7) Markov Chain. A physicist, testing the laws of chance, flips a coin repeatedly until it lands tails. (a) Treating the two states of the physicist ( still flipping and ( done )) as states in ρflipping a Markov process. The current probability vector then is. Write the ρ done transition matrix P, giving the time evolution P ρ n = ρ n+1, assuming that the coin is fair. (b) Find the eigenvalues and right eigenvectors of P. Which eigenvector is the steady state ρ? Call the other eigenvector ρ. For convenience, normalize ρ so that its first component equals one. (c) Assume an arbitrary initial state is written ρ 0 = Aρ + B ρ. What are the conditions on A and B needed to make ρ 0 a valid probability distribution? Write ρ n as a function of A and B, ρ and ρ. 6