Physics 562: Statistical MechanicsSpring 2002, James P. SethnaHomework 3, due Wednesday Feb. 27Latest revision: February 28, 2002, 9:11ReadingYeomans, chapter 1, 2.1-2.5, 3.1-3.4, 7.Optional Related ReadingDavid Chandler, “Introduction to Modern Statistical Physics”, chapter 8.Problems(3.1) Red and Green Bacteria (from Princeton’s qualifying exam, fall 1977.)A growth medium at time t = 0 has 500 red bacteria and 500 green bacteria. Each hour,each bacterium divides in two. A color-blind predator eats exactly 1000 bacteria per hour.(a) After a very long time, what is the probability distribution for the colors of the bacteriain the growth medium?(b) Roughly how long will it take to reach this final state?∗(c) Assume that the predator has a 1% preference for green bacteria. Roughly how muchwill this change the final distribution?(3.2) Random Walks in Grade Space.Let’s make a simple model of the prelim grade distribution in one of the large physicsservice courses. Let’s imagine a multiple-choice test of ten problems of ten points each.Each problem is identically difficult, and there is a probability of 0.6 for each question thatthe student will get it right. What is the expected mean and standard deviation for theexam? We typically aim for a mean of 70 and a standard deviation of 15. What physicalinterpretation do you make of the ratio of the random standard deviation and the one weaim for?†A typical course will have homework, two prelims, and a final, for a total of 500points. If all the points in the course came from ten-point problems and all students wereequally likely to answer each problem correctly with probability 0.7, what would the RMSstatistical fluctuations be for the course total?∗Within the accuracy of this question, you may assume either that one bacteriumreproduces and then one is eaten 1000 times per hour, or that at the end of each hour allthe bacteria reproduce and then 1000 are consumed.†Baseball teams, I’m told, have smaller win/loss fluctuations than would be expectedfrom statistical fluctuations. Presumably the “short answer” essay problems have muchsmaller fluctuations than the multiple choice.1(3.3) The Ising Model.You should already have a program ising, in the same directory that jupiter is in fromproblem set 1. Or, you can download it from links athttp://mum.physics.cornell.edu/sethna/teaching/sss/ising/ising.htm. The Ising Hamilto-nian isH = −Jij SiSj− HiSi, 3.5.1where Si= ±1 are “spins” on a square lattice, and the sumij is over the four nearest-neighbor bonds (each pair summed once). It’s conventional to set J = 1 and Boltzmann’sconstant kB= 1. which amounts to measuring energies and temperatures in units of J.As noted in class, the Ising model can be viewed as an anisotropic magnet with Sibeing2σzfor the spin at site i, or it can represent the occupancy of a lattice site (atom or noatom for a lattice gas simulation, copper or gold for a binary alloy, ...). Our simulationdoesn’t conserve the number of spins up, so it’s not a natural simulation for a lattice gas.You can think of it as a grand canonical ensemble, or as a model for a lattice gas on asurface exchanging atoms with the vapor above.(a) Phase diagram. Play with it. At high temperatures, the spins should not be stronglycorrelated. At low temperatures the spins should align all parallel, giving a largemagnetization. Can you roughly locate the phase transition? Can you see growingclumps of aligned spins as T → Tc+(i.e., T approaching Tcfrom above)? Draw arough phase diagram in the (H, T) plane, showing (i) the ”spin up” phase, (ii)the”spin down” phase, (iii) the paragnetic phase, (iv) the phase boundary between spinup and spin down, (v) the critical point, where at H = 0 the system develops anon-zero magnetization.(b) Correlations and Susceptibilities: Analytical. The partition function for theIsing model is Z =nexp(−βEn), where the states n run over all 2Npossibleconfigurations of the spins, and the free energy F = −kT log Z. Show that the averageof the magnetization M =iSiequals − (∂F/∂H) |T. Derive the formula relatingthe susceptibility χ =(∂M/∂H) |Tin terms of (M −M )2 = M2 −M 2. (Hint:remember our derivation of the formula (E −E )2 = kT2cv?)Notice that the program outputs, at each temperature and field, averages of several quan-tities: |M | , (M −M )2 , E , (E −E )2 . Unfortunately, E and M in these formulasare measured per spin, while the formulas in the class are measured for the system as awhole. I’ll write the total values in boldface, and the values per spin without boldface:you’ll need to multiply the squared quantities by the number of spins to make a compar-ison. To make that easier, change the system size to 100×100, using configure. Whileyou’re doing that, increase speed to ten or twenty to draw the spin configuration fewertimes. To get good values for these averages, equilibrate for a given field and temperature,“reset”, and then start averaging.(c) Correlations and Susceptibilities: Numerical. Check your formulas for part (b)at zero external field, for T = 3, by measuring the fluctuations and the averages,2and then changing H or T by a small amount and measuring the averages again.Check them for T = 2, making sure you don’t change the field so much as to flip themagnetization.(d) Low Temperature Expansion. What is the energy for flipping a spin antiparallel toits neighbors? Equilibrate at low temperatures (you may need to increase and decreasethe external field to avoid domain walls). Notice that the primary excitations are singlespin flips. In the low temperature approximation that the flipped spins are dilute (sowe may ignore the possibility that two flipped spins touch or overlap), write a formulafor the magnetization. Check it numerically; graph theory versus experiment.(e) High Temperature Expansion. At high temperatures, we can ignore the couplingto the neighboring spins. Calculate a formula for the susceptibility of a free spincoupled to an external field. Compare it to the susceptibility you measure at hightemperatures for the Ising model.Response Functions and the Fluctuation-Dissipation Theorem. The responsefunction χ(t) gives the change in magnetization due to an infinitesimal impulse in theexternal field. By superposition, we can use χ(t) to generate the linear response to anyexternal