Physics 4488/6562: Statistical Mechanics Material for Week 2 Exercises due Monday Feb 5 Last

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1 Physics 4488/6562: Statistical Mechanics Material for Week 2 Exercises due Monday Feb 5 Last correction at January 26, 218, 11:5 am c 217, James Sethna, all rights reserved Pre-class Preparation All exercises are from Version 2. of the text: sethna/ StatMech/v2EntropyOrderParametersComplexity.pdf Wednesday Read: Chapter 2, Sec. 2.3 (Currents and forces) Pre-class question: 2.17: Local conservation (Submit electronically by 9:3 Tuesday evening.) Friday Read: Chapter 2, Sec. 2.4 (Solving: Fourier & Green) Pre-class question: 2.18: Absorbing boundary conditions (Submit electronically by 9:3 Thursday evening.) Monday Read: Chapter 3, Sec. 3.1 (Microcanonical) and 3.2 (Ideal Gas) Pre-class question: 3.13: Weirdness in high dimensions. (Submit electronically by 9:3 Sunday evening.) Exercises Those in 4488 may choose two of the four exercises. 2.5: Generating random walks. (Hints are available in Python and Mathematica at http: //pages.physics.cornell.edu/ sethna/statmech/computerexercises.html.) 8.4: Red and green bacteria, treated as a random walk in the number of red bacteria. Full credit for sensible arguments that get within a factor of two of the right answer. 2.11: Stocks, volatility, and diversification. (Hints are available in Python and Mathematica at sethna/statmech/computerexercises.html.) Class choose one 2.19: Random walks, generating functions, and diffusion. 2.2: Continuous time random walks: Ballistic to diffusive transition : Diffusion equation and universal scaling functions. (This exercise covers material in Chapter 12, which is largely independent of the other parts of the book, but is very sophisticated. To do the problem, you will at least need to read some of that chapter to find out what relevant, marginal, and irrelevant mean. Read more of the chapter to find out why universal scaling functions and power laws are important.)

2 In-class exercises 2.2 Photon diffusion in the Sun. (Astrophysics) i If fusion in the Sun turned off today, how long would it take for us to notice? This question became urgent some time back when the search for solar neutrinos failed. Neutrinos, created in the same fusion reaction that creates heat in the Solar core, pass through the Sun at near the speed of light without scattering giving us a current picture. The rest of the energy takes longer to get out. (The missing electron neutrinos, as it happened, oscillated into other types of neutrinos.) Most of the fusion energy generated by the Sun is produced near its center. The Sun is km in radius. Convection probably dominates heat transport in approximately the outer third of the Sun, but it is believed that energy is transported through the inner portions (say to a radius R = m) through a random walk of X-ray photons. (A photon is a quantized package of energy; you may view it as a particle which always moves at the speed of light c. Ignore for this exercise the index of refraction of the Sun.) There are a range of estimates for the mean free path for a photon in the Sun. For our purposes, assume photons travel at the speed of light, but bounce in random directions (without pausing) with a step size of l =.1cm = 1 3 m. About how many random steps N will the photon take of length l to get to the radius R where convection becomes important? About how many years t will it take for the photon to get there? Related formulæ: c = m/s; x 2 2Dt; s 2 n = nσ 2 = n s 2 1. There are π s in a year Modified random walk. i Random walks with a constant drift term will have a net correlation between steps. This problem can be reduced to the problem without drift by shifting to a moving reference frame. In particular, suppose we have a random walk with steps independently drawn from a uniform density ρ(l) on [,1), but with a non-zero mean l = l. Argue that the sums s N = N n (l n l) describe random walks in a moving reference frame, with zero mean. Argue that the variance of these random walks (the squared standard deviation) is the same as the variance (s N s N ) 2 of the original random walks.

3 2.15 Modified diffusion. i Photons diffusing in clouds are occasionally absorbed by the water droplets. Neutrons diffusing in a reactor, or algae diffusing in the sea, may multiply as they move. How would you modify the derivation of the diffusion equation in eqns ( ) to allow for particle non-conservation? Which equation in 2.9 should change? What would the new term in the diffusion equation look like? 2.16 Density dependent diffusion. i The diffusion constant can be density dependent; for example, proteins diffusing in a cell membrane are so crowded they can get in the way of one another. What should the diffusion equation be for a conserved particle density ρ diffusing with diffusion constant D(ρ)? (Hint: see footnote 18 on page 21.) 2.7 Periodic diffusion. 2 Day 3 Consider a one-dimensional diffusion equation ρ/ t = D 2 ρ/ x 2, with initial condition periodic in space with period L, consisting of a δ-function at every x n = nl: ρ(x, ) = n= δ(x nl). (a) Using the Green s function method, give an approximate expression for the the density, valid at short times and for L/2 < x < L/2, involving only one term (not an infinite sum). (Hint: How many of the Gaussians are important in this region at early times?) (b) Using a Fourier series, 1 give an approximate expression for the density, valid at long times, involving only two terms (not an infinite sum). (Hint: How many of the wavelengths are important at late times?) (c) Give a characteristic time τ in terms of L and D, such that your answer in (a) is valid for t τ and your answer in (b) is valid for t τ. Day 3 1 You can use a Fourier transform, but you will find ρ(k, ) is zero except at the values k = 2πm/L, where it is a δ-function.

4 2.6 Fourier and Green. 2 An initial density profile ρ(x, t = ) is perturbed slightly away from a uniform density ρ, as shown in Fig. 1. The density obeys the diffusion equation ρ/ t = D 2 ρ/ x 2, where D =.1 m 2 /s. The lump centered at x = 5 is a Gaussian exp( x 2 /2)/ 2π, and the wiggle centered at x = 15 is a smooth envelope function multiplying cos(1x). (a) Fourier. As a first step in guessing how the pictured density will evolve, let us consider just a cosine wave. If the initial wave were ρ cos (x, ) = cos(1x), what would it be at t = 1 s? Related formulæ: ρ(k, t) = ρ(k, t ) G(k, t t ); G(k, t) = exp( Dk 2 t). (b) Green. As a second step, let us check how long it would take to spread out as far as the Gaussian on the left. If the wave at some earlier time t were a δ-function at x =, ρ(x, t ) = δ(x), what choice of the time elapsed t would yield a Gaussian ρ(x, ) = exp( x 2 /2)/ 2π for the given diffusion constant D =.1 m 2 /s? Related formulæ: ρ(x, t) = ρ(y, t )G(y x, t t ) dy; G(x, t) = (1/ 4πDt) exp( x 2 /(4Dt))..4 ρ(x, t = ) - ρ Fig. 1 Initial profile of density deviation from average. (c) Pictures. Now consider time evolution for the next ten seconds. The initial density profile ρ(x, t = ) is as shown in Fig. 1. Which of the choices (A) (E) represents the density at t = 1 s? (Hint: Compare t = 1 s to the time t from part (b).) Related formulæ: x 2 2Dt..4 ρ(x, t = 1) - ρ (A)

5 .4 ρ(x, t = 1) - ρ (B) ρ(x, t = 1) - ρ (C) ρ(x, t = 1) - ρ (D) ρ(x, t = 1) - ρ (E)