Fig. 1 Cluster flip: before. The region inside the dotted line is flipped in one Wolff move. Let this configuration be A.

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1 Physics 6562: Statistical Mechanics In Class Exercises Last correction at March 25, 2017, 1:37 pm c 2017, James Sethna, all rights reserved 1. Detailed balance. 2 In an equilibrium system, for any two states α and β with equilibrium probabilities ρ α and ρ β, detailed balance states (eqn 8.14) that P β α ρ α = P α β ρ β, (1) that is, the equilibrium flux of probability from α to β is the same as the flux backward from β to α. It is both possible and elegant to reformulate the condition for detailed balance so that it does not involve the equilibrium probabilities. Consider three states of the system, α, β, and γ. (a) Assume that each of the three types of transitions among the three states satisfies detailed balance. Eliminate the equilibrium probability densities to derive P α β P β γ P γ α = P α γ P γ β P β α. (2) Viewing the three states α, β, and γ as forming a circle, you have derived a relationship between the rates going clockwise and the rates going counter-clockwise around the circle. It is possible to show conversely that if every triple of states in a Markov chain satisfies the condition 2 then it satisfies detailed balance (that there is at least one probability density ρ which makes the probability fluxes between all pairs of states equal), except for complications arising when some of the rates are zero. (b) Suppose P is the transition matrix for some Markov chain satisfying the condition 2 for every triple of states α, β, and γ. Assume that there is a state α 0 with nonzero transition rates from all other states δ. Construct a probability density ρ δ that demonstrates that P satisfies detailed balance (eqn 1). (Hint: Assume you know ρ α 0 ; use eqn (1) to write a formula for ρ δ to ensure detailed balance for the pair. Solve for ρ α 0 to make the probability distribution normalized. Then use the cyclic condition eqn (2) to show that ρ satisfies detailed balance for any two states β and δ.)

2 2. Metropolis. (Mathematics, Computation) 1 The heat-bath algorithm described in the text thermalizes one spin at a time. Another popular choice is the Metropolis algorithm, which also flips a single spin at a time: (1) pick a spin at random; (2) calculate the energy E for flipping the spin; (3) if E < 0 flip it; if E > 0, flip it with probability e β E. Show that Metropolis satisfies detailed balance. Note that it is ergodic and Markovian (no memory), and hence that it will lead to thermal equilibrium. Is Metropolis more efficient than the heat-bath algorithm (fewer random numbers needed to get to equilibrium)? 3. Wolff. (Mathematics, Computation) 3 Near the critical point T c where the system develops a magnetization, any single-spinflip dynamics becomes very slow (the correlation time diverges). Wolff [7], improving on ideas of Swendsen and Wang [5], came up with a clever method to flip whole clusters of spins. Wolff cluster flips (1) Pick a spin at random, remember its direction D = ±1, and flip it. (2) For each of the four neighboring spins, if it is in the direction D, flip it with probability p. (3) For each of the new flipped spins, recursively flip their neighbors as in (2). Because with finite probability you can flip any spin, the Wolff algorithm is ergodic. As a cluster flip it is Markovian. Let us see that it satisfies detailed balance, when we pick the right value of p for the given temperature. (a) Show for the two configurations in Figs 1 and 2 that E B E A = 2(n n )J. Argue that this will be true for flipping any cluster of up-spins to down-spins. Fig. 1 Cluster flip: before. The region inside the dotted line is flipped in one Wolff move. Let this configuration be A.

3 The cluster flip can start at any site α in the cluster C. The ratio of rates Γ A B /Γ B A depends upon the number of times the cluster chose not to grow on the boundary. Let Pα C be the probability that the cluster grows internally from site α to the cluster C (ignoring the moves which try to grow outside the boundary). Then Γ A B = α P C α (1 p) n, (3) Γ B A = α P C α (1 p) n, (4) since the cluster must refuse to grow n times when starting from the up-state A, and n times when starting from B. Fig. 2 Cluster flip: after. Let this configuration be B. Let the cluster flipped be C. Notice that the boundary of C has n = 2, n = 6. (b) What value of p lets the Wolff algorithm satisfy detailed balance at temperature T? Unless you plan to implement the Wolff algorithm yourself (Exercise 8.9, download the Wolff simulation from the computer exercises section of the text web site [4]. Run at T = 2.3, using the heat-bath algorithm for a system or larger; watch the slow growth of the characteristic cluster sizes. Now change to the Wolff algorithm, and see how much faster the equilibration is. Also notice that many sweeps almost completely rearrange the pattern; the correlation time is much smaller for the Wolff algorithm than for single-spin-flip methods like heat bath and Metropolis. (See [3, sections 4.2 3] for more details on the Wolff algorithm.) 4. Symmetries and wave equations. 3 We can use symmetries and gradient expansions not only for deriving new free energies (Exercise 9.5), but also for directly deriving equations of motion. This approach (sometimes including fluctuations) has been successful in a number of systems that are strongly out of equilibrium [1, 2, 6]. In this exercise, you will derive the equation of motion for a scalar order parameter y(x, t) in a one-dimensional system. Our order parameter might represent the height of a string vibrating vertically, or the horizontal displacement of a one-dimensional crystal, or the density of particles in a one-dimensional gas.

4 Write the most general possible law. We start by writing the most general possible evolution law. Such a law might give the time derivative y/ t =... like the diffusion equation, or the acceleration 2 y/ t 2 =... like the wave equation, or something more general. If we take the left-hand side minus the right-hand side, we can write any equation of motion in terms of some (perhaps nonlinear) function G involving various partial derivatives of the function y(x, t): ( G y, y x, y t, 2 y x, 2 y 2 t, 2 y 2 x t,..., 7 y ) x 3 t,... = 0. (5) 4 Notice that we have already assumed that our system is homogeneous and time independent; otherwise G would explicitly depend on x and t as well. First, let us get a tangible idea of how a function G can represent an equation of motion, say the diffusion equation. (a) What common equation of motion results from the choice G(a 1, a 2,... ) = a 3 Da 4 in eqn 5? Restrict attention to long distances and times: gradient expansion. We are large and slow creatures. We will perceive only those motions of the system that have long wavelengths and low frequencies. Every derivative with respect to time (space) divides our function by a characteristic time scale (length scale). By specializing our equations to long length and time scales, let us drop all terms with more than two derivatives (everything after the dots in eqn 5). We will also assume that G can be written as a sum of products of its arguments that it is an analytic function of y and its gradients. This implies that f + g y ( ) 2 y t + h y x + i + + n 2 y = 0, (6) t t x where f, g,..., n are general analytic functions of y. (b) Give the missing terms, multiplying functions j(y), k(y),..., m(y). Apply the symmetries of the system. We will assume that our system is like waves on a string, or one-dimensional phonons, where an overall shift of the order parameter y y + is a symmetry of the system (Fig. 3). This implies that G, and hence f, g,..., n, are independent of y. Let us also assume that our system is invariant under flipping the sign of the order parameter y y, and to spatial inversion, taking x x (Fig. 4). More specifically, we will keep all terms in eqn 6 which are odd under flipping the sign of the order parameter and even under inversion. 1 1 All terms in the equation of motion must have the same dependence on a symmetry of the system. One could concoct inversion-symmetric physical systems whose equation of motion involved terms odd under inversion.

5 y(x,t) + y(x,t) Fig. 3 Shift symmetry. We assume our system is invariant under overall shifts in the order parameter field. Hence, if y(x, t) is a solution, so is y(x, t) +. y(x,t) y(x,t) y(x,t) y( x,t) Fig. 4 Flips and inversion. We assume our system is invariant under flipping (y y) and inversion (x x). Hence, if y(x, t) is a solution, so are y( x, t) and y(x, t). (c) Which three terms in eqn 6 are left after imposing these symmetries? Which one is not part of the wave equation 2 y/ t 2 = c 2 2 y/ x 2? This third term would come from a source of friction. For example, if the vibrating string was embedded in a fluid (like still air), then slow vibrations (low Reynolds numbers) would be damped by a term like the one allowed by symmetry in part (c). Systems with time inversion symmetry cannot have dissipation, and you can check that your term changes sign as t t, where the other terms in the wave equation do not. This third term would not arise if the vibrating string is in a vacuum. In particular, it is not Galilean invariant. A system has Galilean invariance if it is unchanged under boosts: for any solution y(x, t), y(x, t) + vt is also a solution. 2 The surrounding fluid stays at rest when our vibrating string gets boosted, so the resulting friction is not Galilean invariant. On the other hand, internal friction due to bending and flexing the string is invariant under boosts. This kind of friction is described by Kelvin damping (which you can think of as a dashpot in parallel with the springs holding the material together). (d) Show that your third term is not invariant under boosts. Show that the Kelvin damping term 3 y/ t x 2 is invariant under boosts and transforms like the terms in the wave equation under shifts, flips, and inversion. References [1] Hodgdon, J. A. and Sethna, J. P. (1993). Derivation of a general three-dimensional crack-propagation law: A generalization of the principle of local symmetry. Physical Review B, 47, This is a non-relativistic version of Lorentz invariance.

6 [2] Kardar, M., Parisi, G., and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Physical Review Letters, 56, [3] Newman, M. E. J. and Barkema, G. T. (1999). Monte Carlo methods in statistical physics. Oxford University Press. [4] Sethna, J. P. and Myers, C. R. (2004). Entropy, Order Parameters, and Complexity computer exercises: Hints and software. StatMech/ComputerExercises.html. [5] Swendsen, R. H. and Wang, J.-S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters, 58, 86. [6] Toner, J. and Tu, Y. (1995). Long-range order in a two-dimensional dynamical XY model: How birds fly together. Physical Review Letters, 75, [7] Wolff, U. (1989). Collective Monte Carlo updating for spin systems. Physical Review Letters, 62, 361.