Chapter 8, Spin dynamics Video Solutions, A Kinetic View of Statistical Physics

Problem 1

Consider the voter model in one dimension.
(a) Suppose that the initial state consists of two $+1$ voters in a sea of "uncommitted" voters, namely, voters that may be in either opinion state with the same probability: $S(x, t=0)=\delta_{x,-1}+\delta_{x, 1}$.
(i) Determine $S(x, t)$ for all integer $x$.
(ii) Show that $S(0, t)=2 e^{-t} I_{1}(t)$. Using the definition of the Bessel function $I_{1}(t)$ prove the validity of the (physically obvious) inequality $S(0, t)<1$.
(b) Consider now the "dipole" initial condition, $S(x, t=0)=\delta_{x, 1}-\delta_{x,-1}$, namely one $+1$ voter and one $-1$ voter in a sea of uncommitted voters. For this initial condition show that
$$
S(x, t)=\frac{2 x}{t} e^{-t} I_{x}(t)
$$

Laszlo Zalavari

Numerade Educator

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Problem 2

Use the correlation functions (8.12) for the voter model in the consensus criterion (8.15) to derive the asymptotic results quoted in (8.16).

Victor Salazar

Numerade Educator

02:40

Problem 3

This problem concerns the derivation of the exact solution for the pair correlation function in the voter model in arbitrary spatial dimension $d$.
(a) Deduce the leading asymptotic (8.23).
(b) Show that the strength of the source varies according to Eq. (8.25) in the longtime limit.
(c) Complete the derivation of Eq. (8.27). You may need to compute integrals of the form
$$
\int_{0}^{\infty} e^{-x} I_{0}(x) d x, \quad \int_{0}^{\infty} e^{-x} I_{1}(x) d x, \quad \int_{0}^{\infty} e^{-2 x} I_{0}(x) I_{1}(x) d x
$$
For these computations, you may find the following integral representations helpful:
$$
I_{0}(x)=\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{x \cos q} d q, \quad I_{1}(x)=\frac{1}{2 \pi} \int_{0}^{2 \pi} \cos q e^{x \cos q} d q
$$

Sanjeev Kumar

Numerade Educator

08:27

Problem 4

Investigate Ising-Glauber dynamics in the mean-field approximation. Let the number of spins be finite but very large.
(a) For zero-temperature dynamics determine $P_{M}(t)$, the probability to have $M$ up spins and $N-M$ down spins.
(b) Determine $P_{M}(t)$ for critical dynamics.
(c) In the low-temperature regime $\left(\beta_{c}<\beta<\infty\right)$, the distribution $P_{M}(t)$ is bimodal with peaks of width of the order of $\sqrt{N}$ around $M_{\pm}=\frac{1}{2} N\left(1 \pm m_{\infty}\right) .$ The system spends almost all the time near one of the peaks but occasionally makes transitions from one peak to the other. Estimate the transition time.

Amit Srivastava

Numerade Educator

12:04

Problem 5

Consider Ising-Glauber dynamics at zero temperature in one dimension with the antiferromagnetic initial state.(a) Compute the pair correlation function $G_{k}(t)$.
(b) Verify that the domain-wall density is given by $(8.60)$.
(c) Show that the density for domain-wall doublets is given by $\rho_{2}=e^{-2 t}\left[I_{0}(2 t)-\right.$ $\left.I_{1}(2 t)\right]$. Verify that its asymptotic behavior is the same as that for the uncorrelated initial condition, Eq. (8.61).

Saeeda Aman

Numerade Educator

00:57

Problem 6

Consider the one-dimensional Ising-Glauber model at zero temperature starting from the uncorrelated initial condition (with $m_{0}$ ).
(a) Write evolution equations for the three-spin correlation function $\left\langle s_{i} s_{j} s_{k}\right\rangle$, where $i \leq j \leq k$ but otherwise arbitrary.
(b) Verify that the solution to these equations is trivial: $\left\langle s_{i} s_{j} s_{k}\right\rangle=0$ (here the initial condition is important).
(c) Write down evolution equations for the four-spin correlation function $\left\langle s_{i} s_{j} s_{k} s_{\ell}\right\rangle$ where $i \leq j \leq k \leq \ell$
(d) Verify that the four-spin correlation function can be expressed via the pair correlation functions:
$$
\left\langle s_{i} s_{j} s_{k} s_{\ell}\right\rangle=\left\langle s_{i} s_{j}\right\rangle\left\langle s_{k} s_{\ell}\right\rangle+\left\langle s_{i} s_{\ell}\right\rangle\left\langle s_{j} s_{k}\right\rangle-\left\langle s_{i} s_{k}\right\rangle\left\langle s_{j} s_{\ell}\right\rangle
$$
(e) Show that the previous formula reduces to $(8.63)$ for four consecutive spins.

Manik Pulyani

Numerade Educator

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Problem 7

Show that the density of domain walls $\rho$ and the density of domain-wall doublets $\rho_{2}$ are related by $\dot{\rho}=-2 \rho_{2}$. Verify that the densities $(8.59)$ and (8.61) agree with $\dot{\rho}=-2 \rho_{2}$

Victor Salazar

Numerade Educator

05:53

Problem 8

Compute the asymptotic behavior of the density of domain-wall triplets defined in Eq. (8.62). Explain heuristically why both $\rho_{3}$ and $\left(\rho_{2}\right)^{2}$ decay as $t^{-3}$.

Issa Dababneh

Numerade Educator

08:28

Problem 9

Suppose that the Ising chain is endowed with the dynamics of Eq. (8.46) with parameters $\gamma=2, \delta=1$
(a) Show that this dynamics corresponds to zero temperature.
(b) Verify that both energy-increasing and energy-conserving spin flips are forbidden for this dynamics.
(c) Show that the system falls into a jammed state where the distances between adjacent domain walls is greater than the lattice spacing.
(d) Map the spin evolution onto a random sequential adsorption process. Show that the antiferromagnetic initial state maps onto an empty system, so the final density of domain walls can be read off the results from Chapter $7 .$
(e) Compute the final density of domain walls in the case when the initial state is uncorrelated.

Amit Srivastava

Numerade Educator

01:38

Problem 10

Consider the Ising chain in a non-zero magnetic field with zero-temperature Glauber dynamics. Solve for the density of down domains of length $n$, for the initial condition of alternating domains of up and down spins that are all of length $L$.

Penny Riley

Numerade Educator

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Problem 11

Starting with the master equation (8.73), check that the total density of domains $\rho=\sum_{k} P_{k}$ satisfies (8.71) and that the total domain length is conserved, $\sum_{k} k d P_{k} / d t=0$

Victor Salazar

Numerade Educator

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Problem 12

Starting with the master equation (8.73), check that the total density of domains $\rho=\sum_{k} P_{k}$ satisfies (8.71) and that the total domain length is conserved, $\sum_{k} k d P_{k} / d t=0$

Victor Salazar

Numerade Educator

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Problem 13

For the Ising model on the square lattice at a positive temperature, show that the spin-flip rate can be written in a form similar to $(8.79)$, with coefficients
$$
\frac{1}{4} \tanh (2 \beta J)+\frac{1}{8} \tanh (4 \beta J) \quad \text { and } \quad \frac{1}{4} \tanh (2 \beta J)-\frac{1}{8} \tanh (4 \beta J)
$$
instead of $3 / 8$ and $1 / 8$

Victor Salazar

Numerade Educator

01:44

Problem 14

Generalize Eq. (8.79) and derive the spin-flip rate for the Ising model on the cubic lattice $(d=3)$ with zero-temperature Glauber dynamics.

Lottie Adams

Numerade Educator

01:10

Problem 15

Consider the Ising model on an $L \times L$ square lattice. For free boundary conditions, compute the total number of stripe states that the system can reach by zero-temperature Glauber dynamics. (Note: the answer is related to the Fibonacci numbers.) Also, estimate the corresponding number of stripe states for periodic boundary conditions.

Hast Aggarwal

Numerade Educator

08:27

Problem 16

Study the Ising chain with zero-temperature Kawasaki dynamics.
(a) Write the evolution equation for the mean spin at site $i, S_{i}$.
(b) Show that the correlation functions obey an infinite hierarchy of equations.
(c) Solve for the domain-wall density for random initial conditions. Compare your result with that of Eq. (8.92) for the antiferromagnetic initial condition.
(d) Obtain the number of frozen configurations.

Amit Srivastava

Numerade Educator

05:50

Problem 17

Consider the Ising-Kawasaki chain at zero temperature. Consider a domain of up spins of length $L$ that contains a single down spin at position $x$. Compute the exit probability $E(x)$ for this diffusing down spin to reach the right boundary without touching the left boundary by the following steps:
(a) Determine the equations that are obeyed by the exit probability, paying particular attention to the cases $x=1$ and $x=L-1$.
(b) Solve these equations and show that $E(x)=(2 x-1) /(2 L-2)$ for all $0<$ $x<L$

Robin Corrigan

Numerade Educator

05:16

Problem 18

Consider zero-temperature Swendsen-Wang dynamics for the $q=\infty$ Potts model, in which only energy-lowering events can occur. In this case, the flipping of a domain causes it to merge with only one of its neighbors. Determine the density of domains and the domain length distribution.

Anthony Ramos

Numerade Educator