Chapter 5, Aggregation Video Solutions, A Kinetic View of Statistical Physics

Problem 1

5.1 Consider constant-kernel aggregation with the initial mass distribution $c_{j}(0)=0$ for $j \geq m+1$ and with $m>4$. Deduce the asymptotic behavior of the cluster mass distribution by generalizing the method of Example $5.2$ (page 144$)$ as follows:
(a) Although it is not possible to compute the roots of the generating function explicitly, show that the initial generating function $\mathcal{C}(z, t=0)$ can be written as
$$
\mathcal{C}\left(z_{1}, 0\right)=c_{1}(0) z_{1}+c_{2}(0) z_{1}^{2}+\cdots+c_{m}(0) z_{1}^{m}=1+M(0) \epsilon+\mathcal{O}\left(\epsilon^{2}\right)
$$
where $c_{1}(0)+c_{2}(0)+\cdots+c_{m}(0)=1$ is the initial cluster density and $c_{1}(0)+$ $2 c_{2}(0)+\cdots+m c_{m}(0)=M(0)$ is the initial mass density. Here $z_{1}=1+\epsilon$ is the smallest root of $\mathcal{C}\left(z_{1}, 0\right)$
(b) Use the above result for $\mathcal{C}(z, t=0)$ to show that the generating function can be written as
$$
\mathcal{C}(z, t) \simeq \frac{1}{t^{2} M(0)} \frac{1}{1-z / z_{1}}
$$
where $z_{1}=1+1 /[M(0) t]$ is the smallest root in the generating function. Expand this expression in a power series in $z$ and show that the scaling form for the mass distribution is given by Eq.

Victor Salazar

Numerade Educator

04:03

Problem 2

Solve for the cluster concentrations $c_{k}(t)$, for constant-kernel aggregation with the initial condition $c_{k}(0)=2^{-k}$.

Narayan Hari

Numerade Educator

08:48

Problem 3

Show that the third and fourth moments of the cluster mass distribution for productkernel aggregation with the monomer-only initial condition are given by
$$
M_{3}(t)= \begin{cases}(1-t)^{-3}, & \text { for } \quad t<1 \\ e^{2 g t}\left(e^{g t}-t\right)^{-3}, & \text { for } \quad t>1\end{cases}
$$
and
$$
M_{4}(t)= \begin{cases}(1+2 t)(1-t)^{-5}, & \text { for } \quad t<1 \\ \left(e^{4 g t}+2 t e^{3 g t}\right)\left(e^{g t}-t\right)^{-5}, & \text { for } \quad t>1\end{cases}
$$
More generally, show that near the gel point the moments $M_{n}$ diverge as
$$
M_{n} \simeq \frac{2^{n-2} \Gamma(n-3 / 2)}{\Gamma(1 / 2)}|1-t|^{-(2 n-3)}
$$

Sagar Singh

Numerade Educator

09:01

Problem 4

Consider sum-kernel aggregation in which $K_{i j}=i+j$.
(a) Show that the master equations are
$$
\frac{d c_{k}}{d t}=\frac{k}{2} \sum_{i+j=k} c_{i} c_{j}-k c_{k}\left(k M_{0}+1\right)
$$
where the mass density equals one: $M_{1}=\sum_{k \geq 1 k} c_{k}=1$
(b) Write the rate equations for the moments. Show that for the monomer-only initial conditions the first few moments are $M_{0}=e^{-t}, M_{2}=e^{2 t}, M_{3}=3 e^{4 t}-$ $2 e^{3 t}, M_{4}=15 e^{6 t}-20 e^{5 t}+6 e^{4 t}$
(c) Solve the above master equations for the monomer-only initial conditions and obtain the exact solution
$$
c_{k}(t)=\frac{k^{k-1}}{k !}\left(1-e^{-t}\right)^{k-1} e^{-t} e^{-k\left(1-e^{-t}\right)}
$$

Keshav Singh

Numerade Educator

01:17

Problem 5

Consider the island growth model in the absence of input in which $A_{1}+A_{k} \rightarrow A_{k+1}$, where the rate of each reaction equals 1, except for the reaction between monomers, $A_{1}+A_{1} \rightarrow A_{2}$, whose rate equals 2 (why?). Suppose that the system initially contains only monomers.
(a) Introduce an auxiliary time variable that linearizes the master equations for this reaction.
(b) Determine the island size distribution at infinite time.
(c) Show that the final state is approached exponentially in time.

Carson Merrill

Numerade Educator

07:44

Problem 6

Consider the addition process of the previous problem but assume that the re kernel is $K_{i j}=i \delta_{j, 1}+j \delta_{i, 1}$. Show that
$$
c_{k}(t)=\left[\left(1-e^{-t}\right)^{k-1}-k^{-1}\left(1-e^{-I}\right)^{k}\right]\left(2-e^{-t}\right)^{-k}
$$

Anthony Ramos

Numerade Educator

05:41

Problem 7

5.7 Use the scaling approach to determine the mass distribution for constant-kernel aggregation; that is, solve Eq. (5.59) for the scaling function.

Laurent Bergeron

Numerade Educator

02:37

Problem 8

$5.8$ Consider constant-kernel aggregation with steady monomer input. $^{18}$
(a) Show that the monomer density is given by
$$
c_{1}=\frac{1}{2}\left[\frac{t}{\cosh ^{2} t}+\tanh t\right]
$$
(b) Solve Eqs (5.64) recursively and derive the formal solution (for $k>1$ )
$$
c_{k}(t)=\frac{1}{\cosh ^{2} t} \int_{0}^{t} d t^{\prime} \cosh ^{2} t^{\prime} \sum_{i+j=k} c_{i}\left(t^{\prime}\right) c_{j}\left(t^{\prime}\right)
$$

Vikash Ranjan

Numerade Educator

07:12

Problem 9

5.9 Consider sum-kernel aggregation with input.
(a) Verify that the total density of clusters is given by $N(t)=\int_{0}^{t} d t^{\prime} e^{\left(t^{\prime 2}-t^{2}\right) / 2}$. Show that $N(t)$ initially grows, reaches a maximum, and then decays to zero as $N \simeq t^{-1}$ when $t \gg 1$
(b) Show that the density of monomers also initially increases, then decreases, and asymptotically decays as $c_{1} \simeq t^{-1}$.
(c) Show that the density of dimers decays as $c_{2} \simeq t^{-3} .$
(d) Verify that generally $c_{k} \simeq A_{k} t^{-(2 k-1)}$ in the long-time limit.
(e) Find the recursion for the amplitudes $A_{k}$ and show that these amplitudes form the sequence A088716 from The On-Line Encyclopedia of Integer Sequences.

Khoobchandra Agrawal

Numerade Educator

01:00

Problem 10

lnvestgate proudct-kernel aggregation witn input.
(a) Show that in the pre-gel regime, the second moment is $M_{2}=\tan t$. Use this solution to argue that gelation occurs at $t_{g}=\pi / 2$.
(b) Show that $N=t-t^{3} / 6$ in the pre-gel regime.
(c) Show that $M_{3}=\tan t+\frac{2}{3} \tan ^{3} t$ in the pre-gel regime.
(d) Show that the monomer density is $c_{1}(t)=\int_{0}^{t} d t^{\prime} e^{\left(t^{\prime 2}-t^{2}\right) / 2}$ for all times.

Ajay Singhal

Numerade Educator

03:45

Problem 11

Starting with the evolution equations (5.87) for the island growth model, determine the time dependence of the island density $I(t)$ and the monomer density $c_{1}(t)$ by the following steps:
(a) Show that the island density monotonically increases with time and that $c_{1} \rightarrow 0$ as $t \rightarrow \infty$. Hint: Suppose the opposite; namely, $c_{1} \rightarrow$ const. as $t \rightarrow \infty$, and show that this behavior leads to a contradiction.
(b) Using $c_{1} \rightarrow 0$ for $t \rightarrow \infty$ determine the asymptotic solution for the master equation for $c_{1}$. Use this result to derive Eq. (5.88).

Sid Wan

University of Louisville

01:18

Problem 12

Analyze the improved continuum description (5.93) of the island growth model.
(a) Show that this approach cures the singularity of the wave solution (5.92), namely, the singular behavior is replaced by a finite-width peak.
(b) Determine the height of the peak, that is, the density $c_{k}(\tau)$ when $k=\tau$.
(c) Estimate the width of the peak.

Liuxi Sun

Numerade Educator

08:27

Problem 13

$5.13$ Investigate the island growth model with unstable dimers, namely each dimer can break into two mobile adatoms with rate $\lambda$, while all larger islands are stable.
(a) Write and solve the master equations for this process, and show that asymptotically $I=F^{3 / 4}(4 t / \lambda)^{1 / 4}, c_{1}=F^{1 / 4}(4 t / \lambda)^{-1 / 4}$, and $c_{2}=F^{1 / 2}(4 t \lambda)^{-1 / 2}$
(b) Show that the maximal island density, which is reached at the end of the submonolayer regime, scales with $F$ as $I_{\max } \sim F^{1 / 2}$

Amit Srivastava

Numerade Educator

02:51

Problem 14

5.13 Investigate the island growth model with unstable dimers, namely each dimer can break into two mobile adatoms with rate $\lambda$, while all larger islands are stable.
(a) Write and solve the master equations for this process, and show that asymptotically $I=F^{3 / 4}(4 t / \lambda)^{1 / 4}, c_{1}=F^{1 / 4}(4 t / \lambda)^{-1 / 4}$, and $c_{2}=F^{1 / 2}(4 t \lambda)^{-1 / 2}$
(b) Show that the maximal island density, which is reached at the end of the submonolayer regime, scales with $F$ as $I_{\max } \sim F^{1 / 2}$
5.14 Assume that islands of mass $\geq n$ are stable, while lighter islands are unstable.
(a) Show that $c_{1} \sim t^{-1 /(n+1)}$.
(b) Show that $c_{n} \sim t^{-(n-1) /(n+1)}$.
(c) Show that $I_{\max } \sim F^{(n-1) /(n+1)}$

Sana Riaz

Numerade Educator

02:10

Problem 15

5.15 Consider "greedy" exchange in which the larger of the two interacting clusters always gains a monomer, while the smaller loses a monomer. Symbolically, the reaction is $(j, k) \rightarrow(j+1, k-1)$ for $j \geq k$.
(a) Show that the evolution of the system is described by
$$
\dot{c}_{k}=c_{k-1} \sum_{j=1}^{k-1} c_{j}+c_{k+1} \sum_{j \geq k+1} c_{j}-c_{k} N-c_{k}^{2}
$$
(b) Verify that the total density of clusters obeys $d N / d t=-c_{1} N$.
(c) Show that in the continuum limit the master equations reduce to
$$
\begin{aligned}
\frac{\partial c(k, t)}{\partial t} &=-c_{k}\left(c_{k}+c_{k+1}\right)+N\left(c_{k-1}-c_{k}\right)+\left(c_{k+1}-c_{k-1}\right) \sum_{j \geq k} c_{j} \\
& \simeq 2 \frac{\partial c}{\partial k}\left[\int_{k}^{\infty} d j c(j)\right]-N \frac{\partial c}{\partial k}
\end{aligned}
$$
(d) Use the scaling ansatz $c_{k} \simeq N^{2} \mathcal{C}(k N)$ to separate, in the continuum limit, the master equations into
$$
\frac{d N}{d t}=-\mathcal{C}(0) N^{3}, \quad \mathcal{C}(0)\left[2 \mathcal{C}+x \mathcal{C}^{\prime}\right]=2 \mathcal{C}^{2}+\mathcal{C}^{\prime}\left[1-2 \int_{x}^{\infty} d y \mathcal{C}(y)\right]
$$
Solve these equations. Hint: Use the equation for $\mathcal{B}(x)=\int_{0}^{x} d y \mathcal{C}(y)$ to show that the scaled distribution $\mathcal{C}(x)=\mathcal{B}^{\prime}(x)$ coincides with the zero-temperature Fermi distribution,
$$
\mathcal{C}(x)= \begin{cases}\mathcal{C}(0), & x<x_{*} \\ 0, & x \geq x_{*}\end{cases}
$$
Determine $x_{*}$ using "mass" conservation.

Surendra Kumar

Numerade Educator