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A Kinetic View of Statistical Physics

Pavel L. Krapivsky, Sidney Redner, Eli Ben-Naim

Chapter 6

Fragmentation - all with Video Answers

Educators

Chapter Questions

01:45

Problem 1

6.1 Obtain the leading asymptotic behavior of the moments for the random scission model.

Carson Merrill
Carson Merrill
Numerade Educator
01:02

Problem 2

Solve the random scission model for an initial exponential length distribution and show that this distribution remains exponential.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:49

Problem 3

6.3 Consider the homogeneous breakup kernel $R(x)=1$, which has the homogeneity index $\lambda=0$, and hence lies on the borderline between scaling and shattering.
(a) Show that for the monodisperse initial condition $c(x, t=0)=\delta(x-\ell)$, the fragment mass distribution is
$$
c(x, t)=e^{-r} \delta(x-\ell)+\frac{2 t e^{-t}}{\ell} \sum_{n \geq 0} \frac{[2 t \ln (\ell / x)]^{n}}{n !(n+1) !}
$$
(b) Express the series solution from (a) in terms of the modified Bessel function $I_{1}[2 \sqrt{2 t \ln (\ell / x)}]$
(c) Show that the moments of the mass distribution are given by the expression $M_{n}=\ell^{n} \exp [(1-n) t /(1+n)]$

Amany Waheeb
Amany Waheeb
Numerade Educator
03:01

Problem 4

Obtain the steady-state solution of the master equation (6.1b) for arbitrary breakup crate $R(x) \neq 0$

Srikar Pasumarthy
Srikar Pasumarthy
Numerade Educator
03:17

Problem 5

Consider the three-dimensional generalization of rectangular fragmentation from Section 6.2. Assume that the initial fragment is a $1 \times 1 \times 1$ cube and that each crack nucleates a planar cut in any of the three mutually orthogonal directions.
(a) Show that the Mellin transform satisfies
$$
\frac{\partial c\left(s_{1}, s_{2}, s_{3}, t\right)}{\partial t}=\left[\frac{2}{3}\left(\frac{1}{s_{1}}+\frac{1}{s_{2}}+\frac{1}{s_{3}}\right)-1\right] c\left(s_{1}+1, s_{2}+1, s_{3}+1, t\right)
$$
(b) Seek the asymptotic behavior of the Mellin transform in the form $c\left(s_{1}, s_{2}, s_{3}, t\right) \sim t^{-\alpha\left(s_{1}, s_{2}, s_{3}\right)}$ and determine the exponent $\alpha\left(s_{1}, s_{2}, s_{3}\right)$.
(c) Verify that the volume distribution admits a scaling form $F(V, t) \simeq t^{2} e^{-V I}$, again the same as in the random scission model.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
23:27

Problem 6

Consider the geometric fragmentation process in which a rectangle splits into four smaller fragments, as illustrated below.
(a) Verify that the Mellin transform $c\left(s_{1}, s_{2}, t\right)$ obeys
$$
\frac{\partial c\left(s_{1}, s_{2}, t\right)}{\partial t}=\left(\frac{4}{s_{1} s_{2}}-1\right) c\left(s_{1}+1, s_{2}+1, t\right)
$$
(b) Check that area conservation is satisfied by the above master equation and show that the average number of fragments is equal to $1+3 t$.
(c) Using the equation for $c\left(s_{1}, s_{2}, t\right)$ from (a) and the asymptotic behavior $c\left(s_{1}, s_{2}, t\right) \sim t^{-\alpha\left(s_{1}, s_{2}\right)}$, show that
$$
\alpha\left(s_{1}, s_{2}\right)=\left[\left(s_{1}+s_{2}\right)-\sqrt{\left(s_{1}-s_{2}\right)^{2}+16}\right] / 2
$$
(d) Show that the area distribution admits a scaling form
$$
F(A, t) \simeq t^{-2} f(A t), \quad f(z)=6 \int_{0}^{1} d \xi\left(\xi^{-1}-1\right) e^{-z / \xi}
$$
(e) Using the results from (c) show that the area distribution weakly diverges in the limit of small area, $f(z) \simeq 6 \ln (1 / z)$. Verify that in the opposite limit the scaled area distribution exhibits an exponential decay reminiscent of the onedimensional random scission model, but with an algebraic correction: $f(z) \simeq$ $6 z^{-2} \exp (-z)$ as $z \gg 1$

Chris Trentman
Chris Trentman
Numerade Educator
03:50

Problem 7

Consider a three-state system with non-zero transition rates $w_{i \rightarrow j} \neq 0$ for all $i \neq j$ What condition or conditions must these rates meet for the steady state to satisfy detailed balance $(6.45) ?$

Leon Druch
Leon Druch
Numerade Educator
09:59

Problem 8

Consider combined polymerization/fragmentation in the continuum limit, where the polymer length is a continuous variable.
(a) Show that the master equation for this continuum model is
$$
\begin{aligned}
\frac{\partial c(\ell)}{\partial t}=& \int_{0}^{\ell} c(x) c(\ell-x) d x-2 c(\ell) \int_{0}^{\infty} c(x) d x \\
&+\lambda\left[2 \int_{\ell}^{\infty} c(x) d x-\ell c(\ell)\right]
\end{aligned}
$$
(b) Verify that the stationary solution to this master equation is given by $c(\ell)=$ $\lambda e^{-\ell / \ell_{0}}$ and determine $\ell_{0}$.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
04:49

Problem 9

Extend the previous problem to determine the stationary length distribution of closed strings by combined aggregation/fragmentation. In a fragmentation event of a closed string, two cuts at random locations are made and the two resulting segments immediately form closed loops. In aggregation event, two loops join to create a larger loop whose length equals the sum of these initial lengths.
(a) Argue that the master equation is
$$
\begin{aligned}
\frac{\partial c(\ell)}{\partial t}=& \int_{0}^{\ell} x(\ell-x) c(x) c(\ell-x) d x-2 \ell c(\ell) \int_{0}^{\infty} x c(x) d x \\
&+\lambda\left[2 \int_{\ell}^{\infty} x c(x) d x-\ell^{2} c(\ell)\right]
\end{aligned}
$$
The first two terms on the right-hand side account for aggregation events while the last two terms account for fragmentation events.
(b) Show that the above master equation admits the stationary solution $c=$ $(\lambda / \ell) e^{-\ell / \ell_{0}}$
(c) Show that the same length distribution follows from the detailed balance condition.

Michael Nartey
Michael Nartey
Numerade Educator
01:38

Problem 10

Show that for the chipping model the fragment density is equal to $N=\lambda / 2$ in the gel phase $(\lambda<1)$, thus establishing the second line of $(6.56)$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:27

Problem 11

Verify the expressions (6.59) and (6.61) for the mass density of the chipping model in the sol and gel phases.

Ahmed Ali
Ahmed Ali
Numerade Educator
06:19

Problem 12

6.12 Consider the collisional fragmentation model from Example $6.4$, where one of the two colliding particles splits in two.
(a) Show that near the shattering transition, the Poisson form $(6.65)$ for mass density approaches a Gaussian distribution.
(b) Show that the typical mass $m_{*}$ (corresponding to the peak of the mass distribution) decays to zero as
$$
m_{*} \sim(1-t)^{\sigma} \quad \text { with } \quad \sigma=2 \ln 2 \doteq 1.386294
$$

Philomena Marfo
Philomena Marfo
Numerade Educator
02:21

Problem 13

Consider the collisional fragmentation model from Example 6.5, where the larger of the two colliding particles splits in two.
(a) Show that in terms of the cumulative densities, Eq. (6.66) simplifies to
$$
\frac{d A_{n}}{d t}=2 A_{n-1}^{2}-A_{n}^{2}
$$
(b) Normalizing the cumulative distribution, $F_{n}(\tau)=N^{-1} A_{n}(t)$, recast the master equation from (a) into
$$
\frac{d F_{n}}{d \tau}=2 F_{n-1}^{2}-F_{n}^{2}-F_{n}
$$
(c) Argue that the master equation for $F$ admits a traveling wave solution $F_{n}(\tau) \rightarrow$ $f(n-v \tau) .$ Substitute this wave form into the above equation for $F$ to deduce
$$
v \frac{d}{d x} f(x)=f(x)+f^{2}(x)-2 f^{2}(x-1)
$$
(d) Show that the boundary conditions for the above equation are $f(-\infty)=1$ and $f(\infty)=0 .$ Try to guess the approach to final asymptotics in the $x \rightarrow \pm \infty$ limits.
(e) To determine the speed $v$, seek the asymptotic behavior of $f(x)$ far behind the front in the form $1-f(x) \sim e^{\lambda x}$ as $x \rightarrow-\infty$. Substitute this form into the equation given in part (c) to deduce the "dispersion" relation $v=\left(3-4 e^{-\lambda}\right) / \lambda$ between the speed $v$ and the decay coefficient $\lambda .$ Plot $v=v(\lambda)$ for $\lambda>0$ and show that the spectrum of possible velocities is bounded from above: $v \in$. $\left(-\infty, v_{\max }\right]$
(f) Out of the spectra of possible velocities $v \in\left(-\infty, v_{\max }\right]$, the maximal value is selected. Show that $v=v_{\max } \fallingdotseq 1.52961$
(g) Show that the typical mass decays to zero according to Eq. (12) with $\sigma=$ $v \ln 2 \doteq 1.06024$

Sana Riaz
Sana Riaz
Numerade Educator
04:50

Problem 14

Consider a collisional fragmentation model where a randomly chosen collision partner splits stochastically, namely a particle of masses $m$ splits into two fragments of mass $m^{\prime}$ and $m-m^{\prime}$ with $m^{\prime}$ chosen uniformly in $[0, m] .$
(a) Show that the mass density $c(m, \tau)$ satisfies
$$
\frac{\partial}{\partial \tau} c(m, \tau)=-c(m, \tau)+2 \int_{m}^{\infty} \frac{d m^{\prime}}{m^{\prime}} c\left(m^{\prime}, \tau\right)
$$
(b) Solve the master equation from (a) using the Mellin transform. Show that for the initial condition $c(m, 0)=\delta(m-1)$, the mass distribution is
$$
c(m, \tau)=e^{-\tau} \delta(m-1)+e^{-\tau} \sqrt{\frac{2 \tau}{\ln (1 / m)}} I_{1}[\sqrt{8 \tau \ln (1 / m)}]
$$
with $I_{1}$ the modified Bessel function.
(c) Show that, aside from the logarithmic factor, $c(m, t) \sim 1-t$ near the shattering transition. (Essentially the same behavior was found for collisional fragmentation with a deterministic splitting rule.)

Rehmat Kazmi
Rehmat Kazmi
Numerade Educator