Chapter 13, Diffusive reactions Video Solutions, A Kinetic View of Statistical Physics

Problem 1

Fill in the details of solving for the survival probability of a diffusing particle in the absorbing interval $[0, L]$
(a) First solve the diffusion equation in one dimension, subject to the boundary conditions $c(0, t)=c(L, t)=0$ and the initial condition $c(x, t=0)=$ $\delta\left(x-x_{0}\right)$ and thereby obtain the particle concentration given in Eq. (13.9).(b) Next, integrate over all final points of the particle and average over all starting positions to arrive at the integral expression for the survival probability given in Eq. (13.11).

Nathan Silvano

Numerade Educator

06:47

Problem 2

Apply the Laplace method to the integral in Eq. (13.11) and show that the long-time asymptotic behavior is indeed given by (13.12).

Aidan Mcnabb

Numerade Educator

01:13

Problem 3

Consider a diffusing particle that starts at the origin of an absorbing sphere of radius a. Using spherical symmetry and separation of variables, show that the solution to the radial equation has the form of Eq. (13.17). Compute the amplitudes $A_{n}$. Show that, in three dimensions, the Bessel functions can be expressed via trigonometric functions.

Raj Bala

Numerade Educator

02:38

Problem 4

Simulate aggregation in one dimension assuming that the diffusion coefficient varies as the inverse cluster mass, $D_{k}=1 / k$. Show that the cluster density decays as $t^{-1 / 3}$ in agreement with heuristic arguments, Eq. (13.46) at $v=1$.

Madi Sousa

Numerade Educator

02:52

Problem 5

Compute the integral in Eq. (13.72) and verify the results of Eq. (13.73).

Mary Wakumoto

Numerade Educator

01:38

Problem 7

Consider aggregation with a spatially localized source. Complete the discussion in the text for a general spatial dimension and show the total number of clusters grows asymptotically according to
$$
\mathcal{N} \sim \begin{cases}t, & d>4 \\ t / \ln t, & d=d^{c}=4 \\ t^{(d-2) / 2}, & 2<d<4 \\ \ln t, & d=d_{c}=2 \\ \text { const., } & d<2\end{cases}
$$

Manik Pulyani

Numerade Educator

01:53

Problem 8

Study aggregation in the half-space $x>0$ with absorption on the plane $x=0$. (The true spatial dimension is $d=3$, so that mean-field rate equations hold.)
(a) Show that the cluster density $N(x, t)$ satisfies the reaction-diffusion equation
$$
\frac{\partial N}{\partial t}=D \nabla^{2} N-K N^{2}
$$
with boundary condition $N(x=0, t)=0$ and initial condition $N(x, t=0)=$ 1 for $x>0$
(b) Argue that the solution must have a scaling form and that the proper scaling ansatz is $N(x, t) \simeq(K t)^{-1} f(\xi)$, with $\xi \equiv x / \sqrt{D t}$.
(c) Show that the ansatz from (b) reduces the governing partial differential equation, namely, the reaction-diffusion equation in (a), to an ordinary differential equation
$$
f^{\prime \prime}+\frac{1}{2} \xi f^{\prime}+f(1-f)=0
$$
(d) Determine the scaling function $f$ numerically,

Sana Riaz

Numerade Educator

02:10

Problem 9

Consider two-species annihilation $A+B \rightarrow \emptyset$ in the domain $[-L, L]$, with a steady input of $A$ at $x=L$ and input of $B$ at $x=-L$. The input rates of the two species are equal. The appropriate boundary conditions are
$$
\left.D c_{A}^{\prime}\right|_{x=L}=-\left.D c_{B}^{\prime}\right|_{x=-L}=-J,\left.\quad D c_{A}^{\prime}\right|_{x=-L}=\left.D c_{B}^{\prime}\right|_{x=L}=0
$$
corresponding to constant fluxes of $A$ at $x=L$ and $B$ and $x=-L$, and reflection at the opposite sides. Starting with the reaction-diffusion equations (13.86) in the steady state, subject to the above boundary conditions, develop an argument analogous to that in the transient case to show that the width of the reaction zone is given by
$$
w \sim\left(\frac{D^{2}}{J K}\right)^{1 / 3}
$$

Surendra Kumar

Numerade Educator

02:14

Problem 10

Consider again the steady behavior of the reaction-diffusion equations for twospecies annihilation $A+B \rightarrow \emptyset$, with a steady input of $A$ at $x=L$ and input of $B$ at $x=-L$.
(a) Consider the combinations $c_{\pm}=c_{A} \pm c_{B}$. Show that the solution for $c$ - that satisfies the boundary conditions is $c_{-}(x)=j x / D$.
(b) Use the result for $c_{-}$to rewrite the equation for $c_{+}$as
$$
c_{+}^{\prime \prime}=\frac{k}{2 D}\left[c_{+}^{2}-\left(\frac{j x}{D}\right)^{2}\right]
$$
where the prime denotes differentiation with respect to $x$ and the boundary conditions are $D c_{+}^{\prime}=j$ at $x=L$ and $D c_{+}^{\prime}=-j$ at $x=-L$. Show that forlarge flux, defined as $j>j_{0}=D^{2} /\left(K L^{3}\right), c_{+}$has the Taylor series expansion
$$
\frac{c_{+}(z)}{c_{0}}=\left(\frac{4}{5}\right)^{1 / 3}+\frac{z^{2}}{(10)^{2 / 3}}-\frac{z^{4}}{40}+\cdots
$$
where $z \equiv x / x_{0} \ll 1, x_{0}=\left(D^{2} / j K\right)^{1 / 3}$, and $c_{0}=\left(j^{2} / K D\right)^{1 / 3}$
(c) Show that the concentration of the minority species outside the reaction zone is asymptotically given by
$$
c_{B}(x) / c_{0} \sim z^{-1 / 4} \exp \left(-2 z^{3 / 2} / 3\right)
$$
To obtain this result, substitute the large- $x$ approximation $c_{A} \cong c_{B}+(j x / D)$ in the steady-state equation for $c_{B}$ to give $D c_{B}^{\prime \prime}=k c_{B}\left(c_{B}+j x / D\right)$ and determine the approximate solution of this equation for large $x$.

Sana Riaz

Numerade Educator

06:47

Problem 12

Apply the Laplace method to the integral in Eq. (13.11) and show that the long-time asymptotic behavior is indeed given by $(13.12)$.

Aidan Mcnabb

Numerade Educator

01:13

Problem 13

Consider a diffusing particle that starts at the origin of an absorbing sphere of radius $a$. Using spherical symmetry and separation of variables, show that the solution to the radial equation has the form of Eq. (13.17). Compute the amplitudes $A_{n}$. Show that, in three dimensions, the Bessel functions can be expressed via trigonometric functions.

Raj Bala

Numerade Educator