A Kinetic View of Statistical Physics

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

Chapter 2 Diffusion - all with Video Answers

Educators

Chapter Questions

Problem 1

A discrete random walk hops to the right with probability $p$ and to the left with probability $q=1-p$ at each step. Let $P_{N}(x)$ be the probability that the particle is at $x$ at the $N^{\text {th }}$ time step.
(a) Write the master equation for this occupation probability.
(b) For the initial condition $P_{0}(x)=\delta_{x, 0}$, show that the combined Fourier transform and generating function
$$
P(k, z)=\sum_{N \geq 0} z^{N} \sum_{x=-\infty}^{\infty} e^{i k x} P_{N}(x)
$$
is given by $P(k, z)=[1-z u(k)]^{-1}$, where $u(k)=p e^{i k}+q e^{-i k}$ is the Fourier transform of the single-step hopping probability.
(c) Invert the Fourier transform and the generating function to determine the probability distribution of the discrete random walk.

Surendra Kumar

Numerade Educator

07:31

Problem 2

Consider a discrete random walk in one dimension in which a step to the right of length 2 occurs with probability $1 / 3$ and a step to the left of length 1 occurs with probability $2 / 3$. Investigate the finite-time corrections to the asymptotic isotropic
Gaussian probability distribution. Hint: Study the behavior of moments beyond second order, $\left\langle x^{k}\right\rangle$ with $k>2$

Ameer Said

Numerade Educator

05:14

Problem 3

Solve Eq. (2.4) for the continuous-time random walk by first Laplace-transforming to give the difference equation
$$
s P(n, s)-P(n, t=0)=P(n+1, s)+P(n-1)-2 P(s)
$$
for the Laplace transform $P(n, s)=\int_{0}^{\infty} P_{n}(t) e^{-s t} d t .$ Show that $P(n, s)=$ $\lambda_{-}^{n} /\left(s+1-\lambda_{-}\right)$, where $\lambda_{\pm}=\left(1 \pm \sqrt{1-4 a^{2}}\right) / 2 a$ and $a=(s+2)^{-1}$. Show that in the long-time limit, corresponding to $s \rightarrow 0$ in the Laplace transform, inversion of the Laplace transform recovers the Gaussian distribution (2.6) for $P_{n}(t)$.

Arpit Gupta

Numerade Educator

05:03

Problem 4

Generalize Eq. (2.4) for the hypercubic lattice in $d$ dimensions and find the probability for the walk to be at a given site. Verify that the probability for being at the origin is given by Eq. (2.46).

Sana Riaz

Numerade Educator

01:39

Problem 5

Continuum limit of the master equation ( $2.1$ ).
(a) For the case of symmetric hopping, $p=q=\frac{1}{2}$, derive the diffusion equation $(2.13)$ with $D=\frac{1}{2}$ by expanding the governing equation in a Taylor series to lowest non-vanishing order.
(b) Generalize the derivation to the case of non-symmetric hopping $(p \neq q)$ and show that the continuum limit is the convection-diffusion equation
$$
\frac{\partial P(x, t)}{\partial t}+v \frac{\partial P(x, t)}{\partial x}=D \frac{\partial^{2} P(x, t)}{\partial x^{2}}
$$
Determine the diffusion coefficient $D$ and the bias velocity $v$ in terms of the microscopic parameters ( $p$ and $q$ ) of the random walk.
(c) For the initial condition $P(x, t=0)=\delta(x)$, show that the solution to the convection-diffusion equation is
$$
P(x, t)=\frac{1}{\sqrt{4 \pi D t}} e^{-(x-v t)^{2} / 4 D t}
$$

Sana Riaz

Numerade Educator

03:50

Problem 6

Solve the convection-diffusion equation by the Laplace transform method and show that the Laplace transform of the probability distribution, $P(x, s)$, is
$$
P_{\pm}(x, s)=\frac{1}{\sqrt{v^{2}+4 D s}} e^{-\alpha_{\mp}|x|}
$$
where the subscript $\pm$ denotes the distribution in the regions $x>0$ and $x<0$, respectively, and $\alpha_{\pm}=\left(v \pm \sqrt{v^{2}+4 D s}\right) / 2 D .$ Invert this Laplace transform to

Vikash Ranjan

Numerade Educator

01:40

Problem 7

Solve the convection-diffusion equation on the half-line $x>0$, subject to the absorbing boundary condition $P(x=0, t)=0$ and the initial condition $P(x, t=$ $0)=\delta(x-\ell)$

Manik Pulyani

Numerade Educator

06:05

Problem 8

Suppose that $N$ independent identically distributed random variables are drawn from the probability distribution $p(x)$. Estimate the largest of these variables for
the cases: (a) $p(x)=e^{-x}$ for $x$ positive; (b) $p(x)=\mu x^{-(1+\mu)}$ for $x>1$ and $p(x)=0$ for $0<x<1 ;$ and (c) $p(x)=1$ for $0<x<1$ and $p(x)=0$ otherwise. As a corollary, estimate the speed of the most energetic molecule in a typical classroom.

Willis James

Numerade Educator

16:36

Problem 9

Investigate a discrete-time random walk with a symmetric and broad step length distribution that has an algebraic tail: $p(x) \simeq A|x|^{-1-\mu}$ when $|x| \rightarrow \infty$. Assume that $0<\mu<1$
(a) Write the Fourier transform of the step length distribution in the form
$$
p(k)=1+\int_{-\infty}^{\infty}\left(e^{i k x}-1\right) p(x) d x
$$
and show that the second term behaves as $-C|k|^{\mu}$ when $k \rightarrow 0$, where $C=$ $-A \sqrt{\pi} \Gamma(-\mu / 2) / \Gamma((1+\mu) / 2)$
(b) Substitute $p(k)=1-C\left|k^{\mu}\right|$ into $(2.18)$ and show that the displacement distribution attains the scaling form
$$
P_{N}(X) \rightarrow N^{-1 / \mu} L_{\mu, 0}(z), \quad z=X / N^{1 / \mu}
$$
with $L_{\mu, 0}(z)$ given by $(2.27)$.

Abhirup Pal

Numerade Educator

View

Problem 10

Suppose that a discrete-time random walk has the step length distribution $p(x)=$ $\pi^{-1}\left(1+x^{2}\right)^{-1}$ (the Cauchy distribution).
(a) Show that the corresponding Fourier transform is $p(k)=e^{-|k|}$.
(b) Compute the integral in $(2.18)$ when $p(k)=e^{-|k|}$ and show that
$$
P_{N}(X)=\frac{1}{\pi} \frac{N}{N^{2}+X^{2}}
$$
for any integer $N \geq 1$. Thus the Cauchy distribution is not merely a stable law in the $N \rightarrow \infty$ limit, it is stable for all $N$, a feature that characterizes the Gaussian distribution.

Shu Naito

Numerade Educator

02:28

Problem 11

This problem concerns the derivation of the amplitude in (2.37).
(a) Use $\mathbf{f}$ instead of $\mathbf{r}$ as an integration variable in $(2.36) .$ Show that $d \mathbf{r}=$ $-\frac{1}{2} f^{-9 / 2} d \mathbf{f}$, so that $(2.36)$ becomes
$$
\Phi(\mathbf{k})=\frac{1}{2} \int \frac{d \mathbf{f}}{f^{9 / 2}}\left(1-e^{i \mathbf{k} \cdot \mathbf{f}}\right)
$$
(b) Show that the above integral is unaffected if we replace $\mathrm{f}$ by $-\mathbf{f}$. Taking the arithmetic mean of these two integrals show that
$$
\Phi(\mathbf{k})=\frac{1}{2} \int \frac{d \mathbf{f}}{f^{9 / 2}}[1-\cos (\mathbf{k} \cdot \mathbf{f})]
$$
An alternative derivation is to notice that the integral in (a) must have a real value.
(c) Introduce spherical coordinates in $\mathbf{f}$-space with $z$ axis along $\mathbf{k}$, so that $d \mathbf{f}=$ $2 \pi \sin \theta d \theta f^{2} d f .$ Show that the integral from (b) becomes
$$
\Phi(\mathbf{k})=\pi \int_{0}^{\infty} \frac{d f}{f^{5 / 2}} \int_{-1}^{1} d \mu[1-\cos (k f \mu)], \quad \mu=\cos \theta
$$
(d) Perform the integration over $\mu$ and show that the integral from (c) simplifies to
$$
\Phi(\mathbf{k})=2 \pi k^{3 / 2} \int_{0}^{\infty} d z \frac{z-\sin z}{z^{7 / 2}}, \quad z=k f
$$
(e) Compute the integral from part (d) and recover $(2.37)$ with amplitude $a=$ $\frac{4}{15}(2 \pi)^{3 / 2}$

Adriano Chikande

Numerade Educator

04:54

Problem 12

Suppose that stars are distributed randomly (no correlations) and uniformly with density $n$. What is the average distance between a star and its closest neighbor?
(a) On dimensional grounds show that the average distance scales as $n^{-1 / 3}$, or more generally as $n^{-1 / d}$ in $d$ dimensions.
(b) Show that in $d$ dimensions the average distance equals $\Gamma(1 / d)\left(V_{d} n\right)^{-1 / d}$, where $V_{d}$ is the volume of a unit ball in $d$ dimensions.
(c) Show that $V_{d}=\pi^{d / 2} / \Gamma(1+d / 2)$.
(Hint: Work with the integral $J_{d}=\int e^{-r^{2}} d \mathbf{r}$ over all space $\mathbb{R}^{d}$. Compute this integral by two methods. First, write $r^{2}=x_{1}^{2}+\cdots+x_{d}^{2}, d \mathbf{r}=d x_{1} \ldots d x_{d}$ to recast $J_{d}$ into the product, $J_{d}=\left(J_{1}\right)^{d}$, where $J_{1}=\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi} .$ Second, using rotational invariance and spherical coordinates, reduce the integral to $J_{d}=\int_{0}^{\infty} e^{-r^{2}} \Omega_{d} r^{d-1} d r$ and express this integral via the gamma function. Comparing the two answers will give the surface "area" $\Omega_{d}$ of the unit sphere; the volume of the unit ball is then determined from $\left.\Omega_{d}=d V_{d} .\right)$

Khoobchandra Agrawal

Numerade Educator

01:41

Problem 13

For $\mathbf{r} \neq 0$ and $t>0$, explicitly verify Eq. (2.44) in one dimension.

Nick Johnson

Numerade Educator

01:28

Problem 14

Derive $(2.54)$ from $(2.53)$

Manik Pulyani

Numerade Educator

01:26

Problem 15

Explain the connection between the flux and the exit time probability distribution and thereby complete the derivation of $(2.64) .$

Bon Zapata

Numerade Educator

04:54

Problem 16

Show that the solution to Eq. (2.72) on the interval $(0, L)$ is $t(x)=x(L-x) /(2 D)$.

Nidhi Singhi

Numerade Educator

05:41

Problem 17

Generalize Eq. (2.72) by deriving the differential equations for the higher moments, $\left\langle T^{2}(x)\right\rangle$ and $\left\langle T^{3}(x)\right\rangle .$ Solve these equations on the interval $(0, L)$

Sarvesh Somasundaram

Numerade Educator

07:47

Problem 18

Consider Eq. (2.77) on the interval $(-L, L)$
(a) Show that the Laplace transform of the exit time is given by
$$
\Pi(s, x)=\frac{\cosh [x \sqrt{s / D}]}{\cosh [L \sqrt{s / D}]}
$$
Suppose that a random walker starts at the origin, $x=0 .$ Expand $\Pi(s, x=0)$ in powers of $s$ and deduce from this expansion that $\left\langle T^{n}\right\rangle=n ! A_{n} \tau^{n}$, where
$\tau=L^{2} / D$ and the coefficients $A_{n}$ are
$$
\begin{aligned}
&A_{1}=\frac{1}{2}, \quad A_{2}=\frac{5}{24}, \quad A_{3}=\frac{61}{720}, \quad A_{4}=\frac{277}{8064} \\
&A_{5}=\frac{50521}{3628800}, \quad A_{6}=\frac{540553}{95800320}, \ldots
\end{aligned}
$$
(b) Show that the general expression for the $n$th moment of the exit time is
$$
\left\langle T^{n}\right\rangle=\frac{n !}{(2 n) !}\left|E_{2 n}\right| \tau^{n}
$$
where $E_{n}$ are the Euler numbers that are defined by the expansion
$$
\frac{1}{\cosh z}=\sum_{n \geq 0} \frac{E_{n}}{n !} z^{n}
$$

Sam Low

Numerade Educator

49:08

Problem 19

Suppose that a particle initially at $x=\ell$ undergoes unbiased diffusion with diffusion constant $D .$ The particle is confined to the half-line $\left(x_{*}, \infty\right)$, in which the boundary $x_{*}(t)$ is moving. Suppose that $x_{*}=V t$, i.e. the boundary is initially at the origin and it approaches the particle with velocity $V>0$. The exit time is the time when the particle first crosses the moving boundary.
(a) Show that the average exit time does not depend on the diffusion coefficient: $\langle T\rangle=\ell / V$
(b) Show that the mean-square exit time is $\left\langle T^{2}\right\rangle=(\ell / V)^{2}+2 \ell D / V^{3}$
(c) Show that the Laplace transform of the exit time $\left\langle e^{-s T}\right\rangle$ is given by $\exp \left[-\ell\left(\sqrt{V^{2}+4 s D}-V\right) /(2 D)\right]$
Examine the same problem when the boundary is receding: $x_{*}=-V t$.
(d) Show that the exit time is infinite with probability $\operatorname{Prob}(T=\infty)=1-e^{-\ell V / D}$.
(e) Show that the Laplace transform of the exit time is $\exp \left[-\ell\left(\sqrt{V^{2}+4 s D}+\right.\right.$ $V) /(2 D)]$

Dr. Rajveer Singh

Numerade Educator

22:39

Problem 20

A diffusing particle is confined to the interval $[0, L]$. The particle has a diffusivity $D$ and it is also experiences a constant bias, with bias velocity $v$. The endpoint $x=L$ is reflecting, while if the particle reaches $x=0$ it is adsorbed.
(a) Calculate the mean and the mean-square time for the particle to get trapped at $x=0$ when it starts at $x=L .$ Examine in detail the three cases of $v>0$ $v<0$, and $v=0$
(b) Compute the Laplace transform of the distribution of trapping times for the cases of $v>0, v<0$, and $v=0$, and discuss the asymptotic behaviors of these distributions in the limits $t \rightarrow 0$ and $t \rightarrow \infty$

Dr. Rajveer Singh

Numerade Educator

10:23

Problem 21

A Brownian particle begins at the center of the three-dimensional ball of radius $R$.
(a) Show that the Laplace transform of the exit time is
$$
\left\langle e^{-s T}\right\rangle=\frac{\sqrt{s \tau}}{\sinh \sqrt{s \tau}}, \quad \tau=R^{2} / D
$$
(b) Derive the general expression for the integer moments of the exit time
$$
\left\langle T^{n}\right\rangle=\left(2^{2 n}-2\right) \frac{n !}{(2 n) !}\left|B_{2 n}\right| \tau^{n}
$$
in terms of Bernoulli numbers $B_{m}$ that are defined by the expansion
$$
\frac{z}{e^{z}-1}=\sum_{n \geq 0} \frac{B_{n}}{n !} z^{n}
$$

Ruirui Liu

Numerade Educator

03:31

Problem 22

Consider the gambler's ruin problem: two players $A$ and $B$ gamble repeatedly between themselves by betting $\$ 1$ in each trial. The probability of player $A$ winning in a single bet is $p$. Assume that player $A$ starts with $n$ dollars and player $B$ with $N-n$ dollars, and the game ends when one player has no money.
(a) What is the probability $\mathcal{E}_{n}$ that player $A$ eventually wins?
(b) How long does the game last?
(c) Given that player $A$ wins, how long does the game last? This setting defines the conditional exit time.

Nick Johnson

Numerade Educator

00:10

Problem 23

Solve the diffusion equation exterior to an absorbing sphere of radius $a$ in one and two dimensions and thereby compute the exact flux to this sphere. $^{27}$ That is, solve
$$
\frac{\partial c}{\partial t}=D \nabla^{2} c \quad \text { with } \quad c(a=t, t)=0, \quad c(r>a, t=0)=1
$$
and then compute the time-dependent flux onto the sphere,
$$
K(t)=D \int_{r=a} \nabla c(\mathbf{r}, t) \cdot d \sigma
$$

Frank Lin

Numerade Educator

04:10

Problem 24

A gas of spherical non-interacting particles of radii $a$ are captured by a ball of radius $R$ centered at the origin. In each capture event the volume of this ball increases by the volume $4 \pi a^{3} / 3$ of the captured particle. $^{28}$ Far from the absorbing ball, the concentration is $c_{\infty}$.
(a) Using the stationary value of the reaction rate, $K=4 \pi D R c_{\infty}$, show that this capture process results in a "diffusive" growth of the radius $R(t)$ of the absorbing ball:
$$
R=\sqrt{\mathcal{A D} t}, \quad \mathcal{A}=\frac{8 \pi}{3} c_{\infty} a^{3}
$$
(b) Argue that the usage of the stationary reaction rate would be correct if the predicted growth was slower than diffusive. Justify the validity of the scaling ansatz
$$
c(r, t)=c_{\infty} f(\eta), \quad \text { where } \quad \eta=\frac{r}{R} \quad \text { and } \quad R=\sqrt{A D t}
$$
with as-yet unknown amplitude $A$.
(c) Substitute the above scaling ansatz into $(2.83)$ and show that the scaled concentration is
$$
f(\eta)=1-\frac{g(\eta)}{g(1)}, \quad \text { with } \quad g(\eta)=\int_{\eta}^{\infty} \xi^{-2} e^{-A \xi^{2} / 4} d \xi
$$
(d) Use volume conservation to verify that the amplitude is an implicit solution of the equation
$$
\frac{8 \pi}{3} c_{\infty} a^{3}=A e^{A / 4} \int_{1}^{\infty} \xi^{-2} e^{-A \xi^{2} / 4} d \xi
$$
(e) In applications, the volume fraction of external particles is usually very low, $c_{\infty} a^{3} \ll 1$, which implies that $A \ll 1 .$ Show that in this situation the above relation for $A$ approximately becomes
$$
\frac{8 \pi}{3} c_{\infty} a^{3}=A\left(\ln \frac{4}{A}-\gamma_{E}\right)
$$

Khoobchandra Agrawal

Numerade Educator

07:09

Problem 25

Solve for the asymptotic survival probability of a diffusing particle inside an infinite two-dimensional wedge spanned by the radii $\theta=0$ and $\theta=\phi$, with absorbing boundary conditions on both radii. Instead of solving for the full Green's function, consider the initial condition, $c(r, \theta, t=0)=\pi \sin (\pi \theta / \phi) \delta\left(r-r_{0}\right) /(2 \phi)$, that gives the long-time angular dependence. For this restricted problem, only the solution for the radial coordinate is needed.

Guilherme Barros

Numerade Educator

01:35

Problem 26

Complete the derivations of Eqs (2.99) and (2.102).

Suzanne W.

Numerade Educator

05:29

Problem 27

A harmonically bound Brownian particle with non-zero mass obeys the equation of motion
$$
\ddot{x}(t)+\gamma \dot{x}(t)+\omega_{0}^{2} x(t)=\eta(t)
$$
where $\eta(t)$ is zero-mean white noise with $\left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle=\Gamma \delta\left(t-t^{\prime}\right)$. For an arbitrary initial condition $x(0)$ and $\dot{x}(0)$, find $\langle x(t)\rangle$ and $\left\langle x(t)^{2}\right\rangle .$ In the long-time limit $t \gg$ $\gamma^{-1}$, show that your result is consistent with equilibrium statistical mechanics for a suitable choice of $\Gamma$.

Khoobchandra Agrawal

Numerade Educator

08:41

Problem 28

Verify Eq. (2.113) for the two-point correlation function in two dimensions.

Chris Trentman

Numerade Educator

07:52

Problem 29

Fill in the missing steps to compute the exact form of mean-square width $w^{2}$ in Eq. (2.108) for the Edwards-Wilkinson model.

Will Erickson

Numerade Educator

02:17

Problem 30

Confirm the result given in (2.115) for the two-point correlation function $C(x, t)$ in one dimension.

Gennady Notowidigdo

Numerade Educator

01:19

Problem 31

Analyze the Edwards-Wilkinson equation in three dimensions. Confirm the prediction of Eq. (2.115) for the two-point correlation function. Show that, when $x \ll \sqrt{D t}$, the two-point correlation function simplifies to
$$
C(\mathbf{x}, t)=\frac{\Gamma}{4 \pi D} \frac{1}{x}
$$

Ameer Said

Numerade Educator

01:03

Problem 32

Consider the Mullins equation (2.120).
(a) Use dimensional analysis and linearity to show that the width scales as $w \sim$ $\Gamma^{1 / 2} v^{-d / 8} t^{(4-d) / 8} .$ Therefore noise roughens the interface in the physically relevant case $d=2$ and even in $d=3$
(b) Compute the width and the two-point correlation function in dimensions $d=1$, 2, and 3

Raj Bala

Numerade Educator

03:27

Problem 33

Show that the probability distribution (2.126) is a solution of the Fokker-Planck equation (2.125) in one dimension.

Ren Jie Tuieng

Numerade Educator

03:49

Problem 34

Show that, for $d>1$, the probability distribution (2.126) is not a solution to $(2.125)$. (The integrand in Eq. (2.126) should contain $(\nabla h)^{2}$ in higher dimensions.)

Amany Waheeb

Numerade Educator