Chapter 1, Aperitifs Video Solutions, A Kinetic View of Statistical Physics

Problem 1

Consider the SEP in the extreme limit where the lattice is fully occupied, apart from one vacancy that is initially at the origin. The problem is to investigate the displacement of the tracer particle.
(a) Show that in one dimension if the tracer particle is initially to the right of the vacancy, say at site $n>0$, then at any time the tracer particle will be either at its original position $n$ or at site $n-1 .$ Estimate the typical number of hops of the tracer particle during the time interval $(0, t)$. Show that the probability that the tracer particle has not moved decays as $t^{-1 / 2}$ in the long-time limit.
(b) Try to answer the same questions as in (a) in the two-dimensional case. Argue that the typical number of hops scales as $\ln t$ and that the typical displacement grows $^{9}$ as $\sqrt{\ln t}$
(c) Consider the same problem in three dimensions. Show that the tracer particle will never move with a positive probability. Argue that the number of hops throughout the entire evolution is finite.

Sikandar Baig

Numerade Educator

04:47

Problem 2

Consider the inviscid Burgers equation $(4.22)$ with the initial condition $\rho(x, 0)=$ $\left(1+x^{2}\right)^{-1}$. Show that a smooth solution exists up to time $t_{*}=4 / \sqrt{27}$, and that a shock wave originates at $x_{*}=-5 / \sqrt{27}$ at time $t_{*}$

Khoobchandra Agrawal

Numerade Educator

03:52

Problem 3

Consider the interaction of a shock and a rarefaction wave that are generated starting with initial condition (4.29). Assume that $\rho_{L}<\rho_{M}$ and $\rho_{M}>\rho_{R}$. In contrast to Example 4.5, consider the case when $\rho_{L}>\rho_{R}$.

Prachita Kush

Numerade Educator

02:25

Problem 4

Describe the evolution that arises from the initial condition (4.29) when $\rho_{L}>\rho_{M}>$ $\rho_{R}$, in which two rarefaction waves interact.

Khoobchandra Agrawal

Numerade Educator

03:44

Problem 5

Give a heuristic derivation of the Burgers equation (4.34) by the following steps:
(a) Start with the master equation
$$
\frac{\partial \rho(x, t)}{\partial t}=\rho(x-1, t)[1-\rho(x, t)]-\rho(x, t)[1-\rho(x+1, t)]
$$
Try to justify its validity using the exact evolution equation (4.46). What are the caveats in writing local currents as products?
(b) Expand the densities in Taylor series
$$
\rho(x \pm 1, t)=\rho(x, t) \pm \frac{\partial \rho(x, t)}{\partial x}+\frac{1}{2} \frac{\partial^{2} \rho(x, t)}{\partial x^{2}}+\cdots
$$

Supratim Pal

Numerade Educator

04:58

Problem 6

Consider the Burgers equation in the canonical form $u_{I}+u u_{x}=D u_{x x} .$ The goal of this problem is to solve the Burgers equation for an arbitrary initial condition.
(a) Use the Cole-Hopf transformation
$$
u=-2 D(\ln \phi)_{x}=-2 D \phi_{x} / \phi
$$
and show that the auxiliary function $\phi(x, t)$ satisfies the diffusion equation, $\phi_{t}=D \phi_{x x} .$ Thus the Cole-Hopf transformation recasts the nonlinear Burgers equation into the linear diffusion equation.
(b) Consider an arbitrary initial condition $u_{0}(x) .$ Show that
$$
\phi_{0}(x)=\exp \left[-(2 D)^{-1} \int_{0}^{x} u_{0}(y) d y\right]
$$
can be chosen as the initial condition for the diffusion equation.
(c) Starting with solution to the diffusion equation subject to the initial condition $\phi_{0}(x)$
$$
\phi(x, t)=\frac{1}{\sqrt{4 \pi D t}} \int_{-\infty}^{\infty} \phi_{0}(y) e^{-(x-y)^{2} / 4 D t} d y
$$
show that the solution of the Burgers equation is
$$
u(x, t)=\frac{\int_{-\infty}^{\infty}[(x-y) / t] e^{-G / 2 D} d y}{\int_{-\infty}^{\infty} e^{-G / 2 D} d y}, \quad G(x, y ; t)=\int_{0}^{y} u_{0}\left(y^{\prime}\right) d y^{\prime}+\frac{(x-y)^{2}}{2 t}
$$

Saman Zulfiqar

Numerade Educator

12:23

Problem 7

Specialize the general solution described in the previous problem to the Burgers equation (4.34) with initial condition $\rho_{0}(x)=1$ for $x \leq 0$ and $\rho_{0}(x)=0$ for $x>0$. Show that in the bulk region $|x|<t$ the solution approaches $\rho=\frac{1}{2}(1-x / t)$. Describe the behavior in the internal layers around $x=\pm t$

Victoria Dollar

Numerade Educator

03:11

Problem 8

Consider an open system of size $N=3$. Verify that the matrix ansatz gives the stationary solution.

Jingyun Wang

Numerade Educator

02:48

Problem 9

Verify the relation $(4.54)$.

Harmender Singh Yadav

Numerade Educator

03:54

Problem 10

Consider the configuration $\underbrace{1 \ldots 1}_{M} \underbrace{0 \ldots 0}_{N-M}$ and show that its stationarity is equivalent to
$$
\left\langle W\left|D^{M} E^{N-M}\right| V\right\rangle=\alpha\left\langle W\left|E D^{M-1} E^{N-M}\right| V\right\rangle+\beta\left\langle W\left|D^{M} E^{N-M-1} D\right| V\right\rangle
$$
Check the validity of the above relation using $(4.52)$.

Zulfiqar Ali

Numerade Educator

07:47

Problem 11

Verify the identity (4.57) on the line $\alpha+\beta=1$ where we can replace the operators by scalars: $E=\alpha^{-1}, D=\beta^{-1} .$ Then prove $(4.57)$ in the general case using mathematical induction.

Sriparna Bhattacharjee

Numerade Educator

01:04

Problem 12

The derivation of the expressions for the current and density in different phases, Eqs (4.59)-(4.60), is lengthy as it requires the asymptotic analysis of various sums like the one that appears in (4.58). It is useful to get an idea of what to expect by considering the line $\alpha+\beta=1$ where the solution greatly simplifies (since the operators $E$ and $D$ are scalar) and the major results are independent of $N$. Show that, on the line $\alpha+\beta=1,\left\langle n_{i}\right\rangle=\alpha$ independent of both $i$ and $N$. Verify that this result agrees with the general prediction of $(4.60)$.

Dominador Tan

Numerade Educator

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Problem 13

Consider the ASEP in an open system, with equal input and output rates, $\alpha=\beta$.
(a) Using l'Hôpital's rule simplify the sum in (4.58) to yield
$$
\frac{\left\langle W\left|C^{N}\right| V\right\rangle}{\langle W \mid V\rangle}=\sum_{p=1}^{N} p \frac{(2 N-1-p) !}{N !(N-p) !} \frac{p+1}{\alpha^{p}}
$$
(b) Show that when $\alpha>1 / 2$, the terms with $p=\mathcal{O}(1)$ dominate. Use this fact to compute the leading asymptotic
$$
\frac{\left\langle W\left|C^{N}\right| V\right\rangle}{\langle W \mid V\rangle} \simeq \frac{\alpha^{2}}{\sqrt{\pi}(2 \alpha-1)^{3}} \frac{4^{N+1}}{N^{3 / 2}}
$$
(c) Show that for $\alpha<1 / 2$, the terms with $p$ close to $N(1-2 \alpha) /(1-\alpha)$ dominate. Use this fact to compute the leading asymptotic
$$
\frac{\left\langle W\left|C^{N}\right| V\right\rangle}{\langle W \mid V\rangle} \simeq \frac{(1-2 \alpha)^{2}}{(1-\alpha)^{2}} \frac{N}{\alpha^{N}(1-\alpha)^{N}}
$$
(d) Show that when $\alpha=1 / 2$,
$$
\frac{\left\langle W\left|C^{N}\right| V\right\rangle}{\langle W \mid V\rangle}=4^{N}
$$
(e) Use the results of (b)-(d) to confirm the predictions of (4.59) on the diagonal $\alpha=\beta$
(f) Confirm the predictions of $(4.60)$ on the diagonal $\alpha=\beta$.

Victor Salazar

Numerade Educator