Chapter 10, Disorder Video Solutions, A Kinetic View of Statistical Physics

Problem 1

This problem concerns the moments of total number of weak bonds.
(a) Start with the explicit formula (10.13) for the generating function $G_{N}(\omega)=$ $\sum_{\Omega \geq 0} e^{\omega \Omega} \Pi_{N}(\Omega)=\left\langle e^{\omega \Omega}\right\rangle$ and expand it in powers of $\omega$.
(b) Using the expansion from part (a), extract the variance:
$$
\left\langle\Omega^{2}\right\rangle-\langle\Omega\rangle^{2}=\frac{2}{45} N \quad \text { as } \quad N \rightarrow \infty
$$
(c) Derive the variance directly without using $(10.13)$ by generalizing the probabilistic argument that led to $\langle\Omega\rangle=N / 3$.

Raj Bala

Numerade Educator

12:37

Problem 2

Using $(10.19)$ and $(10.20)$ show that for the uniform distribution of coupling strengths, $\psi(x)=1$ for $0 \leq x \leq 1$ and zero otherwise, the ground-state energy is $\mathcal{E}_{\mathrm{GS}}=-\frac{1}{2}$, while the metastable-state energy is $\mathcal{E}_{\mathcal{M}}=-\frac{17}{36}$. Similarly, show that for the exponential distribution, $\psi(x)=e^{-x}$, the energies are given by $\mathcal{E}_{\mathrm{GS}}=-1$ and $\mathcal{E}_{\mathcal{M}}=-\frac{26}{27}$.

Ren Jie Tuieng

Numerade Educator

01:43

Problem 3

Consider the cluster size distribution that appears in (10.18). Determine this distribution using the transfer matrix approach. You should get
$$
X_{\ell}=2^{\ell} \frac{(\ell-1)(\ell+2)}{(\ell+3) !}
$$
Notice that this distribution is also independent of the distribution of interaction strengths. Verify the sum rules: $\sum_{\ell \geq 2} X_{\ell}=1 / 3$ and $\sum_{\ell \geq 2} \ell X_{\ell}=1$.

Manik Pulyani

Numerade Educator

01:27

Problem 4

Think of Fig. $10.2$ as a one-dimensional landscape: each site has a certain height, the heights are independent identically distributed random variables, and the height distribution is irrelevant as long as it does not contain delta functions. (For computations, it is convenient to draw heights from the uniform distribution on the unit interval $(0,1)$.) A weak bond corresponds to a valley in the landscape language, and a strong bond corresponds to a peak.
(a) Show that the probability to have a sequence of $\ell$ consecutive ascending heights is $1 / \ell !$.
(b) Show that the probability $P_{i j}$ that $i+j+1$ consecutive heights contain a valley which is separated by distance $i$ from the peak on the left and distance $j$ from the peak on the right is
$$
P_{i j}=\frac{1}{(i+1) !(j+1) !}\left[\frac{i j}{i+j+1}+\frac{1}{i+j+3}\right]
$$
(c) Let $X_{\ell}$ be the length distribution between neighboring peaks, or equivalently neighboring valleys. Show that
$$
X_{\ell}=\sum_{i, j \geq 1, i+j=\ell} P_{i j}
$$
Substitute the expression for $P_{i j}$ determined above and compute the sum to give the expression for $X_{\ell}$ from the previous problem.

Manik Pulyani

Numerade Educator

View

Problem 5

Consider a two-dimensional landscape in which each site of the square lattice has a certain height, and the distribution of these heights has the same assumptions as in the previous problem.
(a) Show that the density of valleys is $1 / 5$. (The density of peaks is the same.)
(b) Show that the density $M_{3}$ of points with three directions of descent is $M_{3}=1 / 5$. (The density of points with one direction of descent is the same: $M_{1}=1 / 5$.)
(c) Show that the density $M_{2}$ of points with two directions of descent is also $M_{2}=1 / 5$
(d) Show that the density of saddles is $1 / 15$.
(e) Show that the variance of the total number of valleys is
$$
\left\langle\mathcal{V}^{2}\right\rangle-\langle\mathcal{V}\rangle^{2}=\frac{13}{225} N \quad \text { as } \quad N \rightarrow \infty
$$
The average number of valleys is proportional to the density, i.e. $\langle\mathcal{V}\rangle=N / 5$.

Shu Naito

Numerade Educator

02:14

Problem 6

Consider the one-dimensional random walk in a random environment that is governed by Eqs $(10.21)-(10.23)$. Repeat the argument that led to $(10.25)$ in a dimensionally correct form and show that
$$
L \sim L_{*}\left(\ln \frac{t}{t_{*}}\right)^{2}, \quad L_{*}=\frac{D^{2}}{\Gamma}, \quad t_{*}=\frac{D^{3}}{\Gamma^{2}}
$$
Deduce $t_{t}=D^{3} / \Gamma^{2}$ and $L_{*}=D^{2} / \Gamma$ using dimensional analysis.

Manik Pulyani

Numerade Educator

01:39

Problem 7

Consider the one-dimensional random walk in a random environment with an overall bias, namely, instead of $(10.23)$, we have
$$
\langle F\rangle=F_{0}, \quad\left\langle F(x) F\left(x^{\prime}\right)\right\rangle-F_{0}^{2}=2 \Gamma \delta\left(x-x^{\prime}\right)
$$
Show that there is one dimensionless combination of the three parameters $D, F_{0}$, and $\Gamma$, namely $\mu=D F_{0} / \Gamma$ or any function thereof. As a side note, we mention that the random walker moves ballistically when $\mu>1$,
$$
\lim _{t \rightarrow \infty} \frac{\langle x\rangle}{t}=V, \quad V=F_{0}\left(1-\mu^{-1}\right)
$$
(Note that the speed $V$ is smaller than the "bare" speed $V_{0}=F_{0}$ which the walker would acquire in the absence of disorder.) When $0<\mu<1$, the movement is sub-ballistic: $\langle x\rangle \sim t^{\mu}$.

Dominador Tan

Numerade Educator

03:21

Problem 8

Deduce the amplitudes for the moments given in Eq. (10.39).

Hafiz Shahzaib

Numerade Educator