Pb (5.10): Entropy increases: diffusion. We see in Exercise 5.7 that entropy technically does not increase for a closed system, for any Hamiltonian, either classical or quantum. However, we can show that entropy increases for most of the coarse-grained effective theories that we use in practice; when we integrate out degrees of freedom, we provide a means for the information about the initial condition to be destroyed. Here you will show that entropy increases for the diffusion equation.

Let $\rho(x, t)$ obey the one-dimensional diffusion equation $\partial \rho/\partial t= D\partial^2 \rho/\partial x^2$. Assume that the density $\rho$ and all its gradients die away rapidly at $x = \pm\infty$.

Derive a formula for the time derivative of the entropy $S = -k_B \int \rho(x) \log\rho(x) dx $ and show that it strictly increases in time. (Hint: Integrate by parts. You should get an integral of a positive definite quantity.)

Pb (5.12): Rubber band. Figure 5.19 shows a one-dimensional model for rubber. Rubber is formed from long polymeric molecules, which undergo random walks in the undeformed material. When we stretch the rubber, the molecules respond by rearranging their random walk to elongate in the direction of the external stretch. In our model, the molecule is represented by a set of $N$ links of length $d$, which with equal energy point either parallel or antiparallel to the previous link. Let the total change in position to the right, from the beginning of the polymer to the end, be $L$. As the molecule extent $L$ increases, the entropy of our rubber molecule decreases.

(a) Find an exact formula for the entropy of this system in terms of $d$, $N$, and $L$. (Hint: How the external world, in equilibrium at temperature $T$, exerts a force pulling the end of the molecule to the right. The molecule must exert an equal and opposite entropic force $F$.)

(b) Find an expression for the force $F$ exerted by the molecule on the bath in terms of the bath entropy. (Hint: The bath temperature $1/T = \partial S_{bath}/\partial E$, and force times distance is energy.) Using the fact that the length $L$ must maximize the entropy of the Universe, write a general expression for $F$ in terms of the internal entropy $S$ of the molecule.

( c ) Take our model of the molecule from part (a), the general law of part (b), and Stirling’s formula $\log(n!) \approx n \log n − n$, write the force law $F(L)$ for our molecule for large lengths $N$. What is the spring constant $K$ in Hooke’s law $F = −KL$ for our molecule, for small L?

(d) If we increase the temperature of our rubber band while it is under tension, will it expand or contract? Why?

Pb 5.18: Undistinguished particles and the Gibbs factor. If we have $N$ particles in a box of volume $V$ and do not distinguish between the particles, then the effective configurational entropy is $S_Q = k_B \log(V^N/N!)$, where $N!$ is sometimes called the Gibbs factor. We saw that this factor keeps the entropy of mixing small for undistinguished particles.

(a) Use Stirling’s formula to show that this configurational entropy is related to $N$ times the entropy of a particle in a region roughly given by the distance to neighboring particles.

(b) In eqn 3.58 for the ideal gas entropy, $S = Nk_B(5/2 − \log(\rho \lambda^3))$, what part corresponds to this configurational entropy? (Hint: Expand the logarithm.)

Pb 5.20: Loaded dice and sticky spheres.

(a) A loaded three-sided die has probability $1/2$ of rolling a one, and probability $1/4$ of rolling a two or a three. What is the entropy (in units of $k_B$) per roll?

(b) $Let \theta , \phi$ be the latitude and longitude on the Earth, which we assume to be a sphere. If comets hit the earth with uniform probability density $\rho_c(\theta, \phi) = 1/(4\pi)$ and asteroids hit the earth with probability density $\rho_a(\theta, \phi) = \cos(\theta)/\pi^2$ that is maximum at the equator and zero at the poles, which distribution has higher entropy? What is the entropy difference between the distribution of asteroids and comets? (Hints: The integral of $f(\theta)$ over a sphere is $\int_{-\pi/2}^{\pi/2} \cos^2(\theta)f(\theta)d\theta$. You may also want to know that $\int_{-\pi/2}^{\pi/2} \cos^2(\theta)\log(\cos(\theta))d\theta = \pi(1-\log(4))/4$.)