Physics 4311: Thermal Physics – Homework 12

Problem 1: Model of DNA (10 points)

A simple model of the DNA double helix molecule is analogous to a zipper: a chain of $N$ links each of which can be open or closed. A closed link has energy ${𝜖}_0$, and an open link has energy ${𝜖}_1>{𝜖}_0$. Replication of the DNA starts with the opening of the “zipper”. Assume that it can only open from one end (say the left), i.e., a link can only be open if all links left of it are also open.

a)

Calculate the partition function for this DNA model.

b)

Find the average number $n$ of open links as a function of $N$ and the temperature $T$ .

c)

Discuss the behavior of $n$ in the limits of high and low temperatures.

Problem 2: Ideal gas in rotating cylinder (15 points)

Consider a non-relativistic classical ideal gas of $N$ indistinguishable particles (mass $m$) at temperature $T$ in a cylindrical vessel of radius $R$ and height $H$. The cylinder is rotating around its vertical axis with angular velocity $\omega$.

a)

Compute the partition function [Hint: Work in a rotating reference frame and neglect the Coriolis force.]

b)

Calculate the internal energy and the specific heat of the gas as functions of temperature.

c)

Calculate how the particle density $n(r)$ changes with the distance $r$ from the rotation axis. (Hint: the particle density $n(r)$ is a reduced probability density of the phase space density $\rho (r,p$).)

Problem 3: Ultra-relativistic classical ideal gas (15 points)

Consider a gas of $N$ non-interacting, indistinguishable, classical particles at temperature $T$ in a cubic box of linear size $L$. The energy-momentum relation is ultra-relativistic, $E=c\left|p\right|$, where $c$ is the speed of light.

a)

Calculate the partition function and the free energy of the gas.

b)

Calculate the pressure as function of $N$, $T$, and $V$.

c)

Find the internal energy $U$ and the specific heat $C_V$ at constant volume.

d)

Also determine the specific heat at constant pressure, and compare the ratio of to that of the nonrelativistic case.