## Entropy and Temperature

• Energy Temperature Ideal gas Entropy
• assignment Quantum harmonic oscillator

assignment Homework

##### Quantum harmonic oscillator
Entropy Quantum harmonic oscillator Frequency Energy Thermal and Statistical Physics 2020
1. Find the entropy of a set of $N$ oscillators of frequency $\omega$ as a function of the total quantum number $n$. Use the multiplicity function: $$g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}$$ and assume that $N\gg 1$. This means you can make the Sitrling approximation that $\log N! \approx N\log N - N$. It also means that $N-1 \approx N$.

2. Let $U$ denote the total energy $n\hbar\omega$ of the oscillators. Express the entropy as $S(U,N)$. Show that the total energy at temperature $T$ is $$U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}$$ This is the Planck result found the hard way. We will get to the easy way soon, and you will never again need to work with a multiplicity function like this.

• assignment One-dimensional gas

assignment Homework

##### One-dimensional gas
Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of $N$ particles, each of mass $M$, confined to a one-dimensional line of length $L$. The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature $T$. You may assume that the temperature is high enough that $k_B T$ is much greater than the ground state energy of one particle.
• assignment Free energy of a two state system

assignment Homework

##### Free energy of a two state system
Helmholtz free energy entropy statistical mechanics Thermal and Statistical Physics 2020
1. Find an expression for the free energy as a function of $T$ of a system with two states, one at energy 0 and one at energy $\varepsilon$.

2. From the free energy, find expressions for the internal energy $U$ and entropy $S$ of the system.

3. Plot the entropy versus $T$. Explain its asymptotic behavior as the temperature becomes high.

4. Plot the $S(T)$ versus $U(T)$. Explain the maximum value of the energy $U$.

• assignment Paramagnetism

assignment Homework

##### Paramagnetism
Energy Temperature Paramagnetism Thermal and Statistical Physics 2020 Find the equilibrium value at temperature $T$ of the fractional magnetization $$\frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N}$$ of a system of $N$ spins each of magnetic moment $m$ in a magnetic field $B$. The spin excess is $2s$. The energy of this system is given by \begin{align} U &= -\mu_{tot}B \end{align} where $\mu_{tot}$ is the total magnetization. Take the entropy as the logarithm of the multiplicity $g(N,s)$ as given in (1.35 in the text): $$S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N}$$ for $|s|\ll N$, where $s$ is the spin excess, which is related to the magnetization by $\mu_{tot} = 2sm$. Hint: Show that in this approximation $$S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N},$$ with $S_0=k_B\log g(N,0)$. Further, show that $\frac1{kT} = -\frac{U}{m^2B^2N}$, where $U$ denotes $\langle U\rangle$, the thermal average energy.
• face Equipartition theorem

face Lecture

30 min.

##### Equipartition theorem
Contemporary Challenges 2022 (4 years)

This lecture introduces the equipartition theorem.
• assignment Free energy of a harmonic oscillator

assignment Homework

##### Free energy of a harmonic oscillator
Helmholtz free energy harmonic oscillator Thermal and Statistical Physics 2020

A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with $\varepsilon_n = n\hbar\omega$, where $n$ is an integer $\ge 0$, and $\omega$ is the classical frequency of the oscillator. We have chosen the zero of energy at the state $n=0$ which we can get away with here, but is not actually the zero of energy! To find the true energy we would have to add a $\frac12\hbar\omega$ for each oscillator.

1. Show that for a harmonic oscillator the free energy is $$F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right)$$ Note that at high temperatures such that $k_BT\gg\hbar\omega$ we may expand the argument of the logarithm to obtain $F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)$.

2. From the free energy above, show that the entropy is $$\frac{S}{k_B} = \frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1} - \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right)$$

This entropy is shown in the nearby figure, as well as the heat capacity.

• face Entropy and Temperature

face Lecture

120 min.

##### Entropy and Temperature
Thermal and Statistical Physics 2020

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.

assignment Homework

AIMS Maxwell 2021 (2 years)

Task: Draw a right triangle. Put a circle around the right angle, that is, the angle that is $\frac\pi2$ radians.

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• assignment Energy of a relativistic Fermi gas

assignment Homework

##### Energy of a relativistic Fermi gas
Fermi gas Relativity Thermal and Statistical Physics 2020

For electrons with an energy $\varepsilon\gg mc^2$, where $m$ is the mass of the electron, the energy is given by $\varepsilon\approx pc$ where $p$ is the momentum. For electrons in a cube of volume $V=L^3$ the momentum takes the same values as for a non-relativistic particle in a box.

1. Show that in this extreme relativistic limit the Fermi energy of a gas of $N$ electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where $n\equiv \frac{N}{V}$ is the number density.

2. Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}

• face Introducing entropy

face Lecture

30 min.

##### Introducing entropy
Contemporary Challenges 2022 (4 years)

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
• Thermal and Statistical Physics 2020

Suppose $g(U) = CU^{3N/2}$, where $C$ is a constant and $N$ is the number of particles.

1. Show that $U=\frac32 N k_BT$.

2. Show that $\left(\frac{\partial^2S}{\partial U^2}\right)_N$ is negative. This form of $g(U)$ actually applies to a monatomic ideal gas.