## Physics 151 Statistical Physics I

**Prerequisite:**

Physics 126 or COI (2018)

**SUBMISSION FORM for Problem Set 2 and Finals Notebook (deadline 6PM 22 Dec 2023)**

**Please upload proof of satisfying prerequisite here: Prerequisite proof submission link**

** Announcements:**

**Notebook checklists: **

**W12 4-8 Dec: Example: Black-body radiation**References RF 9.13–9.15

- Briefly describe the significance of the study of black-body radiation in the history and development of quantum physics (read on the ultraviolet catastrophe).

- Show that the mean energy density of photons in a cavity at thermal equilibrium increases as T^4 (Stefan-Boltzmann law).

**Week 11 Dec 1**Indistinguishable particles II: Quantum Ideal Gases

References RF 9.1–9.8, Reichl 7.H

- Do Problem Set 2

**Week 10 Nov 20-24**Indistinguishable particles II: Quantum Ideal Gases

References RF 9.1–9.8, Reichl 7.H

In a quantum gas of identical particles, the particles are indistinguishable so that permuting any two particles gives the same state.

We consider a system of identical non-interacting particles (a gas). Since the particles are identical, they have the same Hamiltonians. Since they are non-interacting, each particle can be in one of the single-particle eigenstates of that Hamiltonian. Let a single-particle state labeled by r have energy eigenvalue ϵ_r, and the state of the whole gas be denoted by the set of labels R = {r_1,r_2, ...}. Let the occupation number n_r be the number of particles in the single-particle state r. The average occupation number gives the Bose-Einstein and Fermi-Dirac quantum distribution functions. The Bose-Einstein and Fermi-Dirac distributions approach the classical Maxwell-Boltzmann distribution at high temperatures.

- Give the distinguishing feature of bosonic and fermionic many-particle wavefunctions under a permutation of two particles.

- The average occupation number of particles at thermal equilibrium gives rise to different number distribution functions and different statistical behavior of gases. Give the number distribution functions for (a) classical particles, (b) bosons and photons, and (c) fermions at temperature T and chemical potential μ.

- At low temperatures, show that the Bose-Einstein distribution predicts Bose-Einstein condensation of particles in the lowest energy state.

- At zero temperature, show that the Fermi-Dirac distribution exhibits Fermi exclusion so that the gas is built up of particles occupying different low energy single-particle states up to a Fermi energy. The highest occupied energy states form a Fermi surface. At low temperatures, make a sketch of how the Fermi surface is smeared as particles become thermally excited.

- Begin Problems I and II of PS2. Download PS2 **here**.

**Week 9 Nov 13-17**Indistinguishable particles I: Semiclassical description of the ideal gas

References RF 7.3–7.4, 7.2

The classical derivation of the entropy of an ideal gas (distinguishable particles) fails at low temperature because it becomes negative and eventually diverges as T goes to 0 violating the third law of thermodynamics (cf. RF Eq. 7.2.16). Further the entropy should be extensive (proportional to system volume). The classical description fails because atoms are indistinguishable and a quantum approach is needed.

- In a gas of non-interacting atoms, briefly describe how Gibbs paradox is resolved by taking into consideration the indistinguishability of atoms.

- For this ideal gas, show how the semiclassical correction Zsc = Zcl/N! of the classical canonical partition function Zcl yields the Sackur-Tetrode expression for an extensive entropy.

**Week 8 Nov 6-10**Equipartition theorem; Kinetic theory of dilute ideal gas

References RF 7.5–7.6, 7.9–7.10

- The equipartition theorem applies to systems described by classical statistical mechanics. Enumerate the conditions needed for the equipartition theorem to hold given a Hamiltonian H(qi;pi), and summarize the main result. Apply the theorem to some examples of quadratic Hamiltonians (for example, the harmonic solid, monatomic ideal gas, diatomic ideal gas, and others).

- In a dilute ideal gas, the interactions between atoms are negligible. At thermal equilibrium, give the functional form of the Maxwell speed distribution F(v). Describe the distribution in words. Show that the moments of this distribution can be expressed as Gaussian integrals.

**COURSE GUIDE**

**Class requirements and assignment guide**

References:

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, 2007).

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists: A Comprehesive Guide (Academic Press, 2011).

L. E. Reichl, A Modern Course in Statistical Physics (John Wiley & Sons, Inc., 1998).

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, Inc., 1965).

M. Plischke and B. Bergersen, Equilibrium Statistical Physics (World Scientific Publishing Co. Pte. Ltd., 2006).

R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, 1996).

J. M. Yeomans, Statistical Mechanics of Phase Transitions (Oxford University Press, Inc., 1992).