Statistical Physics - Dr. W. Allison

Quick links:    Lectures and handouts : (1)  Statistical Methods - Lectures 1-7;
                                                           (3) Applications in Condensed Matter and Collective Phenomena - Lectures 8-16.
                      Examples sheets. .

Statistical Physics provides an introduction to thermal physics using the methods of statistical mechanics.  The lectures assume the knowledge of IA courses and the IB Physics course on Classical Thermodynamics.  The aim is to emphasise the important insights into the world that statistical methods offer, as well as giving quantitative illustrations.  I have divided the course into two, largish blocks: (1)  Statistical methods (nine lectures) in which the theoretical ideas and methods are set out, and, (2) Applications of the statistical approach, including condensed matter systems and collective phenomena (eight lectures).

Lecture notes and Examples sheets are available as Portable Document Files (PDF files).  Access is restricted to computers in cambridge (.cam.ac.uk domain). The format of the material is the same as that of the individual lecture handouts except that lecture overheads are complete and in colour.

Examples sheets:

Examples 1 (pdf)
 
Examples 2 (pdf)

Lecture material and suggestions for reading:

(1) Statistical Methods:

• Lecture 1 , handout (pdf) available.   The material reviews the zeroth and first Laws, which are covered more fully in the IB Physics course on Classical Thermodynamics. Bowley and Sanchez Introduction to Statistical Mechanics (OUP, 1996) Chapters 1 and 2 cover the same matreial.
• Lecture 2 , handout (pdf) available.   The lecture introduces Boltzmann's definition of entropy.  The equation, S = k ln g , is one of the great icons in Physics.  We also deal with counting the number of configurations and show how to deal with large numbers (Stirling's approximation).  See Bowley and Sanchze, Section 3.6 for material on conuting the number fo arrangements; and sections 4.1 - 4.3 for a Boltzmann entropy..
• Lecture 3 , handout (pdf) available.  The main aim of the lecture is to derive the Boltzmann distribution, which follows from a discussion of a small system in equilibrium with a heat reservoir.  A similar derivation can be found in most books: see, for example, Mandl Statistical Physics (Wiley, 2nd Ed, 1988), section 2.5.  Chapter 3 of Waldram's book covers some of the same ground as the lecture.   Discussion of the Canonical ensemble and derivation of the formulae for the Free Energy ( F ) in terms of the Partition Function ( Z ) can be found in Bowley and Sanchez  in chapter 5, section 5.1-5.6.  See also, Mandl section 2.5, and Waldram section 9.2.
• Lecture 4 , handout (pdf) available. An ideal monatomic gas is treated correctly, and quantum  mechanically, using the methods of Statistical Mechanics.  The important new idea, which will recur in this course and others at the the Part II and Part III level, is that of the density of states.  In this lecture we calculate the density of states, g(E ).  Once the density of states is known, g ( E)d E gives the number of states between E and E+d E .  For a 3-D system of massive particles, we prove that g ( E )~E ^½ (to be used later in the context of electrons in solids).  Some other cases are dealt with on the examples sheet.  For example,  sheet 1, Q10 repeats the derivation for a 2-D gas of massive particles and Q11 is a 3-D relativistic gas.  In these two cases the approach is the same as the lecture but with a different integration over states in k -space (because of the dimensionality) or a different relation between E and k ( NB, E ~ k ² only for non relativistic, massive particles).  The books by Waldram, section 10.1, and Mandl, appendix B, repeat the derivation but for mass less particles (photons).  Heat and Work in Statistical Mechanics is discussed by Bowley and Sanchez section 5.10.
• Lecture 5 , handout (pdf) available. The Partition Function for a complex system is seen to be a product of separate Partition Functions and the thermal energy the sum of each contribution.  Results for rotation and vibration in a diatomic gas, including Planck's famous result for a quantised harmonic oscillator are derived and the result is applied to black body radiation. Finally we discuss the heat capacity of a solid in Einstein's treatment.  We will return to this latter problem towards the end of the course after a proper treatment of the lattice vibrations in a periodic solid.  See Waldram chapter 9 and Mandl chapter 10 for similar treatment of the material.  Chapter 10 in Waldram's book gives lots of examples of "Waves in a box" including Einstein's result for the lattice heat capacity. Bowley and Sanchez sections 5.11-5.14 cover the main points discussed at the start of the lecture.  In chapter 8 they deal comprehensively with black body radiation and Einsteins model for vibrations in a solid.
• Lecture 6 , handout (pdf) available.  The methods developed in the previous lectures are applied to some simple, but important systems.  First we deal with paramagnetism arising from electron spin.  A spin 1/2 particle has two orientations and, hence, we have a simple two level system to consider.  A book with the treatment near to that in lectures is Mandl Chapter 3 (specifically sections 3.1 and 3,2).  There is a small sectuion in Bowley and Sanchez (p 100). Secondly, we look at the remarkable elastic properties of rubber.  Something on this topic was covered in the Classical thermodynamics course in IB Physics. In the present lecture, we see that the origin of the elastcicity is in the configurational entropy of the rubber molecules.  For reading: see Bowley and Sanchez section 4.4.3.
• Lecture 7 , handout (pdf) available.  The lecture consists of two separate topics: firstly, adiabatic demagnetisation is illustrated as a methods of achieving low temperatures; secondly, the origins of irreversible behaviour is discussed briefly.  A discussion of cooling using adiabatic processes, similar to that in the lecture and handout, can be found in Mandl Section 5.6.  A full treatment of the origin of reversibility is more complex (see for example Waldram's book, chapter 19) and is beyond ther scope of this course.

(2) Applications in Condensed Matter and Collective Phenomena:

• Lectures 8 and 9 , handout (pdf) available.  We begin an exploration of condensed matter systems with a discussion of indistinguishability and the inherent quantum properties of all matter.  First, consider the entropy of a gas of many particles.  We demonstrate that the assumption that particles are "distinguishable" leads to an incorrect counting of states in the partition function and an entropy (see page 3) that does not scale correctly with the size of the system (Voluime divided by number of particles,V/N).  We also show how to correct the counting (at least in the low density limit) and obtain the correct entropy function (the equation at the bottom of page 4 is know as the Sackur Tetrode equation). The matreial in this part of the lecture is discussed in Mandl (section 7.1).  Sanchez and Bowley also cover this material, respectively, in section 5.9, for distinguishable particles, and in section 6.5, leading to the Sackur Tetrode equation, (eq. 6.5.5)   The material we need in order to discuss fermions and bosons will also be given in the last lectures of the Quantum Physics lectures in IB Physics.  The derivation of Fermi-Dirac (F-D) statistics and Bose-Einstein (B-E) statistics (the latter, not for examination) follows the argument given in Mandl (section 11.3) and the Classical limit is treated in Mandl (section 11.4).   Bowley and Sanchez cover F-D and B-E statistics using using the grand canonical ensemble, which is beyond the scope of the course.
• Lecture 10 , handout (pdf) available.  The basic properties of degenerate Fermi gases are discussed.  The material inculdes the heat capacity of an electron gas, Pauli paramagnetism and degeneracy pressure.  Much of the material can be found in Bowley and Sanchez, section 10.3 and 10.4.
• Lecture 11 , handout (pdf) available.  The handout completes the discussion of Fermi systems by looking at degeneracy pressure in white dwarf stars.  The main part of the lecture deals with waves in periodic systems and, particularly, the vibrational properties of crystalline solids.  Similar material is presented in the new book by Blundell and Blundell (Concepts in Thermal Physics, OUP, 2006), Chapter 24.
• Lecture 12 , handout (pdf) available.  The ideas from lecture 11 are extended to include a diatomic chain and the dispersion characteristics of optical and acoustic phonons.  The measured properties of real, 3-D, monatomic and diatomic systems are discussed.  The book by MT Dove (Structure and Dynamics, OUP, 2003) deals withsimilar material in Chapter 8.  Again, Blundell and Blundell, chapter 24, covers the same ground at an appropriate level.
• Lecture 13 , handout (pdf) available.  Lecture 13 deals with the thermal properties of a crystalline solid and specifically the Debye Model.  Rosenberg's book (The Solid State) is a good source; however, most matrerial is also in Blundell and Blundell, chapter 24.
•  Lecture 14 ,  Lecture 15 ,  Lecture 16   handouts (pdf) available.  Collective Phenomena:  So far we have treated matter as if there were either no interactions (perfect classical and quantum gases), or on the basis of harmonic interactions (phonons).  In the final three lectures we deal with phenomena where interactions between the constituents dominate the behaviour.  The new effects come under the general heading of collective phenomena and include processes such as phase transitions, phase equilibria etc.,  The lectures applying the methods of statistical mechanics, particularly caluclations of the Helmholtz function, to a variety of  simple situations.  Lecture 14 deals with phase separation in a diatomic alloy; lecture 15 deals with the mean-field theory of continuous phase transitions; lecture 16 shows that the ferromagnetic-paramegnetic transition has similar behaviour and  hence we come to the ideas of Universality.  The final topic, not for examination, is a brief and qualitative introduction the ideas behind Landau's general  theory of phase transitions. Bowley and Sanchez, chapters 11 and 12 cover all the material, and more.