Statistical Thermodynamics

PYU33P/A15

Statistical Thermodynamics

Michaelmas Term

Prof. Graham Cross

PYU33P/A15 Statistical Thermodynamics

• 15 lectures equivalent – split into smaller videos on Blackboard

• Blackboard for lecture notes in PDF

• Two questions on annual exam (must do at least one)

• I will try to recommend some problems from textbooks

Textbooks

Kittel and Kroemer (K&K) Mandl Woolfson

• Count states in classical and quantum systems

• Understand fundamental assumption of statistical physics, concept of ensembles

• Model of a 2-state system (spin)

• Two systems in equilibrium: Stat mech versions of entropy, temperature, and chemical potential

• Partition functions and relations to classical thermodynamic quantities

• Third Law of Thermodynamics

• Fermi-Dirac and Bose-Einstein statistics

• Quasi-classical statistics: Equipartition of energy

• Application of quantum statistics to photons, gases and solids

Overview and Course Objectives

Our central objective is to show how a simple assumption of equal statistical weights

allows the properties of individual quantum particles to be combined together to calculate macroscopic thermodynamic quantities that can be compared with experiment.

Overview and Course Objectives

Introduction

PYU33P/A15 Statistical Thermodynamics

• You have an introduction to classical thermodynamics, ie. PYU22P10

• This is a logical/mathematical development of experimentally based laws

• Its predictions are tested experimentally

• It is powerful in that it makes wide range predictions of great generality

• It has limits however

• The origin of the Laws of Thermodynamics is unclear – and some quantities like entropy are completely abstract

• It applies to macroscopic phenomena only

• It does not allow a priori calculations of system properties, eg. from lower level physics of individual particles, etc.

• For example: A general equation of state which relates macroscopic variables and distinguishes one state from another:

Introduction

( , , ) 0 f p V T  pV  nRT

Ideal gas R  8.314JK mol^{1 1} is experimentally determined constant

• In statistical thermodynamics (stat mech), eg. PYU33P/A15 this course

• We assume an underlying atomic (particle) nature of systems

• We deduce macroscopic quantities directly from the atomic properties

• For macroscopic systems consisting of moles (~10²³) of material, we use powerful statistical methods

• In stat mech the following is found

• The behaviour of collections of particles can be predicted accurately without knowing detailed behaviour of any one particle

• Quantum systems turn out to be easier to handle (!)

• Eg. We need the postulates of QM combined with one extra postulate about probability to create a function that behaves as thermodynamic entropy

• Stat mech is more computationally more complex however

Introduction

Foundations

PYU33P/A15 Statistical Thermodynamics

• System Macrostate: Properties at large scale of the system when we know constraining thermodynamic parameters such as

• These are often only well defined and will always be time-independent at equilibrium

• System Microstate: Particle scale properties of system

• Quantum microstate: Each quantum state is a separate and distinct microstate of the system

• Solutions to eg. the Schrödinger equation

• Classical microstate: An abstract phase space description in q-p space (position-momentum)

• Uses classical Hamiltonian mechanics and Liouville’s Theorem

• We will use the quantum approach in PYU33P/A15

Concepts and Terminology

, , , p V T

• We assume weakly coupled systems:

• Consider an isolated system such that total energy, volume and number of particles is constant

• Weak coupling implies that the energy levels of a single particle are effectively unchanged by particle interactions… BUT (!) the interaction is sufficiently large to allow energy exchange and equilibrium reached at a common temperature.

• This assumption allows the QM to be solved easily:

• For example for a gas of hydrogen (H₂) molecules, QM calculations of electronic,

vibrational, rotational and translational eigenvalues of a single H₂ (ie. the single particle state or “orbital”) can be solved analytically.

• Once we have this, Statistical Thermodynamics simply counts the number microstates in a macrostate – that’s it (!)

• Quantum microstates are often discrete and easy to count (particle in box, etc.)

Concepts and Terminology

• To count the microstates in a macrostate:

• Specify the constraining parameters of the macrostate

• Determine the single particle states (orbitals)

• Determine if the particles are localized

• If the particles are localized they are distinguishable

• Eg. The magnetic nuclei of atoms in a solid

• These are weakly coupled with

• Distinguished by label-able position within solid lattice

• If the particles are non-localized they are indistinguishable

• Eg. Gas molecules, where the full wavefunction of system then depends on whether particles are bosons or fermions

Concepts and Terminology

external

Nucleus Nucleus Nucleus B

E _  E _

• Consider two indistinguishable particles 1 and 2, and two single particle states (orbitals) a and b

• Say we have and

• Due to weak system coupling, we can write the combined system wavefunction as a product

Bosons and Fermions

a(1)

 _b(2)

(1, 2) _a(1)_b(2)

 

(1, 2) _a(2)_b(1)

 

(1, 2) _a(1)_b(2) _a(2)_b(1)

  

Events are ~independent Indistinguishable property But we can also write

Quantum mechanics give linear combination as most general:

The form is symmetric with respect to a particle exchange:

• Bosons with Bose-Einstein statistics (1, 2) _a(1)_b(2) _a(2)_b(1)

  

The form is anti-symmetric with respect to a particle exchange:

• Fermions with Fermi-Dirac statistics (1, 2) _a(1)_b(2) _a(2)_b(1)

  

• All particles in nature are either bosons or fermions

• A single particle state can be occupied by any number of bosons, but only one fermion (Pauli exclusion principle)

Bosons and Fermions

• The total particle spin determines whether we have bosons or fermions

• Photons with spin 1: Boson

• Electrons, protons, neutrons with spin ½: Fermion

• Bosons have integral spin

• Example of ⁴He atoms with spin sum of 3

• Fermions have half-integral spin

• Example of ³He atoms with spin sum of 2½

Time averages and ensemble averages

• We deal with the large number of microstates accessible to the system statistically

• Imagine r successive observations of the system microstate at times t₁, t₂, t₃,…, t_r

• Let n(l) be the number of times the system is found in a particular microstate l

• The probability P of finding the system in the microstate l is ( ) n l( )

P l r

 r   ( ) 1

l

P l 



l

n l  r



Note that because

• The average of some physical parameter of a system A is

( ) ( ) 1 ( ) ( )

l l

A A l P l A l n l







• To realize this property, observations must be on a time scale that is long in comparison with the time for the system to randomize (come to equilibrium), known as the relaxation time.

• Relaxation times depend on the property A in question.

Time averages and ensemble averages

• An ensemble average or thermal average is the average over a large number of replicas of the entire system

• A replica realizes one microstate of the system

• The collection of replicas is an ensemble

• The ensemble can also be used to determine P(l), and is postulated to be equivalent to the time sampling of a single system (this is famously known as the ergodic hypothesis)

Ensemble averages produce results which agree with experiment

Fundamental assumption of statistical thermodynamics (stat mech):

• A system in thermal equilibrium is equally likely to be in any of the microstates accessible to it

• This is the assumption of equal a priori probabilities

Summary

• A macrostate is defined when we know the constraining thermodynamic state variables like p, V, T,… of the system

• A microstate is a quantum state of the system

• Statistical mechanics is based on counting the microstates that make up a macrostate

• Each microstate is given the same statistical weight (assumption of equal a priori probabilities)

• An ensemble is the collection of microstates (replicas) of the system

• Ensembles enable microstates to be counted

• Ensemble averages are equivalent to time averages

Counting States

PYU33P/A15 Statistical Thermodynamics

A simple model system: K&K Chapter 1

• No interaction between spins, no external magnetic field.

• Assume each has magnetic moment where^,   ^{eћ 2 ,m} the Bohr magneton

• Each spin has two distinct possible values giving magnetic moment or (spin up or spin down).1 2 , ^ ^

3





Example N=3. As particles are localised,    etc., we have 2³ = 8 possible arrangements of spin (microstates):



  

  







3









(N + 1) = 4 distinct values of M

M Total magnetic moment – a macrostate

• Set of distinct elementary magnets: spin ½ particle, spin angular momentum at N fixed points on a lineћ 2 ,

These are localised and distinguishable.

• Each arrangement is a separate and distinct microstate of the system.

• For N=3 , from our fundamental assumption of stat mech, the probability of finding a given microstate is 1/8.

• We can define a generating function by (◊+●)³ = ◊ ◊ ◊ + ◊ ◊ ● + ◊ ● ◊ + … + ● ● ●

In General: N spins

2^N distinct microstates

probability of finding any single state is 1/2^N generating function (◊ + ●)^N

• The macrostate here is M, the total magnetic moment of the system

• From the language of last year (PYU22P10), this is just a bulk thermodynamic state variable

• Summing spin vectors, M goes from to in steps of (for each reverse of a spin).

• M does not depend on arrangement of spins  M has (N+1) distinct values.

• The magnetisation, which is used to describe an intrinsic property of a macroscopic magnetic material, is just the

^

N N 2^

A simple model system

• Math is easier if we take N as even (N is very large, e.g. a microgram of material has N~10¹⁶, so a difference of one is not significant)

• Binomial theorem (Ch. 1, Woolfson):

• M is given by the excess of one type of spin over the other, so this spin excess is important:

2m is the spin excess

^











m N m M m

N ) spins,( ) spins 2

( 2

1 2

1

Finding the number of microstates in a macrostate



   



        

4^

 2^

 4^

2^



M

0

m

0

1

2 1 2

Eg. For N=4 (even):

Example: (x y)³  x³ 3x y² 3xy²  y³ - compare to our 3 spin system, above Note that 0! 1

0

( )

N

N N N i i

i i

x y C x ^ y



 



Re-expressing the binomial theorem in terms of spin excess, with :i  ¹₂ N  m

 

1

1 1

2

2 2

1

1 2

2 N

N m N m

N N

N m

m N

C

  

 

   



2

! !

1 1 ! !

( )!( )!

2 2

N

N m

N N

C N N

N m N m

  

 

 

where

 

2

m m

m m

C

  





   



m m

 

 





where For example, for N=4 (even):

Finding the number of microstates in a macrostate

• generates all possible microstates.

• An ensemble comprises all microstates of a given macrostate, M (and hence spin excess 2m).

Eg. For N=4, the m=0, M=0 ensemble is

• The Binomial Coefficient gives the number of microstates in the macrostate of given N, M (and hence m ).

• This coefficient is called the multiplicity function ( in K&K) or the statistical weight (Mandl):

)N

(  

Summary of the model

1 2

( , ) !

1 1

( )!( )!

2 2

N

N m

N m C N

N m N m

   

  ⁽²⁾

     

( , )N m

 g N s( , )

• With no applied magnetic field, all microstates have the same energy  degenerate (QM lingo)

• If a field B is applied, then the total potential energy of the system

(so the system is more stable if spins are aligned parallel to the field, “spin-up”.)

• The overall degeneracy is lifted, but states of the same m are still degenerate.

B m m

U( )  M B  2 ^

• Is the most probable (its 6 / 2^4 = 6/16 = 3/8)

• Can be in any one of 6 microstates

Thus, with no external magnetic field, the macrostate with m = 0:





Example for most likely state: Eg. N=4 (even):

(N 4,m 2) 1

    

(N 4,m 1) 4

    

(N 4,m 0) 6

   

Model interaction with surroundings (B-field)

• This is a Gaussian distribution with a maximum at , and the most probable macrostate has an equal number of up and down spins.

is an extremely (!!) sharply peaked function (width ) (See K&K pp. 18-21) m 0

( , )N m ~ N

The shape of  (N,m) for large N  10 ²⁰

We obtain

2 2

ln ln ( , 0) m

N N

   

2 2

( , ) ( , 0) exp m

N m N

N

  

    

 

  ²

! 2

( , 0) 2

2 ! N N

N N N

  

 

 

and

where

(3)

1 2

( , ) !

1 1

( )!( )!

2 2

N

N m

N m C N

N m N m

   

 

The mathematical derivation is set out in Appendix A. Briefly here:

• Take the natural logarithm to convert products to sums

• Split out constant terms

• Recognize that near the peak 𝑚 ≪ 𝑁

Example: Consider a system of ~1 g (10²²) spins and no magnetic field. What is the chance of measuring the

magnetisation and finding a spin excess of 1 part in 10¹⁰, compared to the most probable value of zero spin excess?

So only 1 in 10²² probability of finding a macrostate with a spin excess of only 1 in 10¹⁰ spins, in 1 g of spins.

Statisticians commonly call this distribution the normal distribution

and, because of its curved flaring shape, social scientists refer to it as the bell curve. The distribution P(x) is properly normalized.

~1.18s

The Gaussian probability distribution with mean and standard deviation s is a normalized Gaussian function of the form

(See Ch. 32.3, Woolfson) ²

2

1 ( )

( ) exp

2 2

P x x 

s s

  

  

 

1022

N  ^{22 12}

10

2 1 10 10

m 10  

 

22 11

12

22 2

11 2

22 2 22

( , ) ( , ) (10 , 5 10 )

(2 10 )

( , 0) ( , 0) (10 , 0) ( , 0) exp 2

2 5 10

exp exp( 50) 10

2 10 ( , 0) exp

N m N m

P m N N

N m

N N m



     

   

  

  

       

    

 

  

    

We are considering excess spins

For large N we have

(N is taken as even)

Appendix A: Approximation of the multiplicity function (N,m) for large N, eg.  10²⁰

1 2

( , ) !

1 1

( )!( )!

2 2

N

N m

N m C N

N m N m

   

  ^{ln( , )N m  ln N!}^{ln(N 2m)! ln( N 2m)!}

       

(N 2m)! N 2 ! 1 N 2  2 N 2   m N 2

 

 

1

ln ( 2 )! ln 2 ! ln ( 2 )

m

i

N m N N i



    



 

 

1

ln ( 2 )! ln 2 ! ln ( 2 1)

m

i

N m N N i



    



   

1

( 2 )

ln ( 2 )! ln ( 2 )! 2 ln 2 ! ln

( 2 1)

m

i

N i

N m N m N

N i



  

      



and so

(N 2  i 1) N 2i

  ^{2 1}

! 1 2

ln ( , ) ln ln

2 ! 1 2

m

i

N i N

N m N ^ i N

  

    



       

(N 2m)! N 2 ! /  N 2  N 2 1   N 2 m 1  and

See K&K pp. 18-21 Note that the approximation is only good for m<<N, near the peak of (N,m)

Letting and noting near the peak, we have to first order

This is a Gaussian distribution with a maximum at , and the most probable macrostate has an equal number of up & down spins.

m 0

Appendix A: Approximation of the multiplicity function (N,m) for large N  10²⁰

3 5

ln1 2

1 3 5

x x x

x x

 

      

  

1 2 4

ln 1 2

i N i i N N

   

  

 

We obtain

2 2

ln ln ( , 0) m

N N

   

2 2

( , ) ( , 0) exp m

N m N

N

  

    

 

  ²

( , 0) !

2 ! N N

N

 

 

 

and

where

(3) 1

x 

For we have the Taylor expansion

2

x  i N i  m N

  ²

! 2

2 2 !

N N

N ^ N

 

 

See K&K pp. 18-21 to further show

2

1 2

m

i

i m



