1. Statistical Physics Notes
1. Probability
Discrete distributions
A variable x takes n discrete values, {xi , i = 1,…, n}
(e.g. throwing of a coin or a dice)
After N events, we get a distribution, {N1, N2, …, Nn}
Probability:
Normalization:
Markovian assumption: events are independent

2. Continuous distributions
A variable x can take any value in a continuous interval [a,b]
(most physical variables; position, velocity, temperature, etc)
Partition the interval into small bins of width dx.
If we measure dN(x) events in the interval [x,x+dx], the probability is
Normalization:

3. Examples: Uniform probability distribution
Gaussian distribution
A: normalization constant
x0: position of the maximum
s: width of the distribution
P(x)
a x
1/a 2s

4. Normalization constant
Substitute
Normalized Gaussian distribution

5. Properties of distributions
Average (mean)
discrete distribution
continuous distribution
Median (or central) value: 50% split point
Most probable value: P(x) is maximum
For a symmetric distribution all of the above are equal (e.g. Gaussian).
Mean value of discrete
an observable f(x) continuous

6. Variance: measures the spread of a distribution
Root mean square (RMS) or standard deviation:
Same dimension as mean, used in error analysis:

7. Examples: Uniform probability distribution, [0,a]
Gaussian distribution
For variance, assume

8. Detour: Gaussian integrals
Fundamental integral:
Introduce

9. Addition and multiplication rules
Addition rule for exclusive events:
Multiplication rule for independent variables x and y
discrete
continuous
Examples: 2D-Gaussian distribution If

10. In cylindrical coordinates, this distribution becomes
Since it is independent of , we can integrate it out
Mean and variance

11. 3D-Gaussian distribution with equal ’s
In spherical coordinates
We can integrate out  and 
Here r refers to a vector physical variable, e.g. position, velocity, etc.

12. Mean and variance of 3D-Gaussian distribution

13. Most probable value for a 3D-Gaussian distribution
Set
Summary of the properties of a 3D-Gaussian dist.:

14. Binomial distribution
If the probability of throwing a head is p and tail is q (p+q=1), then the
probability of throwing n heads out of N trials is given by the binomial
distribution:
The powers of p and q in the above equation are self-evident.
The prefactor can be found from combinatorics. An explicit construction
for 1D random walk or tossing of coins is shown in the next page.

15. Explicit construction of the binomial distribution
There is only 1 way to get all H or T,
N ways to get 1T and (N-1)H (or vice versa),
N(N-1)/2 ways to get 2T and (N-2)H,
N(N-1)(N-2)…(N-n+1)/n! ways to get nT and (N-n)H (binomial coeff.)

16. Properties of the binomial distribution:
Normalization follows from the binomial theorem
Average value of heads
For

17. Average position in 1D random walk after N steps
For large N, the probability of getting is actually quite small.
To find the spread, calculate the variance

18. Hence the variance is
To find the spread in position, we use

19. Large N limit of the binomial distribution:
Mean collision times of molecules in liquids are of the order of picosec.
Thus in macroscopic observations, N is a very large number
To find the limiting form of P(n), Taylor expand its log around the mean
Stirling’s formula for ln(n!) for large n,

20. Substitute the derivatives in the expansion of P(n)

21. Thus the large N limit of the binomial distribution is the Gaussian dist.
Here is the mean value, and the width and normalization are
For the position variable we have

22. How good is the Gaussian approximation?
Bars: Binomial distribution with p=q
N= N=14
Solid lines: Gaussian distribution

23. 2. Thermal motion Ideal gas law: Macroscopic observables:
P: pressure, V: volume, T: temperature
N: number of molecules
k = 1.38 x J/K (Boltzmann constant)
At room temperature (Tr = 298 K), kTr = 4.1 x J = 4.1 pN nm
(kTr provides a convenient energy scale for biomolecular system)
The combination NkT suggests that the kinetic energy of individual
molecules is about kT. To link the macroscopic properties to molecular
ones, we need an estimate of pressure at the molecular level.

24. Derivation of the average kinetic energy from the ideal gas law
Consider a cubic box of length L filled with N gas molecules
The pressure on the walls arises from the collision of molecules
Momentum transfer to the y-z wall
Average collision time
Force on the wall due a single coll.

25. In general, velocities have a distribution, so we take an average
Average force due to one molec.
Average force due to N molec’s
Pressure on the wall
Generalise to all walls
Average kinetic energy
Equipartition thm.: Mean energy associated with each deg. of fredom is:

26. Distribution of speeds
Experimental
set up
Experimental results for Tl atoms
○ T=944 K
● T=870 K
▬▬ 3D-Gaussian dist.
v is reduced by
Velocity filter

27. Velocities in a gas have a Gaussian distribution (Maxwell)
The rms is
Distribution of speeds (3D-Gaussian)
This is the probability of a molecule
having speed v regardless of direction

28. Example: the most common gas molecule N2
Oxygen is 16/14 times heavier, so for the O2 molecule, scale the above
results by
Hydrogen is 14 times lighter, so for H2 scale the above results by

29. Generalise the Maxwell distribution to N molecules
(Use the multiplication rule, assuming they move independently)
This is simply the Boltzmann distribution for non-interacting N particles.
In general, the particles interact so there is also position dependence:
Universality of the Gaussian dist. arises from the quadratic nature of E.

30. Speed dist. for boiling water at 2 different temperatures
Activation barriers and relaxation to equilibrium
Speed dist. for boiling water at 2 different temperatures
Removing the most energetic molecules creates a non-equilibrium state.
When boiled, water molecules with sufficient kinetic energy evaporate
Arrhenius rate law: Ebarrier : Activation barrier
Those molecules with K.E. > Ebarrier can escape

31. Equilibrium state (i. e. Gaussian dist
Equilibrium state (i.e. Gaussian dist.) is restored via molecular collisions
Injecting very fast molecules
in a box of molecules results
in an initial spike in the
Gaussian distribution
Gas molecules collide like billiard balls
(Energy and momentum are conserved)
Thus in each collision, the fast molecules
lose energy to the slower ones (friction)

32. 3. Entropy, Temperature and Free Energy
Entropy is a measure of disorder in a closed system.
When a system goes from an ordered to a disordered state, entropy
increases and information is lost. The two quantities are intimately
linked, and sometimes it is easier to understand information loss or gain.
Consider any of the following 2-state system
Tossing of N coins
Random walk in 1-D (N steps)
Box of N gas molecules divided into 2 parts
N spin ½ particles with magnetic moment m in a magnetic field B
Each of these systems can be described by a binomial distribution

33. There are 2N states in total, but only N+1 are distinct.
Introduce the number of states with a given (N1, N2) as
We define the disorder (or information content) as
For the 2-state system, assuming large N, we obtain
(Shannon’s formula)

34. Thus the amount of disorder per event is
I vanishes for either p1=1 or p2=1 (zero disorder, max info) and
It is maximum for p1=p2=1/2 (max disorder, min info)
Generalization to m-levels:
Max disorder when all pi are equal, and zero disorder when one is 1.

35. Entropy
Statistical postulate: An isolated system evolves to thermal equilibrium.
Equilibrium is attained when the probability dist. of microstates has the
maximum disorder (i.e. entropy).
Entropy of a physical system is defined as
Entropy of an ideal gas
Total energy:
Radius of the sphere in
3N dimensions

36. Area of such a sphere is proportional to r3N-1 ≈ r3N
Hence the area of the 3N-D volume in momentum space is (2mE)3N/2
The number of allowed states is given by the phase space integral
Sakure-Tetrode formula
Area of a unit
sphere in 3N-D
Planck’s const.

37. Temperature:
If the energies are not equal, and we allow exchange of energy via
a small membrane, how will the energy evolve? (maximum disorder)
Isolated system
Total energy is conserved

38. Example: NA = NB= 10
Fluctuations in energy are proportional to

39. Definition of temperature
At equilibrium:
In general
Free energy of a microscopic system “a” in a thermal bath
(zeroth law of thermodynamics)
Average energy
Partition function

40. Example: 1D harmonic oscillator in a heat bath
In 3D:
Equipartition of energy: each DOF has kT/2 of energy on average

41. Free energy of the harmonic oscillator
Entropy:

42. Harmonic oscillator in quantum mechanics
Energy levels
Free energy:
Entropy:

43. To calculate the average energy let
Using <Ea> yields the same entropy expression.
Classical limit: