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Microscopic definition of temperature

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Some key results for QPM

Distribution of microstates

Microscopic definition of entropy

This applies to an isolated system for which all the microstates are equally probable

Boltzmann distribution

Partition Function

This gives the normalised probability of finding the system in a state with energy Ei when in thermal equilibrium at temperature T.

Calculating thermal averages

Canonical Entropy

Starting from the results

leads to a connection between the Helmholtz Free Energy and the partition function:

It follows that

We have a route for calculating these from Z

Standing wave solutions of the form

so k= (kx,ky,kz) with each component quantised:

Constant energy surface

This gives a cubic mesh with mesh size π/L .

In the case of electromagnetic radiation we have a linear “dispersion relation”, since E = ħω = ħck.

with k= E/ħc and dk= dE/ħc:

Density of single particle states (3d)

Suppose we are dealing with non-relativistic particles where

Since ħk = mv, we can define the number of states that have speeds in the range v to v+dv as:

The probability that an atom has a speed in the range v to v+dv equals the number of states in that range times the Boltzmann probability of having the corresponding energy:

Substituting the result for Z we get:

Velocity distribution for a nitrogen molecule at three temperatures

Quantum Harmonic Oscillator

The mean number of oscillators is:

The mean energy per oscillator is:

(ignoring zero point energy)

This applies to any system that can be regarded as a collection of QHOs, e.g. photons in a cavity (black body radiation), lattice vibrations, etc.

Stefan’s Law

The energy density of EM radiation in a cavity in the frequency range ωtoω+dω is

- Wien’s law ωmax ≈ T
- The area under the curve increases asT 4 .
- Power radiated by a black body is

= 5.6710-8 Wm-2K-4

Debye approx. for lattice vibrations

3NkB

ω

ωD

ω=ck

Molar heat capacity of copper compared to Debye theory

kD

k

The dispersion surface ωj(k) (where j=1,2,3 labels the branches) is replaced by the simple linear dispersion ω = ck, where c is an average sound velocity.

Debye introduced a cut-off wavevector kD which is chosen so that there are exactly 3N normal modes. The cut-off in frequency is then ωD = ckD,

- CV is proportional to T 3 as T→0
- CV →3NkB for kBT>> ħωD

Systems with variable numbers of particles

The Grand (Gibbs) Distribution

where

This is the equivalent of the Boltzmann distribution for the situation where the number of particles in the system is not fixed.

kBT/μ=0

kBT/μ=0.1

<n>

kBT/μ=0.25

kBT=1.0

kBT/μ=0.5

ε/μ

<n>

kBT=0.5

kBT=0.1

kBT=0.25

ε

Fermi-Dirac Distribution

Bose Einstein Condensation

To calculate the chemical potential for a Bose gas:

We find that μ→ 0 at a finite temperature. This leads to a macroscopic number of particles in the ground state. These particles cannot contribute to CV or the pressure, since they have zero momentum.

In real BEC systems the condensate forms a macroscopic quantum state that displays superfluidity etc.

Rearranging

We need N/2 k states to accommodate N fermions. The total number of fermions is the integral of the occupancy over all the states:

Ground state energy of the fermi gas

Using our result for the Fermi energy:

we find for the ground state energy:

This is called the degeneracy pressure.

It contributes to the bulk modulus of metals and is the main source of stability of white dwarf and neutron stars.

Heat capacity of fermi gas

- Heat capacity is linear as long as T << TF.
- At high temperatures CV approaches the classical value (3/2)NkB above T ~ 2TF.

Expect <n (ε)> << 1

In the limit where , we may ignore the 1 in the denominator, compared to the exponential. This leads to:

Classical Distribution Function:

→T

μ

↑

We find that for both BE and FD gases the chemical potential becomes large and negative in the high temperature limit. This is the criterion that defines the classical limit.