Microscopic definition of temperature

1. Some key results for QPM Microscopic definition of temperature Distribution of microstates Microscopic definition of entropy This applies to an isolated system for which all the microstates are equally probable

2. Systems in contact with a heat bath 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

3. Connection to thermodynamics 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

4. k-states in three dimensions 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 .

5. Density of states for EM modes 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

6. Distribution of speeds for atom in a gas 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:

7. Maxwell-Boltzmann distribution Velocity distribution for a nitrogen molecule at three temperatures

8. Quantum Harmonic Oscillator The Planck Distribution 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.

9. Stefan’s Law Planck’s Black Body Radiation 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  = 5.6710-8 Wm-2K-4

10. 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

11. Systems with variable numbers of particles Chemical Potential 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.

12. Bose-Einstein Distribution 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

13. 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.

14. Rearranging The Fermi energy: εF = μ(T=0) 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:

15. This is called the degeneracy pressure. Degeneracy pressure of fermi gas 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.

16. Expect <n (ε)> << 1 The Classical Limit 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.