1. Statistical Mechanics
Some conclusions from
Statistical Mechanics

2. Special issues for nanoscale systems
Fluctuations are important because the number of particles in a system is much less than Avogadro’s number;
Importance of the surface properties. Thermodynamic quantities no longer scale with the number of atoms in a system becuase the energy associated with the surface may be a significant fraction of the total.

3. Microscopic view of bulk properties
Nanosystems generally contain too many atoms to be thought of as simple mechanical systems, but too few to be described by bulk properties.
Microscopic view of bulk properties
Equilibrium thermodynamic properties are well defined because fluctuations are negligible in large (N=1023) systems.

4. Copyright (c) Stuart Lindsay 2008
The Boltzmann Distribution
P(1,2)=P(1)P(2).
E(1,2)=E(1)+E(2.). ,
Normalized Boltzmann distribution
kB = 1.381·10-23 J·K-1 = 8.62·10-5 eV·K-1
Copyright (c) Stuart Lindsay 2008

5. Copyright (c) Stuart Lindsay 2008
Values for kT
kBT at room temperature (300K) = J
= 25 meV (Much smaller than most bonds)
= 0.6 Kcal·mol-1
kBT at room temperature in terms of force· distance
= 4.14 pN·nm
- molecular motors produce forces ca. ten times more over nm distances
Copyright (c) Stuart Lindsay 2008

6. Normalized Boltzmann distribution
partition function
In case of degeneracy g(r):
The partition function enumerates all the states as a function of energy, so all the equilibrium properties of a system can be derived from it.
Copyright (c) Stuart Lindsay 2008

7. Copyright (c) Stuart Lindsay 2008
The degeneracy of a state leads naturally to a statistical definition of Entropy:
Entropy is proportional to the number of ways (statistical weight, ) a given macrostate r can occur.
In terms of the probability of the rth state:
Copyright (c) Stuart Lindsay 2008

8. The Equipartition Theorem
Average thermal energy of e.g., a harmonic oscillator
For a classical system (all energies allowed) replace the Boltzmann sum with an integral and calculate the product of E and p(E), e.g. for potential energy:
Copyright (c) Stuart Lindsay 2008

9. With a change of variables and a standard integral we find
Similarly
The thermal average of any quantity that appears in the classical Hamiltonian as a quadratic term is
Equipartition
theorem

10. The Equipartition theorem assumes that all degrees of freedom are in equilibrium with the heat bath and are independent.
However, it takes coupling between the degrees of freedom to ‘spread’ the thermal energy out evenly and this requires a non linear response.
Ex. It takes an anharmonic potential to couple vibrational and translational degres of freedom (V-T energy transfer).

11. Thermodynamics and Statistical mechanics
Thermodynamic potentials (“Free Energies”) can be minimized to obtain the equilibrium properties of a system.
Isolated system
Closed system (V,T)
Closed system (p,T)
Copyright (c) Stuart Lindsay 2008

12. Thermodynamic potentials in terms of partition functions

13. Grand Canonical ensemble
Open systems
Grand Canonical ensemble
Each system is enclosed in a container whose walls are both heat conducting and permeable to the passage of molecules.
→ transport of matter allowed, N variable
V,T,μ

14. aNr = number of systems in the ensemble that contain N molecules and are in the state r.
The set of occupation numbers {aNr} is a distribution.
Each possible distribution must satisfy the balance equations:
Number of systems in the ensemble
Total energy of the ensemble
Total number of molecules

15. For any possible distribution, the number of states is given by:
The distribution that maximizes W is:

16. Grand Partition function
 is the chemical potential
Gibbs Distribution
Grand Partition function

17. Canonical partition function
Summing over r states, it is possible to write:
Canonical partition function
activity

18. Ideal Gas: Z for one free particle (N=1)
For large systems:

19. Quantum concentration (one particle per wavelength3)
The de Broglie wavelength for a free particle is:
So, one particle occupies a quantum volume of about λ3:
Quantum concentration (one particle per wavelength3)

20. Ex. Quantum volume for a free electron at 300K
A sphere of a radius of 2.7nm!
For N non-interacting particles:

21. Quantum statistics Expliciting Q(N,V,T) (i.e. energy distribution):
Ej(N,V) = energy states available to a system containing N molecules
εk= molecular quantum states
nk= number of molecules in the kth molecular state when the system energy is Ej.

22. This last passage originates from the fact that we are summing over all values of N and that nk ranges over all possible values.

23. Fermi-Dirac statistics: ni=0 or 1, ni,max=1

24. Bose-Einstein statistics: ni=0, 1, 2,… nmax=∞
Where we used:

25. + = FD
- = BE

26. Classical limit (λ→0):
At the classical limit (high temperatures or low density) the number of available molecular quantum states is much greater than the number of particles.
The average number of molecules in any state is very small (nk→0, λ→0).
Thermodynamically:

27. Maxwell-Boltzmann
distribution
Summing over i:

28. Quantum gasses
We consider just an ideal gas (non-interacting particles) now subject to restrictions on how states are counted:
Bosons
Fermions
Using the Grand partition function:
With the Gibbs distribution

29. Single particle distribution
Writing numerator and denominator as products:
Single particle distribution
Fermi-Dirac (FD) statistics: ni=1 or 0

30. Fermi Dirac thermal average occupation
The chemical potential at T=0 is called the Fermi energy.
The electronic properties of most conductors are dominated by quantum statistics.

31. For metals  is several eVs!!
Ex. Fermi energy of Na is 3.24 eV.
For εi=μ:

32. Copyright (c) Stuart Lindsay 2008
Bose-Einstein (BE) statistics: ni=0, 1, 2, 3….
Summing Zi from n = 0 to :
<i
As ε approaches μ in the BE distribution, the occupation number approaches infinity, i.e. bosons condense into one quantum state at very low temperatures (Bose condensation).
Copyright (c) Stuart Lindsay 2008

33. Phonons are bosons with no chemical potential (μ=0), so that the occupation number goes to zero as temperature approaches zero.
μ=0