1. Lecture 1 – Introduction to Statistical Mechanics

2. Biological systems are complex macroscopic systems
Composed of many components or particles (atoms, molecules, etc.)
Thermodynamics: properties of complex systems at equilibrium
Described by state variable (P, V, T, etc … Chem 452)
Classical mechanics or quantum mechanics
Properties of ‘simple systems’
Position, velocity, momentum etc
We execute measurements on macroscopic biological systems
(1023 atoms or molecules)
We want to learn about their microscopic properties
Structure, mobility, interactions

3. Relating the microscopic properties of biomolecules to their macroscopic behavior
Statistical mechanics relates the average properties of a complex system containing many molecules to the individual microscopic properties of the molecules that compose the system under investigations
Observables - what we measure macroscopic
- what we want to know - microscopic

4. Relating the microscopic properties of biomolecules to their macroscopic behavior
Example 1 –RNA melting and its three-dimensional structure
Example 2 – Separation of DNA molecules by size
Example 3 – Separation of protein molecules by size and charge
Example 4 – Helix-coil transitions in polypeptides

5. Relating the microscopic properties of biomolecules to their macroscopic behavior
Example 1 –RNA structure/function
Example 2 – DNA sequencing
Example 3 – Proteomics
Example 4 – Protein folding and engineering

6. Relating the microscopic properties of biomolecules to their macroscopic behavior
Observables - what we measure macroscopic
- what we want to know - microscopic
Energy, entropy, pressure, temperature - macroscopic
Speed of molecules - microscopic
Statistical mechanics relates the average actions of many individual molecules to measurable macroscopic properties

7. Why not simply use classical mechanics?
Mechanical properties
Coordinates (x, y, z)i
Momenta (p)i
Masses mi
Kinetic energy Ei
Potential energy Ui

8. Relating the microscopic properties of biomolecules to their macroscopic behavior
Mechanical properties Thermodynamic properties
Coordinates (x, y, z)i Temperature T
Momenta (p)i Pressure P
Masses mi Mass M
Kinetic energy Ei Entropy S
Potential energy Ui Free Energy G

9. Relating the microscopic properties of biomolecules to their macroscopic behavior
Mechanical properties Thermodynamic properties
Coordinates (x, y, z)i Temperature T
Momenta (p)i Pressure P
Masses mi Mass M
Kinetic energy Ei Entropy S
Potential energy Ui Free Energy G

10. Two fundamental description of atoms and molecules
Classical Mechanics Quantum mechanics
Kinetic energy Kinetic energy
Potential energy etc Potential energy
Can take on any value Can only take certain values
(are quantized!)
Description of certain statistical mechanical properties is more natural in terms of quantized energy values

11. Classical and quantum mechanical particle-in-a-box

12. Classical particle-in-a-box: velocity and energy are continuous
E - total energy
Ek - kinetic energy
p - momentum
U - potential energy
U=0 if the particle is in the box
U=infinity if the particle is outside of the box

13. Quantum particle-in-a-box: energy is quantized
h - Planck constant
Another example, the hydrogen atom from Chem 152 (we shall see later where this comes from)

14. Distributions
A fundamental assertion of equilibrium statistical mechanics is that a thermodynamic property is an average over all the individual microscopic states of the system

15. Distributions
Consider a system composed of N molecules (for example of the order of 1023) that do not interact with each other
Each of the N gas molecules in the system (note N is a very large number) can have energies Ei, (i=1, 2, 3, etc.)
Let us assume for simplicity the energy levels Ei are equally spaced, as shown to the left

16. Distributions
At any instant, N1 molecules in the system have energy E1, N2 have energy E2, etc.
The average energy is then defined as:

17. Distributions
Ni – occupations number of level i
Pi – fraction of molecules with energy Ei
- probability of a molecule having
energy Ei
{P1, P2, P3 … Pi ..} – Distribution
- how the particles are distributed
among energy levels

18. Distributions
No matter how complex the energy levels are, we can always calculate total energy and average energy from the distribution
Example, the quantum harmonic oscillator (which we shall see later)

19. Distributions
If we know the energy levels (which may be very difficult) we can calculate all thermodynamic properties of a system provided we know the distribution
A fundamental statistical mechanics/statistical thermodynamics is the distribution of a system at thermodynamic equilibrium (Boltzmann)