Lecture 1 – Introduction to Statistical Mechanics

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

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

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

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

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

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

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

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

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

Classical and quantum mechanical particle-in-a-box

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

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)

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

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

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

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

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)

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)