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Lecture 1 – Introduction to Statistical Mechanics
Ch 22
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
- what we want to know
- macroscopic
- 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
- what we want to know
- macroscopic
- microscopic
Energy, entropy, pressure, temperature
Speed of molecules
- macroscopic
- microscopic
Statistical mechanics relates the average actions of many
individual molecules to measurable macroscopic
properties
Why not simply use classical mechanics?
Mechanical properties
Coordinates
Momenta
Masses
Kinetic energy
Potential energy
(x, y, z)i
(p)i
mi
Ei
Ui
Relating the microscopic properties of biomolecules to their
macroscopic behavior
Mechanical properties
Thermodynamic properties
Coordinates
Momenta
Masses
Kinetic energy
Potential energy
Temperature
Pressure
Mass
Entropy
Free Energy
(x, y, z)i
(p)i
mi
Ei
Ui
T
P
M
S
G
Relating the microscopic properties of biomolecules to their
macroscopic behavior
Mechanical properties
Thermodynamic properties
Coordinates
Momenta
Masses
Kinetic energy
Potential energy
Temperature
Pressure
Mass
Entropy
Free Energy
(x, y, z)i
(p)i
mi
Ei
Ui
T
P
M
S
G
Two fundamental description of atoms and molecules
Classical Mechanics
Quantum mechanics
Kinetic energy
Potential energy etc
Kinetic energy
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
2
p
1 2
E  EK  U 
 mv
2m 2
E Ek p U U=0
U=infinity
total energy
kinetic energy
momentum
potential energy
if the particle is in the box
if the particle is outside of the box
Quantum particle-in-a-box: energy is quantized
p2
h2 n2
En 

2m 8m a 2
h -
where n  1, 2,3, 4, etc.
Planck constant
Another example, the hydrogen atom
from Chem 152 (we shall see later where
this comes from)
En 
2 2 e 4 m 1
....n  1,2,3, etc.
2
2
h
n
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:
N E
i
E 
i
N
i
Ni
  Ei   Pi Ei
N
i
i
Distributions
N E
i
E 
i
N
i
Ni
  Ei   Pi Ei
N
i
i
• 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
N E
i
E 
i
N
i
Ni
  Ei   Pi Ei
N
i
i
E   N i Ei
i
2
h 2  n x2 n y n z2 
E (n x , n y , nz ) 
 2 2
2

8m  a
b
c 
• 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)