STATISTICAL MECHANICS
BEH.410 Tutorial
Maxine Jonas
February 14, 2003
Why Statistical Mechanics?
Understand & predict the physical properties of
macroscopic systems from the properties of their
constituents
Deterministic approach
- need of 6N coordinates at t0: ri and vi
- but typically N ≡ moles (1023) !
“Ensemble” rather than microscopic detail
… and its surroundings
¾ microcanonical, canonical, grand canonical
What With Statistical Mechanics?
Averages, distributions, deviation estimates…
… of microstates: specification of the complete set of
positions and momenta at any given time (points on
the constant energy hypersurface for Hamiltonian
dynamics)
Ensemble average & ergodic hypothesis:
A system that is ergodic is one which, given an infinite amount of
time, will visit all possible microscopic states available to it.
The First Law – Work
Work, heat & energy = basic concepts
Energy of a system = capacity to do work
¾ At the molecular level, difference in the surroundings
Energy transfer that makes use of…
Heat
… chaotic molecular motion
Work
… organized molecular motion
state function – independent of how state was reached
Second Law – Gibbs
ƒ Spontaneous processes increase
the overall “disorder” of the
universe
™ Reasoning through an example
- microstates to achieve macrostate
Gibbs postulate: for an isolated system, all microstates
compatible with the given constraints of the macrostate
(here E, V and N) are equally likely to occur
- Here 2N ways to distribute N molecules into 2 bulbs
Second Law - Probability
ƒ Number of (indistinguishable)
ways of placing L of the N
molecules in the left bulb:
L
ƒ Probability WL / 2N maximum if L = N / 2
9 With N = 1023, p ( L = R ± 10-10 ) = 10-434
possible but extremely unlikely
Second Law - Entropy
Boltzmann’s constant
k = 1.38 x 10-23 J.K-1
Principle of Fair
Apportionment
Multiplicity of outcomes
Second Law - Entropy
The absolute entropy
is never negative
S≥0
S max at equilibrium
pi
1
0
n
e
0 0
s w
1.33
Flat distribution ≡ high S
1/3
0.69
0
0rder
1/6 1/6
1/3
1/2 1/2
0
0
1.39
1/4 1/4 1/4 1/4
The Boltzmann Distribution Law
ƒ Maximum entropy principle + constraints
E3
E2
⇒ exponential distribution
E1
Partition function
The Boltzmann Distribution Law (2)
E
ƒ More particles have low energy:
more arrangements that way
high T
E
ƒ Q ≡ connection between microscopic models &
macroscopic thermodynamic properties
and
medium T ƒ Q ≡ number of states effectively accessible to system
E
low T
The Helmholtz Free Energy
ƒ Systems held at constant T → minimum free energy (≠ Smax)
Equilibrium if F (T, V, N) minimum (T fixed at boundaries)
Internal energy
F = U - TS
Entropy
™ Example of ‘dimerization’
0
-ε
dim
mon
F (T)
Fundamental Functions
U (S, V, N)
min
(S and V at boundaries)
S (U, V, N)
max
(U and V at boundaries)
H (S, p, N)
min
(S and p at boundaries)
calorimetry
F (T, V, N)
min
(T and V at boundaries)
Internal energy
vs.entropy
G (T, p, N)
min
(T and p at boundaries)
Enthalpy
vs. entropy
calorimetry
cal.
Macromolecular Mechanics
ƒ Why study the mechanics of biological macromolecules?
-
provide structural integrity and shape
coupling of geometry & dynamics ⇒ what is possible
importance of conformation for ion channels, pumps…
motility
mechanotransduction, signaling
The Gaussian Chain Model (Kuhn)
ƒ Long floppy chain made of N rigid links of length b
(free to swivel about joints, overlapping & crossing allowed)
ƒ Valid for small displacements from equilibrium, not large
extensions
b
Entropic reasoning ⇒ mechanical spring
(straightening out ≡ decrease of entropy)
R
The Worm-like Chain Model
ƒ Self-avoiding linear chains (Flory, 1953)
ƒ Freely-jointed chain model (Grosberg & Khoklov, 1988)
ƒ Worm-like chain model: Bending stiffness of polymer on
short length scales
(Kratky-Porod)
(diverges for x → L)
s=L
Fx
x
bending
Persistence length
thermal
Experimental Validation of Models
100
ƒ Single-molecule studies of
DNA mechanics:
ƒ WLC interpolated:
Marko & Siggia (1995)
Macromolecules, 28: 8759
10
Force (pN)
Bustamante et al. (2000) Curr. Op.
Struct. Biol., 10: 279
dsDNA
fit to WLC model
1
FJC
0.1
Hooke’s law
0.01
0
0.2
0.4
0.6
0.8
1.0
1.2
Extension (x/L)
After Bustamante et al., Current Opinion in Structural Biology, 2001
Effect of Force on Equilibrium
F
1
F
2
x2
x1
G
ƒ Force tilts energy profile
⇒ favors configuration
no force
1
2
Reaction coordinate
force
Sources
ƒ Boal D. (2002) Mechanics of the cell. Cambridge University Press
ƒ Bustamante C. et al. (2000) Curr. Op. Struct. Biol., 10: 279
ƒ Dill K.A. & Bromberg S. (2003) Molecular driving forces: Statistical
thermodynamics in chemistry and biology. Garland Science.
ƒ Leland T.W. Basic principles of classical and statistical thermodynamics
http://www.uic.edu/labs/trl/1.OnlineMaterials
ƒ Mahadevan L. Macromolecular mechanics, class material.
ƒ Marini D. (2002) Some thoughts on statistical mechanics (notes)
ƒ Marko J.F. & Siggia E.D. (1995) Macromolecules, 28: 8759
ƒ Stanford encyclopedia of philosophy: plato.stanford.edu/entries
ƒ Tuckerman’s lecture notes: www.nyu.edu/classes/tuckerman/stat.mech
ƒ www.biochem.vt.edu/courses/modeling/stat_mechanics.html