Transcript Lecture #6 09/14/04

Thermodynamics II
I. Ensembles
II. Distributions
III. Partition Functions
IV. Using partition functions
V. A bit on gibbes
Ensembles
Formally, an ensemble is virtual construct of many copies of a system of interest.
Each member of an ensemble has some mechanic or thermodynamic variables fixed,
but all states corresponding to these fixed variables all allowed.
Each state is represented equally in an ensemble; or alternatively an isolated
system is equally likely to be found in any one possible quantum state
Suppose we had a single protein molecule in a box of water molecules of fixed
volume at a single temperature. Then an ensemble is a collection of these molecules
This is why we often call a collection of proteins in a test-tube an ensemble,
even though it is not be formally correct.
Ensembles
Depending on which variables are fixed, we have different ensembles:
If every member of the ensemble has a fixed volume, number of particles, and E:
then we are in the microcanonical ensemble
Practical Example of the microcanoical?
If every member of the ensemble has a fixed volume, number of particles, and
Temperature, then we are in the canonical ensemble
We will be simulating members of the canonical ensemble
Other Ensembles
Depending on which variables are fixed, we have different ensembles:
Grand canonical has fixed V and T, and can be constructed by considering copies of
canonical ensembles connected by permeable membranes
This is used when simulating systems with variable number of particles
e.g. particles absorbed on a surface
If every member of the ensemble has a fixed pressure, number of particles, and
Temperature, then we are in the isothermal-isobaric ensemble
The isothermal-isobaric ensemble is another common one to simulate
Ensembles and Reality
If a system has a large enough number of molecules, then the behavior of the
system will coincide with that predicted from a statistical consideration of the individual
molecules.
If we wait long enough, a single particle will sample all possible states.
These are the erogodic hypothesis.
Theory and experiment both rely on different aspects of the erogodic hypothesis
Dice and Distributions
Suppose we roll two dice: What is the most probable number ?
What are the least probable numbers? What are the odds of getting the least
probable numbers?
Suppose we roll twenty dice: What is the most probable number ?
What are the least probable numbers? What are the odds of getting the least
probable numbers?
So as we increase the number of identical particles, the
probability of seeing extreme events decreases.
The most probable state dominates, and the range of fluctuations decrease.
When the number of members of the ensemble become very large, all macroscopic
quanitites are essentially fixed!
Ensembles and
Distributions
We when have an ensemble, there are fixed quantities, and all states which
meet these constraints are allowed, and are equally probable for any particle.
The most probable distribution is that corresponding to the largest number of ways
of arranging particles in a given configuration
The number of ways of arranging particles is the statistical weight; for a constant
energy and particle number this is W or W
Does W look familiar ?
Note: the particles need not be molecules; they can be atoms inside molecules.
All the force-fields terms discussed last time have entropic contributions
Boltzmann
Distribution
The Boltzmann distribution is the most probable distribution for a large system
near equilibrium
i
ni  n1e
1
 (  i 1 )
k BT
Notes: This can be proven; those of you in physics/physical chemistry probably
will do so or have done so already. This is also the high T limit of the fermi-dirac and
Bose-Einstein distributions
Distribution and
Conformations
Experiment: the thermodynamics of HEWL and a mutant missing a disulfide bond
were studied with scanning microcalorimetry.
Both proteins have the same enthaply of unfolding, and had “two-state” behavior, but
there is a difference in the entropy of unfolding.
The x-ray structures show essentially the same structures and interactions.
Where does the entropy change come from?
Distribution and
Conformations
S
conf
WT  M
 S
M
Folding
 S
WT
Folding
 25cal / Kmol
 N A kB  ln  M u  ln M  f    ln WT u  ln WT f 
Assume the folded states have the same entropy
 N A kB  ln 
M
 ln
WT
u

u   N kB ln 
A
M
u
/
WT
u
There are 129 residues in the protein, and if we calculate per residue:

M
u /
WT
u
 1.1
What does this mean?
What if we hadn’t divide by 129?


Probability and the
partition function
Using the Boltzmann distribution, the probability at a given temperature of a
particular state is:
 (  j 1 )
Pj 
nj
N

 je
k BT
 e
i
 (  i 1 )
k BT
i
The bottom is the canonical partition function.
Ensemble averages
We can write macroscopic variables in terms of the parition function
Energy, pressure, entropy, helmholz free energy, and more
 ln Q
S  k ln Q  kT
T
A   kT ln Q
 ln Q
p  kT
V
Utility of Other
Ensembles:
Microcanonical:
S  k ln W
Isothermal/Isobaric:
G  kT ln 