Hello, bonjour, merhaba, assalomu alaykum,
welcome to the third week of Statistical Mechanics: Algorithms and Computations
from the Physics Department of Ecole Normale Superieure.
Last week we discussed how the statistical approach provided an exact description
for physical systems.
As you found out by yourself in Homework Session 2,
this statistical approach captures the system even for finite number of particles.
The equivalence between Newton and Boltzmann
is one of the great miracles of the natural sciences.
It is also a breakthrough in mathematics, because there are now mathematical theorems
that back up our numerical findings,
at least for hard spheres systems.
This week we will study phenomena in the physics of liquids
and this object will accompany us
a giant clothes-pin
that resembles the object that many people use to hang up their clothes.
For us, believe it or not, it will be the key for deep insights into physics.
We place a first one randomly on a washing line
between two poles
like this
and a second one like this
and a third one like this
and so on, until N clothes-pins have been placed onto the line.
Each arrangement of N clothes-pins is equally likely
and we use the tabula rasa rule
to wipe-out a configuration that presents an overlap.
What is the probability for a pin to be at position x?
This simple question will get us to the heart of liquid and soft-matter physics.
We will discover that although apparently the pin positions are independent,
pins in fact experience a strong force:
a force that can be seen in neutron scattering experiments or in X-ray diffraction,
and that can be measured.
This famous interaction was discovered by Asakura and Oosawa in 1954,
and it is so important that some people take it to be the fifth force in nature.
In this week's lecture we will study this interaction.
It comes out of nowhere
and it exists even though there are no charges,
no currents, no mechanical strains.
This is why it is called an entropic interaction.
It can be so strong
that it introduces phase transitions from the liquid to the solid.
The clothes-pins model corresponds to one-dimensional hard disks.
In this week's tutorial - Tutorial 3 -
we will study an ingenious algorithm
that allows to sample assemblies of tens, hundreds, thousands even millions of clothes-pins on a line.
Remember: last week we had trouble simulating four hard disks in a square..
This week we can study so large systems
that we can even approach the infinite-system limit:
the thermodynamic limit.
As you will see several times in this course,
a great algorithmic success
(the fact that we can directly sample thousands or millions of particles without rejections)
is mirrored by the fact that we can solve this model analytically.
This is what you will do in the tutorial of this week.
The analytic solution allows us to drive home essential messages
about liquids and solids, and about the phase transition between them.
The providential Monte Carlo algorithm that we alluded to
works only in one dimension.
In two dimensions and higher (for hard disks or hard spheres)
we must resort to the tools of last week:
Markov chain Monte Carlo algorithms or Molecular Dynamics.
At small densities, everybody understands that the system of hard disks or spheres
is a liquid.
But will it solidify, at higher density?
This is a question that has motivated some of the best work
in computational physics
and that has also brought about the revolution in numerical algorithms:
in Monte Carlo and in Molecular dynamics.
In this week's homework session
you will retrace these classic papers.
Our algorithms and our very simple Python implementations
are just good enough
to get a flavor of what is going on in the system
and to observe our first phase transition.
So let's get started with week 3 of Statistical Mechanics: Algorithms and Computations.
We take up the random clothes-pin model
and place pins on positions x_0, x_1, x_2 and so on
as shown in this program
Each time, we check for the distances of the new pin
with all the pins already present
and if one of these distances is below (2 sigma)
we have an overlap and we reject the configuration through the tabula rasa rule.
This is what allows us to sample configurations x_0, .., x_(N-1)
with a probability pi=constant if the configuration is legal, and pi=0 if it is illegal.
As we are on a line (that is, one dimension)
it is more efficient to sample the position x_0, x_1, x_2, x_3 and so on
then to sort them such that x_0 < x_1 < x_2 < ..
and then check only the (N-1) overlaps between neighboring pins.
So we check for the distance between x_1 and x_0, x_2 and x_1, x_3 and x_2, and so on.
This is written in the program direct_pins_improved.py.
Output of this program for 5 pins
of radius sigma=0.075 on a line of length 1,
(that is with 75% of the line occupied by clothes-pins) is shown here.
Each line shows an accepted configuration
with the 5 positions x of the centers of the clothes-pins.
It has taken about 500 trials
each time to get one accepted configuration.
As we discussed, for N=5, we'll have one acceptance out of about 500.
For N=10, it will be really interesting to see that rejection rate will be so high
that you accept only one out of 500000 samples.
Output will be really interesting for N=15,
but don't try to obtain it with this algorithm
you will accept only one legal configuration
out of 500000000 trials.
Wait until Michael - in this week's tutorial - shows you how to sample this configuration
for any N (as you like: hundred, thousand and millions) or any density, without any effort.
Output of this program (direct_pins_noreject.py) is shown here for N=15
Again, you see the 15 x positions in ordered way, for all the legal configurations that we obtained.
Now, let us analyze these data
and the way to do it is to do a histogram of all the x-positions in this table.
This is shown here.
Look in what a paradoxical situation we are
We place our pins uniformly on an interval, yet the distribution is oscillatory
Many people.. everybody finds this outcome counterintuitive
and difficult to understand.
How can it be that close to the poles we have 4 to 5 times more particles than in the center?
The poles actually attract the particles.
Furthermore, what you see here is not a mere boundary effect.
Particles also attract each other, even though this is more difficult to see.
What we see here (attraction of particles to a wall or the attraction of particles with the others)
is what Asakura and Oosawa found in 1954 in their famous paper
and this is what we will study in a moment.
Before doing so, please take a moment to download, run and modify the two programs we discussed in this section.
On the Coursera website, you'll find the program direct_pins.py
that generates random configurations of N clothes-pins using the tabula rasa rule.
Please check out also the program direct_pins_improved.py
that uses the sorting trick.
And please be patient, for the direct sampling algorithm without rejections
(direct_pins_noreject.py) that we will discuss in this week's tutorial.
When drawing configurations of pins
we must realize that there are two types of excluded regions.
The first type is the core, the space occupied by the pin itself,
no other particles can penetrate into this core.
But we must realize that the position of the pin is given by its center
and this creates a second type of excluded region that we call the halo.
The center of another particle can penetrate neither into the halo
nor into the core.
The poles also have a halo, of size sigma.
Because all pins must be at x > sigma
and x < L - sigma.
The total core area of our N pins is fixed:
it is equal to (2 N sigma).
The total area of the halo is not fixed
If we put the pin into the center (far from the poles) the total halo area is equal to 4 sigma
if we put the pin to one of the poles
the total halo area is equal to 2 sigma
it follows that the configuration on the right
has more accessible space for other particles to be put,
and a smaller chance to undergo a tabula rasa wipe-out.
This means that the configuration to the right has a higher probability than the configuration to the left
or that the particle is attracted to the boundaries.
You can also say that there is a force between the boundary and the pin.
Let us now see for ourselves whether the halo picture
predicts interactions between particles.
We will consider a line of length L, with a pole to the left and a pole to the right
and two pins, with radius sigma, which means core area 2 sigma.
Question 1
what is the available space for other particles
if we place our two pins far from each other and far from the poles?
Answer: for two pins that are far from each other and are far from the poles
the available space for other particles is equal to (L - 10 sigma), as you can see from this figure.
Question 2
what is the available space for other particles
if we place the two pins close together but far from the poles?
Answer:
the available space for the other particles is equal to L - 8 sigma, as you see in this figure.
Third question:
what is the available space for the other particles
if we put one pin close to the left pole and another pin close to the right pole?
Answer:
the available space for the other particles is equal to (L - 6 sigma), as you see here.
Final question:
the halo picture, can it fully describe the density profile that we obtained earlier?
The answer is no,
the halo picture nicely describes the increase of density close to the boundaries
but it cannot account for the intricate oscillations of the profile.
Please be patient for this week's tutorial, where we will derive an analytical formula
which exactly describes this density profile that we obtained from numerical simulations.
The halo picture, although it is approximate, applies in any dimension
and it explains why in Homework Session 2 (last week's homework)
you obtained an inhomogeneous density profile for the 4 hard disks in a box.
For a disk of radius sigma,
there is a ring of radius sigma that forms the halo
No other center of disks can penetrate into the halo or the core area.
But notice that the halo itself can penetrate into another halo
or into another core area, isn't that curious?
Here, on the screen, you see three configurations of two disks in a box.
In configuration c, the two disks are in the corners
and the available space for other particles is much larger than for configuration b,
but configuration a has the least accessible space for other particles.
If we have two particles already in configuration c, that means in the corners,
we have much more available space for the other particles,
and a lower chance to undergo a tabula rasa wipe-out.
This means we expect a higher density in the corner of the box than in the center
and this is exactly what you observed in last week's homework
but you probably concluded that these inhomogeneities were a boundary effect.
This misses the main point.
Consider the configuration b: the two particles attract each other.
In the one dimensional clothes-pin model, the oscillations that you see here decay exponentially
on the scale of a few clothes-pins.
This means that the boundary effects and the pair correlations decay on the scale of a few sigma
and this implies that the Asakura-Oosawa interaction
leads to local modulations of densities and pair correlations.
On long length scales, the system is completely homogeneous
and this is what defines the liquid state.
The clothes-pin model is an example
of a very general class of physical systems with short range interactions.
Powerful mathematical theorems exclude the possibility of a phase transition in this systems in one spatial dimension.
The question is now whether these interactions that we discussed
can be strong enough in 2 and higher dimensions to be seen on arbitrary length scales
and whether they can introduce phase transitions.
This will be the subject of the next section and also of this week's homework session.
In this week's tutorial, we will continue our detailed study of the random clothes-pin model,
but here let me put what we have done and what we will do in the tutorial into a wider prospective.
And I will start from a discussion of the two extremes
the close-packed limit and the dilute system
considering systems with periodic boundary conditions.
In a one dimensional system, for random clothes-pin the close-packed limit eta=1 is special
We have one pin at x_0, x_0 + 2 sigma, x_0 + 4 sigma, x_0 + 6 sigma, and so on..
even though, because of the periodic boundary conditions, we do not know where is x_0
we have position long-range order in the system
but we also know (in fact we will prove, in Tutorial 3)
that for all densities lower than eta=1
the system is different, it is liquid-like
like it is in a very low density limit.
In two dimensions, that is for systems of hard disks,
the close-packed density is equal to pi / (2 sqrt(3))
and this is about 0.907.
And the close-packing configuration is hexagonal, as shown here in the picture.
The mathematician Laszlo Fejes Toth proved in 1940
that no other configuration than the hexagonal packing
exists at this density pi / (2 sqrt(3)).
So we are sure to have long range positional and orientational order
in this system, at the close-packing density.
There are no other mathematical results
about this system, besides the fact that at very small density
the system is liquid.
To find out what is going on at intermediate densities
we have to resort to numerical simulations:
Markov chain Monte Carlo simulations and Molecular Dynamics simulations allow us to sample this.
For example here you see a system of 256 disks
at density 0.48, again with periodic boundary conditions in x and y.
Here, the Asakura-Oosawa depletion interaction is at work,
and it produces some unconnected local arrangements of disks,
that resemble the close-packing limit
at density 0.907. Look at the highlighted spots in the picture.
At higher density, look here, the system all of a sudden changes.
What you see here on the right is qualitatively different,
from the configuration on the left.
A phase transition has taken place.
The discovery of this phase transition from the configuration on the left (the liquid one)
to a denser phase is a computational fact,
a computational achievement, and it also been observed in many experiments.
But it is not backed by mathematical theorems.
This transition at a finite density, below close-packing,
takes place in two dimensions, but also in three dimensions for hard spheres.
The existence of such a transition was first suggested in theoretical works by Kirkwood and Monroe in 1941.
In conclusion, please note that in Lecture 3 of Statistical Mechanics: Algorithms and Computations
we have remained within the tight conceptual framework of the equal probability principle.
What a simple principle
but what far reaching consequences and rich and surprising phenomena!
And what a nice interplay between algorithms and theory!
In coming weeks, we will continue to develop the conceptual framework,
the consequences, and the algorithms.
I hope to keep up your interest in Statistical Mechanics: Algorithms and Computations.
In the meantime, let me thank you for listening.