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Mathematicians,
those adorable and nerdy creatures... Not many people know what they
actually do... or even if what they do is useful, but almost
everybody has a mental picture of what they look like. Some people
imagine bearded men walking aimlessly in circles while muttering
words to themselves; others picture men with thick glasses making
sums and multiplications all day long with a powerful mental skill;
the most generous ones think of 'beautiful minds'.
I am a mathematician myself, my name is Sara Merino and currently I
carry out a PhD in Mathematics. Just to answer some of the questions
that people usually ask me: no, I do not work with numbers all day
long, actually I 'see' numbers very rarely... basically when I pay my
bills; I work with equations... not of this kind...
but more of this kind...
and, no, my ultimate goal is
not to obtain a number. No, I do not do a PhD because I want to
become a high school teacher, but because I want to do research. No,
not everything has been discovered in Mathematics. Actually there is
still a lot to be discoreved. I hope I do not need to tell you that I
am not a man. And yes,... I wear thick glasses.
INTRODUCTION
In this entry I want to give you a flavour of the kind of problems I
try to solve and the mathematical tools I use, namely differential
equations and Probability. Don't worry, I will not get technical, I
promise not to show any symbol... except for stetical purposes...
Let's get to it!
I
work in a field of Mathematical Physics called Statistical
Mechanics. In this document we will see how Statistical Mechanics
was born to solve problems that Classical Mechanics could not solve
and how Mathematics played a fundamental role.
MATHEMATICAL
MODELING: Patterns in Physics
Physicists
recognise patterns in nature and describe them mathematically to make
predictions.
For
example, in Classical Mechanics, through Newton's
equations, we can predict the trajectories of the planets around
the sun.
The mathematical tool used to describe the physical law
governing the orbits is called differential equation.
Newton's equations are just a particular instance of a differential
equation.
Lets
consider now another example. With Newton's equations, it can also
be modelled the
behaviour of a gas. However, it is impossible
to make predictions from them. Why is this so?
There is a problem of lack of information when we study the
evolution of a gas; we need to know the position and velocity of
the particles at a given time to make the prediction, but this
measurement is technically impossible. Moreover, even if we could
make the measurements, the differential equations are so complex
(due to the large amount of particles) that cannot be studied
mathematically.
Summarizing, Newton's equations work well to predict the planetary
movement but it becomes intractable when studying a gas. It does
not mean that Newton's model is wrong, simply that it is not
practical for the study of a gas.
Since we are lacking information, we have to work with guesses.
This is how a new mathematical field, Probability, entered
the study of Physics. We can predict general features of physical
phenomena despite lacking information.
For example, we do not know the outcome of tossing a coin, but we
know that if we toss it a lot of times, roughly half of the time
we will get tails and the other half, heads.
Ludgwig Boltzmann founded Statistical Mechanics by
studying the dynamics of gases using this new approach based on
Probability and random (or stochastic) models.

Ludwig Boltzmann 
Thanks to Probability and stochastic models, specially one called
Brownian motion, we can not only study a gas but also other
physical phenomena like the erratic trajectories of nano particles
in water
or the following sound which is known as white noise.
What is a stochastic model or process? In mathematics, which is the
difference between a deterministic process and a stochastic one?
A deterministic process is, for example, when we know
exactly the trajectory of a particle; so Newton's laws state that
a particle will move in a straight line at a constat velocity if
there are no other interactions with the particle. A stochastic or
random process would be one in which we cannot know exactly how
the particle will move but we know some properties of its
behaviour.
An example of stochastic process
is the socalled random walk,
which is the following: imagine that you want to take a
walk. You allow yourself to only move to the left or to the right,
one step at a time, and to determine in which direction to go, you
toss a coin; if it is head, you turn right; if it is tails, you
turn left. At the beginning of your walk, you do not know which is
the path that you are going to take, but, roughly, half of the
times you will turn right, and the other half you will turn left.
Your trajectory is the stochastic process called random walk, and
mathematicians study this kind of processes and are able to prove
properties about them. We will come back to stochastic processes
when we talk about a very special one that we have mentioned
before, Brownian motion,
which is a generalisation of a random walk.
Thanks
to this new mathematical tools and the ideas behind them, Boltzmann
entered a new conception in Physics with which he was able to
explain, among other things, why the world is irreversible, namely,
why we move from the past to the future without the time never
going backwards. We will see this later, after explaining the new
model that Boltzmann proposed for the study of gases, called
Boltzmann equation.
THE
BOLTZMANN EQUATION AND THE DIFFERENT SCALES OF DESCRIPTION
What is the Boltzmann equation?
What is this model different from Newton's equations?
To
understand what the Boltzmann equation is, we need to put it into a
context. The Boltzmann equation is a point of view.
Let me explain this. If we observe each particle of a gas with its
exactly position and velocity, then we use Newton's equations.
However, not always we want to have so much detail; sometimes we
just want to know the general behaviour of the gas, namely, what can
be observe by the naked eye. For that, we have hydrodynamical
equations. The difference between the two models is the point of
view of description; Newton's equations have all the detailed
information of the microscopic system, while hydrodynamical
equations is a rough description of what we observe. Nevertheless,
keep in mind that the physical phenomena is the same; the dynamics
of a gas. And here is where the Boltzmann equation comes in; it is a
model between these two levels of description. Instead of knowing
exactly what which particle does, we know the proportion
of particles that does it; so this model gives
less information than the Newton's model but more information than
the hydrodynamical ones,
you could think of it is as a blurry image of Newton's model.
WHY
IS IT IMPORTANT? Applications
The
Boltzmann equation has important practical and theoretical
applications.
Some
of the practical applications are in aeronautics at high altitude or
interactions in dilute plasmas. Also, it allows to make predictions
in specific situations in which the ones provided by hydrodynamical
equations are not accurate enough. For more information on the
practical applications look at the book of Cercignani 'Rarefied Gas
Dynamics'.
The
theoretical applications of the Boltzmann equation help us to
understand better the world. Here is an example.
Thanks
to his probabilistic approach, Boltzmann was able to give an
explanation for the irreversibility in physical phenomena.
Irreversibility is associated with the fact that time goes in
one direction, hence we cannot go back to the past.
For
example, a manifestation of irreversibility in the physical world
is the box with two types of sugar [reference
here]. Imagine that you have a box with the lower part
filled with white sugar and the upper part filled with brown sugar.
If we shake the box for a while, we expect the two types of sugar
to mix uniformly. We will not expect that, if we keep shaking, at
some point we will have the initial configuration of brown sugar on
top, white sugar at the bottom, i.e., the process will not
reverse to its initial state^{2}.
In
the same way, irreversibility appears when observing a gas. For
example, in this video we have a box divided in two. In each side
there are gas particles at different temperature (and color). When
the wall disappears between the two compartments, we expect the
blue and red particles to start mixing, becoming in the end,
homogeneously distributed in the room, reaching an equilibrium
and making the temperature of the box uniform. We do not expect to
have again, in the future, the blue particles on the lefthand side
and the red ones on the righthand side, i.e., we do not expect
reversibility. However, Newton's laws tell us that that is
possible.
Newton's
equations are reversible, meaning that if we invert the velocities of the gas particles
at a given time, then they will go back to its initial position; it
will look like time runs backwards. However, this does not happens
with the Boltzmann equation; it is not reversible.
Newton's
equation and the Boltzmann equation are models for the same physical
phenomena, but the first is reversible and the second not.
How can this apparent contradiction be explained?
Boltzmann
explained it using, as we said, Probability. In Classical Mechanics
everything is deterministic and a particular phenomena is possible
or impossible to happen. In Statistical Mechanics, since we work
with uncertainties, the concepts of possible and impossible are
transformed into probable and highly
improbable. In
this way, to observe reversibility becomes highly improbable but
not impossible.
How did Boltzmann use this
difference of concept to explain the irreversibility that we
observe around us? He said that the number of configurations,
i.e., the number of possible positions and velocities of the
particles that make us observe, to the naked eye, uniformity of
particles or equilibrium, is infinitely bigger than the number of
microscopic configurations that will make us observe
reversibility. Hence, it is much more probable that the
configuration of the particles 'fall' into one that will make us
observe equilibrium than one that make us observe reversibility.
To
make an analogy, imagine that we toss a coin and let it fell to
the floor. We always consider the outcome to be heads or tails,
however, there is another possibility: that it stands on its edge.
The probability of that is so low that we do not consider it; we
do not expect to experience it. In the same sense, expecting to
observe reversibility is like expecting to get the coin on its
edge; not impossible, but highly improbable.
On one hand, Newton's
equations, since they have all the possible information of a gas,
consider all the microscopic configurations (in the analogy, it
considers also the possibility of getting the coin on its edge).
On the other hand, Boltzmann's equation does not have all the
information and, hence, gathers together different microscopic
states that give the same macroscopic picture and consider only
the macroscopic pictures that are highly probable to happen, i.e.,
the ones that reach an equilibrium (in the analogy, in discards
the possibility of getting the coin on its edge); this makes his
equation non reversible.
Thanks
to the introduction of Probability, Boltzmann was able to explain
physical phenomena that could not be explained in Classical
Mechanics, like irreversibility, existence of equilibrium and
entropy^{3}.
Here you have an excellent clip in which Brian Cox explains the concept of entropy.
GOOD
MATHEMATICAL MODELS? THE PROBLEM OF COHERENCE (Hilbert's 6^{th}
problem)
Which
is the kind of questions I am trying to solve? Now that I have
explained you all this. Let me explain you which are the kind of
questions I am trying to give an answer to.
Remember
that physicists recognise patterns in nature and find mathematical
models to describe them and make predictions. Afterwards,
mathematicians have to analyse these models to check their
coherence, validity and information that can be obtained from them.
Let's
go back to the gas dynamics and the different mathematical models
that we have for it. We have different mathematical models at
different levels of description, namely, Newton's, Boltzmann's and
hydrodynamical equations. Each model, though, was derived
independently from each other using physical intuition. However, if
the models are correct, we expect some coherence between them
since the physical phenomena that they model is the same; the
dynamics of a gas.
This
coherence between the models means that we expect to be able to
derive, mathematically, the models at a larger scale from the ones
at a lower scale; the behaviour of atoms determines what we observe
by the naked eye. This is called Hilbert's 6^{th}
problem, proposed by Hilbert, one of the greatest mathematicians
of the XX century in the International Congress of Mathematics in
1900.
Partial
answers to Hilbert's 6^{th} problem have been given and I
am currently working in this direction; I am trying to derive
hydrodynamical models from the models in Statistical Mechanics. For
example, it has been proven that a simplified version of the
Boltzmann equation derives at macroscopic level into a Heat
Equation^{4},
which is the equation that models how the temperature in a room
evolves over time.
Allow
me to give you a small flavour of how this link between the models
was proven. As we saw before, in Probability, we use random
processes, like the Brownian motion and Stochastic differential
equations, which are the analog of differential equations for
random processes instead of deterministic processes.
Brownian motion is a
generalization of the random walk that we saw before. In the plane
(two dimensions), it will look as follows. Imagine that, instead
of walking only to right or left, we also move forwards of
backwards, one step at a time, and we decide which direction to
take randomly, having each direction the same probability to
happen. The video here shows one possible trajectory that such random
walk could produce. This is approximately, a Brownian motion.
It has been seen that the
trajectory of a particle which follows the Heat equation
corresponds to a Brownian motion.
The derivation of the Heat
equation from the simplified Boltzmann equation is done using
Brownian motion. Observe the following video in which appears a
gas with a singled out particle. The trajectory followed by the
singled out particle seen at a larger scale and speeding up in
time produces a Brownian motion, which corresponds,
as we just said, to the trajectory of a particle under
the Heat equation.
This
kind of problems are fundamental, among other reasons, because the
models need to be validated, i.e., we need to check their
correctness; that they provide a good description of the physical
phenomena. For example, there was a huge controversy when Boltzmann
presented his equation. An important part of the scientific
community, including Poincare [add link], did not accept his model.
Boltzmann had a hard time
defending his theory. However, if Boltzmann would have obtained
his equation from Newton's one, there would have been no
controversy and would have been able to explain, from the very
beginning, the apparent incoherences that appeared in his theory,
including the irreversibility of his equation, that we have
mentioned before^{5}.
SUMMARY
Summarizing,
to describe and predict the physical world around us, physicists use
mathematical models. Newton's equations, in Classical Mechanics, are
a particular type of model called differential equation. It is based
on deterministic processes and has proven to be very useful to
describe particular physical phenomena, like the planetary
movements. However, differential equations requires an initial
amount of information that cannot be provided in particular physical
systems, like when studying a gas. To work with this lack of
information, Boltzmann proposed a new model based on random
processes instead of deterministic ones where the lack of
information was dealt with the use of Probability.
The
Boltzmann equation has proven to be both, practical and
theoretically, useful for physicists and engineers. For example, by
introducing Probability and random processes to the study of
Physics, Boltzmann provides a new conception in which he can explain
phenomena like the irreversibility in our world.
How
can we be sure that a model is “correct”? The mathematical
derivation of models having less information from the ones having
more information (Hilbert's 6^{th} problem) is fundamental
towards the understanding of these models and proving their
validity. Mathematicians have been able to do so for some particular
cases. The use of probabilistic tools, like Brownian motion, help us
to make and understand the link between these models.
I hope
you have enjoyed this because a consequence of irreversibility is that...
Further
reading