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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.

What is a differential equation? How does a differential equation works? A differential equation is a special type of equation. To explain how a differential equation works lets take as an example Newton’s one applied to the movements of the planets.

A differential equation requires certain information, in our example, the position and velocities of the planets at a given time. With this information, if the differential equation can be “solved”^{1}, it provides the positions and velocities of the planets in the future, i.e., it predicts their trajectories.



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.



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 wateror 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

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.

Remember that all this started because in Classical Mechanics we have lack of information, the Boltzmann equation deals with less information by working with proportions (or sets) of particles instead of dealing with the exact particles.

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. [put this with videos that compare the microscopic with what we observe to explain this]



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}.
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.

The tools to prove this link are differential equations and Probability.


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
Cercignani. ‘Lugwig Boltzmann. The man who trusted atoms.‘
1Differential equations can be solved in few cases. In the major part of the cases, it is necessary to carry out, on one hand, a mathematical analysis to find quantitative and qualitative properties of the equations, and, on the other hand, computer simulations to approximate the solution of the equation.
2This example was given by someone else to explain the concept of entropy. I think that the author is Cercignani, though I am not sure.
3Entropy is a fundamental concept in the theory of thermodynamics. Due to lack of space, we do not deal here with it.
4For a technical description of this see Chapter 4 of ‘Multiscale analysis: from the linear Boltzmann equation to the diffusion equation and fractional diffusion equation‘
5The interested reader can look for ‘Locksmith Paradox’ and ‘Poincaré recurrence theorem’.
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