“I have hopes of being able to achieve
something of value through my current studies or
with any new ideas that come in the future.”
J. F. Nash
The bridge over the Arno
In 1999 I was wandering in Pisa with a booklet in the pocket of a worn coat, too short for my frame. It was dark blue on the outside, green and red inside, with a mended sleeve and an austere cowl: I was much like a young monk, with his holy book (figure 1). I can remember neither its title nor the author, though. It was an introduction to statistical thermodynamics, with beautiful figures, a coloured cover, and less than 100 pages. It contained the work by Maxwell on the kinetic theory of ideal gasses, along with other material. I borrowed it from the University Library because I was fascinated by the way in which Maxwell was able to describe the properties of gasses with just a few hypotheses and some relatively easy mathematical passages. I felt that there was an enormous attractive in these methods, I realized with pleasure that math could give the power to completely understand and hold in hand physical systems and even, I started speculating, biological ones.
My second favourite composer back then was Gustav Mahler (the favourite one being Basil Poledouris): he represented my own way to classical music and I chose him because he wasn’t among the musicians my father and my brother shared a love for. I felt, during my teens, that I had to find my private space, and I met it one day on a used book stand: a cassette tape of Das Lied von The Erde, with a few sheets containing the translation to Italian of the songs. Mahler was born in 1860, a few weeks after Maxwell published his pivotal work about ideal gasses in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (R) (figure 2). But in 1999 I was repeatedly listening to a CD with a collection of songs sung by Edith Piaf and Charles Trenet, because I was studying French, and I was having a hard time with pronunciation. So, imagine a secular monk in his prime who listens to old French songs while keeping one hand on a book of statistical thermodynamics hidden in his pocket, wherever he goes, wandering in the streets of Pisa, a city which gave birth to Galileo Galilei. This seems a beautiful story, much like a dream, right? Wrong.
I had already started my struggle against the mysterious disease that would have completely erased my life in the following years. In the beginning, it had a relapsing-remitting course, so I could sometimes host the hope that I was recovering, only to find myself caught by the evil curse again. At the end of the year 1999, I was losing my mind, I knew that and I was also aware that my holy book couldn’t save me. I clearly remember one evening, I was walking on Ponte di Mezzo, a simple and elegant bridge above the Arno, and I felt that I couldn’t feel sorrow for the loss of my mind: I realized that not only the functions of my brain assigned to rational thinking were gone, but my feelings couldn’t function properly either. In fact, I noted without a true manifestation of desperation that I had lost my emotions. One day, after spending in vain about eleven hours on a single page of a textbook of invertebrate palaeontology, I accepted that I couldn’t read anymore, at least for the moment.
Had I known for sure that I wouldn’t have recovered in the following twenty years, I would have quite certainly taken my own life, jumping from a building; a fate that I have been thinking about almost every day ever since. I considered this possibility during the endless sequence of days in which there has been nothing other than the absence of my thoughts.
The distribution of velocities of an ideal gas and the one hundred years gap
In the already mentioned paper by Maxwell, he derived the probability density of the speed of the molecules of a gas, granted that the three components of the vector of speed are independent random variables (hyp. 1) and that they share the same density (hyp. 2), let’s say f. Moreover, the density of the speed has to be a function only of its module (hyp. 3). These three hypotheses together say that there is a function Φ such that
This is a functional equation (i.e. an equation in which the unknown is a function) whose solution is not detailed in Maxwell’s work. But it can be easily solved moving to polar coordinates (see figure 3) and deriving with respect to θ both members (the second one gives naught since it depends only to the distance from the origin).
Another way to solve the functional equation is to use the method of Lagrange’s multipliers, searching for the extremes of the density of the velocity, when its module is fixed. In either case, we obtain the differential equation:
which leads to the density for each component of the speed:
where σ can’t be determined using only the three hypotheses mentioned above. Considering then the well-known law of ideal gasses (pV=nRT) and an expression for p derived from the hypothesis that the collisions between the molecules of gas and the container are completely elastic, Maxwell was able to conclude that:
where m is the mass of the molecule of gas, T is the absolute temperature and K_B is the Boltzmann’s constant. It was 1860, Mahler’s mother was going to deliver in Kaliště, Charles Darwin had just released his masterpiece “On the origin of species”, forced to publish much earlier than what he had planned because of the letter he had received from Wallace, in which he described about the same theory Darwin had been working on for the previous 20 years. In the same point in time, Italy was completing its bloody process of unification, with the Mille expedition, led by Giuseppe Garibaldi.
But the functional equation I have mentioned at the beginning of this paragraph has brought with it a mystery for years, until 1976, when an employee at General Motors Corporation published a short note in the American Journal of Physics [R] in which he showed how Maxwell’s functional equation is, in fact, an example of the well known Cauchy’s functional equation:
In order to prove that, you just have to consider the following definition:
The name of the mathematician who made this observation is David H. Nash, and he has the merit of finding something new in one of the most known equation of physics, an equation that is mentioned in every book of thermodynamics, an equation that has been considered by millions of students in more than a century. It was 1976, my mother was pregnant with my brother; Alma, Gustav Mahler’s wife, had died about ten years before.
Module of random vectors
Once Maxwell found the density of probability for each component of the speed of the molecules of an ideal gas, he searched for the density of the module of the speed. There is a relatively simple way of doing that. With the following notation
we have that the repartition function of Z is given by the integrals of the density of X within the sphere in figure 4. We have:
The second expression is the same as above but in polar coordinates. Then we can obtain the density of Z by derivation of the repartition function. And this method can be extended to an m-dimensional space. This was the method used by Maxwell in his paper. And yet, there is another way to obtain the expression of the module of a random vector: I have explored it in the last months, during the rare hours in which I could function. By the way, only in the Summer of 2007 I was able to study the kinetic theory by Maxwell, eight years after I borrowed the holy book. Such a waste.
The hard way towards the density of the module of a random vector
When a student starts studying statistics, she encounters a list of densities: the normal distribution, the gamma distribution, the exponential distribution etc. Then there are several derived distributions that arise when you operate sums, roots extractions etc. on random variables. In particular, if f_X is the densty of X and Y = X², then we have
On the other hand, if Y = √X we have
Another important result that we have to consider is that given
By using these results I have been able to find that the expression of the density for the module of an m-dimensional random vector is:
In particular, for m = 3 we have
The case of normal random vectors: the modified Bessel function
In particular, if the random vector has dimension 3 and its components are normal random variables with the same expected value and variance, we have that the density of its module is given by
where I_0 is the modified Bessel function, which is one solution of the differential equation:
whose name is modified Bessel equation. The integral expression of the modified Bessel function is:
I have coded a script in Matlab which integrates numerically this function (available here for download) which plots the surface in figure 5 and also gives the following table of values for this function.
The following is the flowchart of the script I coded.
The case of normal random vectors with a naught expected value: the upper incomplete gamma function
If we consider random variables with an average that is zero (this is the case with the components of speed in ideal gasses), then the density is given by
which is a Chi distribution with 3 degrees of freedom, scaled with a scale parameter given by s = 1/σ. In the expression of the repartition function, ϒ is the lower incomplete gamma function, which is defined as follows:
I have written a code for its numerical integration (available here for download), the output of which is in figure 6.
Conclusion, twenty years later
The density of the module of the velocity of the molecules of an ideal gas is, in fact, a scaled Chi distribution with 3 degrees of freedom, and it is given by
It can be numerically integrated with the following script I made for Octave/Matlab, which gives the plot in figure 7. Another similar script gives the plot in figure 8. These plots represent the Maxwell-Boltzmann distribution, the centre of the holy book that an unfortunate boy was carrying in his pocket, all alone, some twenty years ago. He could have easily died by his own hand in one of the several thousand days of mental and physical disability that he had to face alone. Instead, he has survived. Had it been only for finding the Maxwell-Boltzmann distribution following another path, it would have been worth it. But he has found much more, including a bright girl, the magnificent next stage of evolution of human beings.
% file name = legge_Maxwell-Boltzmann_2
% date of creation = 22/02/2019
% it plots the density and the distribution function for the
% Maxwell-Boltzmann distribution considered as a function of temperature
% and speed
% we define some parameters
K_B = 1.381*10^(-23) % Boltzmann’s constant
m = 4.65*10^(-26) % mass of the molecule N_2
% we define the array of temperature from 0° C to 1250° C
T (1) = 273.15
for i = 2:1:1250
T (i) = T (i-1) + 1.;
% it defines f_gamma in 3/2
f_gamma = sqrt(pi)/2.
% delta of integration
delta = 1.0
% it defines the array for the abscissa
z (1) = 0.;
for i = 2:1:2500
z (i) = z(i-1)+delta;
% it defines the density
for j = 1:1:1250
% it defines a constant
c = ( m/(K_B*T(j)) );
for i = 1:1:2500
f (j,i) = ( c^1.5 )*sqrt(2./pi)*( z(i)^2. )*( e^( -0.5*c*z(i)^2. ) );
% it calculates the ripartition function for He
F (j,1) = 0.;
F (j,3) = F (j,1) + delta*( f(j,1) + ( 4*f(j,2) ) + f(j,3) )/3;
F (j,2) = F (j,3)*0.5;
F (j,k+2) = F (j,k)+delta*( f(j,k)+( 4*f(j,k+1) )+f(j,k+2) )/3;
% It plots f and F
mesh(z(1:100:2500), T(1:100:1250), f(1:100:1250,1:100:2500));
mesh(z(1:100:2500), T(1:100:1250), F(1:100:1250,1:100:2500));