# Probability Turns Material: The Boltzmann Equation

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## Abstract

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## 1. Introduction

“It has thus been rigorously proved that, whatever may be the initial distribution of kinetic energy, in the course of a very long time it must always necessarily approach the one found by Maxwell. The procedure used so far is of course nothing more than a mathematical artifice employed in order to give a rigorous proof of a theorem whose exact proof has not previously been found. It gains meaning by its applicability to the theory of polyatomic gas molecules”.

“The applicability of probability theory to a particular case cannot of course be proved rigorously. If, out of 100,000 objects of a certain kind, about 100 are annually destroyed by fire, then we cannot be sure that this will happen next year. On the contrary, if the same conditions could be maintained for ${10}^{{10}^{10}}$ years, then during this time it would often happen that all 100,000 objects would burn up on the same day; and likewise there will be entire years during which not a single object is damaged. Despite this, every insurance company relies on probability theory. It is even more valid, on account of the huge number of molecules in a cubic millimetre, to adopt the assumption (which cannot be proved mathematically for any particular case) that when two gases of different kinds or at different temperatures are brought in contact, each molecule will have all the possible different states corresponding to the laws of probability and determined by the average values at the place in question, during a long period of time. These probability arguments cannot replace a direct analysis of the motion of each molecule; yet if one starts with a variety of initial conditions, all corresponding to the same average values (and therefore equivalent from the viewpoint of observation), one is entitled to expect that the results of both methods will agree, aside from some individual exceptions which will be even rarer than in the above example of 100,000 objects all burning on the same day. The assumption that these rare cases are not observed in nature is not strictly provable (nor is the entire mechanical picture itself) but in view of what has been said it is so natural and obvious, and so much in agreement with all experience with probabilities, from the method of least squares to the dice game, that any doubt on this point certainly cannot put in question the validity of the theory when it is otherwise so useful. It is completely incomprehensible to me how anyone can see a refutation of the applicability of probability theory in the fact that some other argument shows that exceptions must occur now and then over a period of eons of time; for probability theory itself teaches just the same thing”.

## 2. Probability: Obscure and Practical

“In whatever way we may define the concept of probability, or whatever axiomatic formulations we choose: so long as the binomial formula is derivable within the system, probability statements will not be falsifiable. Probability hypotheses do not rule out anything observable; probability estimates cannot contradict, or be contradicted by, a basic statement; nor can they be contradicted by a conjunction of any finite number of basic statements; and accordingly not by any finite number of observations either”[17].

My thesis, paradoxically, and a little provocatively, but nonetheless genuinely, is simply this:

PROBABILITY DOES NOT EXIST

The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, …, or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs[19].

**finitely**

**many**possible events, N say, the classical probability is objectively defined by counting as

- given two points on the circumference, consider the chord joining them, and take one of the two points as the vertex of one inscribed equilateral triangle. If the chord lies outside the triangle, it is shorter than $\sqrt{3}$; if it intersects the triangle, it is longer than or equal to $\sqrt{3}$. Now, the vertices of the triangle delimit three arcs of equal length, and only one of them corresponds to a chord longer than $\sqrt{3}$. Therefore, the probability of L is $1/3$.
- consider a radius of the circle, and choose a point on it. There is a chord perpendicular to this radius, and its length is larger than $\sqrt{3}$ if the point is closer to the center of the circle than to the circumference. Therefore, the probability of L is $1/2$.
- take a point anywhere within the circle and construct a chord with this point as its midpoint. The chord is longer than $\sqrt{3}$ if the point lies within a concentric disk of radius $1/2$ the radius of the larger circle. The area of the smaller circle is $1/4$ of the full disk; therefore, the probability of L is $1/4$.

They say that Understanding ought to work by the rules of right reason. These rules are, or ought to be, contained in Logic; but the actual science of Logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true Logic for this world is the Calculus of Probabilities, which takes account of the magnitude of the probability (which is, or which ought to be in a reasonable man’s mind). This branch of Math., which is generally thought to favour gambling, dicing, and wagering, and therefore highly immoral, is the only ‘Mathematics for Practical Men’, as we ought to be[21].

- There is a set $\mathsf{\Omega}$ called sample space, which contains all the “elementary” events $\omega $;
- there is a collection of subsets of $\mathsf{\Omega}$, $\mathcal{F}\left(\mathsf{\Omega}\right)$ that has the structure of a $\sigma -$ algebra of subsets $E\subset \mathsf{\Omega}$, representing all (elementary and combined) events to which a probability is to be assigned;
- for all $E\in \mathcal{F}$, $P\left(E\right)\ge 0$, $P(\varnothing )=0$, and $P\left(\mathsf{\Omega}\right)=1$, where ∅ is the empty set;
- for ${E}_{1}\cap {E}_{2}=\varnothing $ one has $P({E}_{1}\cup {E}_{2})=P\left({E}_{1}\right)+P\left({E}_{2}\right)$;
- given an infnite sequence of disjoint events, ${\left\{{E}_{i}\right\}}_{i=1}^{\infty}$ with ${E}_{i}\cap {E}_{j}=\varnothing $ if $i\ne j$, one has$$P\left(\phantom{\rule{0.166667em}{0ex}}\bigcup _{i=1}^{\infty}{E}_{i}\right)=\sum _{i=1}^{\infty}P\left({E}_{i}\right)$$

## 3. Mass vs. Probability

**(E1)**- For every integrable function $\mathcal{O}:\mathcal{M}\mapsto \mathbb{R}$, time average and phase space average coincide for $\mu $ -almost every x, which means for all $x\in \mathcal{M}$ apart from a set E of vanishing measure, i.e., such that $\mu \left(E\right)=0$:$$\overline{\mathcal{O}}\left(x\right)={\mathbb{E}}_{\mu}\left(\mathcal{O}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\mu -\mathrm{a}.\mathrm{e}.\phantom{\rule{4pt}{0ex}}x\in \mathcal{M}$$
**(E2)**- For every measurable $A\subset \mathcal{M}$ and for $\mu $ -a.e. $x\in \mathcal{M}$, the fraction of time a trajectory spends in A is given by:$${\tau}_{A}\left(x\right)=\mu \left(A\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\mu -\mathrm{a}.\mathrm{e}.\phantom{\rule{4pt}{0ex}}x\in \mathcal{M}$$$${\tau}_{A}\left(x\right)=\underset{t\to \infty}{lim}\frac{1}{t}{\int}_{0}^{t}{\chi}_{A}\left({\mathsf{\Phi}}^{s}x\right)\mathrm{d}s$$
**(E3)**- There are no non-trivial integrals of motion. In other words, let $\mathcal{O}$ be a function of phase, such that $\mathcal{O}\left({\mathsf{\Phi}}^{t}x\right)=\mathcal{O}\left(x\right)$ for all t and for $\mu $ -a.e. $x\in \mathcal{M}$, and let $\mathcal{O}$ be integrable. Then,$$\mathcal{O}\left(x\right)=\mathrm{constant}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\mu -\mathrm{a}.\mathrm{e}.x\in \mathcal{M}$$
**(E4)**- The dynamical system is metrically indecomposable. In other words, let A be an invariant measurable set, i.e., ${\mathsf{\Phi}}^{t}A=A$ for all t. Then, either $\mu \left(A\right)=1$ or $\mu \left(A\right)=0$. When that is the case, the expression $\mathcal{M}=A\cup (\mathcal{M}\setminus A)$ is called metrically trivial decomposition of $\mathcal{M}$.

**(M1)**- For every pair of measurable sets in the phase space, $A,B\subset \mathcal{M}$, the following holds:$$\underset{t\to \infty}{lim}\mu ({\mathsf{\Phi}}^{-t}A\cap B)=\mu \left(A\right)\mu \left(B\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\mathrm{or}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\underset{t\to \infty}{lim}\mu (A\cap {\mathsf{\Phi}}^{t}B)=\mu \left(A\right)\mu \left(B\right)$$
**(M2)**- for every pair of observables, e.g., $\mathcal{O},\mathcal{P}\in {L}_{2}(\mathcal{M},\mu )$, the following holds:$$\underset{t\to \infty}{lim}{\int}_{\mathcal{M}}(\mathcal{O}\circ {\mathsf{\Phi}}^{t})\mathcal{P}d\mu ={\int}_{M}\mathcal{O}d\mu {\int}_{M}\mathcal{P}d\mu \phantom{\rule{0.166667em}{0ex}},$$

“It seems at first that the orderless collisions of this innumerable dust can only engender an inextricable chaos before which the analyst must retire. But the law of great numbers, that supreme law of chance, comes to our assistance. In face of a semi-disorder we should be forced to despair, but in extreme disorder this statistical law re-establishes a kind of average or mean order in which the mind can find itself again”[25].

## 4. Hamiltonian Particle Systems and Ensembles

“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes”.

“Studying the thermodynamical state of a homogeneous fluid of given volume at given temperature […] we observe that there is an infinite number of states of molecular motion that correspond to it. With increasing time, the system exists successively in all the dynamical states that correspond to the given thermodynamical state. From this point of view we may say that a thermodynamical state is the ensemble of all the dynamical states through which, as a result of the molecular motion, the system is rapidly passing”[27].

**(a)**- macroscopic systems are made of very many particles: $N\u22d91$;
**(b)**- only several and special phase functions are physically relevant;
**(c)**- it does not matter if ensemble averages disagree with time averages on limited sets of trajectories.

## 5. Boltzmann Equation from the Probabilistic BBGKY Hierarchy

**q**and momentum

**p**, labelled by 1, to that of another particle of momentum ${\mathbf{p}}_{\ast}$; the range of integration ${S}_{-}$ is the hemisphere within which $(\mathbf{p}-{\mathbf{p}}_{\ast})\xb7\mathbf{n}<0$, i.e., particles directed toward each other before collision; and $(N-1)$ is the number of particles with which the test particle, can collide, assuming they all have the same probability of doing it, having integrated over all their momenta ${\mathbf{p}}_{\ast}$. Moreover, having denoted by ${f}_{N}^{\left(2\right)}$ the two-particle distribution function, whose expression immediately before and after the collision respectively takes the form

## 6. Boltzmann Equation from Mass Balance in Real Space

**p**are distributed as a Maxwellian probability distribution.

## 7. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Rondoni, L.; Di Florio, V.
Probability Turns Material: The Boltzmann Equation. *Entropy* **2024**, *26*, 171.
https://doi.org/10.3390/e26020171

**AMA Style**

Rondoni L, Di Florio V.
Probability Turns Material: The Boltzmann Equation. *Entropy*. 2024; 26(2):171.
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**Chicago/Turabian Style**

Rondoni, Lamberto, and Vincenzo Di Florio.
2024. "Probability Turns Material: The Boltzmann Equation" *Entropy* 26, no. 2: 171.
https://doi.org/10.3390/e26020171