A few weeks ago, a preprint by Yu Deng, Zaher Hani and Xiao Ma appeared on the Arxiv website announcing the resolution of the so-called Hilbert’s sixth problem, i.e., the problem of deducing a macroscopic, continuous description of physics from atomistic models. Sergio Simonella tells us what it is and what we should expect from this contribution. The original Italian version of this article can be foud at this link. 

In 1872, Ludwig Boltzmann started from the atomistic model of matter, at the time only partially accepted, and wrote a transport equation (hereafter Boltzmann Equation BE) for the distribution function \(f = f_t(x,v)\) of a particle in a rarefied gas:

(BE)                  \((\partial_t + v \cdot \nabla_x) f = Q(f_t,f_t)\)

The function \(f_t(x,v)\)  describes the density of particles in a gas, that is, the number of molecules that, at time \(t\), occupy a unit volume  \(dx dv\) of the phase space of a single particle. Because of the rarefaction hypothesis, a single particle in the Boltzmann gas moves in uniform rectilinear motion covering long distances and, from time to time, it experiences seemingly random collisions. Such collisions are described by a “collision operator”  \(Q\)  in (BE), in which the dependence on \(f\)  is “bilinear”: \(Q\)  depends on the product of two identical functions computed in different particle states.

In modern terms, Boltzmann argued that, in a collision, the two incoming velocities are statistically independent. This crucial assumption leads to the closed equation (BE), which turns out to be an effective and accurate description of how the gas evolves over time, rapidly approaching thermodynamic equilibrium states.

With the (BE), Boltzmann initiated the field of research known as kinetic theory of gases and an evergreen series of studies, in which mathematicians were and are key players.

By writing the equation and studying its properties, Boltzmann fulfilled a program that had involved him since he was very young. At the age of twenty-two (in 1866), the Austrian physicist had already stated that his goal would be to obtain the first and second laws of thermodynamics as mathematical theorems starting from mechanical systems.

I will try to explain this enterprise in a non-technical way. Physical reality looks extremely different depending on the scale on which it is observed. Let us start with the microscopic scale, that is, the angstrom (\(10^{-8}\) cm) scale. On this scale, the fundamental model is the particle model in which small units of matter, molecules or atoms, move incessantly and interact mutually with each other according to precise mechanical laws. A classical description of such laws of motion dates back to the 17th century and was synthesized in Isaac Newton’s Principia. Newton’s law is completely deterministic: this means that if we know the state of the system at some initial time (\(t=0\)), then we will be able to determine uniquely the state of the system for all times, past and future. Moreover, Newton’s law (like all currently established fundamental physical laws) makes no distinction between past and future. For this reason, the law is called “reversible”. On the atomic scale, what we see happening for increasing times can happen identically going back in time, like a movie displayed backwards. More precisely, suppose we observe a mechanical system consisting of \(N\)  particles evolving from time \(0\)  to time \(t\): if at this point we could reverse all the velocities, letting the system evolve for another time \(t\), we would obtain exactly the initial state with the reversed velocities, i.e., the collection of positions \(x\)  and velocities  \(-v\)  (modulo the sign) from which we had started. This character of the mechanical laws is in apparent contradiction with the most familiar observations of physical reality on a macroscopic scale. In particular, the laws of thermodynamics are in apparent contradiction with time reversibility. Such “contradiction” constituted a scientific puzzle for a long time. It is resolved by investigating the relationship between small and large scales. For we know very well that, going forward in time, one ages but does not rejuvenate; heat does not spontaneously pass from a cold body to a warm body; a drop of ink in a glass of water spreads by darkening the liquid, but does not spontaneously return to the small, round shape it originally had. All these phenomena, however, have in common that they involve a very large number of particles. The secret is lying in this disparity between scales: a single macroscopic state, observed in an experiment, can be realized by a huge number of equivalent microscopic states. In practice, most initial configurations (which have a large probability of being realized) provide the correct macroscopic behavior in the transition from time \(0\) to time \(t>0\), but do not provide the correct macroscopic behavior in the reverse transition, from time \(t>0\) to time \(0\). In other words, configurations that are good for the future are not good for the past.

The second law of thermodynamics explains the irreversibility of processes in terms of the entropy \(S\) – a peculiar quantity that can be thought of as a measure of the system complexity. In spontaneous processes, the entropy of a system can never decrease. Boltzmann proposed to interpret entropy by tying it to the number of microstates compatible with a certain macrostate (hence the formula “\(S = k \log W\)” carved on his grave in the cemetery of Vienna). He further asserted that, in a particularly simple physical system, such as a very dilute gas, entropy has an elementary formulation in terms of the density function \( f\), which is expressed by the formula

$$S = – \int f \log f$$

and that its growth over time can be easily demonstrated from the equation (BE). In this way, Boltzmann was able to reconcile mechanical laws, entropy increase and approach to equilibrium. His work was revolutionary, but also inevitably controversial, precisely because of the problem of compatibility between reversible and irreversible behavior of matter. Colleagues of Boltzmann, even distinguished ones, doubted the rigorous validity of his theory.

The “sixth problem” of David Hilbert fits historically into this compelling scientific affair. Its statement, however, is vague (especially when compared with other problems on his long list): “…To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.” ^{[1 ]}This text is the translation of the official one proposed by Hilbert. It should be noted that Hilbert gave, for each problem, a title and one or more statements. In the case of the sixth problem, the statement is unique. The original German text can be accessed online through the repository of the SUB Göttingen.

In the historical context of the International Congress of Mathematicians in 1900, Hilbert quotes a number of contemporary physicists and, in particular, provides due prominence to Boltzmann – he could not refrain from doing it. He suggested that mathematicians consider the kinetic theory of gases as a concrete field of investigation. But his sixth problem goes far beyond kinetic theory and, to this day, may well be seen as a brilliant foreshadowing of the modern role of mathematical physics.

Specifically, the Hilbert problem can be declined into at least two monumental programs. illustrated in the mid-20th century by Harold Grad and Charles B. Morrey, respectively. The first program concerns the rigorous derivation of the Boltzmann equation from Newtonian particle systems [N → B]; the second concerns the derivation of hydrodynamic equations (Euler equations and their corrected version, the Navier-Stokes equations) from the same particle systems [N → H]. At the turn of the century when Hilbert wrote, no one could distinguish, nor properly formalize the two programs. Nowadays, they are a classic of mathematical physics and it is well known that the two of them are deeply different. In common, they have the ambition to explain the transition from the microscopic to the macroscopic world in a comprehensive and convincing way.

To illustrate the difference between the two problems, it should be noted that the (BE) can only be used to describe gases under extremely rarefied conditions – so-called “perfect gases”. A typical example of the application of the (BE) is in aerodynamic problems, which involve modeling the atmosphere at high altitudes. Under such extreme conditions, hydrodynamic equations may be inadequate: one then resorts to the (BE) to explain the observed behavior. The number of molecules per \(cm^3\) is \(10^{16}\) or less, compared to at least \(10^{23}\) normally present in a solid, fluid or gas at moderate density. In the latter, the typical distances between molecules are comparable with the magnitude of intermolecular forces, and the resulting physical properties are drastically different. In particular, the laws concerning density, pressure, and temperature are not those of a perfect gas, but must be calculated through local equilibrium states (Gibbs measures associated with the interaction potential of the system). Morrey clarified the mathematical significance of the validity of the hydrodynamic laws, but the problem is unsolved to this day.

Hydrodynamic equations can also be obtained directly from Boltzmann’s equation [B → H]. Again, it should be emphasized that the problems [N → H] and [B → H] are very different. In the second limit, the starting point is no longer atomistic and reversible, but is already a continuous and irreversible description. Note also that [B → H] leads to the (reversible) Euler equations. The latter are nothing but local conservation laws of mass, energy and momentum. [N → H] also produces the same equations: but the law linking density, energy and pressure is very different and describes the physical nature of the fluid under observation (atmospheric gas, river or ocean water, stellar plasma…).

The problem [B→ H] is interesting because it explains that the (BE) can be used to obtain equations quite analogous in structure to the hydrodynamics of systems of particles, albeit with thermodynamic laws corresponding only to the case of extreme rarefaction. Hilbert himself became passionate about the problem [B→ H]. In 1912, he derived the famous “Hilbert expansion”, which describes the solutions of the (BE) in power series of a parameter inversely proportional to the density of the gas. In connection to this he said: ‘I recognize in the theory of gases the most splendid application of the theorems concerning integral equations’. The problem [B → H], conceptually far from Morrey’s program but far more tractable on a technical level, would rightly excite mathematicians for a long time.

Returning to the case of kinetic theory, namely the [N → B] problem, progress was very slow. In 1949, Grad proposed the precise limit in which the Boltzmann equation becomes exact, starting with a system of classical particles evolving in a container according to Newton’s laws. In 1972, Carlo Cercignani observed that Grad’s limit fits perfectly with the structure of the evolution equations for the moments of a probability measure on the system of hard spheres. Hard spheres constitute an idealized particle dynamics, which we should imagine as small billiard balls free to move in three-dimensional space. The dynamics is thus governed by free transport and elastic, binary collisions, which occur only when two of the spheres come into contact, or at the edges of the container. The Boltzmann-Grad limit (BG-limit) corresponds to the regime in which the total number of spheres tends to infinity, and simultaneously the diameter of the spheres tends to zero, such that the time of flight of a sphere between one collision and the next is typically equal to 1 (the spheres travel a distance comparable to the length of the container in which they move). With this insight, at the 1974 international Battelle Rencontres conference in Seattle, Oscar Lanford presented a first complete proof of the transition from Newton’s laws the Boltzmann’s equation: valid at least for a short time. ^{[2 ]}Lanford, O. E., “Time evolution of large classical systems”, in Dynamical Systems, Theory and Applications, vol. 38, 1975, pp. 1–111. doi:10.1007/3-540-07171-7_1..

On Monday March 3, 50 years after the printing of Lanford’s famous paper on the validity of the Boltzmann equation, an extraordinary result appeared on the ArXiv preprints in mathematics ^{[3 ]}Deng, Y., Hani, Z., and Ma, X., “Long time derivation of the Boltzmann equation from hard sphere dynamics”, arXiv e-prints, Art. no. arXiv:2408.07818, 2024. doi:10.48550/arXiv.2408.07818... The authors are Yu Deng (University of Chicago), Zaher Hani and Xiao Ma (University of Michigan). In Theorem 1 of the manuscript, the authors consider regular solutions of the Boltzmann equation in a finite spatial domain with periodic boundary conditions. The theorem states that such a solution can be constructed from the model of hard spheres, in the Boltzmann-Grad limit, as in Lanford’s theorem. More precisely, by hypothesis, the spheres are randomly distributed at time zero, according to a specific and simple law. For such an initial microstate, the Deng-Hani-Ma theorem asserts the convergence of the system of hard spheres to the solution of the Boltzmann equation, up to arbitrary times. The striking feature of the Deng-Hani-Ma theorem lies precisely in the absence of restrictions on the time of validity. In other words, the authors reduce the problem of the derivation of the equation, to that of existence. It must be remembered, however, that the Cauchy problem for the Boltzmann equation, namely the existence of a solution for all times given an initial \(f_0\) function, still remains open more than 150 years after its inception. But there are special situations, of great physical interest, in which the solution exists globally. Hence the Deng-Hani-Ma theorem makes it possible to obtain, from first principles (Newton’s laws), some nonlinear phenomena of approach to thermodynamic equilibrium, for the first time on physically reasonable time scales.

The long proof of the theorem relies heavily on a preliminary manuscript of more than 150 pages from last August and is presented to the reader as a climb of exceptional technical level. The key ideas of the new advance do not emerge easily, even for “insiders”. The proof does not resort to new a priori estimates, nor to a new guiding principle based on global properties of the system (energy, entropy…); but to a new technique based on an iterative argument that allows a global control of the error terms.

As I write this text, no one outside the authors has been able to reproduce the iterative algorithm. To seriously carry out such work will take a long time. As with other undertakings in contemporary mathematics, the technical level of research demands singular effort and prudence.

If the proof is correct, the Deng-Hani-Ma Theorem would be a most important step in mathematical physics, as it would fulfill Grad’s program for the justification of kinetic theory, at least under the assumptions mentioned above.

As mentioned above, the proper hydrodynamics, observed in physical experience, have laws of state that cannot be predicted via the Boltzmann-Grad limit. At most, kinetic theory can aspire to capture the value of physical observables in an extrapolation of experimental results to virtually zero density. Current mathematical techniques have not, to date, allowed the microscopic justification of the most common hydrodynamics for systems of interacting particles, as desired by Morrey.

As you may have guessed, what is commonly called “Hilbert’s sixth problem” is actually a majestic area of research. And as such it is certainly not yet finished posing its challenges for our scientific future.

Sergio Simonella
Sapienza Università di Roma