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# Kinetic theory without pain: an unpublished paper by Jean Ginibre

*Advances in Continuous and Discrete Models*
**volume 2023**, Article number: 25 (2023)

## 1 Introduction by Y. Pomeau

Let us first precise that the references quoted in this section are the ones postponed at the end the paper.

The Notes by Jean Ginibre on kinetic theory (“La théorie cinétique sans peine”) were written by him when he was still a student, slightly before he got his PhD. I thought it worthwhile to add a few comments to help understand this remarkable piece and to put it in the more general framework of theoretical physics at the time of their writing, this being based partly on personal memory and would likely be quite different when seen by others. Moreover, it is not intended to be a thorough review of progresses in the field made since, like in a standard review paper.

The situation of theoretical physics at the time, fifties to early sixties, was still dominated by the outstanding success of calculations of various effects due to the quantum interaction between electrons and electromagnetic radiation. This had given access [1] to phenomena like the Lambshift and to the small \((g-2)\) difference. All this relied on elaborate strategies of calculation allowing to get rid of a priori diverging “corrections” with strategies invented by Heisenberg and Dirac in the thirties. Even though the results were certainly correct, the whole strategy of the calculation was not free of difficulties. Therefore there was a natural tendency for the most mathematically inclined theoreticians, as certainly Jean was, to try to put things on a safer ground. This is surely one of the basis of his use of a mathematical formalism aiming at putting things in a fully rigorous framework, somewhat unfamiliar to many physicists. Such a formalism allows to do equilibrium thermodynamics in an infinitely extended homogeneous system without dealing first with a finite system and then making its size grow to infinity at constant density and temperature. This tendency was strengthened by the successful approach of equilibrium problems like the understanding of phase transitions in various models of spin systems. However, as well understood by Jean, the understanding of nonequilibrium process was (and remains) far less advanced than the one of physics “at equilibrium”. This is partly because outside of equilibrium there is nothing like a Gibbs ensemble. The only available “mathematical object” describing nonequilibrium systems is the so-called BBGKY hierarchy, a quite cumbersome theory relating between themselves an infinite set of functions with an increasing number of arguments as one climbs the hierarchy. Because of its complexity and as emphasized by Jean, one has to make approximations to get something out of this BBGKY hierarchy. The most remarkable result in the theory of nonequilibrium systems is the Boltzmann kinetic equation that requires that the system is at very low density so that the collisions (namely the strong interaction between particles) occur between a pair of particles only, or almost so if they interact by a smooth potential. Jean explains in the Notes how Boltzmann kinetic equation can be derived from the hierarchy, something done first by Bogoliubov [2] as he indicates. He does not mention one resounding success that took place slightly before the writing of the Notes, namely the derivation by Balescu and Lenard [3] of the kinetic equation for a neutral plasma. This case is not covered by the standard Boltzmann kinetic theory because the long range interaction between electric charges makes diverge the cross section of two body collisions and so the kinetic operator. The Balescu–Lenard kinetic operator introduces a kind of out-of-equilibrium screening of the interaction between electric charges like the one found at equilibrium in the Debye–Huckel theory of electrolytes.

The derivation by Bogoliubov [2] of Boltzmann kinetic equation used a fully coherent analysis of the dynamics of a dilute gas in a state close to uniform in space. This is no trivial matter because the original derivation by Boltzmann is based on many assumptions not showing very clearly what are the assumptions needed to make this kinetic equation valid and in particular what could be the “next order” corrections taking into account, for instance, finite density effects and/or of long range 2-body interactions. This “formal” derivation by Bogoliubov is done by Jean here and leads to a very compact form of Boltzmann kinetic equation, as written in his equation (3.12).

Section 5 is devoted to the outstanding question of irreversibility. There Jean explains in a simple way how irreversibility shows up in Boltzmann kinetic theory, it is because his operator \(C(t)\) has to be taken at *t* tending to plus infinity to derive the kinetic equation, whereas reversibility applies only for finite times. Jean makes an interesting connection between the Mayer expansion at equilibrium and the calculation of the correlations out of equilibrium at low densities.

Jean insists well that various assumptions necessary to derive even the standard Boltzmann kinetic equation may be invalid. Actually it was discovered [4] slightly after the writing of the Notes that, out of equilibrium, a specific phenomenon occurs, absent at equilibrium, namely the long range propagation of disturbances via the hydro-dynamical modes. Because of that there is no local kinetic theory in 2D fluids (and in 1D also!). There is an allusion to this kind of long range propagation of nonequilibrium correlations in the Notes when referring to the article by Cohen and Dorfman [5] on page 26 in the conclusion section without saying precisely what is involved and by referring to an “opinion”. From this reference it seems that Jean refers to the discovery of long range cycles of collisions between particles interacting with short range forces, like hard spheres (in 3D) or discs (in 2D). It had been discovered that, for instance in 3D, four hard spheres can make a cycle of collisions of size as big as wished because of the possibility that two spheres making an initial encounter can meet again by being thrown back to each other by collisions with a few other spheres. At second collision such a re-collision event makes invalid the Stosszahlansatz of Boltzmann necessary to derive his kinetic equation, because this ansatz implies that the two colliding particles are statistically independent before the collision.

This makes the situation of nonequilibrium gases far more complex than at equilibrium where the range of correlation is the same as the range of the interaction potential. As was shown slightly later than the paper [5], there is another mechanism of long range propagation of correlations in nonequilibrium fluids. It results from the hydrodynamic modes, always present in a fluid. Because of that the transport coefficients in 2D fluids diverge (logarithmically), whereas the density expansion of transport coefficients in 3D gets beyond a finite order contributions depending logarithmically on the density.

At about the time where the Notes were written, other attempts were made to develop a formalism enabling to study nonequilibrium situations, based on molecular dynamics like Boltzmann kinetic theory but without assuming a dilute gas and so—hopefully—giving access to chemical physics of dense media and to a more global understanding of the status of irreversibility. Such attempts were made by Prigogine and his group in Brussels. This is developed for instance in the book by Balescu [6]. This book includes a highly original part on quantum systems and derives from scratch the Boltzmann–Nordheim kinetic equation for quantum gases, fermionic and bosonic. The general philosophy of the approach of problems of kinetic theory by the Brussels school can perhaps be summarized by saying that besides the change of the mean values, changes in the correlations are also taken into account if their time scale is smaller than the one of the mean values. In other terms, the correlations are created by quick events like the two body collisions in Boltzmann kinetic theory. This constraint is quoted in the Notes, but, as we just explained, it may be of dubious validity outside of the case of dilute gases because of either the long cycles of re-collisions [5] or the long range propagation of hydrodynamic modes [4].

## 2 English Summary of Jean Ginibre’s manuscript, presented by Y. Pomeau

This summary of the unpublished Notes written by Jean Ginibre is numbered according to the sections of the original text (written in french) which is presented in Sec. 3. Moreover the references quoted here are postponed at the end of the paper, most of them are taken from the original text.).

### 2.1 Introduction

As rightly said by the author, Liouville equations are too complex to be useful. Kinetic theory yields a “simplified” picture of gas/fluid dynamics involving a distribution function for one particle. Boltzmann kinetic equation was derived by him by using a statistical argument foreign to a purely mechanical picture. References [2], [7] to [10] did look at possible derivations of Boltzmann kinetic equation directly from the Liouville equation and its BBGKY consequence. The most systematic (and earliest) derivation is by Bogoliubov [2], and it relies on various conjectures of validity hard to check. This essay is to formulate clearly the assumptions behind such a derivation by displaying as concisely as possible the algebraic structure of the theory.

Section 1 explains the notations, based on ref [11]. Section 2 introduces the BBGKY hierarchy and solves it formally. Section 3 explains the principle of the theory in the simplest possible way. Section 4 displays Bogoliubov derivation of the Boltzmann kinetic equation. Section 5 explains how irreversibility sets in by starting from formally reversible dynamical equations. Section 6 explains the relationship between the theory of Bogoliubov and the works of Green [9] and Cohen [10].

### 2.2 Preliminaries

This section introduces the formalism used in the Notes. Compared to the expositions of BBGKY theory and particularly to Bogoliubov [2], this formalism is quite different. This goes far beyond trivial changes of notations but implies also an attempt to put things in a far more abstract framework than usual in this field.

The fundamental object there is denoted as \(E_{m}\) and is the set of real functions depending on positions and momenta of *m* classical particles. Moreover, the functions are symmetrical under permutation of particle indices. The functions in \(E_{m}\) are measurable by the Liouville measure, which implies that integrals of functions in \(E_{m}\) over space and momenta converge. *E*, the direct product of the \(E_{m}\)’s, is endowed of a structure of algebra thanks to the ⋆ product defined in Eq. (1-2), an interesting and I believe original idea that streamlines many calculations done later. Thanks to this algebraic structure, one can define an application in *E*, called Γ and defined in Eq. (1-5).

The next five examples have partly a pedagogical purpose, well deserved, and give, if possible, some agility for the algebraic manipulations to come. Examples 1 to 3 are devoted to the operator Γ. Example 4 introduces a new operator *P*, and Example 5 introduces others, denoted as Δ, Φ, \(H_{0}\), *R*, *P*, and \(P^{-1}\) used in later calculations.

### 2.3 The equation of motion and its formal solution

The equation solved is the Liouville equation. An added complexity there is that the constraint of a fixed number of particles is dropped, which somehow extends the usual grand canonical ensemble of equilibrium thermodynamics to nonequilibrium. The basic element of the analysis becomes the function \(\rho _{\Lambda}(X)\) with the dynamical equation given in Eqs. (2-5) and (2-6). The Liouville equation is Eq. (2-8). It assumes an interaction potential \(V(.)\) fairly general that may involve more than two particles.

The fundamental operator of evolution is \(U^{t}\) defined in Eq. (2-30) and so to speak computed in Eqs. (2-32) and (2-33). As noticed a bit later (at the very end of Sect. 2), the convergence of the developments introduced in Eq. (2-32) is not certain, by far! The final result of this section is Eq. (2-47), completed by (2-48). This is an outstanding result that extends formally relations known at equilibrium to out of equilibrium situations, as pointed out by the author. Indeed the well-known limitation [12] in the validity of this kind of extension is that outside of equilibrium correlations tend to be far more long ranged than at equilibrium because of the propagation of perturbations at distances of order of the mean free path in dilute gases or even at infinite distances by hydrodynamic fluctuations in denser media.

### 2.4 The correlation equation and the kinetic equation

This section is devoted to the derivation of “the kinetic equation”. Its range of applicability is limited by the assumption that the number density is small, meaning practically that some sort of Boltzmann kinetic equation will be the final result.

The beginning of this section outlines what is meant by “kinetic equation”. This implies (even though the word is not used) an adiabatic assumption of synchronization of solutions of the full dynamical problem with the one body distribution denoted as \(f(x)\). It states also that this is true in the limit of a small density, where practically the kinetic equation is Boltzmann kinetic equation. Differently from the usual derivations of Boltzmann kinetic equation, the author begins by computing a correlation operator, denoted as *C*, which yields the full distribution *ρ* as a function of F and lastly of f (Eq. 3-1). The author derives then the closed equation (3-10) for the time derivative of *C*. By integration this gives *C*, and by inserting the result into Eq. (3-4) one should get the kinetic equation.

Even though this calculation is formally exact, at the end the kinetic equation is derived by inserting into Eq. (3-4) the operator \(C(t)\) computed for *t* tending to +∞. In standard derivations of Boltzmann kinetic equation, this amounts to putting in the collision operator the scattering data of the two body problem to get changes of the velocity distribution at time scales much longer than the duration of a two body collision.

The last paragraph explains the relationship between the method of derivation of the Notes and that of Zwanzig [13]. The difference lies in the linearity assumed by Zwanzig that does not hold true.

### 2.5 The interaction representation and the point of view of Bogoliubov

This section is about the effective derivation of the kinetic equation in the aforementioned limits. This is done in interaction representation, where the leading order is the hamiltonian for noninteracting particles and the perturbation comes from the two body interaction. The operator \(C'\) is derived from Eqs. (4-9) and (4-10). After integration with respect to time, one obtains an expression of C(∞), which is the same as the one given by Bogoliubov, and finally agrees at the end with Boltzmann kinetic operator. At the end of Sect. 4, there is a table of correspondence between the equation of this section and the equations in Bogoliubov paper [2]. This comparison should be helpful for everybody wishing to understand better the link between the original derivations of Boltzmann kinetic equation and the ones based on the BBGKY hierarchy or directly the Liouville equations as in the Notes.

### 2.6 The problem of irreversibility

The way irreversibility does appear in kinetic theory is of course central to the subject since the discussions following Boltzmann’ s work. A good introduction to the historical aspects of this can be found in the second book of S.G. Brush on kinetic theory [14] that includes the original papers or English translation of articles by Poincaré, Zermelo and the answers by Boltzmann himself. Technically, and as noticed in this section, irreversibility sets in because of the substitution of C(∞) for \(C(t)\) in the equations, something necessary to get an explicit equation at the end that is local in time, namely such that one integrates over time the pair correlation created by binary collisions, assuming there is none *before* the collision. This is the well-known “Stosszahlansatz” by Boltzmann, equivalent to assuming that before they meet particles are uncorrelated or have no knowledge of each other. The way this Stosszahlansatz is formulated (the word however does not appear) is interesting because it displays very explicitly how irreversibility sets in in the analysis, although generally it is only seen as a final result of all the assumptions made without linking it precisely to the step where it appears.

### 2.7 Formal solution of the equation for correlations

The purpose of this section is to explain, by formal analogy of the theory developed so far with equilibrium statistics, that the correlations created by the interaction of particles can be found explicitly.

This section exposes the principle of the derivation of the Boltzmann equation from the result derived in Sect. 5. This requires the computation of the operator C(∞), which is done in this section, with the final result given in Eq. (6-18).

### 2.8 Conclusion

This concluding section is very far reaching, given the time where the Notes were written. It says rightly that the convergence of the formal expansions introduced in the derivation of the kinetic equation is at least dubious and concludes that “an important work” remains to be done in this respect. Looking retrospectively it is right that the Notes and other works done later show that so-called long tails pop up in kinetic theory as soonas one goes beyond the limit of a very dilute gas, where Boltzmann kinetic theory applies. But this is another story, and we refer the interested reader to ref [12] that lists relevant publications on this topic.

## 3 The manuscript by Jean Ginibre

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## References

Todorov, I.: From Euler’s play with infinite series to the anomalous magnetic moment. IHES, January 2018 and references therein

Bogoliubov, N.N.: Problems of a dynamical theory in statistical physics. In: de Boer, J., Uhlenbeck, G.E. (eds.) Studies in Statistical Mechanics, vol. 1. North-Holland, Amsterdam (1962)

Balescu, R.: Irreversible processes in ionized gases. Phys. Fluids

**3**, 52–63 (1960)Pomeau, Y.: A new kinetic theory for a dense classical gas. Phys. Lett.

**27A**, 601–602 (1968)Cohen, E.G.D., Dorfman, J.R.: Generic long-range correlations in molecular fluids. Phys. Lett. A

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Green, H.S.: The Molecular Theory of Fluids. Amsterdam (1952)

Green, M.S.: J. Chem. Phys. 25, p. 836 (1956), Physica 24, p. 395 (1958) and Lectures in theoretical physics, Boulder (1960) Interscience, New York (1961)

Cohen, E.G.D.: Physica 28 p. 1025, 1045 and 1060 (1962); J. Math. Phys. 4 p. 183 (1963)

Ruelle, D.: Lectures in Theoretical Physics. Univ. of Colorado press, Boulder (1963), Univ. of Colorado Press, Boulder (1964)

Pomeau, Y., Résibois, P.: Time dependent correlation functions and mode-mode coupling theories. Phys. Rep.

**19**, 63–139 (1975)Zwanzig, R.: Lectures in theoretical physics, Boulder (1960) published by Interscience, New York (1961)

Brush, S.G.: Kinetic Theory, Vol. 2 Irreversible Processes. Pergamon, London (1966)

## Acknowledgements

Dr. Minh-Binh Tran helped to put this article in form and is thanked for that. The text in the first section was elaborated with the invaluable help of Martine Le Berre and Paul Clavin. Both are warmly thanked.

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All authors read and approved the final manuscript. YP wrote the introduction(sec.1) and the english summary (sec.2) . Section 3 is the original manuscript of JG, written in French.

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Ginibre, J., Pomeau, Y. Kinetic theory without pain: an unpublished paper by Jean Ginibre.
*Adv Cont Discr Mod* **2023**, 25 (2023). https://doi.org/10.1186/s13662-023-03767-3

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DOI: https://doi.org/10.1186/s13662-023-03767-3