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Stochastic dynamics and Boltzmann hierarchy. (English) Zbl 1199.82001
Pratsi Instytutu Matematyky Natsional’noï Akademiï Nauk Ukraïny. Matematyka ta ïï Zastosuvannya 74. Kyïv: Instytut Matematyky NAN Ukraïny (ISBN 978-966-02-4783-3). 400 p. (2008).
The monograph is devoted to one of the most important trends in contemporary mathematical physics, the investigation of evolution equations of many-particle systems of statistical mechanics. This work systematizes rigorous results obtained in this field in recent years and presents contemporary methods for the investigation of evolution equations of infinite-particle systems.
In mathematical and statistical physics it was generally accepted that the classical Boltzmann equation is based on the Hamilton equations. The fact of irreversibility of the Boltzmann equation and reversibility of the Hamilton equations leads to well-known paradoxes (see J. Loschmidt [Wien. Ber. LXXV. 287-299, LXXXVI. 209-225, Wien. Anz. 1877. 27–28 (1877; JFM 09.0759.01)], H. Poincaré [Acta Mathematica, 13, 1–270 (1890; JFM 22.0907.01)] and [Revue de Metaphysique et de Morale, 1, 534–537 (1983)], E. Zermelo [Annalen der Physik, 54, 485–494 (1896; JFM 27.0759.03)]). At the same time, some arguments concerning the use of stochastic dynamics in deducing the Boltzmann equation were advanced by L. Boltzmann [Wien. Ber. LXVI. 275–370 (1872; JFM 04.0566.01)] himself, and P. Ehrenfest and T. Ehrenfest [Physik. Zs. 8, 311–314 (1907; JFM 38.0931.01)]. As early as 1935, M. Leontovich [Zh. Ehksper. Teor. Fiz. 5, 211–231 (1935; Zbl 0012.26802)] proposed a stochastic dynamics of point particles in the phase space, postulated the Itô-Liouville equation for this dynamics, and introduced a hypothesis according to which the corresponding one-particle correlation function satisfies the Boltzmann equation in the thermodynamic limit.
For the spatially homogeneous Boltzmann equation in which the one-particle correlation function depends solely on the momentum and is independent of the position of the particle, M. Kac [(Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, Colorado, 1957 Vol. I.) London and New York: Interscience Publishers. XIII, 266 p. (1959; Zbl 0087.33003)] pro posed a stochastic dynamics and, in the mean-field approximation, deduced the Boltzmann equation in the thermodynamic limit.
A. V. Skorokhod [Mathematics and its applications: Soviet Series, 13. Dordrecht (Netherlands) etc.: D. Reidel Publishing Company. xvii, 175 p. (1988; Zbl 0649.60062)] proposed a stochastic dynamics in the phase space and deduced a nonlinear Boltzmann-type equation for the one-particle correlation function in the mean-field approximation in the thermodynamic limit. In all cases the physical meaning of various types of stochastic dynamics and their relationship with the Hamiltonian dynamics was not clarified.
N. N. Bogolyubov [Stud. Statist. Mech. 1, 1–118 (1962; Zbl 0116.45101)] also indicated that, in deducing the Boltzmann equation, the dynamics of particles “is interpreted as a random process …, and the efficient cross sections appearing in the equation of the random process are calculated by using the equations of classical mechanics.” For the first time, he showed that the Boltzmann equation can be derived from the Hamiltonian dynamics as a result of a certain limit transition in a special solution of the hierarchy for correlation functions depending on time through the one-particle correlation function. Moreover, the cluster properties of the correlation functions and low densities are essentially used in this case. The problem of transformation (degeneration) of the Hamiltonian dynamics in the Bogolyubov limit was not studied.
The mathematical substantiation of the Bogolyubov method and the mechanism of appearance of stochastic dynamics from the Hamiltonian dynamics was absent for a fairly long period of time. For this reason, it was quite natural to try to solve this problem for a maximality simplified but still nontrivial model. To this end, H. Grad [Commun. Pure Appl. Math. 2, 331–407 (1949; Zbl 0037.13104)] studied a system of hard spheres and showed that, in the (thermodynamic) limit, as the diameter of the spheres tends to zero but the length of the free path of particles remains constant, all correlation functions turn into products of one-particle correlation functions, and the latter are solutions of the Boltzmann equation. This limiting procedure is called the Boltzmann-Grad limit.
At present time, the mathematical procedure of deducing the Boltzmann equation in the Boltzmann-Grad limit can be regarded as, to a certain extent, completed due to the works by O. E. III Lanford [Dyn. Syst., Theor. Appl., Battelle Seattle 1974 Renc., Lect. Notes Phys. 38, 1–111 (1975; Zbl 0329.70011)], C. Cercignani, R. Illner and M. Pulvirenti [Applied Mathematical Sciences. 106. New York, NY: Springer-Verlag. vii, 347 p. (1994; Zbl 0813.76001)], R. Illner and M. Pulvirenti [Commun. Math. Phys. 105, 189–203 (1986; Zbl 0609.76083)], R. Illner and M. Pulvirenti [Transp. Theory Stat. Phys. 16, 997–1012 (1987; Zbl 0629.76088)], H. Spohn [Texts and Monographs in Physics. Berlin etc.: Springer-Verlag. xi, 342 p. with 19 fig. (1991; Zbl 0742.76002)], Gerasim and Petrina. The detailed rigorous proofs can be found in the works by D. Ya. Petrina and V. I. Gerasimenko [Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 5, 1–52 (1985; Zbl 0605.70012)] (See also the monograph by C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina [Mathematics and its Applications (Dordrecht). 420. Dordrecht: Kluwer Academic Publishers. viii, 244 p. (1997; Zbl 0933.82001)]).
At the same time, the following question remained open: What dynamics does serve as a basis of the Boltzmann equation and the limiting BBGKY hierarchy (now called the Boltzmann hierarchy)? In the works by D. Ya. Petrina and K. D. Petrina [Ukr. Math. J. 50, No. 2, 224-240 (1998) and from Ukr. Mat. Zh. 50, No. 2, 195–210 (1998; Zbl 0933.76078)], D. Ya. Petrina and K. D. Petrina [Ukr. Math. J. 50, No. 3, 425–441 (1998) and Ukr. Mat. Zh. 50, No. 3, 372–387 (1998; Zbl 0933.76079)], D. Ya. Petrina and K. D. Petrina [Ukr. Math. J. 50, No. 4, 626–645 (1998) and Ukr. Mat. Zh. 50, No. 4, 552–569 (1998; Zbl 0933.76080)] and M. Lampis and D. Ya. Petrina [Ukr. Math. J. 54, No. 1, 94–111 (2002) and Ukr. Mat. Zh. 54, No. 1, 78–93 (2002; Zbl 1027.82022)], M. Lampis and D. Ya. Petrina [Ukr. Mat. Zh. 56, No. 12, 1629–1653 (2004) and Ukr. Math. J. 56, No. 12, 1932–1960 (2004; Zbl 1083.82020)], M. Lampis, D. Ya. Petrina and K. D. Petrina [Ukr. Math. J. 51, No. 5, 686–706 (1999) and Ukr. Mat. Zh. 51, No. 5, 614–635 (1999; Zbl 1029.82506)] it is shown that the Hamiltonian dynamics of hard spheres in the Boltzmann-Grad limit degenerates into a certain stochastic dynamics of point particles. According to this stochastic dynamics, point particles move as free ones until they collide.
However, in this case, we encounter the problem of determination of the corresponding correlation functions because the indicated stochastic dynamics differs from the free dynamics of noninteracting point particles on hypersurfaces of lower dimensionality neglected in traditional classical statistical mechanics. For this reason, it is necessary to introduce a new concept of correlation functions taking into account, in a certain way, the contributions of the hypersurfaces where the interaction of stochastic particles is specified. It can be shown that the solutions of the Boltzmann equation are also expressed via the contributions of these hypersurfaces. It is quite surprising that this fact was not discovered earlier.
For these correlation functions, the stochastic Boltzmann hierarchy is deduced with boundary conditions on the hypersurfaces where the positions of pairs of particles coincide. Note that, earlier, these boundary conditions were neglected in the ordinary Boltzmann hierarchy. The stochastic Boltzmann hierarchy is also obtained from the BBGKY hierarchy for a system of hard spheres in the Boltzmann-Grad limit if the boundary conditions are properly taken into account.
Thus, the stochastic Boltzmann hierarchy is deduced on the basis of the stochastic dynamics in exactly the same way as the BBGKY hierarchy is deduced on the basis of the Hamiltonian dynamics.
It is proved that the local (in time) solutions of the stochastic Boltzmann hierarchy exist for the initial data bounded in coordinates and exponentially decreasing in squared momenta. The global (in time) solutions exist for the initial data exponentially decreasing in the squared momenta and coordinates. If the initial data satisfy the condition of chaos, i.e., admit a representation in the form of products of one-particle correlation functions, then, outside the hypersurfaces of interaction of stochastic particles, the solutions of the stochastic hierarchy also satisfy the condition of chaos and the one-particle correlation function is a solution of the Boltzmann equation. The ordinary Boltzmann hierarchy without boundary conditions is solved in the entire phase space by the correlation functions represented in the form of the product of one-particle correlation functions satisfying the Boltzmann equation. Thus, the Boltzmann equation is deduced rigorously. It is shown that the Boltzmann equation is, in fact, based on the irreversible stochastic dynamics and, hence, there are no contradictions with the irreversibility of solutions of the Boltzmann equation.
The stochastic dynamics is very simple and possesses numerous properties of the Hamiltonian dynamics, namely, the trajectories with fixed random parameters, the operators of shift along the trajectories and the hierarchy of equations for correlation functions with fixed random parameters in the boundary conditions. This enables us to use the results obtained for the BBGKY hierarchy for a system of hard spheres to prove the existence of solutions of the stochastic Boltzmann hierarchy and the properties of chaos.
The stochastic dynamics proposed by Kac in the momentum space is obtained from our stochastic dynamics in the phase space as a result of averaging over the coordinates. This clarifies its physical meaning. Note that the spatially homogeneous Boltzmann equation is derived from the stochastic Boltzmann hierarchy without using the mean-field approximation. All results can be generalized to the case of Boltzmann equation with general differential cross section.
We now briefly describe the content of the monograph. It comprises the introduction and eight chapters.
In the first chapter, a critical survey of the results concerning the existence of solutions of the BBGKY hierarchy for a system of hard spheres and the justification of the Boltzmann-Grad limit is presented. Special attention is given to the boundary conditions for both the BBGKY hierarchy and the stochastic Boltzmann hierarchy.
In the second chapter, the stochastic dynamics is derived from the Hamiltonian dynamics of hard spheres in the Boltzmann-Grad limit. We deduce the Itô-Liouville equation and introduce the principle of duality according to which an ordinary function is associated with a generalized function concentrated on hypersurfaces of interaction of stochastic particles. These generalized functions are used to compute the contributions of the hypersurfaces to the correlation functions.
In the third chapter, the stochastic Boltzmann hierarchy with boundary conditions is derived from the stochastic dynamics of point particles.
In the fourth chapter, the existence of solutions of the stochastic Boltzmann hierarchy is proved and the property of chaos is established. These results are used to deduce the Boltzmann equation.
In the fifth chapter, the stochastic Kac dynamics in the momentum space is obtained from our stochastic dynamics in the phase space. It is shown that the spatially homogeneous Boltzmann equation can be derived from the stochastic Boltzmann hierarchy in the phase space without using the mean-field approximation.
In the sixth chapter, the results obtained for a system of hard spheres are generalized to systems of particles with arbitrary scattering cross section.
In the seventh chapter, we study a system of spheres with inelastic scattering used as a model of granular flows. A hierarchy of equations for correlation functions is deduced. This hierarchy contains the squared Jacobian of the phase trajectory unequal to one.
In the eighth chapter, we construct the solution of the Cauchy problem for the hierarchy in the space of sequences of summable functions. The group of evolution operators is obtained in the explicit form. The stochastic dynamics for granular flows corresponding to the Boltzmann equation is introduced.
The monograph contains two types of references: references of the first type are directly related to the problems analyzed in the book and are mentioned in it. References of the second type cover some other important problems of statistical mechanics.
See also the review of the translation [Stochastic dynamics and Boltzmann hierarchy. Translated from the Ukrainian by Dmitry V. Malyshev and Peter V. Malyshev. de Gruyter Expositions in Mathematics 48. Berlin: Walter de Gruyter. xiii, 296 p. EUR 99.95; $ 140.00 (2009; Zbl 1178.82002)].

MSC:
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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