The Random Gas of Hard Spheres
Abstract
:1. Introduction
2. The Hard Sphere Collision Model and the Boltzmann Equation
2.1. The Liouville Problem for Two Spheres
- If the distance between the spheres is less than their diameter (that is, the spheres are overlapped), then the density is set to zero:This condition ascertains that the spheres are rigid and may not overlap.
- Observe that in the open set , also solves the Liouville equation in Equation (9)—albeit with zero initial and boundary conditions. This leads to a possible discontinuity of F at the collision surface , which is taken into account below.
- Assuming that there is a solution of Equation (9) in both open regions and (with boundary conditions given via zero and Equation (10), respectively), on the collision surface itself we assign F to be equal to the “outer” boundary condition in Equation (10). This definition of F at the collision surface means that F is continuous as , and possibly discontinuous as .
2.2. The BBGKY Identity for Two Spheres
2.3. The Boltzmann Hierarchy and Boltzmann Equation
3. Inconsistencies in the Derivation of the Boltzmann Equation
- (1)
- The Liouville equation in Equation (9) by itself does not have any collision effects. The effect of collision is imposed separately for every pair of spheres with coordinates and , on the surface .
- (2)
- Due to the effect on the collision surface, the probability density of states F is discontinuous on this surface—it is zero for all according to Equation (8) (as the spheres are impenetrable) and is generally nonzero otherwise.
- (3)
- Due to the discontinuity of F on the collision surface, the collision surface integrals emerge according to the Gauss theorem and the Leibniz rule, when the BBGKY hierarchy is constructed. The resulting identity is given by Equation (23)—observe that it is valid for any F, which satisfies Equations (8) and (9), and is discontinuous on the collision surface .
- (4)
- In addition to the discontinuity, the velocity deflection condition in Equation (10) is imposed on F. This condition allows rewriting the collision integral in Equation (23) in the form of Equation (26). Note that it is the same exact integral as obtained originally in Equation (23) via Gauss theorem and Leibniz rule, only written in a different manner thanks to Equation (10). Similarly, one can obtain the K-sphere BBGKY identity in Equation (27) [9,11].
- (a)
- (b)
3.1. A Contradiction between the Liouville Problem and the Boltzmann Hierarchy
3.2. A Contradiction between the Liouville Problem and the Boltzmann Closure
3.3. Reversibility and Loschmidt’s Objection
3.4. Our Proposal to Correct the Inconsistency
4. Random Dynamics of Hard Spheres
- Collision configuration. The collision configuration is given by the following two criteria, which must hold concurrently:
- (a)
- The distance between the centers of spheres satisfies
- (b)
- The distance between the centers of spheres also diminishes in time, that is,This condition signifies that the spheres are approaching each other.
In the collision configuration, the velocities may be randomly transformed according to Equation (2), with a specified probability. For now, we write informally that in the collision configuration the velocities evolve according to
4.1. A Dynamical System Driven by an Inhomogeneous Point Process
4.2. Random Dynamics of Two Spheres
- If is away from , or if is growing in time (that is, the spheres are escaping each other), then the triggering point process is not present, and the spheres are moving with constant velocities according to Equation (1). Accordingly, the intensity in Equation (70) is zero in the infinitesimal generator of the process, and thus the generator consists of the free-flight term only.
- Once is close enough to and, at the same time is decreasing in time, the spheres enter the contact zone (both conditions in Equations (43) and (44) are satisfied). In this case, the collision-triggering point process becomes present, with the intensity being strictly greater than zero. Then, there are two possibilities:
- -
- A jump in the point process arrives so that the spheres “collide” according to Equation (2). In this case, the spheres start escaping each other (so that Equation (44) no longer holds) and the triggering point process is no longer present. In the infinitesimal generator, the Heaviside function becomes zero and so does the intensity .
- -
- A jump does not arrive, so that eventually decays back to zero together with the intensity of the point process; in this case, the spheres pass through each other without interaction.
In either scenario, the point process becomes dormant until the spheres approach each other again.
4.3. Extension to Many Spheres
5. Some Properties of the Random Sphere Dynamics
5.1. A Two-Sphere Solution along a Characteristic
5.2. A Steady Solution for Many Spheres
- is constant outside contact zones (that is, the regions in the coordinate space where the mollifier ), and its transitions within contact zones are given via the exponents of anti-derivatives of the mollifier in Equation (71).
- Outside contact zones, the following condition holds:
5.3. Entropy Inequality
5.4. The Steady Solution for a System with Independently Distributed Initial States
5.5. The Structure of Marginal Distributions of the Physical Steady State
5.6. The Marginal Distributions in the Limit of Infinitely Many Spheres
6. The forward Equation for the Marginal Distribution of a Single Sphere
6.1. Approximating the Two-Sphere Marginal via One-Sphere Marginals
6.2. Thin Contact Zone and Impenetrable Spheres
7. The Fluid Dynamics of the Enskog Equation in a Physical Hydrodynamic Limit
7.1. The Mass-Weighted Equation and the Hydrodynamic Limit
7.2. The Euler Equations
7.3. The Newton and Fourier Laws
7.4. The Navier–Stokes Equations
8. Summary
Funding
Acknowledgments
Conflicts of Interest
References
- Lennard-Jones, J. On the Determination of Molecular Fields.—II. From the Equation of State of a Gas. Proc. R. Soc. Lond. A 1924, 106, 463–477. [Google Scholar] [CrossRef]
- Vlasov, A. On vibration properties of electron gas. J. Exp. Theor. Phys. 1938, 8, 291. [Google Scholar]
- Bogoliubov, N. Kinetic Equations. J. Exp. Theor. Phys. 1946, 16, 691–702. [Google Scholar]
- Born, M.; Green, H. A General Kinetic Theory of Liquids I: The Molecular Distribution Functions. Proc. R. Soc. A 1946, 188, 10–18. [Google Scholar]
- Kirkwood, J. The Statistical Mechanical Theory of Transport Processes I: General Theory. J. Chem. Phys. 1946, 14, 180–201. [Google Scholar] [CrossRef]
- Chapman, S.; Cowling, T. The Mathematical Theory of Non-Uniform Gases, 3rd ed.; Cambridge Mathematical Library, Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]
- Cercignani, C. Theory and Application of the Boltzmann Equation; Elsevier Science: New York, NY, USA, 1975. [Google Scholar]
- Cercignani, C. The Boltzmann Equation and Its Applications. In Applied Mathematical Sciences; Springer: New York, NY, USA, 1988; Volume 67. [Google Scholar]
- Cercignani, C.; Illner, R.; Pulvirenti, M. The Mathematical Theory of Dilute Gases. In Applied Mathematical Sciences; Springer: Basel, Switzerland, 1994; Volume 106. [Google Scholar]
- Grad, H. On the Kinetic Theory of Rarefied Gases. Commun. Pure Appl. Math. 1949, 2, 331–407. [Google Scholar] [CrossRef]
- Gallagher, I.; Saint-Raymond, L.; Texier, B. From Newton to Boltzmann: Hard Spheres and Short-Range Potentials; European Mathematical Society: Zürich, Switzerland, 2014. [Google Scholar]
- Boltzmann, L. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitz.-Ber. Kais. Akad. Wiss. (II) 1872, 66, 275–370. [Google Scholar]
- Golse, F. The Boltzmann Equation and its Hydrodynamic Limits. In Handbook of Differential Equations: Evolutionary Equations; Elsevier: Amsterdam, The Netherlands, 2005; Volume 2, Chapter 3; pp. 159–301. [Google Scholar]
- Loschmidt, J. On the Thermal Equilibrium of a System of Bodies under Gravity Force, I. Proc. Imp. Acad. Sci. 1876, 73, 128–142. [Google Scholar]
- Kullback, S.; Leibler, R. On information and sufficiency. Ann. Math. Stat. 1951, 22, 79–86. [Google Scholar] [CrossRef]
- Bellomo, N.; Lachowicz, M. Kinetic Equations for Dense Gases: A Review of Physical and Mathematical Results. Int. J. Mod. Phys. B 1987, 01, 1193. [Google Scholar] [CrossRef]
- Enskog, D. Kinetische Theorie der Wärmeleitung, Reibung und Selbstdiffusion in gewissen verdichteten Gasen und Flüssigkeiten. Kungl. Svenska Vet.-Ak. Handl. 1922, 63, 3–44. [Google Scholar]
- Gapyak, I.; Gerasimenko, V. On Rigorous Derivation of the Enskog Kinetic Equation. Kinet. Relat. Models 2012, 5, 459–484. [Google Scholar]
- Lachowicz, M. On the Hydrodynamic Limit of the Enskog Equation. Publ. RIMS, Kioto Univ. 1998, 34, 191–210. [Google Scholar] [CrossRef]
- van Beijeren, H.; Ernst, M. The modified Enskog equation. Physica 1973, 68, 437–456. [Google Scholar] [CrossRef]
- Rezakhanlou, F. A Stochastic Model Associated with Enskog Equation and its Kinetic Limit. Commun. Math. Phys. 2003, 232, 327–375. [Google Scholar] [CrossRef]
- Miyazaki, K.; Srinivas, G.; Bagchi, B. The Enskog Theory for Self-Diffusion Coefficients of Simple Fluids with Continuous Potentials. Cond. Mat. Phys. 2001, 4, 315–323. [Google Scholar] [CrossRef]
- Clementi, E.; Raimondi, D.; Reinhardt, W. Atomic Screening Constants from SCF Functions. II. Atoms with 37 to 86 Electrons. J. Chem. Phys. 1967, 47, 1300–1307. [Google Scholar] [CrossRef]
- Gikhman, I.; Skorokhod, A. Introduction to the Theory of Random Processes; Courier Dover Publications: Mineola, NY, USA, 1969. [Google Scholar]
- Applebaum, D. Lévy Processes and Stochastic Calculus, 2nd ed.; Number 116 in Cambridge Studies in Advanced Mathematics; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- ⌀ksendal, B. Stochastic Differential Equations: An Introduction with Applications, 6th ed.; Universitext; Springer: Heidelberg, Germany, 2010. [Google Scholar]
- Risken, H. The Fokker-Planck Equation, 2nd ed.; Springer: New York, NY, USA, 1989. [Google Scholar]
- Daley, D.; Vere-Jones, D. An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods, 2nd ed.; Springer: New York, NY, USA, 2003. [Google Scholar]
- Papangelou, F. Integrability of Expected Increments of Point Processes and a Related Random Change of Scale. Trans. Am. Math. Soc. 1972, 165, 483–506. [Google Scholar] [CrossRef]
- Feller, W. An Introduction to Probability Theory and Its Applications, 2nd ed.; Wiley: New York, NY, USA, 1971; Volume 2. [Google Scholar]
- Courrège, P. Sur la forme intégro-différentielle des opérateurs de dans C satisfaisant au principe du maximum. Séminaire Brelot-Choquet-Deny. Théorie du potentiel 1965/66, 10, 1–38. [Google Scholar]
- Shannon, C. A Mathematical Theory of Communication. Bell Syst. Tech. J. 1948, 27, 379–423. [Google Scholar] [CrossRef] [Green Version]
- Abramov, R.; Majda, A. Statistically relevant conserved quantities for truncated quasi-geostrophic flow. Proc. Natl. Acad. Sci. USA 2003, 100, 3841–3846. [Google Scholar] [CrossRef] [PubMed]
- Abramov, R.; Majda, A.; Kleeman, R. Information Theory and Predictability for Low Frequency Variability. J. Atmos. Sci. 2005, 62, 65–87. [Google Scholar] [CrossRef]
- Haven, K.; Majda, A.; Abramov, R. Quantifying Predictability Through Information Theory: Small Sample Estimation in a non-Gaussian Framework. J. Comp. Phys. 2005, 206, 334–362. [Google Scholar] [CrossRef]
- Majda, A.; Abramov, R.; Grote, M. Information Theory and Stochastics for Multiscale Nonlinear Systems; CRM Monograph Series of Centre de Recherches Mathématiques, Université de Montréal; American Mathematical Society: Providence, PI, USA, 2005; Volume 25, ISBN 0-8218-3843-1. [Google Scholar]
- Tiefelsdorf, M. (Ed.) Modelling Spatial Processes; Lecture Notes in Earth Sciences; Springer: Berlin/Heidelberg, Germany, 2000; Volume 87, Chapter 5; pp. 59–66. [Google Scholar]
- Abramov, R. Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers. Physica A 2017, 484, 532–557. [Google Scholar] [CrossRef] [Green Version]
- Abramov, R.; Kovačič, G.; Majda, A. Hamiltonian Structure and Statistically Relevant Conserved Quantities for the Truncated Burgers-Hopf Equation. Commun. Pure Appl. Math. 2003, 56, 1–46. [Google Scholar] [CrossRef]
- Majda, A.; Kleeman, R.; Cai, D. A Framework for Predictability through Relative Entropy. Meth. Appl. Anal. 2002, 9, 425–444. [Google Scholar]
- Abramov, R.; Majda, A. Quantifying uncertainty for non-Gaussian ensembles in complex systems. SIAM J. Sci. Comp. 2003, 26, 411–447. [Google Scholar] [CrossRef]
- Batchelor, G. An Introduction to Fluid Dynamics; Cambridge University Press: New York, NY, USA, 2000. [Google Scholar]
- Levermore, C. Moment closure hierarchies for kinetic theories. J. Stat. Phys. 1996, 83, 1021–1065. [Google Scholar] [CrossRef]
- Grad, H. Note on N-Dimensional Hermite Polynomials. Comm. Pure Appl. Math. 1949, 2, 325–330. [Google Scholar] [CrossRef]
- Lachowicz, M. From Kinetic to Navier-Stokes-Type Equations. Appl. Math. Lett. 1997, 10, 19–23. [Google Scholar] [CrossRef]
© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Abramov, R.V. The Random Gas of Hard Spheres. J 2019, 2, 162-205. https://doi.org/10.3390/j2020014
Abramov RV. The Random Gas of Hard Spheres. J. 2019; 2(2):162-205. https://doi.org/10.3390/j2020014
Chicago/Turabian StyleAbramov, Rafail V. 2019. "The Random Gas of Hard Spheres" J 2, no. 2: 162-205. https://doi.org/10.3390/j2020014