Light impurity in an equilibrium gas

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 83, Issue 1, 2011, Pages

  D'Alessio, L ^a   Krapivsky, P L ^a,b  

^a Boston University   (United States)

^b Centre National de la Recherche Scientifique   (France)

Author keywords

[No Author keywords available]

Indexed keywords

DENSITY OF GASES; VELOCITY DISTRIBUTION;

ARBITRARY DIMENSION; ASYMPTOTIC GROWTH; ATOM INTERACTION; INTERACTION EXPONENTS; LIGHT IMPURITIES; PARTICLE DISPLACEMENT; THERMAL EQUILIBRIUMS; UNIVERSAL SCALING;

ATOMS;

EID: 78751516938     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.83.011107     Document Type: Article

References (40)

1. L. D'Alessio and P. L. Krapivsky, e-print arXiv: 1007.4589v1.
2. H. A. Lorentz, Proc. R. Acad. Sci. Amsterdam 7, 438 (1905);
3. H. A. Lorentz, Proc. R. Acad. Sci. Amsterdam 7, 585 (1905);
4. H. A. Lorentz, Proc. R. Acad. Sci. Amsterdam 7, 684 (1905).
5. E. H. Hauge, in Transport Phenomena, edited by, G. Kirczenow, and, J. Marro, (Springer-Verlag, Berlin, 1974), Lecture Notes in Physics, Vol. 31, p. 337.
6. P. Résibois and M. De Leener, Classical Kinetic Theory of Fluids (Wiley, New York, 1977).
7. J. R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, Cambridge, 1999).
8. D. Szasz, (ed.) Hard Ball Systems and the Lorentz Gas (Springer, Berlin, 2000).
9. P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010).
10. E. Fermi, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.75.1169 75, 1169 (1949).
11. S. M. Ulam, in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, 1961), Vol. 3, p. 315.
12. M. A. Lieberman and A. J. Lichtenberg, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.5.1852 5, 1852 (1972);
13. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer-Verlag, New York, 1991).
14. F. Bouchet, F. Cecconi, and A. Vulpiani, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.040601 92, 040601 (2004).
15. L. Boltzmann, Lectures on Gas Theory (University California Press, Berkeley, 1964).
16. J. C. Maxwell, Philos. Trans. R. Soc. London PRLTAO 0261-0523 10.1098/rstl.1867.0004 157, 49 (1867).
17. A few very special solutions have been found for so-called Maxwell molecules;
18. see M. H. Ernst, Phys. Rep. PRPLCM 0370-1573 10.1016/0370-1573(81)90002-8 78, 1 (1981);
19. C. Truesdell and R. G. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monoatomic Gas (Academic Press, New York, 1980).
20. P. Résibois, Physica A PHYADX 0378-4371 10.1016/0378-4371(78) 90114-0 90, 273 (1978).
21. A. Gervois and J. Piasecki, J. Stat. Phys. JSTPBS 0022-4715 10.1007/BF01010463 42, 1091 (1986).
22. J. Piasecki and R. Soto, Physica A PHYADX 0378-4371 10.1016/j.physa.2005. 12.045 369, 379 (2006).
23. A. Alastuey and J. Piasecki, J. Stat. Phys. JSTPBS 0022-4715 10.1007/s10955-010-9976-x 139, 991 (2010).
24. The particle velocity distribution satisfies the condition f (v, t) = f (- v, t) at all times if the initial velocity distribution is even, for example, f (v, 0) = δ (v). In the general case, the velocity distribution quickly becomes even.
25. C. Jarzynski and W. J. Šwia̧tecki, Nucl. Phys. A NUPABL 0375-9474 10.1016/0375-9474(93)90327-T 552, 1 (1993);
26. J. Błocki, F. Brut, and W. J. Šwia̧tecki, Nucl. Phys. A NUPABL 0375-9474 10.1016/0375-9474(93)90360-A 554, 107 (1993);
27. C. Jarzynski, Phys. Rev. E NUPABL 1539-3755 10.1103/PhysRevE.48.4340 48, 4340 (1993).
28. Apart from the hard-sphere gas, λ = ∞, the integration measure is explicitly known for the Coulomb gas λ = 1 (the Rutherford formula) and the Calogero gas λ = 2; in these three cases, one can compute the factor A.
29. - 1) in the small separation limit, and the potential actually describes the particle-atom interaction; the atom-atom interaction is irrelevant. In simulations, however, we usually assume that the particle-atom and atom-atom interactions are equal.
30. In the spatially homogeneous case the reflection symmetry is strictly obeyed [19] if it holds for the initial condition. In the spatially inhomogeneous case it is not so, yet the asymmetry is weak; for example, the average velocity ∫ - ∞ ∞ dv vf (x, v, t) remains bounded, while the average speed ∫ - ∞ ∞ dv v f (x, v, t) grows as τ. Physically, in the large time limit when the typical velocity is large, the particle undergoes a great number of collisions so its velocity can be ± v Physical Review C with almost the same probabilities.
31. This expression reproduces correctly the known values of the diffusion coefficients for the Lorentz gas in d = 1, 2, 3 and we believe it holds in any d. In one dimension we recover D = v 2 ρ [7], while in three dimensions Eq. (60) reduces to D = v / (3 π a 2 ρ); see [4,6,7]. The three-dimensional formula is well known. (Usually it is written in the form D = v ℓ / 3, with ℓ being the mean-free path.) The expression (see, e.g., Ref. [6]) for the diffusion coefficient in two dimensions, D = 3 v / (16 a ρ), is less known since the hard-sphere scattering is nonisotropic in two dimensions, so in that situation the Lorentz gas is harder to analyze than in three dimensions.
32. We used The On-Line Encyclopedia of Integer Sequences, [http://www.research.att.com/njas/sequences/].
33. Equation (85b) for R 4 shows that this moment is integer when d = 1, 4, 7, 10, 13,... (generally if d = 1 + 3 p with an arbitrary non-negative integer p) and noninteger in all other dimensions. Equations (85b) and (85c) show that the moments R 4 and R 6 are both integer when d = 1, 13, 16, 28, 31,... (generally when d = 1 + 15 q or d = 13 + 15 q with an arbitrary non-negative integer q). Thus, in some special dimensions a few first moments R 4, R 6, etc., can be integer. It appears that in those dimensions only a few first moments are integer; the only exception is d = 1, where all even moments are integer.
34. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science (Addison-Wesley, Reading, MA, 1989).
35. Since we are only interested in the first collision time, Δ t, this problem can be conveniently thought as a decay process. When the particle with velocity v collides with a background atom its velocity changes and the particle "decays." Each background atom provides a decay channel described as a Poisson process with rate proportional to the absolute value of the relative velocity. The first collision time (i.e., decay time) is distributed as a Poisson process with a rate equal to the sum of the rates of the different channels [Eqs. (135a)-(135b)]. Moreover, the probability of decaying in any channel is given by the ratio of the rate for that channel over the total rate [Eq. (136)].
36. W. C. Swope, H. C. Andersen, P. H. Berens, and K. R. Wilson, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.442716 76, 637 (1982).
37. D. Cohen and D. A. Wisniacki, Phys. Rev. E JCPSA6 1539-3755 10.1103/PhysRevE.67.026206 67, 026206 (2003).
38. B. Vacchini and K. Hornberger, Phys. Rep. PRPLCM 0370-1573 10.1016/j.physrep.2009.06.001 478, 71 (2009).
39. 0 √ T in the initial distributions (C5) and (C8) to ensure the validity of Eq. (26) during the entire time range τ > 0. The same inequality is required for the applicability of Eq. (32) when the initial condition is given by Eq. (C10).
40. Exact solutions (C11) easily follow from the scaling solution [Eqs. (34) and (35)] after noting that the governing kinetic equation (32) possesses the time-translational invariance.