Hard-Sphere Brownian Motion in Ideal Gas : Inter-Particle Correlations, Boltzmann-Grad Limit, and Destroying the Myth of Molecular Chaos Propagation Yuriy E. Kuzovlev Donetsk A.A.Galkin Institute for Physics and Technology of NASU, ul. R. Luxemburg 72, Donetsk 83114, Ukraine ∗ (Dated: 10 January 2010) The BBGKY hierarchy of equations for a particle interacting with ideal gas is analyzed in terms of irreducible many-particle correlations between gas atoms and the particle’s motion. The transition to the hard-sphere interaction is formulated from viewpoint of the recently discovered exact relations connecting the correlations with the particle’s probability distribution. Then the Boltzmann-Grad limit is considered and shown not to lead to the Bolzmann hierarchy and the molecular chaos, since correlations of all orders keep significant.in this limit, merely taking a singular form. PACS numbers: 05.20.Jj, 05.40.Fb In the work [1], as well as in its arXive preprints [2] and premising works [3, 4], based on principles and methods of rigorous statistical mechanics, - first of all, on the Bo- golyubov approach to it [5], - I proved existence of general exact relations (“virial relations”) connecting (i) proba- bility law of random walk of a test “Brownian” particle (BP) in a fluid and (ii) irreducible many-particle statis- tical correlations between molecules of the fluid and the BP’s walk (in particular, BP may be merely a marked molecule). I emphasized also that these relations make it clearly visible that all n –particle correlations always are quantitatively significant, even under the low-density (“Boltzmann-Grad”) gas limit. This fact, in turn, implies that the Boltzmannian kinetics is incorrect even in this limit, and true kinetics of fluids, - including low-density gases! - should take into account all the correlations. The meaning of so universal correlations was explained already in [6] (this article is hardly available on-line, but one can read its my own author’s translation [7] from Russian original, or see preprint [8]). It is sufficient to notice that a fluid as the whole (and first of all dilute gas!) is indifferent to a number of past collisions happened to its given particular molecule. Therefore any molecule has no definite ( a priori predictable) “time-average” rate of collisions [9, 10]. In other words, actual rate of its colli- sions undergoes fluctuations which are indifferent to time averaging. Thus that are scaleless fluctuations with 1/f type spectrum [11]. The mentioned statistical correla- tions directly reflect complicity of particles (via mutual collisions) in these fluctuations and, hence, indirectly de- scribe their statistics (for detail see [6] or [7] or [8]). As far as I know, first such statements about molec- ular motion were put forward in works by G.Bochkov and me [12–17] as conjectures about origin and prop- erties of 1/f noise accompanying charge transport (i.e. We will begin in Sec.2 by formulation of the BBGKY hierarchy for our particular problem and corresponding (above mentioned) exact virial relations [1, 2, 21, 22]. Another our purpose is to perceive falsity of popu- lar treatments of the “hard-sphere BBGKY hierarchy of equations” [23–32] trying to reduce it, in the Boltzmann- Grad limit, to so-called “Boltzmann hierarchy”. With this purpose it is natural to concentrate on the Brownian motion of (molecular-size) particle in ideal gas (see [21] or some of preprints [22] and also [1, 2]), which is most simple “kinetic process” since it produces least amount of inter-particle correlations. Besides, this is good motive to scan the limit transition from smooth in- teraction (between BP and gas atoms) to singular “hard- sphere” one dividing into momentary “collisions”. The aforesaid was confirmed by exact results of [1–3]. Nevertheless, a specially visual analysis of inter-particle correlations may be useful. This is just one of purposes of the present paper. The latter circumstance requires, in view of the well known Marcinkiewicz theorem (see e.g. [18]), to deal with whole infinite chain of cumulants of the fluctuations. On the other hand, according to the later works made at the microscopic level, firstly [6] and then its development in [19], n -order cumulant of fluctuations in the rate of collisions associates with specifically ( n +1)-particle cor- relations (irreducible component of ( n + 1)-particle dis- tribution function). That is why one should not truncate the BBGKY hierarchy of equations! Anyway, neglect of three-particle (and thus higher) correlations means rejec- tion of the fluctuations at all (like neglect of two-particle correlations rejects any collisions at all [20]). Brownian motion of charge carriers) and other transport processes (in generalized sense [8]). We demonstrated once again that correct ideology leads to useful results even at phenomenological level. In particular, in [14, 15] it was shown that fluctuations of rate of collisions (or, equivalently, 1/f fluctuations of diffusivity and mobility) possess essentially non-Gaussian statistics gravitating to- wards power-law probability tails. I. INTRODUCTION ∗ kuzovlev@fti.dn.ua arXiv:1001.1558v1 [cond-mat.stat-mech] 10 Jan 2010