Hard-sphere Brownian motion in ideal gas: inter-particle correlations, Boltzmann-Grad limit, and destroying the myth of molecular chaos propagation

0.17 MB·English

by  Yuriy E. Kuzovlev

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)
0
TheBBGKYhierarchyofequationsforaparticleinteractingwithidealgasisanalyzedintermsof
1
irreducible many-particlecorrelations between gas atoms and theparticle’s motion. The transition
0
tothehard-sphereinteractionisformulatedfromviewpointoftherecentlydiscoveredexactrelations
2
connecting the correlations with the particle’s probability distribution. Then the Boltzmann-Grad
n limitisconsideredandshownnottoleadtotheBolzmannhierarchyandthemolecularchaos,since
a
correlations of all orders keep significant.in this limit, merely taking a singular form.
J

0 PACSnumbers: 05.20.Jj,05.40.Fb
1

] I. INTRODUCTION erties of 1/f noise accompanying charge transport (i.e.
h
Brownianmotionof chargecarriers)and othertransport
c
e Inthework[1],aswellasinitsarXivepreprints[2]and processes (in generalized sense [8]). We demonstrated
m premising works [3, 4], based on principles and methods once again that correct ideology leads to useful results
- of rigorousstatisticalmechanics, - first of all, on the Bo- even at phenomenologicallevel. In particular, in [14, 15]
at golyubovapproachtoit[5],-Iprovedexistenceofgeneral it was shown that fluctuations of rate of collisions (or,
t exact relations (“virial relations”) connecting (i) proba- equivalently, 1/f fluctuations of diffusivity and mobility)
s
. bility law of random walk of a test “Brownian” particle possessessentiallynon-Gaussianstatisticsgravitatingto-
t
a (BP) in a fluid and (ii) irreducible many-particle statis- wards power-law probability tails.
m tical correlations between molecules of the fluid and the The latter circumstance requires, in view of the well
- BP’s walk (in particular, BP may be merely a marked known Marcinkiewicz theorem (see e.g. [18]), to deal
d molecule). I emphasized also that these relations make withwholeinfinitechainofcumulantsofthefluctuations.
n
it clearly visible that all n–particle correlations always Onthe otherhand,accordingtothe laterworksmadeat
o
c arequantitativelysignificant,evenunderthelow-density themicroscopiclevel,firstly[6]andthenitsdevelopment
[ (“Boltzmann-Grad”)gaslimit. Thisfact,inturn,implies in [19], n-order cumulant of fluctuations in the rate of
that the Boltzmannian kinetics is incorrect even in this collisionsassociateswithspecifically (n+1)-particlecor-
1
limit, and true kinetics of fluids, - including low-density relations (irreducible component of (n+1)-particle dis-
v
8 gases! - should take into account all the correlations. tributionfunction). Thatiswhyoneshouldnottruncate
5 Themeaningofsouniversalcorrelationswasexplained the BBGKY hierarchy of equations! Anyway, neglect of
5 alreadyin[6] (this articleishardlyavailableon-line,but three-particle(andthushigher)correlationsmeansrejec-
1 one can read its my own author’s translation [7] from tionofthe fluctuationsatall(like neglectoftwo-particle
1. Russian original, or see preprint [8]). It is sufficient to correlations rejects any collisions at all [20]).
0 noticethatafluidasthewhole(andfirstofalldilutegas!) The aforesaid was confirmed by exact results of [1–3].
0 is indifferent to a number of past collisions happened to Nevertheless, a specially visual analysis of inter-particle
1 itsgivenparticularmolecule. Thereforeanymoleculehas correlations may be useful. This is just one of purposes
v: no definite (a priori predictable) “time-average”rate of of the present paper.
i collisions [9, 10]. In other words, actual rate of its colli- With this purpose it is natural to concentrate on the
X
sionsundergoesfluctuationswhichareindifferenttotime Brownianmotionof(molecular-size)particleinidealgas
r averaging. Thus that are scaleless fluctuations with 1/f (see [21] or some of preprints [22] and also [1, 2]), which
a
type spectrum [11]. The mentioned statistical correla- is most simple “kinetic process” since it produces least
tions directly reflect complicity of particles (via mutual amount of inter-particle correlations. Besides, this is
collisions)inthesefluctuationsand,hence,indirectlyde- goodmotiveto scanthe limittransitionfromsmoothin-
scribe their statistics (for detail see [6] or [7] or [8]). teraction(betweenBPandgasatoms)tosingular“hard-
As far as I know, first such statements about molec- sphere” one dividing into momentary “collisions”.
ular motion were put forward in works by G.Bochkov Another our purpose is to perceive falsity of popu-
and me [12–17] as conjectures about origin and prop- lar treatments of the “hard-sphere BBGKY hierarchy of
equations”[23–32]tryingtoreduceit,intheBoltzmann-
Grad limit, to so-called “Boltzmann hierarchy”.
We will begin in Sec.2 by formulation of the BBGKY
∗[email protected] hierarchy for our particular problem and corresponding
2
(above mentioned) exact virial relations [1, 2, 21, 22]. the thermodynamical limit presumes the “cluster prop-
After their consideration in Sec.3 we will see for our- erty” of DFs. that is vanishing of inter-particle correla-
selvesthatcollisionsconstantlygivebirthto varioussta- tions under large spatial separation of particles, so that
tistical correlations,and not only post-collisionones but Fk → Fk−1Gm(ps) at |rs−R|→∞ ,where 1≤s≤k
also pre-collision correlations. Then, in Sec.4, discuss and Fk−1 doesnotinclude rs and ps. Thatarebound-
andexecutethehard-spherelimitofourBBGKYhierar- ary conditions to Eq.1.
chy, taking in mind that it should stay compatible with In view of these conditions, in order to visually ex-
the virial relations as they are independent on charac- tract inter-particle correlations, it is convenient to make
ter of interactions (interaction potentials). The result the linear change of variables, from the DFs F to new
k
is (i) modified BBGKY equations combined with (ii) fa- functions C , [34] as follow:
k
miliarboundaryconditionstothemathyper-surfaces(in
many-particle phase spaces) corresponding to collisions. F0(t,R,P;n) = C0(t,R,P;n) ,
The second ingredient allows to transform the first into
usual “hard-sphereBBGKYhierarchy”but, at the same F1(t,R,r1,P,p1;n) =
time, it forbids its reduction to the “Boltzmann hierar- = C0(t,R,P;n)f(r1−R,p1)+
chy”[24,25,28,29,31]undertheBoltzmann-Gradlimit. +C1(t,R,r1,P,p1;n) ,
Thus, the correct theory does not present such marvel- (3)
lous simplificationsas “propagationofchaos”andclosed F (t,R,r(2),P,p(2);n) =
2
equation for one-particle distribution function (DF): as = C0(t,R,P;n)f(r1−R,p1)f(r2−R,p2)+
before,onehastosolveaninfinitehierarchyofequations! +C (t,R,r ,P,p ;n)f(r −R,p )+
1 1 1 2 2
This situation will be discussed in Sec.5. +C (t,R,r ,P,p ;n)f(r −R,p )+
1 2 2 1 1
+C (t,R,r(2),P,p(2);n) ,
2
II. BBGKY HIERARCHY, ITS CUMULANT and so on. with f(r,p)=E(r)G (p).
m
REPRESENTATION, AND VIRIAL RELATIONS Clearly,suchdefined C canbenamedcumulantfunc-
k
tions (CF) since represent irreducible correlations be-
We want to consider random walk R(t) of a Brown- tween k gas atoms and total BP’s path R. In their
ianparticle(BP)inthermodynamicallyequilibriumideal termsthe BBGKYhierarchy(1)takesthe form[1,2,22]
gas, assuming that at initial time moment t = 0 BP
∂C ∂
starts from certainly known position R(0) = 0. The k =[H ,C ]+n Φ′(R−r )C +
∂t k k ∂PZ k+1 k+1
BBGKY equations for this problem can be either de- k+1
rived [22] directly from the Liouville equation, following k P ∂
′
Bogolyubov [5], or extracted [21] from general results of +T Gm(pj)E (ρj)(cid:20)MT + ∂P(cid:21) PkjCk−1 , (4)
[1–3]. They reads as jX=1
∂∂Ftk =[Hk,Fk]+n∂∂PZ Φ′(R−rk+1)Fk+1 , (1) wrehpelarecemEe′(nrt)o=f p∇aEir(orf),arρgkum≡enrtks−xRj =, a{nrdj,pPjk}j (mifeiatniss
k+1 present) by x = {r ,p }. The mentioned initial and
k k k
where k =0,1,..., H is Hamiltonian of subsystem “k boundary conditions simplify to
k
atoms+BP”, [...,...] meansthePoissonbrackets, Φ(r)
C (0,R,P; n) =δ(R)G (P) ,
ispotentialof(short-rangerepulsive)interactionbetween 0 M
BPandatoms, Φ′(r)=∇Φ(r), n isgasdensity(mean Ck(0,R,r(k),P,p(k);n)=0 , (5)
concentrationofatoms), F =F (t,R,P;n) isnormal- Ck(t,R,r(k),P,p(k);n)→0 at rj −R→∞
0 0
ized DF of the BP’s coordinate R and momentum P,
F = F (t,R,r(k),P,p(k);n) are (k+1)-particle DFs (1 ≤ j ≤ k). A careful enough scanning of these equa-
k k
tions results in observation [22] that exact relations
forBPand k atoms[33], r and p denotecoordinates
j j
apn(kd) m=o{mpen..t.apof}a,taonmds, res..p.e=ctively..,. drr(k)d=p {.r1...rk}, ∂ Ck(t,R,r(k),P,p(k);n) = (6)
1 k k k k ∂n
Initial conditions to thRese equRatRions, corresponding to
the gas equilibrium, at temperature T , are = Ck+1(t,R,r(k+1),P,p(k+1);n) .
Z
k+1
F | =δ(R) exp(−H /T )=
k t=0 k take place. This is particular case of general “virial re-
(2)
=δ(R)GM(P) kj=1E(rj −R)Gm(pj) , lations” found in [1, 2, 21] as exact properties of solu-
Q tions to BBGKY equations describing molecular Brown-
where M and m are masses of BP and atoms, re- ian motion in fluids (they can be also deduced [3] from
spectively, Gm(p)=(2πTm)−3/2exp(−p2/2Tm) isthe thegeneralizedfluctuation-dissipationrelations[35,36]).
Maxwell momentum distribution of a particle with mass Theserelationsdemonstrateexistenceandsignificance
m, and E(r) = exp[−Φ(r)/T]. Besides, existence of of all many-particle correlations. To see straight away
that they keep significant in the Boltzmann-Grad limit
3
(BGL), let us introduce characteristic interaction radius
r ofthe potential Φ(ρ), correspondingfree-pathlength
0
of BP, λ=(πr2n)−1, and integrated correlations out Cin
0 C
s 1
e 1
W (t,R;λ) ≡ nk ... C (t,R,r(k),P,p(k);n) dP at
k Z (cid:20)Z Z k (cid:21) n
1 k di
r
Thus, W (t,R;λ) = F (t,R,P;1/πr2λ)dP is proba- o C
0 0 0 o
bility distribution of tRhe BP’s path. Then Eqs.6 yield C 2
k
1 ∂
W (t,R;λ)= W (t,R;λ) (7)
k λk(cid:18)∂λ−1(cid:19) 0
These exact relations hold at any values of r and n,
0
hence, under BGL (r → 0, n → ∞, λ =const)
0 Time
too. They show that anyway by order of magnitude
W (t,R;λ)∼W (t,R;λ).
k 0
FIG.1. Asimplest diagram transforming post-collision pair
correlation, C1out, into pre-collision pair correlation, C1in,
III. PRE-COLLISION CORRELATIONS AND through three-particle correlation, C2. The “Brownian par-
FAILURE OF MOLECULAR CHAOS ticle” is represented by circles and entire arrows while gas
atoms by dots, light lines and short arrows.
Is the Boltzmann’s molecular chaos (“Stoßahlansatz”)
compatible with virial relations (6), (7)? May be, cor-
“Boltzmann-Lorentz”equation for the BP’s distribution
relations there are purely post-collision and therefore do
C =F , thus realizing the dream of “molecular chaos”.
not contradict Boltzmann s ansatz proclaiming absence 0 0
However, the last term in Eq.10, for C , quite simi-
of pre-collisioncorrelations? 2
larly (even under neglect of C ) produces three-particle
Unfortunately, this is vain hope, and the answer is no. 3
correlationsoutofthe two-particleone, Cout. Resulting
Tomakesureofthis,letusagreethatpost-collision(out- 1
C contributes,viaintegralinEq.9,backto evolutionof
)andpre-collision(in-)statesofBPandanatomsatisfy 2
C , and produces, in particular, pre-collision pair cor-
(ρ·u)>0 or (ρ·u)<0,respectively,-where ρ=r−R, 1
relations between particles going to meet one another.
u = v − V, with V = P/M and v = p /m being
j j In such way, C acquires pre-collision component Cin,
velocities, - and consider the first three of equations (4), 1 1
thusturningthedreamintoaroughapproximation. This
∂C ∂C ∂ is illustrated schematically by Fig.1.
0 =−V· 0 −n Φ′(ρ )C , (8)
∂t ∂R ∂PZ 1 1
1
∂C ∂C ∂
1 =−V· 1 +L C −n Φ′(ρ )C + IV. HARD-SPHERE LIMIT
∂t ∂R 1 1 ∂PZ 2 2
3
′ P ∂ Forfurther,firstnoticethatEqs.4,8-10canbewritten
+T G (p )E (ρ ) + C , (9)
m 1 1 (cid:20)MT ∂P(cid:21) 0 in equivalent form
∂C ∂C ∂
∂t2 =−V· ∂R2 +(L1+L2)C2−n∂PZ3Φ′(ρ3)C3 + ∂∂Ctk =−V·∂∂CRk −n∂∂PZ Φ′(ρk+1)Ck+1 +
k+1
P ∂
′
+T(1+P )G (p )E (ρ ) + C (10) k
21 m 2 2 (cid:20)MT ∂P(cid:21) 1 + Lj [Ck+Gm(pj)E(ρj)PkjCk−1] (11)
Here, we introduced the Liouville operator jX=1
Since these equations, as well as virial relations (6) and
∂ ∂ ∂
L =(V−v )· +Φ′(ρ )· − (7),arevalidatanysmoothinteractionpotential,wecan
j j ∂ρ j (cid:18)∂p ∂P(cid:19)
j j extendthemtothelimitof“hard-sphere”interaction,for
instance, defined by Φ(r) = (r /|r|)p Φ with p → ∞
and made use of the relative atoms’ coordinates ρ . 0 0
j and Φ =const∼T . Atthat,clearly,theBP’sfreepath
Evidently, the last term in Eq.9 represents generation 0
lengthmusttendtoaconstant, λ→const. Thereforeat
ofpaircorrelationbyBP-atominteraction,andtheresult
any stage of this limit transition all CFs C are equally
is “post-collision correlation” since it passes to finishing k
smooth functions of time (except the very beginning of
“out-state”oftheparticles. Ifweneglectedthree-particle
evolution), momenta and coordinates, excluding “colli-
correlation represented by CF C then solution to Eq.9
2 sionregions” |ρ |<r +ǫ,where ǫ→0 alongwithchar-
wouldbe justthis post-collisioncorrelation: C =Cout. j 0
1 1 acteristicthicknesshofthepotentialwall δ =r /p→0
Substitution of this C to Eq.8 would yield a closed 0
1 (at that, satisfying ǫ/δ →∞).
4
In these collision regions, in opposite, C (k > 0) with ρ = ρ , p = p . v = v = p /m,
k k+1 k+1 k+1 k+1
become more and more sharp functions of ρ , so that and momenta P∗, p∗ being related to P, p in the
j
∂C /∂ρ ∼ δ−1C , similarly to Φ′(ρ ) ∼ δ−1Φ(ρ ) . same way as pre-collision momenta Pin, pin in (14)
k j k j j
Hence, the second and the fourth right-hand terms in are related to post-collision ones, Pout, pout, and the
Eq.(9),Eq.(10),etc.,bothareinfinitelyincreasing ∝δ−1 integration involves pre-collision states only.
in collision regions and, in view of absence of other such What is for the operators L0, in respect to CFs they
j
terms, should compensate one another. Equivalently, are defined as L0 = −(v −V)· ∂/∂ρ at |ρ | > r
j j j j 0
the expressions in square brackets in Eqs.11 should be- and by boundary conditions (13) at |ρ | = r . Thus,
j 0
come eigenfunctions of operators Lj corresponding to importantly, L0j are not mere translation operators: at
zero eigenvalues, that is collision surfaces |ρ | = r they act as creation oper-
j 0
ators, creating (k +1)-order correlations from k-order
Lj [Ck+Gm(pj)E(ρj)PkjCk−1] = 0 (12) ones. Therefore factually C remain coupled with both
k
asymptotically, inside the collision regions. Ck+1 and Ck−1, as in basic Eqs.4. In other words,
due to conditions (13), hierarchy (17) keeps character-
This statement. in its turn, means that expression in
thesquarebracketsrepresentsintegralofmotion,andits istic three-diagonal structure of equations for CFs!
values at beginning and at end of the pair collision,are Return from CFs back to DFs, according to the CFs
equal. Thus, visualizingonly variables whatchange dur- definition (3), transforms these equations to
ing collision, we can write
k
∂F ∂F ∂F
Ck(Pin,pjin) +Gm(pjin)PkjCk−1(Pin) = (13) ∂tk =−V· ∂Rk −jX=1 uj ·∂ρjk − nΓFk+1 (18)
= Ck(Pout,pjout) +Gm(pjout)PkjCk−1(Pout) , b
and conditions (13) to
where ρ =r Ω, with Ω being unit normal vector, and
j 0
in-states and out-states correspond to Ω·uj < 0 and Fk(Pin,pjin) = Fk(Pout,pjout) at |ρj|=r0 (19)
Ω·u >0, respectively. and are connected via the limit
j
along with (14). Their substitution into the “collision
collision dynamics (mirror reflection):
integral’ ΓF gives, similarly to (17),
k+1
Pin+pin = Pout+pout ,
j j b k
Ω·(vjout−Vout) = −Ω·(vjin−Vin) , (14) ∂∂Ftk =−V·∂∂FRk + L0jFk +
(1−Ω⊗Ω)(vout−Vout) = (1−Ω⊗Ω)(vin−Vin) jX=1
j j
1
Ontheotherhand,integrationof(12)overthecollision + dp dΩ ((V−v)·Ω) × (20)
region, - for instance, at j =k, - yields πλZ Z(V−v)·Ω>0
∗ ∗
× [F (ρ=r Ω,P ,p )−F (ρ=r Ω,P,p)]
k+1 0 k+1 0
∂
′
− ∂PZkΦ (ρk)Ck = −ΓCk ≡ (15) Hereagain L0j =−(vj−V)· ∂/∂ρj at |ρj|>r0 butat
b |ρj|=r0 action of the operator L0j onto DFs is defined
≡ r02Z Z ((vk−V)·Ω) Ck(ρk =r0Ω) dpkdΩ by boundary conditions (19).
Substituting this equality intoEqs.11, forcomplement
of the now forbidden regions |ρ |≤r we obtain V. THE BOLTZMANN-GRAD LIMIT AND
j 0
“MATHEMATICAL NON-PHYSICS”
k
∂C ∂C ∂C
k =−V· k − u · k − nΓC , (16)
∂t ∂R j ∂ρ k+1 Equations(16) as combinedwith boundary conditions
jX=1 j b (13)-(14) at collision surfaces |ρj| = r0 (and conditions
where u = v − V. At boundaries of the forbidden (5) at infinity) or, equivalently, equations (18) combined
j j
regions. i.e. at |ρ |=r , these equationsshouldbe sup- with (19) and (14) represent direct analogue of so-called
j 0
plementedwith boundaryconditions (13)-(14). Combin- “hard-sphere BBGKY hierarchy” (see e.g. Eq.2.2 from
ing Eqs.16 with (15) and (13) we come to [28] or Eq.4.1 from [29]).
Importantly, the term “hard-sphere BBGKY hierar-
∂C ∂C k chy” is adequate on the understanding only that when
∂tk =−V· ∂Rk + L0jCk + theboundaryconditionsaresubstitutedinto Fk+1 inside
jX=1 the “collision integral” in equation for F then simul-
k
1 taneously and necessarily they are satisfiedby F
+ dp dΩ ((V−v)·Ω) × (17) k+1
πλZ Z(V−v)·Ω>0 in the next equation for Fk+1 itself (i.e. included into
× [G (p∗)C (P∗)− G (p)C (P) + definition of the operators L0j as above). Otherwise one
m k m k makes some “mathematicalnon-physics”,since resulting
∗ ∗
+Ck+1(ρ=r0Ω,P ,p )−Ck+1(ρ=r0Ω,P,p)] equations will be non-derivable from Liouville equation
for an (infinitely) many hard sphere system!
5
In opposite, fulfilment of the mentioned requirement pre-collisioncorrelationsdoarise,at u ·ρ <0 andagain
j j
guarantees observance of the virial relations (see Ap- at |u ×ρ |/|u | < r , since statistical picture acquires
j j j 0
pendix) and consequently belonging of resulting equa- more and more time-reversalsymmetry.
tions to the class of BBGKY hierarchies (since virial re- At the same time, at |u ×ρ |/|u |>r there are al-
j j j 0
lations do follow alreadyfrommostgeneralpropertiesof most no reasons for correlations. Hence, in directions
many-particle dynamics and Liouville equations [1–3]). perpendicular to u all CFs become more and more
j
In view of these facts, it seems impossible to accept sharp functions of ρ . At that. the integral value of
j
theoldideathatundertheBoltzmann-Gradlimit(BGL) correlation per one atom goes to zero ∝ πr2λ = 1/n.
0
the “hard-sphereBBGKYhierarchy”is equivalentto so- Butthishasnosignificancesincecorrelationsconcentrate
called “hard-sphere Boltzmann hierarchy” (see e.g. [24, just where they are most effective. This is confirmed by
28, 29, 31]). For our system it looks as the virial relations (7). The latter show also that really
significant correlational characteristics are nkC which
k
k
∂F ∂F ∂F turn under BGL into singular generalized functions.
k =−V· k − (v −V)· k +
∂t ∂R j ∂ρ
jX=1 j
1 VI. CONCLUSION
+ dp dΩ ((V−v)·Ω) × (21)
πλZ Z(V−v)·Ω>0
×[F (ρ=0,P∗,p∗)−F (ρ=0,P,p)] , We haveseenthatthe Boltzmann-Gradlimitdoes not
k+1 k+1
lead to any essential simplifications (and in this sense it
wheretherearenoforbiddenregions,thatis ρj cantake does not exist). In particular, it does not lead to the
arbitrary values from the whole R3, and the boundary mythologic Botzmannian kinetics with Boltzmann equa-
conditions (19) at |ρj|→r0 →0 are thrown, that is op- tion, Boltzmann-Lorentz equation, etc.), except non-
erators L0 are replaced by trivial translation operators, interestingspatiallyhomogeneouscase[39]. Physicaland
j
−(vj −V)· ∂/∂ρj. statistical reasonsof this pleasant fact already were over
This means that in corresponding equations for CFs andover againexplained in [6–8] andearlier [12–17] and
any Ck is now connected to Ck+1 only, and not to later[1,3,4,19,22,40,41]. Thefactispleasantbecause
Ck−1. Thus, these equations form two-diagonal hier- it shows once again that real many-particle dynamical
archy qualitatively different from the original one! chaos does not degenerates into miserable stochastics.
As the result of such frivolity, these equations allow
for factored solution with “propagation of chaos”, when
F =F k G (p ), C =0 at k >0. and F =C Appendix: Virial relations for hard-sphere system
k 0 j=1 m j k 0 0
undergoeQs the “Boltzmann-Lorentz equation”.
Evidently, this “Boltzmann hierarchy” contraries to Integrationof (k+1)-thofEqs.16over p and ρ
k+1 k+1
theaboveemphasizedrequirement: itusestheboundary
at |ρ |>r yieldsfor C = C ,inviewof(5),
conditions to write “collision integrals” but neglects the k+1 0 k k+1 k+1
equations R
same conditions in higher equations what determine the e
integrands(asifthelattertookparticlesfroma“parallel k
∂C ∂C ∂C
world”)! As the consequence, the Boltzmann hierarchy k =−V· k − u · k − ΓC − nΓC
∂t ∂R j ∂ρ k+1 k+1
does not satisfy the virial relations. Its wrong is clear e e jX=1 ej
b b e
already from observation that it results when one first
damagesEqs.20byreplacing L0 with −(v −V)·∂/∂ρ At the same time, differentiation of Eqs.16 in respect to
j j j
and only after that goes to r0 =0 [37]. Therefore, it is the density produces for Ck =∂Ck/∂n equations
not surprisingthat the famous Lanford’s attempt [24] to
k
provethementionedideawasinfactunsuccessful[8,38]. ∂C ∂C ∂C
k =−V· k − u · k − ΓC − nΓC
Possibleformalcauseofthisnonsuccessisverysimple: ∂t ∂R j ∂ρ k+1 k+1
the smaller is r0 the less smooth functions of ρj are all jX=1 j b b
theCFs[3,4],sinceconditions(19)ensurecontinuityand
whichareidenticaltothepreviousones. Since,according
smoothness of the probability measures F along phase
k to(5),initialconditionsfor C and C alsoareidentical
trajectories only but not in perpendicular dimensions. k k
(all are zeros), we come to equalities C = C , that is
Indeed, let uj ·ρj > 0, that is BP and an atom scat- e k k
to the virial relations (6), ∂C /∂n = C .
ter one from another. If at that r0 ≪ |ρj| . λ and k k+1 ke+1
additionally |u ×ρ |/|u |<r (where × denotes vec- NoticethatthisconsiderationisvalidRregardlessofcon-
j j j 0
crete form of boundary conditions at |ρ | = r . There-
tor product), i.e. vectors u and ρ are nearly paral- j 0
j j
fore, virialrelations hold under replacement of the hard-
lel, then the particles form an out-state which (almost
sphere “mirror reflection” conditions (14) by some other
certainly) arose from their recent collision (especially if
rule. Correspondingly,equations(16)and(18)canbeex-
|ρ | ≪ λ).According to (13), post-collision correlations
j
ofanyorderbetweensuchparticlesinevitablytakeplace, tended to collision operators Γ with other “scattering
andbyorderofmagnitudealltheyareequalto C =F . laws” than (14).
0 0 b
With time, as was explained in Sec.3, also quite similar
6
[1] Yu.E. Kuzovlev,“Molecular random walk and a sym- ofcourse,statisticalcorrelationsneverinfluenceonsome
metry group of the Bogoliubov equation” Theoret- real processes, since reside in theory.But there (at arbi-
ical and Mathematical Physics 160 (3) 1300-1314 trary density!) they are the only representatives of any
(Sep. 2009) {DOI:10.1007/s11232-009-0117-0} [in Rus- orocesses including particles’ interactions and collisions.
sian: TeoreticheskayaiMatematicheskayaFizika 160(3) [21] Yu.E. Kuzovlev, “OnBrownian motion inidealgasand
517-533 (Sep. 2009)]. related principles”, arXiv: 0806.4157.
[2] Yu.E. Kuzovlev, arXiv: 0810.5383, 0908.0274. [22] Yu.E.Kuzovlev,arXiv: 0902.2855, 0908.3007, 0911.0651.
[3] Yu.E.Kuzovlev, arXiv: 0705.4580, 0710.3831, [23] H.Grad. “Principles of the kinetic theory of gases”. in
0802.0288, 0803.0301, 0804.2023. Handbuch der Physik. V.12 (ed. S.Flugge). Springer,
[4] Yu.E. Kuzovlev, Math.PhysicsArchive (http://www. Berlin, Heidelberg, 1958.
ma.utexas.edu): 07-144, 07-309, 08-209, 09-26. [24] O.E. Lanford, “Time Evolution of Large Classical Sys-
[5] N.N.Bogolyubov. Problems of dynamical theory in sta- tems”. In: “Dynamical Systems. Theory and Applica-
tistical physics. North-Holland, Amsterdam, 1962. tions” (ed. J. Moser),Lecture Notes in Physics 38, p.1-
[6] Yu.E. Kuzovlev:“The BBGKY equations, self-diffusion 111. Springer,Berlin, 1975.
and 1/f noise in a slightly nonideal gas”, Sov.Phys.- [25] C.Cercignani.T˙heory and application of the Boltzmann
JETP 67(12), 2469-2477 (1988). equation. Scottish Academic Press, 1975.
[7] Yu.E.Kuzovlev, arXiv: 0907.3475. [26] P.Resibois and M.deLeener. Classical kinetic theory of
[8] Yu.E.Kuzovlev, arXiv: cond-mat/9903350. fluids. Wiley, NewYork,1977.
[9] Similarly, in the human life, if somebody is so much [27] N.N.Bogolyubov.“Microscopic solutions of the
free that forgets his past expenses, then he can not plan Boltzmann-Enskog equation in kinetic theory for
“mean expenses perunit time”! elastic balls”, Theoretical and Mathematical Physics
24, No.2, 804-807 (1975).
[10] This is realization of possible statistical property of
[28] H.vanBeijeren, O.E.LanfordIII, J.L.Lebowitz, and
many-particledynamicalsystemsforeseenbyN.Krylovin
H.Spohn, J.Stat.Phys. 22, No.2, 237 (1980).
hisremarkablebook[N.S.Krylov. Worksonthefounda-
[29] H. Spohn, “Fluctuation theory of the Boltzmann equa-
tionsofstatisticalphysics. Princeton,1979]whereheun-
tion”, in“Non-equilibriumPhenomenaI:theBoltzmann
derlined that “relative frequencies of some phenomenon
equation”, Amsterdam, 1983.
along a given phase trajectory, generally speaking, in no
[30] V.I.Gerasimenko and D.Ya.Petrina, “Existence of the
wayareconnectedtohypotheticaprobabilities”, inspite
Boltzmann-Grad limit for an infinite system of hard
ofthemixingofphasetrajectories(moreover,inessence,
spheres”, Theoretical and Mathematical Physics 83,
just duetothe mixing).
No.1, 402-418 (1990).
[11] In the low-density gas limit these fluctuations become
[31] D.Ya.Petrina, “Mathematicalproblemsofstatisticalme-
evenmore(maximally)pronouncedsincegetridofdamp-
chanicsofasystemofelasticballs”, RussianMathemat-
ingbycollective(hydrodynamical)interactions. Asthese
ical Surveys 45, No.3, 153 (1990).
fluctuations have no specific reasons, one can say that
[32] H.Spohn, “OntheintegratedhormoftheBBGKYhier-
merely they manifest non-ergodicity of kinetic behavior
archy for hard spheres”, arXiv: math-ph/0605068.
of many-particle dynamical systems [1, 2, 6, 8].
[33] Theprobability densitiesoffindingBPat point R with
[12] Yu.E.Kuzovlev and G.N.Bochkov. “On origin and sta-
momentum P and simultaneously finding out some k
tistical characteristics of 1/f noise.” Preprint No.157,
NIRFI, Gorkii (Nijniy-Novgorod),Russia, 1982. atomsatpoints rj withmomenta pj [1,2,5,21,22].In
respecttoatomvariables,allthesedistributionfunctions
[13] Yu.E.Kuzovlev and G.N.Bochkov, Radiophysics and
QuantumElectronics 26, No.3, 228-233 (1983). can be definedexactly as in [5].
[14] G.N.Bochkov and Yu.E.Kuzovlev, “On probabilistic [34] In my previous papers thece functions were designated
characteristicsof1/fnoise.” RadiophysicsandQuantum as Vk in place of Ck.
Electronics 27, No.9, 811-816 (1984). [35] G.N.BochkovandYu.E.Kuzovlev, Sov.Phys.-JETP 45,
[15] G.N.Bochkov and Yu.E.Kuzovlev. “On the theory of 125 (1977); 49, 543 (1979); 52, No.12 (1980).
1/f noise.” Preprint No.195, NIRFI, Gorkii (Nijniy- [36] G.N.Bochkov and Yu.E.Kuzovlev, Physica A106 443,
Novgorod), Russia, 1985 (in Russian). 480 (1981).
[16] G.N.BochkovandYu.E.Kuzovlev, “Newaspectsin1/f [37] Apparently, just such damaged equations are taken in
noise studies”, Sov.Phys.-Uspekhi 26, 829-853 (1983). mind as “hard-sphere BBGKY hierarchy” by amateurs
[17] Yu.E.Kuzovlev and G.N.Bochkov, Pis’ma v ZhTF of “Boltzmann hierarchy”.
8, No.20, 1260 (1982) [translated into English in [38] Maybe,morecarefulandfineconsiderationslikee.g.[32]
Sov.Tech.Phys.Lett. (1982)]. will makethis comprehensible to all.
[18] E.Lucasc. Characteristic functions. Ch.Griffin & Co., [39] M.Kac˙ Probability and related topics in physical sci-
London,1960. ences. Intersci. Publ., London,New York,1957.
[19] Yu.E.Kuzovlev, arXiv: cond-mat/0609515, 0612325. [40] Yu.E. Kuzovlev, “Relaxation and 1/f-noise in phonon
[20] Unfortunately,sotrivialthingsarebeyondfieldofvision systems”, JETP, 84(6), 1138 (1997).
of reviewers who easy write such nonsenses as: “statis- [41] Yu.E. Kuzovlev, Yu.V. Medvedev, and A.M. Grishin,
tical correlations at low density have little influence on JETP Letters 72, No.11, 574 (2000).
the motion of particles” (how do you like it!). In fact.