aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Why the nature needs 1/f -noise
Yu. E. Kuzovlev
Donetsk Free Statistical Physics Laboratory ∗ Low-frequency 1/f-noise occurs at all levels of the nature organization and became an actual factorof nanotechnologies, but in essence it remains misunderstood by its investigators. Here, once againit is pointed out that such the state of affairs may be caused by uncritical application of probabilitytheory notions to physical random phenomena, first of all the notion of ”independence”. It is shownthat in the framework of statistical mechanics no medium could provide an inner wandering particlewith quite certain value of diffusivity and mobility, thereby producing flicker fluctuations of thesequantities. This is example of realization of universal 1/f-noise origin in many-particle systems:dependence of time progress of any particular relaxation or transport process on the whole system’sdetailed initial microstate.
PACS numbers: 05.20.Jj, 05.40.Fb
CONTENTS
I. Introduction 1A. Root of the question and popular hypothesis 1B. Idea of the answer and plan of doing 2II. Phenomenology of Brownian motion 2A. Formulation of problem 2B. Conditional averaging and continuityequation 3C. Conditional average velocity of Brownianparticle 3D. General form of probabilistic law of diffusionand uncertainty of its coefficient 4III. Microscopic approach 4A. Newton equation and Liouville equation 4B. Equation of friction of Brownian particle 5C. Paradox of Brownian motion: Gaussianstatistics for it is beyond strength of itsmechanics 6D. Thermodynamics of Brownian motion andstatistics of large deviations 7E. Uncertainty and flicker fluctuations ofdiffusivity 8IV. Myths and reality of random walks 9A. Gaussian probability law and two meanings ofindependence of random events 9B. Collisions, chaos and noise in system of hardballs 10C. Paradox of independence 10D. Uncertainty and 1/f -noise of relativefrequency of collisions and rate of diffusion 10E. Game of independences and problems ofstatistical mechanics 11V. Conclusion 11 ∗ [email protected] References 12
I. INTRODUCTIONA. Root of the question and popular hypothesis
Here, as for many years before, the question of 1/f-noise grows in urgency, by extending and deepening to-gether with physical experiments and new technologiesand concerning almost all in the world, from cosmic phe-nomena down to molecular biology and nano-electronics.Nevertheless, there are no modern reviews of the questionproportional to its volume and significance. Seemingly,this is so because investigators do not find an inspira-tional ideas and happy thoughts for that. Though, oneappropriate suggestion, - to be under consideration be-low, - was made already in [1–4], but it had not exciteda response. Somehow or other, today we see reports onmore and more inventive and fine measurements of 1/f-noise, - for example, in films of metals and alloys [5] oratomic layers of graphene [6], - but as before with no un-ambiguous indication in its origin. Citing [6], “... despitealmost a century of research, 1/f noise remains a con-troversial phenomenon and numerous debates continueabout its origin and mechanisms”.It can be added that “debates”, in the author’s expe-rience, not rarely take rather totalitarian forms. Maybe,partly by this reason, from the author’s viewpoint, thepresent situation in general very slightly differs from whatwas outlined in [1] and a little later in [7]. We would liketo compare it with situation in astronomy nearly threehundreds years ago before appearance of the celebratedI. Newton’s work [8].We venture such the comparison not for the sake of wit-ticism but in view of our intention to demonstrate in thepresent notes that just the Newton’s laws of mechanicsmay be the place where solution of the 1/f-noise problemis hidden. More precisely, 1/f-noise is permanent prop-erty of systems of many particles moving and interactingby these laws (in their classical or quantum formulationincluding fields). In order to recognize it, we have onlyto follow the Newton’s advice to avoid unnecessary hy-potheses (“Hypotheses non fingo” [8]).Well, what hypotheses are thought up by physicists inrespect to 1/f-noise? Let us decipher it by example ofelectric current noise in a conductor under fixed voltage.Presence of 1/f-noise there means that the current hasno certain value, in the sense that its averaging (smooth-ing) over time produces an unpredictable result, - that israndomly varying from one experiment to fnother, - witha diversity which practically does not decrease, or evenincreases, when the averaging duration grows (since re-lated narrowing of frequency band contributing to the di-versity is almost, or with excess, compensated by growthof the noise power spectral density inside that band). So,when asking oneself a question about origin of such thephenomenon, one first of all assumes that it is in somespecific fluctuation processes influencing the current, -through e.g. number of charge carriers or their mobil-ity, - while specificity of these processes is in extremelywide variety of their time scales (memory, or life, or re-laxation, or correlation times, etc.) [5, 6, 9]. Just this ismain hypothesis.
B. Idea of the answer and plan of doing
Really, this hypothesis is not necessary, since the me-chanics as it is in no way requires certainty of the cur-rent and, hence, some special reasons for its uncertainty.Indeed, no matter what a concrete mechanism of con-ductivity may be, if it is indifferent in respect to amountof charge early transported through the conductor fromone side of an outer electric circuit to another and thusto past value of time-smoothed current, then later thismechanism also will be indifferent in respect to them,and on the whole it will be unable to set conditions forcertainty of the current. Thus, it by itself serves as mech-anism of 1/f-noise.What is for the indifference, it is supported by theexperiment conditions in themselves which state thatfluctuations of (time-smoothed) current do not meet aback reaction of outer circuit instead passively swallow-ing them.In this reasoning, there is no collections of large char-acteristic times, instead a single time only is present,- usually small in practice, - which indicates ending ofmemory of conductance mechanism. If, for instance, it isless than several hours, then transfer of charge carriers, -with their collisions, scatterings, reflections, etc., - now,at present time interval, is passing indifferently to whatamount of charge was transported yesterday, even if anexperimental device was not switched off before going tobed. Correspondingly, at frequencies lower than inverseday one can find a 1/f-noise.Analogously, if somebody possesses unlimited possibil-ities of profits and expenses and does not keep count ofthem, then he himself could not know how much his ex-penditures may be on average over time, and it can be expected that they will be distributed in time like 1/f-noise.If, returning to the conductor, we short out it, thenthe current’s 1/f-noise disappears along with directedcurrent. But irregular charge displacements in oppo-site directions do continue, at that again indifferently totheir past amount, and thus to their time-average inten-sity. The latter therefore is not aimed at a certain value,which results in 1/f-fluctuations of intensity (power spec-tral density) of thermodynamically equilibrium “white”(thermal) current noise. They are connected to the 1/f-noise in non-equilibrium current-carrying conductor bymeans of the “generalized fluctuation-dissipation rela-tions” [1, 2, 10–12].If one measures equilibrium thermal noise of poten-tial difference between sides of opened conductor, - e.g.electric junction, - then 1/f-fluctuations of intensity ofthis noise can be found too. They say that sum of num-bers (per unit time) of random charge carrier transitionsfrom one side to another and backwards is not trackedand regulated by the system, in contrast to residial ofthat numbers [10]. At that, characteristic time constantof the system (equivalent RC-circuit) determines uppertime scale for fluctuations of the residial and lower one forfluctuations in the sum (while their upper time scale doesnot exist, since they do not change system’s macrostate).The aforesaid can be easy extended, - under non-principal substitutions of particular terms and meanings,- to other manifestations of 1/f-noise in the nature. A lotof various examples was exposed in [1, 2, 4, 10, 11, 13, 21].Our demonstration below will be realized in terms ofequilibrium “molecular Brownian motion” [3, 7, 10, 12,14–20, 22].In a maximally simple way we shall show that as-sumption that Brownian particle obeys a certain diffu-sivity (rate, or coefficient, of diffusion) is incompatiblewith exact equations of statistical mechanics, that is withmechanical dynamical background of Brownian motion.Thus, mechanics inevitably generates 1/f-, or “flicker”,fluctuations of diffusivity and mobility of the particle.Then we shall consider quantitative characteristics ofthis 1/f-noise and, finally, present its explanation inthe language of theory of deterministic chaos in many-particle systems.
II. PHENOMENOLOGY OF BROWNIANMOTIONA. Formulation of problem
Let us imagine a small “Brownian particle” in a three-dimensional statistically uniform isotropic and thermo-dynamically equilibrium medium. Very small particle ofdust or flower pollen, - whose motion in liquid for the firsttime was observed through microscope in [23, 24] and inthe beginning of next century theoretically analysed in[25–27], - are suitable objects. But it will be better totake in mind some “nano-particle” or even merely sepa-rate atom or molecule in liquid or gas [28]. In principle,we may speak even about free charge carrier or point de-fect in a solid, but confine ourselves by a particle whichquite definitely is subject to the classical variant of me-chanics.Let R ( t ) and V ( t ) = dR ( t ) /dt denote vectors ofcentre-of-mass coordinate and velocity of our Brownianparticle (BP) at given time instant, while R and V theirpossible values. We can think that initially at time t = 0BP was placed at definitely known space point. Wherenamely, is of no importance, because of thermodynam-ical equivalence of any BP’s positions. Therefore it isconvenient to choose the coordinate origin: R (0) = 0 .Then later instant current position of BP, R ( t ) , will becoinciding with vector of its total displacement, or path,during all previous observation time.Now, let us ask ourselves what is BP’s “diffusion law”,i.e. probability distribution of BP’s path. Density of thisdistribution will be designated by W ( t, R ) . It can berepresented by expression W ( t, R ) = h δ ( R − R ( t )) i , (1)where the Dirac delta-function figures, R ( t ) is thoughtas result of all the previous interaction vetween BP andthe medium, and the angle brackets designate averagingover the equilibrium (Gibbs [29]) statistical ensemble ofinitial states of the medium and initial values of BP’svelocity.Undoubtedly, a plot (relief) of W ( t, R ) as function of R looks like a “bell” extending and lowering with time.We are interested in what shapes of this bell may beformed in reality. B. Conditional averaging and continuity equation
In fact, (1) is mere identity, but its time differentiationimmediately brings us a food for thought. From it wehave ∂W ( t, R ) ∂t = −∇ · h V ( t ) δ ( R − R ( t )) i ( · will denote scalar product of vectors). By attractingmathematical tools of the probability theory [33], thisequality can be rewritten as ∂W ( t, R ) ∂t = −∇ · V ( t, R ) W ( t, R ) , (2)where V ( t, R ) = h V ( t ) i R is conditional average valueof BP’s instant velocity determined under condition thatits current position, and thus its previous path, is known(measured) to be equal to R ( t ) = R . Generally theoperation of conditional averaging h . . . i R is defined byformula h . . . i R ≡ h . . . δ ( R ( t ) − R ) i / h δ ( R ( t ) − R ) i . Obviously, (2) is “continuity equation” for the proba-bility density W ( t, R ) , and the “field of velocity of prob-ability flow”, V ( t, R ) , containes important informationabout solutions to this equation. Therefore, first of alllet us consider possible constrution of the vector-function V ( t, R ) . C. Conditional average velocity of Brownianparticle
We shall keep in mind that the duration t of our ob-servations of BP is much longer than characteristic re-laxation time τ of (fluctuations of) BP’s velocity.Then, firstly, apply heuristic reasonings as follow. Onone hand, by the condition R ( t ) = R , average valueof BP’s velocity in the past, at time of its preceedingobservation, appears equal to R/t . On the other hand,as far as BP makes a random walk and t ≫ τ , the samecondition R ( t ) = R tells us almost nothing about BP’svelocity in the future, so its average value in equal nexttime interval can be expected to be zero. Hence, since theaverage under question, V ( t, R ) , relates to the presenttime instant “in the middle between past and future”, itseems likely that it is equal to half-sum of the mentionedquantities: V ( t, R ) = R t . (3)We can confirm this conclusion in a more formally rig-orous way, basing on the main distinctive statistical prop-erty of Brownian motion [26]: h R ( t ) i = Z R W ( t, R ) dR = 6 Dt (4)at t ≫ τ , i.e. ensemble average of squared BP’s dis-placement grows proportionally to observation time. Itis sufficient to notice that the continuity equation implies ∂∂t Z R W dR = 2 Z R · V W dR and that this requirement is naturally satisfied togetherwith (4) (with taking account of parallelity V k R ) whenequality (3) is valid.By way, notice that BP’s diffusivity, or diffusion coeffi-cient, D and relaxation time τ always can be connectedvia relation D = V τ ≡ TM τ , where T is temperature of the medium, M is mass ofBP, and V = p T /M its chracteristic thermal velocity.
D. General form of probabilistic law of diffusionand uncertainty of its coefficient
After inserting function (3) into (2), one comes to par-tial differential equation2 t ∂W∂t = − W − R · ∇ W , (5)which clearly indicates scale-invariant character of its so-lutions. We are interested in isotropic (spherically sym-metric) solutions looking as W ( t, R ) = (2 Dt ) − / Ψ( R / Dt ) (6)with some dimensionless function Ψ( z ) of dimension-less argument z = R / Dt . In our context it,since representing probability density, anyway shouldbe non-negative and satisfying normalization condition R W dR = 1 in company with equality (4), which surelycan be done. Then (6) is most general law of diffusional random walk, when typical BP’s displacements are pro-portional to square root of observation time: R ( t ) ∝ t .In particular, taking Ψ( z ) = (2 π ) − / exp ( − z/
2) , oneobtains the commonly known Gaussian diffusion law, W = W D ( t, R ) ≡ (4 πDt ) − / exp ( − R / Dt ) . (7)The corresponding walk is much pleasant for users sincein rough enough, in comparison with τ , time scale itssuccessive increments are mutually statistically indepen-dent. Owing to this, the only parameter of such randomwalk, - its diffusion coefficient, or diffusivity, D - canbe unambiguously determined from observations of anyits particular realization, by means of long enough timeaveraging.However, similar observations and time averaging ofnon-Gaussian random walk, obeying some of general typedistributions (6), will produce every time different valuesof diffusivity [1–4, 7, 10]. Indeed, their coincidence, thatis convergence of all results of time averaging to one andthe same value, would be impossible without statisticalindependence of increments (at least mutually far time-distanced ones) which in turn would mean, in accordancewith respective limit theorem of the probability theory(the “law of large numbers”), that at t ≫ τ probabilitydistribution of total path tends to the Gaussian (“nor-mal”) one [31].This becomes quite obvious if distribution (6) is rep-resented by linear combination of Gaussian “bells”: W ( t, R ) = Z ∞ W ∆ ( t, R ) U (cid:18) ∆ D , ξ (cid:19) d ∆ D .
Such expansions naturally arise in the microscopic theory[12, 14–16]. Correspondingly, in place of Ψ( z ) in (6) wecan writeΨ( z, ξ ) = Z ∞ exp ( − z/ ζ )(2 πζ ) / U ( ζ, ξ ) dζ . (8) Function U ( ζ, ξ ) here plays role of probability distribu-tion of ζ = ∆ /D , i.e. random diffusivity of BP ∆ ex-pressed in units of its mean diffusivity D . The latteris formally defined by equality (4), while practically onemay try to determine it with the help of averaging overmany experiments or many copies of BP.The additional argument ξ in this expansion, - if intro-duced e.g. as ξ ≡ τ /t under convention Ψ( z,
0) = Ψ( z ) ,- allows to take into account violation of ideal scale in-variance of random walk at ξ = 0 . First of all, faron “tails” of diffusion law, where R & V t , that is z & /ξ . There rate of diffusion achieves values of rateof free flight, ∆ ∼ V t = D/ξ .Clearly, a correction of tails of diffusion law maystrongly influence its higher-order statistical momentsand cumulants, even in spite of ξ ≪ W ( t, R ) ’s bell in the main staysalmost unchanged. Accordingly, a change of the function V ( t, R ) , - required by equation (2) and condition (4), - isas small as ξ is, so that the expression (3) remains right.Notice that the very possibility of long-term viola-tion of scale invariance automatically presumes non-Gaussianity of diffusion law, since Gaussian statisticsmerely gives no place for it (since it would contradictthe condition (4)). Already this fact gives evidence thatGaussian law is not completely adequate reflection of re-ality, although in mind of scientists it is firmly associatedwith diffusion of physical particles. At the same time nei-ther general reasonings leading to (3) and (5) nor equa-tion (5) by itself in no way dictate the special Gaussianchoice. Therefore, it is desirable to discuss other possi-bilities and search for criteria of choice among them inthe framework of statistical mechanics. III. MICROSCOPIC APPROACHA. Newton equation and Liouville equation
Further, let us go from kinematics of Brownian mo-tion to its dynamics and directly consider BP’s interac-tion with medium using methods of statistical mechanics.With this purpose we can take for our system quite usualsimple Hamiltonian H = P M + Φ( R, Γ) + H th (Γ) , (9)where P = M V is BP’s momentum, Γ is full set of(canonical) variables of the medium, Φ( R, Γ) is energyof BP-medium interaction, and H th (Γ) is Hamiltonianof medium in itself (or, in other words, that of “thermo-stat”). If BP possesses internal degrees of freedom, thentheir variables will be thought included into the set Γ ,thus being formally treated as a constituent of medium.Let D = D ( t, R, P, Γ) designate density of full proba-bility distribution of microstates of our system. Its evolu-tion is described by the formally exact Liouville equation[29, 34]. Here we can display it partly, writing out itsterms only directly concerning BP: ∂ D ∂t = − V · ∇D − F ( R, Γ) · ∇ P D + . . . . (10)Here F ( R, Γ) = −∇ Φ( R, Γ) is force acting onto BP be-cause of its interaction with medium, and the dots sur-rogate terms with Γ derivatives.Considering probability distribution of displacement(coordinate) of BP, W ( t, R ) = Z Z D ( t, R, P, Γ) d Γ dP , from equation (10) after its integration over Γ and P one comes, of course, to the continuity equation (2). Thesame integration after multiplying (10) by V producesadditional equation ∂∂t V W = −∇ · V ◦ V W + M − F W . (11)Here and below the symbol ◦ denotes tensor product ofvectors, while the over-line means, as before, conditionalaverages under given R ( t ) = R . Namely, in the firstterm on the left V ◦ V ( t, R ) = h V ( t ) ◦ V ( t ) i R = RR V ◦ V D d Γ dPW and in second term there F ( t, R ) = h F ( R ( t ) , Γ( t )) i R = RR F ( R, Γ) D d Γ dPW . Equation (11) describes momentum exchange betweenBP and medium. In essence, - as it can be easy verified,- this is merely the Newton equation
M dV /dt = F afterits conditional averaging: h M dV ( t ) /dt − F ( R ( t ) , Γ( t )) i R = 0 . We shall transform it into relation between functions F ( t, R ) and W ( t, R ) which is able to help selection ofacceptable diffusion laws without more deepening intothe Liouville equation. B. Equation of friction of Brownian particle
Replacing derivative ∂W/∂t in the equation (11) withright-hand side of (2), after simple manipulations onecomes to equivalent exact equation dVdt + ∇ · V ◦ V WW = FM , (12)with “material derivative” of BP’s average velocity, dVdt = ∂V∂t + ( V · ∇ ) V , and the double over-line marking tensor (matrix) of con-ditional quadratic cumulants (second-order cumulants)of velocity: V ◦ V ≡ V ◦ V − V ◦ V .
Next, at first let us consider the latter object.Since we are speaking about thermodynamically equi-librium Brownian motion, we can state that the condi-tional cumulants’ matrix V ◦ V ( t, R ) at t ≫ τ coincideswith matrix of unconditional equilibrium quadratic sta-tistical moments of velocity, h V ( t ) ◦ V ( t ) i , that is re-duces to scalar number V = T /M regardless of R .Indeed, if t ≫ τ , then at any R the condition R ( t ) = R fixes BP’s position occurred after many random stepsand cycles of momentum and energy exchange betweenBP and medium under detail balance in this process.Therefore, the value (variance) of corresponding thermalrandomness of BP’s velocity is not affected by this condi-tion (otherwise, thermal kinetic energy of BP, on averageequal to M V · V / V ◦ V ( t, R ) in case of Gaussian random walk sub-ject to distribution (7), which yields V ◦ V ( t, R ) = V (1 − ξ/ → V (13)at ξ ≡ τ /t → dVdt = − R t = − ξ TM R Dt .
For the second term after insertion of (13) and (6) wehave (cid:18) − ξ (cid:19) TM d ln Ψ( z, ξ ) dz R Dt ∼ − TM R Dt with same shortened notation z = R / Dt as before.Right-hand expression here corresponds to the Gaussiandiffusion law, for which d ln Ψ /dz = − / z ≪ /ξ . It shows that the first term, being approximately2 t/τ times smaller than the second, is negligibly small inthe limit ξ → − (cid:18) TD (cid:20) − d ln Ψ( z, ξ ) dz (cid:21)(cid:19) R t = F . (14)It resembles an equation of viscous friction, with R/ t = V in the role of velocity of a body moving through fluidand the round brackets in the role friction coefficient.One more simplification can be obtained by neglecting,under mentioned limit, violation of scale invariance andtreating Ψ( z, ξ ) as a function of single argument Ψ( z ) .In the next paragraphs we firstly proceed just so.But before that let us once again glance at the van-ishing first left term of(12). If writing its contribution tothe mean force as M dVdt = − ∇ M V , one can say that this is force of reaction of the mediumto addition M V / C. Paradox of Brownian motion: Gaussianstatistics for it is beyond strength of its mechanics
For Gaussian diffusion law, the square bracket in the“friction equation” (14) turns to unit, and the equationbecomes linear: F ⇒ − TD R t = − R | R | √ z T √ Dt . (15)At that, the “friction coefficient” in front of R/ t = V connects to the diffusivity via relation similar to thewidely known “Einstein relation” [26, 28]. Such likeness,however, is not a plus but minus of equality (15).The matter is as follows. Friction force in the true Ein-stein relation represents medium resistance against di-rected motion of a particle. When this particle displacesby distance R , the corresponding force makes work ∼ | R · F | ∼ (cid:18) TD Rt (cid:19) · R ∼ zT , thus producing a heat (recall that z = R / Dt ). Thisquantity, - like the force itself, - in principle can be ar-bitrary large under proper initial value of the particle’skinetic energy.This is clear. But it is strange thing that equality (15)offers the same, also unbounded, characteristic values offorce and work. Such a picture categorically contradictsto sense. Really, - repeating the aforesaid, - in our case the forcewhat figures in (15) represents medium reaction to par-ticle’s displacement achieved along random trajectory ofthermal motion, when initial energy value knowingly isonly ∼ T . At that, the medium creates obstacles to in-ertial free flight of BP but in no way to its unrestrictedmoving off from beginning of its path. In opposite, themoving off proceeds due to medium’s own free will andat the expense of its own equilibrium fluctuations.Therefore in reality, in contrast to (15), the averageforce (14), as a function of passed path R , can not bearbitrary large, instead staying always and everywherebounded. This is required by such factual inherent prop-erty of Brownian motion as translational invariance, thatis indifference of the system in respect to irretrievable de-partures of BP anywhere. Moreover, on this ground it isreasonable to expect that at large | R | the returning forcevanishes at all.Thus, we have to conclude that the Gaussian law isinadequate to physical origin of Brownian motion.Inevitability of this conclusion catches eye when notic-ing that if equality (15) was true then it would meanthat medium returns BP to start of its path with forceproportional to the path, F ∝ − R , i.e. like ideal springwith potential energy zT / ∝ R . From physical pointof view this looks absurdly, since any far BP’s going awayis permitted just because it does not change thermody-namical state of the system.Our conclusion can be denominated as paradoxical, ifrecollecting that Gaussian diffusion law many times is-sued from pen of theoreticians in various physical con-texts and occupies important place in idealized worldof “mathematical physics”. But the paradox resolves invery simple way: the Gaussian statistics always had ap-peared as consequence of clear or implicit hypotheses (orpostulates) about “independences” of random events orvalues. What is for us, we have managed without suchhypotheses and thus showed their fallacy in applicationto Brownian motion.In the past, we too were not connected with themand arrived to the same paradoxical conclusion, in theframework of both phenomenological statistical analy-sis of diffusion and transport processes [1–4, 10] andanalysis based on the full hierarchy of Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) equations [7, 10, 14,16, 17, 22], as well as on the base of exact “generalizedfluctuation-dissipation relations” (FDR) or “dynamicalvirial relations” [12, 15, 16, 20], and by other meth-ods [10, 11, 21], including that for quantum systems[13, 19, 21].In Section IV we shall again touch on “paradox of inde-pendence”. And now, in next paragraph, consider exam-ples of physically correct diffusion law as an alternativeof Gaussian one. D. Thermodynamics of Brownian motion andstatistics of large deviations
From the left expression in (14) it is clear that theboundedness of the force F in general implies relation | F ( t, R ) | ≤ F max ( t ) ∼ T √ Dt , (16)whose right-hand part can be easy guessed for reasonsof dimensionality. Of course, the symbol ∼ here hidessome dimensionless coefficient which reflects particulardistinctions of the system and construction of the func-tion − ln Ψ( z ) .Comparison between (16) and (15) shows that in theregion of “tails” of diffusion law, at z & ∼ √ z times, thus thoroughly falsely describing (stronglyunderestimating) probabilities of large displacements ofBP with z ≫ − ln Ψ( z ) growsnot faster than ∝ √ z , so that ( − ln Ψ( z )) / √ z < ∞ , andconsequently decrease of W ( t, R ) at large | R | → ∞ isalways sub-exponential (anyway, not faster than simpleexponential, not speaking about “Gaussian”).The difference of reality from “Gaussian ideal” be-comes aggravated when not the force itself only isbounded but also value of characteristic energy (work)conjugated with this force: A ( z ) ≡ | R || F | / ≤ A max = A ( ∞ ) ∼ T (17)(with the same remark about ∼ ). It is natural expecta-tion in view of that at result of any walk (any path R )the medium takes from BP not more energy than BP hadbeen able to take from medium before.Boundedness of A ( z ) implies that of the force, more-over, implies that under increase of | R | the force passesthrough a maximum and then decreases down to zero,approximately as | F | ≈ A max / | R | ∝ T / | R | . Thisasymptotic again is prompted already by dimensionalityof quantities we give to disposal of statistical thermody-namics.As the consequence, following equality (14), the tailsof diffusion law and thus probabilities of large deviations( z ≫ z ∼ | R | even much slower than in mere sub-exponential fashion generally dictated by inequality (16).Now they decrease in a power-law fashion:Ψ( z ) ∝ z − A max /T ( z → ∞ ) . It is seen after scalar multiplication of (14) by R , thensolving so obtained differential equation, which yieldsΨ( z ) = Ψ(0) exp (cid:20) − Z z A ( z ) T z dz (cid:21) , and finally applying inequality (17). It must be underlined, besides, that boundedness ofthe force declared by (16) also logically implies vanishingof the force at infinity (excluding border case only when F max ( t ) = | F ( t, ∞ ) | ), so that the medium’s “spring” re-sists to small “stretching” only and always loses elasticityat large stretching.The appropriate example of diffusion law satisfying(17), that is possessing power-law tails, is presented byΨ( z ) = (3 / η )!(2 πη ) / η ! (cid:18) z η (cid:19) − / − η (18)with free parameter η > x ! is standard syn-onym of the gamma-function Γ( x + 1) ). At that, ob-viously, A max = (5 / η ) T . The condition η > η = 1 , for the first timewas obtained in [14] from consideration of Brownianmotion (“self-diffusion” [7]) of test, or “marked”, atomof a gas. Similar distribution was found for molecularBrownian motion in a liquid [15, 16]). Though, strictlyspeaking, this is approximation of formally more exactbut more complicated expressions taking into account,among other factors, violation of the scale invariance.Formula (18) turned out to be a reasonable approxi-mations also for BP whose mass M differs from mass m of medium (gas) atoms. At that, various mathematicalapproaches [17, 20, 22] to the BBGKY equations lead toidentical estimate of the parameter η as a function ofmass ratio, η = M/m .Hence, investigation of complete (infinitely-many-dimensional) Liouville equation qualitatively justifiers re-sults of our semi-heuristic analysis of initial terms of thisequation.We may further lower formal rigor and try to visually“by fingers” interpret mathematical connections betweenstatistics of Brownian motion and its microscopic mech-anism. For instance, namely, let Π be internal pressureof the medium (gas) and the quantity A max = A ( ∞ )be identified with 3 T / m Ω n , - with n being mean concentration ofgas atoms, - just compensate local mass excess M + m accompanying current BP-atom collision. From here wehave Ω = ( M/m + 1) /n and, taking Π /n = T for nottoo dense gas, A max = (5 / M/m ) T .Such reasonings, of course, by themselves are ratherunsafe, but they can be supported by exact results onpair many-particle non-equilibrium statistical correla-tions [12, 15, 16, 20]. In particular, that is a theoremstating that short-range character of one-time spatial pairBP-atom correlation (boundedness of “correlation vol-ume” Ω ) implies long-range (“long-living”) behavior ofmany-time self-correlations in BP’s motion and thus in-validity of Gaussian diffusion law (and, reciprocally, va-lidity of the latter requires non-locality of BP-gas corre-lations in space) [15, 16]. E. Uncertainty and flicker fluctuations of diffusivity
The expansion (8) of non-Gaussian diffusion law (18)over Gaussian ones yields for related probability distri-bution of ζ = ∆ /D expression U ( ζ,
0) = 1 η ! ζ (cid:18) ηζ (cid:19) η +1 exp (cid:18) − ηζ (cid:19) . (19)According to the FDR [1, 4, 12] this distribution trans-mits onto BP’s mobility (at least “low-field” one) andtherefore can be observed in measurements of diversityof “time-of-flight” (time of drift) values of Brownian par-ticles under influence of external force (for example, in-jected electrons or holes in semiconductors) [18].Effects of non-Gaussian statistics were observed alsodirectly in equilibrium, by measuring “fourth cumulants”(irreducible fourth-order correlations) of electric currentor voltage noise [1]. At that, low-frequency, at frequen-cies f ∼ /t , fluctuations of power spectral density ofthermal white noise were under investigation, i.e. inessence uncertainty and fluctuations of rate of chargetransfer and rates (coefficients) of diffusion of charge car-riers.In our example (18)-(19), for η > h ( R ) ih R i − (cid:18) h ∆ ih ∆ i − (cid:19) = 3 η − h ∆ i = D and h R i = 6 Dt . This formula showsnot only degree of uncertainty of diffusion rate but alsodefects of approximation of pure scale invariance: di-vergence of variance of diffusion rate at η ≤ S D ( f ) ∝ D δ ( f ) concentrated at zero frequency.This is usual result of a simplest (though non-trivial) ap-proach to 1/f-noise from microscopic theory [13, 21].Undoubtedly, in a more precise theory beyond idealscale invariance [14, 17] the slow tails of diffusion laware somehow “cut off”, at least at R & V t ( ∆ & D/ξ ∼ V t in (8)), so that the ∆ ’s variance, as wellas all the higher-order statistical moments of R and ∆ ,definitely are finite and hardly exceed values correspond-ing to free BP’s flight: h ( R ) k i . (2 k + 1)!! ( V t ) k and h ∆ k i . (2 k + 1)!! ( V t/ k . What is for the delta-function δ ( f ) , it in a definite way “spreads”, with keep-ing dimensionality and singularity at zero, into ∼ /f ,where ∼ replaces some function of ln ( τ f ) .The first of these corrections is easy describable by re-placing U ( ζ,
0) , - for instance, in (19), - by approximate expression U ( ζ, ξ ) ≈ U ( ζ,
0) Ξ( ζξ ) , in which Ξ(0) = 1and Ξ( · ) in sufficiently fast way tends to zero at infinity.Then instead of (18) one obtains Ψ( z, ξ ) ≈ Ψ( z ) Θ( zξ ) , where scale-invariant factor Ψ( z ) is the same as before,- for instance, in (18), - and also Θ(0) = 1 and Θ( · ) fastdecreases to zero at infinity, thus cutting off the Ψ( z ) ’stail. As the result, the quadratic cumulant of ∆ becomesfinite even at η ≤ ∝ t − η , if η < ∝ t ln ( t/τ )at η = 1 . Correspondingly, the quasi-static spectrum ∝ δ ( f ) transforms to “flicker” spectrum, S D ( f ) ∼ D πf (cid:20) τ f (cid:21) − η , (20)at τ f ≪ ∝ /f , when η = 1 .However, at η > ξ = τ /t → U ( ζ, ξ ) ≈ α ( ξ ) exp + [ − ( ζ − ζ ( ξ )) /c ][ ζ − ζ ( ξ ) + α ( ξ )] , (21)where α ( ξ ) = 1 / ln (1 /ξ ) = [ln ( t/τ )] − , functionexp + ( x ) = exp( x ) at x > + ( x ) = 0 at x < ζ ( ξ ) is determined by condition (4), i.e. R ζ U ( ζ, ξ ) dζ = 1 , while c = r /Dτ with r and τ being minimal space and time scales down to which the“infinite divisibility” of random walk is physically mean-ingful ((21) presumes for simplicity that the constant c is not too small, c ≫ α ( ξ ) ). As it is seen from here,at ξ → U ( ζ,
0) = δ ( ζ −
1) .Thus, the scale-invariant “seed” of such the diffusion lawis purely Gaussian, which motivates to name it “quasi-Gaussian” [10]. In [32] it was considered in detail, includ-ing its generalizations and comparison with experiments[18].For tails of the quasi-Gaussian law at z ≫ z, ξ )Ψ(0 , ξ ) ∼ α ( ξ ) 2 c √ πz / exp − r zc ! , that is tails satisfy the boundedness requirement (16),although lie on boundary of set of diffusion laws permit-ted by (16). And for spectrum of flicker fluctuations ofdiffusion rate, - or, generally, rate of a transport process,- the kernel (21) yields S D ( f ) ≈ D cf (cid:20) ln 1 τ f (cid:21) γ , (22)where γ = − ζ ( ξ ) and 1 in (21) in relative units.More precisely, (22) reflects logarithmically slow decayof this difference with observation time: 1 − ζ ( ξ ) ≈ α ( ξ ) ln ( c/α ( ξ )) .The kernel (19) by its shape is quite similar to (21)(both consist of more or less sharp “wall” on the left andcomparatively gentle slope on the right), but analogousdifference in (19) is fixed. Imparting a time dependenceto it may be one more, parallel, scenario of spreading ofspectrum ∝ δ ( f ) under improved analytical approxima-tions of solutions to the BBGKY equations. From ourpoint of view, this is practically important problem of ofstatistical mechanics.Though, even presently available approximations ofmicroscopic theory are able to realistic quantitative es-timates of 1/f-noise amplitude. Estimates obtained in[7] and in [14, 18] in different approximations ((22) with γ = 1 and (20) with η = 1 or (22) with γ = 0 , re-spectively), although differing one from another by factorln [1 / ( τ f )] , nevertheless, both are in satisfactory agree-ment with experimental data on liquids and gases [1, 18],with taking into account diversity of these data.On the other hand, the scheme of quasi-Gaussian ran-dom walk rather well predicts or explains level of electric1/f-noise in various systems [1–3, 10, 18]. Since trans-ported physical quantity there is charge instead of mass(in view of smallness of mass of usual charge carriers),and interactions of walking charges with medium is es-sentially long-range, it is not surprising that transportstatistics there is non-Gaussian in essentially other man-ner than in case of molecular Brownian motion. Analysisof relation of this statistics to a quantum many-particleLiouville equation or equivalent “quantum BBGKY hi-erarchy”, - for e.g. standard electron-phonon Hamiltoni-ans, - also is actually important problem [19].At today’s stage of development of statistical mechan-ics it is useful to state that unprejudiced treatment of thisscience inevitably discovers flicker fluctuations of ratesof transport processes, even diffusivity (rate of randomwalk) of particle in ideal gas [15, 16, 20, 22] and, more-over, even in the formal Boltzmann-Grad limit (undervanishingly small gas parameter) [30].This fact excellently highlights inconsistency of at-tempts to reduce 1/f-noise and related long-living sta-tistical correlations and dependences to some very longmemory or relaxation times. And thus it highlights in-consistency of the underlying opinion that any statisticalcorrelations between random phenomena gives up someliteral or at least indirect physical correlations betweenthem. In the next Section by means of elementary logics onlywe shall show that in reality in many-particle systemsjust physical disconnectedness of inter-particle collisionsleads to uncertainty and 1/f-noise of relative frequency ofcollisions and rate of wandering of each particle. Thus wefrom a new viewpoint shall justify both the general logicsof Introduction and the following elementary mathemat-ical analysis of molecular random walk. IV. MYTHS AND REALITY OF RANDOMWALKSA. Gaussian probability law and two meanings ofindependence of random events
First, recall why the Gaussian law have appeared andappears in various theoretical models. This is becauseit naturally comes from assumption of statistical inde-pendence of BP’ displacements (increments of randomwalk) at non-intersecting time intervals. And, most im-portantly, because physicists have gotten accustomed toidentify statistical independence of random events in thesense of the probability theory with their independencein the sense of their non-influencing one on another.Both these circumstances have more than three hun-dred years history. A history of the Gaussian law hadtaken beginning from the celebrated “law of large num-bers” [35] discovered by J. Bernoulli who investigatedstatistics of sequences of observations on vicissitudes oflife or, for instance, coin tossing or playing dice, underassumption that unpredictable outcomes of successive“random trials” are mutually independent. To be moreprecise, that their probabilities are independent, that isjoint probability of several random events decomposes(factorizes) into product of their individual probabilities.Exactly in such the way the (statistical) independenceis introduced in modern probability theory [33]. Butthere it is nothing but formal mathematical definition,and therefore, - as A. Kolmogorov warned in [33], - de-duction of this probability property from seeming inde-pendence of physical phenomena as such is possible onlyas a hypothesis to be verified by experiments.In other words, any evidences of independence of phys-ical random events at every concrete their realization,- in the sense, for instance, of absence of cause-and-consequence connections between them, - as such can notbe sufficient ground for declaring statistical independenceof these events in a set (statistical ensemble) of realiza-tions (observations).Logically inverting this thesis, we obtain that evenwhen statistical experiments reveal statistical depen-dence in an ensemble of realizations of random events,this observation does not necessarily mean existence ofsome real interaction of the events. Just such situationsdo occur when one meets 1/f-noise.Hence, identifying of the two meanings of “indepen-dence” is nothing but fallacy. Unfortunately, it tradition-0ally governs relations of physicists to randomness, evendespite its careful disclosure, - from viewpoint of funda-mental statistical mechanics, - by N. Krylov more thansixty years ago [36].
B. Collisions, chaos and noise in system of hardballs
Mathematicians know N. Krylov as one of pioneersof modern theory of dynamical chaos. According to it[37, 38], for instance, motion of N ≥ t/τ N ,where τ is mean free path time of a given ball and thuscharacteristic time of relaxation of its velocity because ofcollisions with other balls [14, 16]. At t/τ ≫ N , clearly, number ∝ t/τ of quantities describing trajectoryof any particular ball is much greater than number ∝ N of quantities establishing initial state of the whole sys-tem, so that each particular trajectory contains one andthe same exhaustive information on the system. More-over, this information is contained even in any small partof the particular trajectory with duration ∼ N τ ≪ t .Due to this circumstance, fluctuations in numbers ofcollisions of given ball from any time sub-interval ∼ N τ to next one behave like statistically independent ran-dom values, or “white noise” (which is well understand-able: presence of some relationship or correlation be-tween them would be recognition of some system’s initialstate specificity yet non-realized on shorter time inter-vals, in contradiction to the condition t ≫ N τ ). Cor-respondingly, relative frequency of the ball’s collisionstime-averaged over whole observation time is almost non-random, that is one and the same for all balls and all ini-tial conditions (at fixed full system’s energy, of course),while statistics of fluctuations in number and rate of col-lisions (of given ball) at intervals t ≫ N τ obeys the lawof large numbers, i.e. is asymptotically Gaussian.
C. Paradox of independence
Such the picture of chaos of collisions like usual noiseis quite pleasant for physicists. But we should not forgetthat it had required the condition t/τ ≫ N establishingrigid (detetrministic) non-local in time and space (non-vanishing at t/τ → ∞ and spanning all the balls) phys-ical (cause-and-consequence) inter-dependence between collisions. Just at the expense of this dependence, - para-doxically! - the statistical independence of time-distantand space-distant events (collisions) was ensured.In other words, interestingly, creation of ideal disor-der, - with which statistical independence is usually as-sociated, - needs vigilant underlying control of it and, inthis sense, global strict order. At this point we invol-untarily remind how Dront in the B. Zakhoder’s Russiantranslation of the L. Carroll’s “Alice’s adventures in won-derland” agitated other personages to “fit into strict dis-order”, or “stand up strictly anyhow”. Along with theselaughable words, comparisons suggest themselves withthe mysterious “quantum non-locality” and “entangledquantum states”. D. Uncertainty and 1/f -noise of relative frequencyof collisions and rate of diffusion
However, in the real world it is not simple to mark offtemporal disorder of random events in so strict way asto subordinate it to the law of large numbers. It is notsimple by those simple reason that real many-particlesystems are characterized by just opposite ratio of du-ration of observations (practically achievable in experi-ments) and number of particles in the system: t/τ ≪ N .
Therefore, the appeal to arbitrary large averaging times,so much beloved in mathematical physics, has no factualgrounds [39].The above inverse inequality is satisfied even for rathersmall volumes of solids and fluids isolated from the restof the world [16]. All the more, this inequality is true ifone takes into account physical impossibility of completeisolation and hence necessity to include to N particles(and generally degrees of freedom) of all huge surround-ings of a system under interest. And definitely this in-equality covers objects of the Gibbs statistical mechanics,in which number of particles N is not limited, and whichwas under N. Krylov’s critical analysis [36].Now, number ∝ t/τ of quantities sufficient for de-scription of observed trajectory of one or another particle(ball) all the time stays small as compared with number ∝ N of independent causes, i.e. variables of system‘s(initial) state, determining the trajectory.But averaging over relatively few number of conse-quences determined by much larger number of causesdefinitely is unable to produce a certain result, since theresult remains dependent on many unknown free parame-ters and does not represent all possible variants of courseof events, all the more can not represent them under somecertain proportion. Therefore, time averaging of observa-tions of particle’s motion in any particular experiment (ateach realization of system’s phase trajectory) inevitablybrings unpredictably new value of relative frequency ofthe particle’s collisions, all the more, new distribution1(histogram) of collisions (or more complex events) in re-spect to their inner characteristics. In other words, anexperimenter meets 1/f-noise (see Introduction).From here we see that, instead of fabrication of hy-potheses on relative frequencies or “probabilities” and“independences” of events constituting random walks itwould be better for all that to follow Newton [8] and de-vote ourselves to investigation of equations of (statistical)mechanics. E. Game of independences and problems ofstatistical mechanics
Just said is just to what N, Krylov called in his book[36] clarifying falseness of the widespread prejudices (ci-tation [42]) “... as if a probability law exists regardlessof theoretical scheme and full experiment” and “... asif “obviously independent” phenomena should have inde-pendent probability distributions”.The “full experiment” here means concrete realizationof system’s phase trajectory considered as a single whole,- as an origin of practical observations, - without its ar-tificial division into “independent” time fragments (thus,we in Sections 2 and 3 above have analyzed just a fullexperiment).As far as, - at N ≫ t/τ , - time-smoothed relative fre-quency, or rate, of a given sort of random phenomena orevents (collisions of a given particle with others) variesfrom one experiment to another, demonstrating non-self-averaging, we can not (have no grounds to) introducefor such event a separately definite (individual) a priori“probability”. This means that all the events occur seem-ing commonly statistically dependent, since, figurativelyspeaking, all equally are responsible for resulting, eachtime new, rate of their appearance (a posteriori proba-bility). This is so in spite of that physically all the eventsare independent, since at N ≫ t/τ are determined byinteractions with different groups from total set of N particles. Consequently, we come to crash (inapplica-bility) of the Bernoulli’s law of large numbers based onpostulate of statistical independence.Here, we clearly see another side of “paradox of inde-pendence”: a true full-value chaos implies infinitely longstatistical dependences and correlations.It is clear also why the molecular Brownian motion,being conjugated with such full-valued chaos, does notwant to go into “Procrustean bed” of Gaussian statisticsand, all the more, Boltzmann’s kinetics. V. CONCLUSION
Unfortunately, the above underlined popular carelessideas of independences and probabilities of random phe-nomena (once again citing [36]) “... are so much habitualthat even a person who had agreed with our argumenta-tion then usually automatically returns to them as soon as he faces with a new question. The origin of stable-ness of these ideas is in that they are based on com-mon intuitive notion about statistical laws, and thereforethey would be permissible and advisable if the talk con-cerned learning of phenomena of empirical reality. How-ever, such ideas turn out to be quite unsatisfactory as abench-mark for substantiation of probability laws whenthe talk is about connections between statistical laws toprinciples of the micro-mechanics”.Fortunately, at present we have understanding of errorsof replacing micro-mechanics by speculative probabilis-tic constructions, let beautiful in themselves and likely.Besides, as we noted above, there is already an experi-ence of consecutive investigation of equations of statis-tical mechanics in application to transport processes. Itclearly shows that mechanics of systems of very manyinteracting particles, or degrees of freedom, in now wayprescribes for the interactions to keep definite rates ofchanging system’s micro-state (transition probabilities),even when molecular chaos takes form of a macroscopicorder (let even thermodynamic equilibrium).The point is that any realization of “elementary” actof interactions in fact is a product of full (initial) micro-state of the system, so that number of causes of visiblerandomness always highly exceeds number of its manifes-tations under time averaging even in most long realisticexperiments. As the consequence, any particular exper-iment presents to researcher’s eyes its own unique as-sortment of relative frequencies (“probabilities”), or timerates, of random events composing a process under ob-servations. That is just the 1/f -noise.Hence, being surprised at 1/f -noise is not more rea-sonable than being surprised at noise in general. TheNature needs 1/f -noise as expression of all inexhaustibleresources of the Nature’s randomness in any particular“irreversible” processes as well as in originality of thewholly observed realization of our Universe’s evolutionat all its time scales. A purely stochastic world, without1/f -noise, in which anything can be easily time-averaged,would be too tedious (and even, possibly, would repressa free will [40]).Unfortunately, as we have seen above, 1/f -noise in-volves a “bad” statistics absolutely alien to the law oflarge numbers and resembling one what sometimes en-forces its observers, - for example, in [41], - to suspectaction of mysterious “cosmic factors”. This fact signifi-cantly complicates theoretical tasks.Fortunately, although an influence from cosmos neveris undoubtedly excluded, a source of randomness quitesufficient for 1/f-noise creation is contained, - as we notedabove, - already in so simple system as molecular Brow-nian particle interacting with ideal gas. And, gener-ally, - as we have demonstrated above, - a source of1/f-noise definitely exists in any medium which allowsBrownian motion. Hence, one has every prospect of suc-cess in building and experimental verification of theory of1/f -noise and accompanying statistical anomalies start-ing from very usual Hamiltonians.2We believe that the presented notes will induce some-body of interested readers to work in this intriguing area of statistical physics. [1] Bochkov G N, Kuzovlev Yu E
Sov. Phys. Usp. UFN
151 (1983)][2] Kuzovlev Yu E, Bochkov G N
On origin and statisticalcharacteristics of 1/f-noise (Preprint NIRFI No.157) (Russia, Nijnii Novgorod: NIRFI, 1982)arXiv:1211.4167[3] Kuzovlev Yu E, Bochkov G N
Radiophys. Quant. Elec-tron. (3) 228 (1983) [in Russian: Izv.VUZov. Ra-diofizika (3) 310 (1983)][4] Bochkov G N, Kuzovlev Yu E Radiophys. Quant. Elec-tron.
811 (1984) [in Russian:
Izv.VUZov. Radiofizika Physics - Uspekhi.
449 (2003) [in Rus-sian:
UFN
465 (2003)][6] Balandin A A
Nature Nanotechnology
549 (Aug. 2013)arXiv:1307.4797[7] Kuzovlev Yu E
Sov. Phys. - JETP (12) 2469 (1988)[ib Russian: ZhETF (12) 140 (1988)] arXiv:0907.3475[8] Newton Isaac The Principia: Mathematical Principles ofNatural Philosophy (Berkeley, Calif.: Univ. of CaliforniaPress, 1999) [Newton I
Philosophia naturalis principiamatematica (Londini: Jussu Societatis Regiae, 1684)][9] Weissman M B
Rev. Mod. Phys.
537 (1988)[10] Kuzovlev Yu E arXiv: cond-mat/9903350[11] Kuzovlev Yu E
JETP ZhETF
Physics - Uspekhi (6)590 (2013) [in Russian: UFN (6) 617 (2013)]arXiv:1208.1202[13] Kuzovlev Yu E, Medvedev Yu V, Grishin A M
JETPLetters
574 (2000);
Phys. Solid State (5) 843 (2002)[in Russian: Pis’ma v ZhETF (11) 832 (2000); FTT (5) 811 (2002)]; arXiv: cond-mat/0010447[14] Kuzovlev Yu E arXiv: cond-mat/0609515[15] Kuzovlev Yu E arXiv:0802.0288 ; arXiv:0803.0301 ;arXiv:0806.4157[16] Kuzovlev Yu E Theor. Math. Phys.
TMF (3) 517 (2009)]; arXiv:0908.0274[17] Kuzovlev Yu E arXiv:1007.1992[18] Kuzovlev Yu E arXiv:1008.4376[19] Kuzovlev Yu E arXiv:1207.0058 ; arXiv:1107.3240 ;arXiv:1110.2502[20] Kuzovlev Yu E arXiv:1209.5425[21] Kuzovlev Yu E arXiv:1302.0373[22] Kuzovlev Yu E arXiv:1311.3152[23] Brown R
Edin. New Phil. J.
358 (1828)[24] Brongniart A
Ann. Sci. Naturelles
41 (1827)[25] Einstein A
Ann. Phys.
549 (1905) [in Russian:
So-branie nauchnykh trudov. T.3 (M.: Nauka, 1966) p.108] [26] Einstein A
Ann. Phys.
289 (1906) [in Russian:
So-branie nauchnykh trudov. T.3 (M.: Nauka, 1966) p.75][27] Einstein A
Ann. Phys.
371 (1906) [in Russian:
So-branie nauchnykh trudov. T.3 (M.: Nauka, 1966) p.118][28] Lifshitz E M, Pitaevskii L P
Physical Kinetics (Oxford:Pergamon Press, 1981)[29] Landau L D, Lifshitz E M
Statistical Physics Vol.1 (Ox-ford: Pergamon Press, 1980)[30] Kuzovlev Yu E arXiv:1411.3162[31] Feller W
An introduction to probability theory and itsapplications. Vol.2 (Wiley, 1971)[32] Bochkov G N, Kuzovlev Yu E
On thery of 1/f-noise(Preprint NIRFI No.195) (Russia, Nijnii Novgorod:NIRFI, 1985) (in Russian)[33] Kolmogorov A N
Foundations of the theory of probability (N-Y: Chelsey, 1956) [in Russian:
Osnovnye ponyatiyateorii veroyatnostei (M.: Nauka, 1974)][34] Arnold V I
Mathematical Methods of Classical Mechan-ics (New York: Springer, 1997) [in Russian:
Matematich-eskie metody klassicheskoi mekhaniki (M.: Nauka, 1989)][35] Bernoulli Jacob and Sylla T D (translator)
Art of con-jecturing (John Hopkins Univ., 2005) [Bernoulli Jakob
Ars conjectandi (Basel: Thurneysen Brothers, 1713); inRussian:
O zakone bol’shikh chisel (M.: Nauka, 1986)][36] Krylov N S
Works on the foundations of statisticalphysics (Princeton, 1979) [Russian original:
Raboty poobosnovaniyu statisticheskoi fiziki (Moscow-Leningrad:USSR Academy of Sciences Publ., 1950)][37] Loskutov A Yu
Physics - Uspekhi Physics- Uspekhi
939 (2010) [in Russian:
UFN (12) 1305(2010);
UFN
989 (2007)][38] Chernov N, Galperin G, Zemlyakov A
The Mathemat-ics of Billiards (Cambridge Univ. Press, 2003) [in Rus-sian: Galperin G, Zemlyakov A
Mathematical Billiards (Moscow, Nauka, 1990)][39] Arnold V I, Avez A
Problemes ergodiques de lamecanique classique. Monogr. Internat. Math. Mod-ernes. Vol. 9 (Paris, Gauthier-Villars, 1967); ErgodicProblems of Classical Mechanics (N-Y: Benjamin, 1968);[in Russian:
Ergodic problems of classical mechanics (Izhevsk: RKhD, 1999)][40] Strugatsky A, Strugatsky B
Definitely Maybe: aManuscript Discovered under Strange Circumstances (Abillion years before the end of the world) (Brooklyn:Melville House Publ., 2014) [in Russian:
Za milliard letdo kontsa sveta (M.: Stalker, 2005)][41] Shnoll S E, Kolombet V A, Pozharskii E V, ZenchenkoT A, Zvereva I M, Konradov A A
Phys. Usp. UFN168