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The Role of Place and Local Knowledge in Ontario's Spring Bear Hunt Debate: Fifteen Years Later
June 26, 2013 June 25, 2013 16 Comments on The Role of Place and Local Knowledge in Ontario's Spring Bear Hunt Debate: Fifteen Years Later
A marauding black bear near the author's home in Sudbury, Ontario. Photograph courtesy of Marthe Brown.
by Mike Commito
Ontario had its last spring black bear hunt fifteen years ago. Dating back to 1937, the province's spring hunt was primarily for non-resident hunters. But spring hunting picked up in 1961 after the Department of Lands and Forests declared the black bear a game animal. By the mid-1990s, spring bear hunting had been well established as a significant revenue generator for northern Ontario communities and an important management tool.
Around this time, however, opposition flared up over the spring hunting season, largely over the ethics of bear baiting and the fate of orphaned cubs after the accidental shooting of mother bears. Consequently, an amalgam of animal welfare, animal rights, and conservationist groups – colloquially known as the "Bear Alliance" – organized a campaign to have the hunt repealed. By 1999, following aggressive marketing and political lobbying, the Bear Alliance succeeded in convincing Ontario's Progressive Conservative government to abruptly cancel the hunt.
In the immediate aftermath of the decision, hunters, outfitters, and residents in northern Ontario charged that emotion and politics had trumped science and conservation. They called supporters of the decision, "bleeding heart liberals" and denounced their opponents' authority on the matter because of their overwhelmingly non-rural residency. Throughout the debate and afterwards, critics referenced "urban southern Ontario" and even "Toronto" in a derisive manner. The idea that people living in areas far removed from bears had influenced government policy was viewed as unacceptable; even more so because it was done using emotionally charged arguments about orphaned cubs. As a result, the idea of local knowledge and place forms an interesting part of the debate that still persists today.Many hunters and outfitters are not university-trained biologists but they still possess an extraordinary amount of technical knowledge about the animals because they reside or work in black bear habitat. Meanwhile, the provincial government also has a cadre of professional scientists with years of technical expertise learned on the ground. Despite the government's competency in deciding wildlife policy, as well as its ability to exercise the authority in cancelling hunting activities that are deemed unethical or problematic, the conversation always seems to boil down to northern Ontario residents versus bureaucrats at Queen's Park. Neither group possesses superior knowledge over the other, but who is in a better position to make policy decisions? People living in bear country or those living outside this realm? Do we need to accord greater significance to the perspectives of people whose understanding has been informed by place? For example, does living in Sudbury versus Mississauga make your pronouncements about bear management more valuable?
Black bear crossing the highway near Killarney, south of Sudbury. Photograph courtesy of Marthe Brown.
Currently, the idea of proximity of place forms an interesting facet to the conversation about bear management in general and spring hunting in particular. People who live, work, or vacation in bear country are obviously more likely to encounter bears than those who primarily reside in urban areas where bears have been extirpated. Although encounters with bears are usually benign, their large, powerful bodies means there is always the remote chance of an attack. In April, a black bear nearly killed Joe Azougar at his cottage in Cochrane. More recently, a bear charged an Ontario Ministry of Natural Resources (OMNR) biologist in Foleyet.
After these respective incidents, many people in northern Ontario took to social media and suggested these negative encounters were proof that the province sorely needed to reinstate the spring bear hunt to keep bear numbers in check. However, no correlation exists between frequency of attack and spring hunting to suggest that this would have mitigated these recent episodes or even others in the past. Yet living in these areas inherently requires a higher degree of personal responsibility and understanding of risk and danger. Consequently, those in close proximity to bears often have a different understanding of the animal and as such, possess different expectations over questions of safety and management.
Although there is no evidence to link a lack of spring hunting to the occurrence of attacks, these concerns should not be trivialized. Rather, such concerns should be carefully analyzed in concert with current scientific studies. Residents that share their homes with bears have genuine concerns about the protection of their property, pets, and family that are informed by their places of residence. These issues, of course, differ from their counterparts elsewhere in the province.
There are also times, however, when one's local knowledge and proximity to wildlife can skew perceptions. Since the hunt's cancellation, a recurring argument from people living in northern Ontario is that the bear population has erupted from decreased hunting pressure. While less bears were harvested in the initial wake of the moratorium, the current number of bears culled annually is just slightly less than the pre-spring bear hunt levels, or roughly six to eight percent of the population per annum. Currently the OMNR estimates the province has around 75,000 to 100,000 black bears, around the same range provided before 1999. Moreover, black bears have very low reproduction rates and high sensitivity to change in adult survival. Even with decreased hunting in the early 21st century, then, it is highly improbable that the population has explode.
Far more likely is that the increase is merely perception. Poor summer berry and fall mast crops, along with unseasonable weather was reported in the early 2000s and that might have driven bears out of the forests in search of alternative food sources.
Humans are also continually encroaching into the home ranges of bears. For example, more people are living in cottage country year round and often in areas that were once prime bear habitat. Therefore, we must also consider the increase in the number of permanent and summer residents in bear country as a factor, as more bear sightings or nuisance bear reports do not necessarily indicate more bears. However, there are also times when local knowledge can be appropriated by groups or individuals that have an overriding agenda. Organizations that favour bear hunting might often tally up anecdotal sighting information from across the province along with junk science to forecast a higher population. Some groups in the province have asserted that the black bear population is closer to 150,000. Unsurprisingly, the claim comes from ardent advocates seeking a reinstatement of the spring hunt.
Issues about the future of bear management in Ontario are important because the province is home to the third-largest black bear population in North America. Fifteen years after the ban, spring bear hunting remains a controversial issue, especially in certain locales where residents view it as a panacea to rising nuisance bear complaints. Many people consider the spring bear hunt be unethical because of the method of shooting bears over bait sites and how cubs are often left orphaned when their mothers are accidentally killed. Indeed, no other large game animal in the province is hunted during the spring when its young are still dependent and vulnerable, so it is only commonsensical that this same logic was applied to black bears. If a particular type of hunting is perceived to be cruel and unsportsmanlike by the majority of hunters and non-hunters in the province, do we not have the moral responsibility to cancel it, provided that doing so will not deleteriously affect ecological sustainability?
The spring bear hunt debate is far from over. While it seems as though the Wynne government is not keen on reopening this issue at the moment, it is not inconceivable that the hunt might be reassessed at a later date. When and if it is, we should remember the significance of local knowledge, tradition, and place. At the same time, we must listen to the opinions and views of those that live beyond the boundaries of bear country.
Mike Commito is a PhD candidate in the history department at McMaster University. His dissertation focuses on the history of big-game management, notably bears, deer and wolves, in Ontario and New York state.
Canadian history, Environment, History and Policy, Local History, Uncategorized	black bears, conservation, game management, hunting, local knowledge, Ontario, Place
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16 thoughts on "The Role of Place and Local Knowledge in Ontario's Spring Bear Hunt Debate: Fifteen Years Later"
Phil August 24, 2013 at 9:56 pm
Not only has the bear population risen but the size of them too. There are many trophy bears around and many monster bears too. The black bear has lost fear of man.They have a different caracter now. I think when you shoot at them in the spring they remember more deeply since they just woke up.LOL
commitz August 25, 2013 at 7:34 pm
Not sure there is definitive proof to suggest that the bear population has increased. The MNR uses a number of techniques to extrapolate the population, so a precise number is impossible to obtain but we can certainly approximate it. As noted in my essay, I believe the government's estimate is still somewhere between 75,000 and 100,000.
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I grew up on the rural fringes of Thunder Bay. Nuisance bears were always an issue in the 1960s and 1970s. But people learned to live with them. Some were shot by neighbours for sure. But it was simply common sense that you paid attention, especially if you had pets or fruit trees near the house. The current claims about the increase in nuisance bears seems to reflect sometimes a naivete about living in areas adjacent to bush. Back in the day we didn't report every nuisance bear sighting or interaction. We just dealt with it. Now that the whole situation is thoroughly politicized by both sides in this debate, there is likely little hope of achieving anything approaching an "objective" understanding of how serious the issue is or is not.
Mike Commito December 6, 2013 at 8:10 pm
Thanks for your comment Tom! Things have definitely changed all around in terms of our attitudes and approaches to black bear management. I would agree, I don't think we will ever find an objective viewpoint on these issues. Like most wildlife management decisions, there really is never a purely objective approach that is guided solely by science or conservation. Often times these policy decisions are very messy and are a combination of political, economic, and social considerations. I think that once we accept that the spring bear hunt will never be a black and white issue, the better we can work towards implementing a policy that benefits the greatest good without doing so at the expense of conservation.
Res Krebs January 21, 2014 at 12:00 pm
Enjoyed the read Mike. Were you aware that the SBH has been reinstated for a 2 year pilot study? I've read quite a few statements from OFAH that human-bear conflict has risen exponentially, but I have a hard time believing their figures. Do you know of any reliable figures on human-bear conflict since the end of the SBH? Are communities tackling this issue proactively (e.g. providing bear-proof garbage containers) as Whistler, BC has done?
commitz January 21, 2014 at 12:11 pm
Thanks Res! Yes, I've been following the announcement pretty closely since November. I've been able to see the MNR's tracking of nuisance bear complaints from 2004-2013 but interpreting these is difficult, especially since it is hard to tell how each complaint varies. My sense is that this year was particularly bad for negative encounters with bears. There was a gentleman in Cochrane that sustained life threatening injuries and an MNR biologist apparently had a standoff with a bear near Folyete. Couple these with the usual nuisance reports then 2013 seems like a pretty notable year, which is why the MNR is marketing the pilot project as a public safety issue. Elliot Lake has actually made itself Bear Wise through the passage of bylaws and I think it's a pretty good example of how communities in northern Ontario can mitigate negative interactions by promoting responsible behaviour and attitudes in bear country. Unfortunately, Bear Wise in the official sense was limited (gutted) in 2012. I believe that a more enhanced program to promote education and responsibility would address some of these issues but this of course would carry a price tag (albeit money well spent, in my opinion) that governments might not be willing to foot the bill for.
marianne February 14, 2014 at 6:50 pm
Good article. Does anyone know of any proposals to ban the practice of bear baiting in Ontario? It is my understanding that it is not allowed in most states that have bear hunting and is also banned in B.C. Since it contributes to the problem of bear habituation, it seems like a logical strategy to eliminate it. It's pretty evident that bears will come to where there is food. I remember going to the dump near Dwight in the 80's on a weekend, night. it was more popular with bears, and the humans watching them, than a drive-in..
commitz February 16, 2014 at 6:27 pm
Hi Marianne,
Thanks for your comment. There have definitely been some proposals to ban baiting in Ontario over the course of a number of years. Members within the MNR have been calling from more stringent regulations or an outright moratorium on baiting as early as the 1970s and animal rights/animal welfare/conservation (some of these) have also advocated for an end to baiting as part of their platforms against spring bear hunting. Baiting is definitely a grey area, for many, in terms of hunter ethics but the artificial feeding of bears under any circumstances is also a significant issue that merits some consideration in this conversation as well.
Eldon Hawton March 27, 2014 at 11:20 am
I read somewhere that habitat 'carrying capacity' (overpopulation) today has two important area's of consideration, 'natural or ecological' (food source, shelter etc.) and 'cultural carrying capacity' (the limit that human populations are willing to accept), Given that sufficient, constant 'food supply' or lack of it is key to bear good/problematic behaviour; natural food supply being dependant on seasonal variable conditions combined with the 'habituation problems' due the fact that most landfill sites are now natural habitat for an extremely large portion of the bear population here in Ontario.
Also I agree with Phil …" Not only has the bear population risen but the size of them too. There are many trophy bears around and many monster bears too. The black bear has lost fear of man. They have a different character now. I think when you shoot at them in the spring they remember more deeply since they just woke up."
It is also stated OMNR population numbers are questionable to say the least, Given their closest best guess remains unexplainably static over the years at between 75,000 -100,000; considering that OMNR funding is gutted and sadly a back burner issue for the Liberal government, arguably the real numbers could in fact be 100,000 -125,000 or indeed 150.000 as the proponents with a vested interest in the return of the spring bear hunt suggest.
I believe the dilemma people living, working, & visiting Ontario's bear country face,in terms of public health & safety; demonstrating their lack of 'due diligence' OMNR and the parliamentarians stemming from their ignorance combined with 'political gamesmanship' are playing Russian roulette with a ticking time bomb!
commitz March 27, 2014 at 11:39 am
Thanks for your comments Eldon, much appreciated. I have from others about the lack of fear that bears are exhibiting and some have even stated that there are more aggressive bears. I do not know if this is true or not or even how we can gauge this so I'm skeptical about this claims. Also, it is still difficult to assert that 150,000 could be the current marker especially because the MNR's annual harvest numbers for the fall hunt in the last few years are almost equal to the pre-cancellation era. The numbers show that on average we are only legally harvesting 883 less bears per year, which would make it difficult to argue that the population has exploded by over a third, especially since black bears have an extremely low reproductive rate. Granted the numbers that the MNR has are not complete, mandatory reporting for residents was only instituted in 2006 and there are an untold number of bears illegally killed each year and legal kills that go unreported. Which is why we should be very cautious when deploying population numbers and harvest numbers since at best, these are educated guesses and often incomplete. If you would like, we can discuss this more in detail via email. You can reach me at commitma (@) mcmaster.ca
Eldon Hawton March 31, 2014 at 2:35 pm
Protecting bears to be ultimately shot and wasted serves no meaningful purpose; many more bears are killed/wasted as "problem bears" than the regulated harvest. Black bears once a valuable big game resource generating social, wildlife management and economic tourist benefits worth millions, bears are now shamefully considered 'nuisance value' and an expensive taxpayer liability. Laws geared to prevent over-harvest are therefore redundant, difficult to enforce, open to abuse and lead to vigil-anti management solutions to bear problems (shoot-shovel-shut-up).
Out of sight, out of mind and to my/public knowledge in unrecorded numbers! Too many bears are killed/wasted and orphaned cub numbers rise; the test of time has proven Ontario's "Bear Wise program" is ineffective, unscientific wildlife management, addresses the symptoms rather than the problem, costs tax-payers millions, promotes the indiscriminate killing/wasting of nuisance males, females and the subsequent increased numbers of orphaned cubs.
In the pages of the following report you will learn …
THE BENEFITS OF FIREARMS OWNERSHIP –
HUNTING AND WILDLIFE MANAGEMENT
Library of Parliament
Bibliothèque du Parlement
"When wildlife populations reach their cultural and natural carrying capacity, hunting becomes even more important.
"To slow the growth of bear populations and reduce conflicts, over half of all states and most provinces have established regulated bear hunting seasons. Many wildlife agencies in jurisdictions without bear seasons, but where bear populations are close to reaching the 'cultural carrying capacity' (the limit that human populations are willing to accept), are beginning to put hunting seasons in their plans. The primary goal is to keep bear populations healthy yet keep their populations within cultural tolerance limits. Wildlife managers do not want bears returning to a nuisance/pest status. Therefore, managers need all of the tools available to them, hunting being one of the most important methods for controlling populations."
An historic example of the negative implications resulting from exceeding the "cultural carrying capacity" has proven devastating for Ontario's bears …
Infamous Government bounty eliminated southern Ontario bears; once a common denizen of Southern Ontario the Government of Upper Canada instituted a ten-dollar bounty on black bear in 1793 to 1796. In many areas bears were relentlessly killed (hunted, trapped and poisoned) until none were to be found in the major agricultural settlements of Ontario; and bounties remained in some areas of Ontario as recently as 1961.
Before we blame the hunters & trappers … remember this is not regulated, recorded harvesting; they are paid by the government to deal with this overpopulation problem … as at this point in time bears are considered a threat to livestock & humans.
In terms of reaching 'cultural carrying capacity' overpopulation … when you consider that many/most municipal councils in Ontario bear country are very concerned and struggling for answers about dealing with problem bears and the serious threat to public safety.
I believe the problem is fast approaching crisis proportions and if history lessons have taught us anything, sadly the ultimate losers will be the bears.
Therefore: "The Role of Place and Local Knowledge in Ontario's Spring Bear Hunt Debate: Fifteen Years Later" remains relevant, crucial and of utmost importance.
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tasse tessier February 26, 2015 at 5:48 pm
What is not mentioned in this article, is that in the spring and summer of 1999 that the MNR under minister JOHN SNOBELEN SHOT AND KILLED OVER 100 BEARS and then dumped them in the bush to rot. Did JOHN SNOBELEM and his MINISTRY fill in the mandatory survey and questionnaire that the ministry wants every bear hunter to fill I think not. As far as the OMNR'S estimates on bear numbers how can they know this when they can't ever predict the actual amount of MOOSE IN NORTHERN ONTARIO. In my opinion if the people of Toronto etc. love bears so much perhaps the MNR can trap them and drop them of on YOUNG STREET. Right that will never happen since the ministry is so short handed do to cutbacks, it's now up to the OPP OR REGENAL POLICE to deal with this matter.
Mike Commito February 26, 2015 at 9:32 pm
I didn't mention that because I've never heard of this, please share your source. The Ministry can "know" estimates because it is just an approximation. No one definitively knows how many bears there are, similar to how no one knows how many moose there are, again the number of moose is just an educated guess. Even if the MNRF had greater resources to trap and relocate, I don't think it would be wise to move the bears to Yonge Street in Toronto, that's a very busy thoroughfare!


Little red plastic spoons from Dairy Queen kept popping up around town.
Educational use only. Copyright Brenda Salat.
Salat, Brenda, "Need a spoon?," Iowa City Public Library, accessed April 19, 2019, http://history.icpl.org/items/show/2649.


text:
Mahler - Totenfeier (piano solo)
Arranged for solo piano
Gustav Mahler (1860-1911) composed the single-movement Totenfeier (Funeral Rites) in 1888 while he was conductor at the Leipzig Theatre, after completion of the First Symphony. It may have been conceived as a symphonic tone poem, a depiction of the hero from the First Symphony being carried to his grave, along with a search for the meaning of life and death. Mahler played the work on the piano to the conductor Hans von Bülow in 1891, when they were both working in Hamburg. After several years, he revised Totenfeier to make the first movement of his Second Symphony. This version by Iain Farrington of Totenfeier for piano solo presents the power and grandeur of the score in a clear, playable arrangement.

redpajama_set_name: RedPajamaCommonCrawl


Lovely Scots. And you can see them every year at the Fergus Scottish festival & Highland Games. 2010 marks the 65th edition of the Fergus Scottish Festival and Highland Games – a Top 50 Ontario Festival Event. It's held at the Centre Wellington Community Sportsplex, known to many still as the "Fergus Arena".
Check out our Calendar of Events or our page about the Fergus Scottish Festival.


Grammy 2013-nominee Frank Ocean and singer Chris Brown got into a fight over a parking space at a Los Angeles recording studio — reported TMZ.
Witnesses told investigating deputies that the fight was over a parking space - and that Brown threw the first punch. [But] those statements conflict with initial reports of the fight, with other witnesses telling TMZ that someone attacked Chris as he tried to shake Ocean's hand.
Ocean, who last year came out as gay, tweeted, "got jumped by chris and a couple guys. lol. i only wish [my dog] everest was there."
He then tweeted, "cut my finger now I can't play w two hands at the grammys."
Meanwhile, police say they will "continue to investigate," adding that they may want to speak with Brown. They also refer to Ocean as "the victim," since he's the one who spoke with the authorities as Brown left.


text:
\section{Introduction}

\subsection{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 together with physical experiments and new
technologies and concerning almost all in the world, from cosmic phenomena
down to molecular biology and nano-electronics.
Nevertheless, there are no modern reviews of the question proportional to its volume
and significance. Seemingly, this is so because  investigators do not find an
 inspirational ideas and happy thoughts  for that. Though, one appropriate suggestion, -
to be under consideration below, - was made already in \cite{ufn,pr157,bk1,bk2},
but it had not excited a response. Somehow or other, today we see reports on
more and more inventive and fine measurements of 1/f-noise, -
for example, in films of metals and alloys \cite{jig}  or atomic layers of graphene
\cite{bal}, -  but as before with no unambiguous indication in its origin.
Citing \cite{bal},
``... despite almost a century of research, 1/f noise remains a
controversial phenomenon and numerous debates continue
about its origin and mechanisms''.

It can be added that  ``debates'', in the author's experience, not rarely take
rather totalitarian forms. Maybe, partly by this reason, from the author's
viewpoint, the present situation in general very slightly differs from what was
outlined in \cite{ufn} and a little later in  \cite{i1}.
We would like to compare it with situation in astronomy nearly three
hundreds years ago before appearance of the celebrated I.\,Newton's
work \cite{new}.

We venture such the comparison not for the sake of witticism but in view
of our intention to demonstrate in the present notes that just the Newton's
laws of mechanics may be the place where solution of the 1/f-noise
problem is hidden.
More precisely, 1/f-noise is permanent property of systems of many particles
moving and interacting by these laws (in their classical or quantum formulation
including fields).
In order to recognize it, we have only to follow the Newton's advice  to avoid
unnecessary hypotheses  (``Hypotheses non fingo''  \cite{new}).

Well, what hypotheses are thought up by physicists in respect to 1/f-noise?
Let us decipher it by example of electric current noise in a conductor under
fixed voltage. Presence of 1/f-noise there means that the current has no
certain value, in the sense that its averaging (smoothing) over time produces
an unpredictable result, - that is randomly varying from one experiment to
fnother, - with a diversity which practically does not decrease, or even increases,
when the averaging duration grows (since related narrowing of
frequency band contributing to the diversity is almost, or with excess, compensated
by growth of the noise power spectral density inside that band).
So, when asking oneself a question about origin of such the phenomenon,
one first of all assumes that it is in some specific fluctuation processes
influencing the current, - through e.g. number of charge carriers or their
mobility, - while specificity of these processes is in extremely wide variety
of their time scales (memory, or life, or relaxation, or correlation times, etc.)
\cite{jig,bal,wei}.  Just this is main hypothesis.

\subsection{Idea of the answer and plan of doing}

Really, this hypothesis is not necessary, since the mechanics
as it is in no way requires certainty of the current and, hence,
some special reasons for its uncertainty.
Indeed, no matter what a concrete mechanism of conductivity may be,
if it  is indifferent in respect to amount of charge early transported
through the conductor from one side of an outer electric circuit to another
and thus to past value of time-smoothed current, then later this mechanism
also will  be indifferent in respect to them, and on the whole it will be
unable to set conditions for certainty of the current.
Thus, it by itself serves as mechanism of 1/f-noise.

What is for the indifference, it is supported by the experiment conditions
in themselves which state that fluctuations of (time-smoothed) current do not
meet a back reaction of outer circuit instead passively swallowing them.

In this reasoning, there is no collections of large characteristic times, instead
a single time only is present, - usually small in practice, - which indicates ending
of  memory of conductance mechanism. If, for instance, it is less than
several hours, then transfer of charge carriers, - with their collisions,
scatterings, reflections, etc., - now, at present time interval, is passing
indifferently to what amount of charge was transported yesterday, even
if an experimental device was not switched off  before going to bed.
Correspondingly, at frequencies lower than inverse day one can find a 1/f-noise.

Analogously, if somebody possesses unlimited possibilities of profits and
expenses and does not keep count of them, then he himself could not
know how much his expenditures 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, then the current's 1/f-noise
disappears along with directed current.  But irregular charge displacements
in opposite directions do continue, at that again indifferently to their
past amount, and thus to their time-average intensity. The latter therefore
is not aimed at a certain value, which results in 1/f-fluctuations of
intensity (power spectral density) of thermodynamically equilibrium
``white'' (thermal) current noise. They are connected to the 1/f-noise
  in non-equilibrium current-carrying conductor by means of the
``generalized fluctuation-dissipation relations''  \cite{ufn,pr157,i2,i3,ufn1}.

If one measures equilibrium thermal noise of potential difference between
sides of opened conductor, - e.g. electric junction, - then 1/f-fluctuations
of intensity of this noise can be found too. They say that sum of numbers
(per unit time)  of random charge carrier transitions from one side to
another and backwards is not tracked and regulated by the system,
in contrast to residial of that numbers \cite{i2}.
At that, characteristic time constant of the system (equivalent RC-circuit)
determines upper time scale for fluctuations of the residial and lower one
for fluctuations in the sum (while their upper time scale does not 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 lot of various examples was exposed
in \cite{ufn,pr157,bk2,i2,i3,kmg,p1302}.
Our demonstration below will be realized in terms of equilibrium
``molecular Brownian motion'' \cite{bk1,i1,i2,ufn1,p1,p0803,tmf,p1007,p1008,p1207,p1209,p1311}.

In a maximally simple way we shall show that assumption that
Brownian particle obeys a certain diffusivity (rate, or coefficient,
of diffusion) is incompatible with exact equations of statistical mechanics,
that is with mechanical 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 of this 1/f-noise
and, finally, present its explanation in the language of theory
of deterministic chaos in many-particle systems.

\section{Phenomenology of Brownian motion}

\subsection{Formulation of problem}

Let us imagine a small ``Brownian particle'' in a three-dimensional
statistically uniform isotropic and thermodynamically equilibrium
medium.  Very small particle of dust or flower pollen, - whose
motion in liquid for the first time was observed through microscope
in \cite{br,fr_br} and in the beginning of next century theoretically analysed
in \cite{ein,ein_,ein__}, - are suitable objects.
But it will be better to take in mind some ``nano-particle'' or even merely
separate atom or molecule in liquid or gas \cite{lp}.
In principle, we may speak even about free charge carrier
or point defect in a solid, but confine ourselves by a particle which quite
definitely is subject to the classical variant of mechanics.

Let \,$R(t)$\, and  \,$V(t)=dR(t)/dt$\, denote vectors of centre-of-mass
coordinate and velocity of our Brownian particle  (BP)  at given time instant,
while  \,$R$\,  and   \,$V$\,  their possible values.
We can think that initially at time \,$t=0$\, BP was placed at definitely known
space point. Where namely, is of no  importance, because of
thermodynamical equivalence of any BP's positions. Therefore it is
convenient to choose the coordinate origin:  \,$R(0)=0$\,.
 Then later instant current position of BP, \,$R(t)$\,, will be coinciding 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 this distribution will be designated by  \,$W(t,R)$\,. It can be
represented by expression
\begin{equation}
 W(t,R) = \langle \, \delta(R-R(t)) \, \rangle \, ,  \label{w0}
\end{equation}
where the Dirac delta-function figures, \,$R(t)$\, is thought as
result of all the previous interaction vetween BP and the medium, and
the angle brackets designate averaging over the equilibrium
(Gibbs \cite{ll}) statistical ensemble of initial states of the medium and
initial values of BP's velocity.

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 be formed in reality.

\subsection{Conditional averaging and continuity equation}

In fact,  (\ref{w0}) is mere identity,  but its time differentiation
immediately brings us a food for thought. From it we have
\[
\frac {\partial W(t,R)}{\partial t} = - \nabla \cdot  \langle V(t)  \,
\delta(R-R(t)) \rangle \,
\]
(\,$\cdot$\, will denote scalar product of vectors).
By attracting mathematical tools of the probability theory \cite{kol},
this equality  can be rewritten as
\begin{equation}
\frac {\partial W(t,R)}{\partial t} = - \nabla\cdot  \overline{V}(t,R) \, W(t,R) \, ,  \label{ce}
\end{equation}
where\, \,$\overline{V}(t,R) =\langle V(t)\rangle _R $\,  is conditional
average value of BP's instant velocity determined under  condition that
its current position, and thus its previous path, is known (measured)
to be equal  to \,$R(t)=R$\,.
Generally the operation of conditional averaging
 \,$ \langle \dots \rangle _R$\, is defined by formula
\[
 \langle \dots \rangle _R \equiv
\langle \, \dots \,\delta(R(t)-R) \rangle / \langle \delta(R(t)-R) \rangle  \, .
\]

Obviously,  (\ref{ce}) is ``continuity equation''
 for the probability density \,$W(t,R)$\,, and
the ``field of velocity of probability flow'',   \,$\overline{V}(t,R)$\,,  containes
important information about solutions to this equation.
Therefore, first of all let us consider possible constrution of
the vector-function  \,$\overline{V}(t,R)$\,.

\subsection{Conditional average velocity of Brownian particle}

We shall keep in mind that the duration \,$t$\,  of our
observations of BP is much longer than characteristic relaxation  time \,$\tau $\,
of (fluctuations of) BP's velocity.

Then, firstly, apply heuristic reasonings as follow.
On one hand, by the condition  \,$R(t)=R$\,,
average value of BP's velocity in the past, at time of its preceeding
observation, appears equal to \,$R/t$\,.
On the other hand, as far as BP makes a random walk and
\,$t \gg \tau $\,, the same condition  \,$R(t)=R$\, tells us almost
nothing about BP's velocity in the future, so its average value
in equal next time interval can be expected to be zero.
Hence, since the average under question, \,$\overline{V}(t,R)$\,,
relates to the present time instant ``in the middle between past
and future'', it seems likely that it is equal to half-sum of the mentioned quantities:
\begin{equation}
\overline{V}(t,R) = \frac R{2t}  \,  . \,  \label{hv}
\end{equation}

We can confirm this conclusion in a more formally rigorous way,
basing on the main  distinctive statistical property of Brownian motion
\cite{ein_}:
\begin{equation}
\langle  R^2(t) \rangle  = \int R^2 \, W(t,R)\, dR =   6Dt  \,  \label{dif}
\end{equation}
at \,$t \gg \tau $\,, i.e. ensemble average of squared BP's displacement
grows proportionally to observation time.
It is sufficient to notice that the continuity equation implies
\[
\frac {\partial }{\partial t} \, \int R^2 \, W \, dR = 2 \int R\cdot \overline{V} \, W\, dR
\]
and that this requirement is naturally satisfied
together  with  (\ref{dif})  (with taking account of parallelity \,$\overline{V} \parallel R$\,) %
when equality (\ref{hv}) is valid.

 By way, notice that BP's diffusivity, or diffusion coefficient,   \,$D$\, and
relaxation time \,$\tau$\,  always can be connected via relation
\[
 D = V_0^2 \, \tau \equiv \frac TM \, \tau  \, ,
\]
where \,$T$\, is temperature of the medium, \,$M$\, is mass of BP,
and \,$V_0 =\sqrt{T/M}$\, its chracteristic thermal velocity.

\subsection{General form of probabilistic law of diffusion
and uncertainty of its coefficient}

After inserting function  (\ref{hv}) into (\ref{ce}), one comes to partial differential  equation
\begin{equation}
2  t \, \frac {\partial W}{\partial t} = -  3 \, W -   R\cdot\nabla W \, ,  \, \label{fce}
\end{equation}
which clearly indicates scale-invariant character of its solutions.
We are interested in isotropic (spherically symmetric) solutions
looking as
\begin{equation}
W(t,R) = (2Dt)^{-3/2} \, \Psi( R^2/2Dt) \, \label{w}
\end{equation}
with some dimensionless function  \,$\Psi(z)$\, of dimensionless
argument \,$z =R^2/2Dt$\,. In our context it, since representing probability
density, anyway should be non-negative and satisfying normalization
condition  \,$\int W \, dR =1$\, in company with equality
(\ref{dif}), which surely can be done.
Then  (\ref{w}) is most general law of  {\it \, diffusional\,} random
walk, when typical BP's displacements are proportional to square root
of observation time:  \,$R^2(t) \propto t$\,.

In particular,  taking \,$\Psi(z)= (2\pi )^{-3/2} \, \exp{(-z/2)}$\,,
one obtains the commonly known Gaussian diffusion law,
\begin{equation}
W = W_D(t,R) \equiv
(4\pi Dt)^{-3/2} \exp{(-R^2/4Dt)} \, .  \label{gw}
\end{equation}
The corresponding walk is much pleasant for users  since in rough enough,
in comparison with \,$\tau$\,, time scale its
successive increments are mutually statistically independent.
Owing to this, the only parameter of such random walk, -
its diffusion coefficient, or diffusivity,  \,$D$\, - can be unambiguously
determined from observations of any its particular realization,
by means of long enough time averaging.

However, similar observations and time averaging of non-Gaussian
random walk, obeying some of general type distributions  (\ref{w}),
will produce every time different values of   diffusivity
\cite{ufn,pr157,bk1,bk2,i1,i2}. Indeed, their coincidence, that is
convergence of all results of time averaging to one and the same value,
would be impossible without statistical independence of increments
(at least mutually far time-distanced ones) which in turn would mean,
in accordance with respective limit theorem of the probability
theory (the ``law of large numbers''), that at \,$t\gg \tau$\,
probability distribution of total path tends to the Gaussian
(``normal'') one \cite{fel}.

This becomes quite obvious if distribution (\ref{w})  is represented
by linear combination of Gaussian ``bells'':
\begin{equation}
W(t,R) = \int_0^\infty W_\Delta(t,R) \,
U \left( \frac {\Delta}{D} , \xi
\right) \, \frac {d\Delta}D \,  .    \nonumber
\end{equation}
Such expansions naturally arise in the microscopic theory \cite{p1,p0803,tmf,ufn1}.
Correspondingly, in place of \,$\Psi(z)$\, in  (\ref{w}) we can write
\begin{equation}
\Psi (z,\xi ) = \int_0^\infty \frac {\exp{( -z/2\zeta )} }{ (2\pi \zeta )^{3/2}} \, \,
U(\zeta ,\xi) \, d\zeta \, .    \label{nip}
\end{equation}
Function  \,$U( \zeta , \xi) $\, here plays role of probability distribution
 of \,$\zeta =\Delta/D$\,, i.e. random diffusivity of BP
\,$\Delta$\, expressed in units of its mean diffusivity \,$D$\,.

 The latter is formally defined by equality (\ref{dif}),  while
practically one may try to determine it with the help of averaging
over many experiments or many copies of BP.

 The additional argument \,$\xi $\, in this expansion,  -
 if introduced e.g. as \,$\xi \equiv \tau/t$\,
 under convention  \,$\Psi(z,0) = \Psi(z)$\,, -
  allows to take into account violation of ideal scale invariance of
random walk at  \,$\xi \neq 0$\,. First of all, far  on ``tails''
of diffusion law, where \,$R^2 \gtrsim V_0^2 t^2$\,,
that is \,$z \gtrsim 1/\xi$\,.  There rate of diffusion achieves values
of rate of free flight, $\Delta \sim V_0^2 t = D/\xi$\,.

Clearly, a correction of tails of diffusion law may strongly
influence its higher-order statistical moments and cumulants,
even in spite of \,$\xi \ll 1$\,. At that, nevertheless,  shaping
of \,$W(t,R)$\,'s bell in the main stays almost unchanged.
Accordingly, a change of the function \,$\overline{V}(t,R)$\,,   -
required by equation (\ref{ce})  and condition  (\ref{dif}), -
is as small as \,$\xi$\, is, so that the expression (\ref{hv}) remains right.

Notice that the very possibility of long-term violation of scale invariance
automatically presumes non-Gaussianity of diffusion law, since
Gaussian statistics merely gives no place for it
(since it would contradict the condition (\ref{dif})).

 Already this fact gives evidence  that Gaussian law is not
 completely adequate  reflection of reality, although
 in mind of scientists it is firmly associated with diffusion of  physical
 particles.
At the same time neither general reasonings leading to (\ref{hv})
and  (\ref{fce}) nor equation  (\ref{fce}) by itself
in no way dictate the special  Gaussian choice.
Therefore, it is desirable to discuss other possibilities
and search for criteria of choice among them in the framework
of statistical mechanics.

\section{Microscopic approach}

\subsection{Newton equation and Liouville equation}

Further, let us go from kinematics of Brownian motion to its dynamics
and directly consider BP's interaction with medium using methods
of statistical mechanics.
With this purpose we can take for our system quite usual simple Hamiltonian
\begin{equation}
H= \frac {P^2}{2M} + \Phi(R,\Gamma)  +
H_{th}(\Gamma)  \, ,  \label{h}
\end{equation}
where  \,$P=MV$\, is BP's momentum,
\,$\Gamma$\, is full set of (canonical) variables of the medium,  %
\,$\Phi(R,\Gamma)$\, is energy of BP-medium interaction,
and \,$H_{th}(\Gamma)$\, is Hamiltonian of medium in itself
(or, in other words, that of  ``thermostat'').
If BP possesses internal degrees of freedom,
then their variables will be thought included into the set \,$\Gamma$\,,
thus being formally treated as a constituent of medium.

Let  \,$\mathcal{D} = \mathcal{D} (t,R,P,\Gamma)\,$\,
designate density of full probability distribution of microstates
of our system. Its evolution is described by the formally exact
Liouville equation  \cite{ll,ar}.
Here  we can display it partly, writing out its terms only directly
concerning BP:

\begin{equation}
\frac {\partial \mathcal{D}}{\partial t} = - V\cdot\nabla \mathcal{D} -
F(R,\Gamma)\cdot\nabla_P\mathcal{D}  + \dots \,  .  \label{le}
\end{equation}
Here\,  \,$F(R,\Gamma)= - \nabla \Phi(R,\Gamma)$\, is
force acting onto BP because of its interaction with medium,
and the dots surrogate terms with  \,$\Gamma$\, derivatives.

Considering probability distribution of displacement (coordinate) of BP,
\[
 W(t,R) = \int \!\! \int \mathcal{D} (t,R,P,\Gamma) \, d\Gamma \, dP  \, ,
\]
from equation (\ref{le}) after its integration over \,$\Gamma \,$
and \,$P$\, one comes, of course, to the continuity equation  (\ref{ce}).
The same integration after multiplying  (\ref{le}) by \,$V$\,
produces additional equation

\begin{equation}
\frac {\partial}{\partial t} \, \overline{V}  W = -\nabla \cdot \overline{V\circ V}
 \, W  + M^{-1} \overline{F} \,W \,  .   \label{efe}
\end{equation}
Here and below the symbol  \,$\circ$\, denotes tensor product
of vectors, while the over-line means, as before, conditional
averages under given \,$R(t)=R$\,. Namely,
in the first term on the left
\[
  \overline{V \! \circ \! V} (t,R) =
 \langle V(t)  \circ  V(t) \rangle _R  =
 \frac {\int \!\! \int  V\!\circ\! V \, \mathcal{D} \, d\Gamma \, dP}{W}
\]
and in second term there
\[
 \overline{F} (t,R) = \langle F(R(t),\Gamma(t)) \rangle _R
= \frac {\int \!\! \int F(R,\Gamma) \,
\mathcal{D}  \, d\Gamma\, dP}{W} \, .
\]

Equation (\ref{efe})  describes momentum exchange between
BP and medium. In essence, - as it can be easy verified, - this is
merely the Newton equation \,$M \, dV/dt = F$\, after its
conditional averaging:
\begin{equation}
\langle \, M \, dV(t)/dt - F(R(t),\Gamma(t)) \,  \rangle_R  = 0 \, .  \nonumber
\end{equation}
We shall transform it into relation between functions
\,$\overline{F}(t,R)$\, and \,$W(t,R)$\, which is able to help
selection of acceptable diffusion laws without more
deepening into the Liouville equation.

\subsection{Equation of friction of Brownian particle}

Replacing  derivative \,$\partial W/\partial t$\, in the equation  (\ref{efe})
with right-hand side of (\ref{ce}), after simple manipulations
one comes to equivalent exact equation
\begin{equation}
\frac {d \overline{V}}{dt}  +  \frac  {\nabla \cdot \overline{\overline{V\circ V}} \, W}{W}  =
 \frac {\overline{F}}{M}  \,  , \,  \label{rel}
\end{equation}
with ``material derivative'' of BP's average velocity,
\[
\frac {d \overline{V}}{dt} = \frac {\partial \overline{V}}{\partial t}  +
(\overline{V}\cdot\nabla ) \, \overline{V} \, ,
\]
 and the double over-line marking tensor (matrixу)
 of conditional  quadratic cumulants (second-order cumulants)
 of velocity:
\[
 \overline{\overline{V\circ V}}  \equiv
 \overline{V\circ V} - \overline{V}\circ\overline{ V}  \, .
\]
Next, at first let us consider the latter object.

Since we are speaking about thermodynamically equilibrium
Brownian motion, we can state that the conditional cumulants' matrix
 \,$\overline{\overline{V\! \circ \! V}} (t,R) $\, at  \,$t\gg \tau$\, coincides with
 matrix of unconditional equilibrium quadratic statistical moments
 of velocity,  \,$\langle V(t) \circ V(t) \rangle$\,,
that is reduces to  scalar number  \,$V_0^2=T/M$\,
regardless of \,$R$\,.
Indeed, if \,$t\gg \tau$\,, then at any \,$R$\, the condition
\,$R(t)=R$\, fixes BP's position occurred after  many random steps
and  cycles of momentum and energy exchange between BP and  medium
under detail balance in this process.
Therefore, the value (variance) of corresponding thermal  randomness
of BP's velocity is not affected by this condition
(otherwise, thermal kinetic energy of BP,
on average equal to \,$M\overline{\overline{V\cdot V}}/2$\,,  would be
dependent on where BP is found).

The said can be confirmed by direct calculation of the matrix
\,$\overline{\overline{V\! \circ \! V}} (t,R) $\, in case of Gaussian random walk
subject to distribution (\ref{gw}), which yields
\begin{equation}
 \overline{\overline{V \!\circ\! V}}(t,R) = V_0^2 \, (1- \xi /2)
\rightarrow V_0^2 \,   \label{var}
\end{equation}
at \,$\xi \equiv \tau/t \rightarrow 0$\,. This result is valid also in case
of non-Gaussian walk obeying  (\ref{w}) and  (\ref{nip}),
since its difference from Gaussian one is contained
in its higher-order cumulants.

Then, let us compare two terms on left side of (\ref{rel}).
For the first of them insertion of expression (\ref{hv})  gives
\begin{equation}
\frac {d \overline{V}}{dt}
= - \frac {R}{4t^2} = - \frac {\xi}2 \,  \frac TM \,
 \frac {R}{2Dt} \,  .  \nonumber
\end{equation}
For the second term after insertion of  (\ref{var}) and (\ref{w})
we have   %
\begin{equation}
    \left(1- \frac {\xi}2 \right )   \frac {T}{M}
 \, \frac {2\, d  \ln  \Psi(z,\xi)}{dz}  \,
\frac R{2Dt}  \, \sim  \, -  \frac TM \,  \frac {R}{2Dt}  \, \nonumber
\end{equation}
with same shortened notation \, \,$z=R^2/2Dt$\, as before.    %
Right-hand expression here corresponds to the Gaussian diffusion law,
for which  \,$ d \ln \Psi /dz = - 1/2$\,, but by order of magnitude
it is true in general case too, at least at  \,$z \ll 1/\xi$\,.
It shows that the first term,
being approximately  \,$2t/\tau $\, times smaller than the second,      %
is negligibly small in the limit \,$\xi \rightarrow 0$\,.

Hence, consideration of long enough time intervals
leads us from (\ref{rel}) to shortened relation

\begin{equation}
-  \left(  \frac {T}{D} \left[ - \frac {2\, d  \ln  \Psi(z,\xi)}{dz} \right] \right) \,
\frac R{2t} = \overline{F}   \, .  \label{rs}
\end{equation}
It resembles an equation of viscous friction,
with  \,$R/2t = \overline{V}$\, in the role of velocity of a body moving
through fluid  and the round brackets in the role friction coefficient.

One more simplification can be obtained by neglecting, under
mentioned limit, violation of scale invariance and treating
\,$\Psi(z,\xi)$\, as a function of single argument  \,$\Psi(z)$\,.
In the next paragraphs we firstly proceed just so.

 But before that let us once again glance at the vanishing first
 left term of(\ref{rel}). If writing its contribution to the mean force as
\begin{equation}
M \, \frac {d \overline{V}}{dt}  = - \, \nabla  \,
\frac {M\overline{V}^2}2 \, ,     \nonumber
\end{equation}
one can say that this is force of reaction of the medium to addition
\,$M\overline{V}^2/2$\, to energy of BP, and system as the whole,
introduced by the very measurement of BP's path,
and therefore this force is not sensible to shape of diffusion law
and thus to concrete peculiarities of medium.
There is evident analogy with perturbing effects of measurements
in quantum mechanics.

In opposite, the remaining, in the large-time limit, part of the force,
(\ref{rs}),  is determined solely by shape of probability distribution
of equilibrium Brownian displacement (``diffusion law'').
Consequently,  this force characterizes inherent, -
unperturbed by observations, - BP-medium interaction.
In particular, it shows characteristic levels of the interaction
forces and energies necessary for realization of one or another
concrete diffusion law.
Now, examine in this respect the Gaussian law (\ref{gw})
and make sure that it is unrealistic.

\subsection{Paradox of Brownian motion: Gaussian statistics for it
is beyond strength of its mechanics}

For Gaussian diffusion law, the square bracket in the
``friction equation''  (\ref{rs}) turns to unit, and the equation
becomes linear:
\begin{equation}
\overline{F} \, \Rightarrow \, - \frac {T}{D} \, \frac R{2t}  =
- \frac R{|R|} \, \sqrt{z} \, \frac T{\sqrt{2Dt}} \, . \,  \label{rs0}
\end{equation}
At that, the ``friction coefficient''  in front of  \,$R/2t = \overline{V}$\,
connects to the diffusivity via  relation similar to the widely known
``Einstein relation'' \cite{ein_,lp}.
Such likeness, however, is not a plus but minus of equality (\ref{rs0}).

The matter is as follows.  Friction force in the true Einstein relation
represents medium resistance against directed motion of a particle.
When this particle displaces by distance \,$R$\,, the corresponding
force makes work
\[
\sim | R\cdot \overline{F}| \sim \left( \frac TD \,
\frac Rt \right) \cdot R  \, \sim \,  zT \, , \,
\]
thus producing a heat (recall that \,$z=R^2/2Dt$\,).
This quantity,  - like the force itself, - in principle
can be arbitrary large under proper initial value of the particle's
kinetic energy.

This is clear. But it is strange thing that equality  (\ref{rs0})
offers the same, also unbounded, characteristic values of force and work.

Such a picture categorically contradicts to sense.

Really, - repeating the aforesaid, - in our case the force what figures
in  (\ref{rs0})  represents medium reaction to particle's displacement
achieved along random trajectory of thermal motion, when initial
energy value knowingly is only \,$\sim T$\,. At that, the medium
creates obstacles to inertial free flight of BP but in no way to
its unrestricted  moving off from beginning of its path.

In opposite, the moving off  proceeds due to medium's own free will
and at the expense of its own equilibrium fluctuations.

Therefore in reality, in contrast to (\ref{rs0}),
the average force  (\ref{rs}), as a function of  passed path
 \,$R$\,, can not be arbitrary large, instead
staying always and everywhere bounded.
This is required by such factual inherent property of Brownian motion
as translational invariance, that is indifference of the system
in respect to  irretrievable departures of BP anywhere.
Moreover, on this ground it is reasonable to expect
that  at large  \,$|R|$\, the returning force  vanishes at all.

Thus, we have to conclude that  the Gaussian law is inadequate
to physical origin of Brownian motion.

Inevitability of this conclusion catches eye when noticing that
if equality (\ref{rs0})  was true then it would mean that
medium returns BP to start of its path with force proportional
to the path,  \,$\overline{F} \propto -R$\,,
i.e. like ideal spring with potential energy \,$zT/2 \propto R^2$\,.
From physical point of view this looks absurdly,
since any far BP's going away is permitted just because it does not
change thermodynamical state of the system.

Our conclusion can be denominated as paradoxical, if recollecting
that Gaussian diffusion law many times issued from pen of
theoreticians in various physical contexts and occupies important place
in idealized world of ``mathematical physics''.
But the paradox resolves in very simple way:\, the Gaussian statistics
always had appeared as consequence of  clear or implicit hypotheses
(or postulates) about ``independences'' of random events
or values. What is for us, we have managed without such hypotheses
and thus showed their fallacy in application to Brownian motion.

In the past, we too were not connected with them and arrived to the same
paradoxical conclusion, in the framework of both phenomenological
statistical analysis of diffusion and transport processes
 \cite{ufn,pr157,bk1,bk2,i2} and analysis based on the full hierarchy
of Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) equations
\cite{i1,i2,p1,p1007,tmf,p1311}, as well as on the base of exact
``generalized fluctuation-dissipation relations'' (FDR) or
``dynamical virial relations''  \cite{ufn1,p0803,tmf,p1209},
and by other methods \cite{i2,i3,p1302},  including
that for quantum systems  \cite{kmg,p1207,p1302}.

In Section IV we shall again touch on  ``paradox of independence''.  %
And now, in next paragraph, consider examples of physically correct
diffusion law as an alternative of Gaussian one.

\subsection{Thermodynamics of Brownian motion
and statistics of large deviations}

From the left expression in (\ref{rs}) it is clear that the boundedness  %
of the force  \,$\overline{F}$\, in general implies relation
\begin{equation}
|\overline{F} (t,R)|\, \leq \, \overline{F}_{max} (t)\, \sim \,
\frac {T}{\sqrt{2Dt}} \, ,  \,  \label{flim}
\end{equation}
whose right-hand part can be easy guessed for reasons of dimensionality.
Of course, the symbol  \,$\sim$\, here hides some dimensionless
coefficient which reflects particular distinctions of the system
and construction of the function  \,$-\ln \Psi(z)$\,.

Comparison between (\ref{flim}) and (\ref{rs0})  shows that
in the region of ``tails'' of diffusion law, at \,$z \gtrsim 1$\,, the Gaussian law
requires to exaggerate the real force value at least \,$\sim \sqrt{z}$\,  times,
thus thoroughly falsely describing (strongly underestimating) probabilities
of large displacements of BP with \,$z \gg 1$\,.

In reality, according to (\ref{flim}), function \,$- \ln \Psi(z)$\,
grows not faster than \,$\propto \sqrt{z}$\,, so that
\,$ (- \ln \Psi(z))/\sqrt{z} < \infty$\,, and consequently decrease
of  \,$W(t,R)$\, at large \,$|R| \rightarrow \infty$\, is always
sub-exponential (anyway, not faster than simple exponential,
not speaking about  ``Gaussian'').

The difference of reality from ``Gaussian ideal'' becomes aggravated
when not the force itself only  is bounded  but also value
of characteristic energy (work) conjugated with this force:
\begin{equation}
A(z)  \equiv |R| |\overline{F}|/2    \, \leq \, A_{max} = A(\infty)\, \sim \,  T \,    \label{alim}
\end{equation}
(with the same remark about \,$\sim$\,).
 It is natural expectation in view of that at result of any
walk (any path \,$R$\,) the medium takes from BP not more energy
than BP had been able to take from medium before.

Boundedness of  \,$A(z) $\,  implies that of the force, moreover,
implies that under increase of \,$|R|$\, the force passes through a maximum
and then decreases down to zero, approximately as
\,$|\overline{F}| \approx 2A_{max} /|R| \propto T/|R|$\,.

 This asymptotic again is prompted already by dimensionality
  of quantities we give to disposal of statistical thermodynamics.

As the consequence, following equality (\ref{rs}), the tails of diffusion law
and thus probabilities of large deviations (\,$z\gg 1$\,)
from typical behavior  (\,$z\sim 1$\,) decrease under growth of  \,$|R|$\,
even much slower than in mere sub-exponential fashion
generally dictated by inequality  (\ref{flim}). Now they decrease
in a power-law fashion:
\[
\Psi(z) \, \propto \, z^{- A_{max}/T} \, \,\,\, \,\,\,
\, \, \, (z \rightarrow \infty)  \, .
\]
It is seen after scalar multiplication of  (\ref{rs}) by \,$R$\,,
then solving so obtained differential equation,
which yields
\[
\Psi(z) = \Psi(0) \,  \exp{ \left[ \,- \int_0^z \frac {A(z)}{Tz} \,
dz \, \right] } \, ,
\]
and finally applying inequality (\ref{alim}).

It must be underlined, besides,  that boundedness of the force declared by
(\ref{flim})  also logically implies vanishing of the force at infinity
(excluding border case only when \,$F_{max}(t) = |F(t,\infty)|$\,),
so that the medium's ``spring'' resists to small
``stretching'' only and always loses elasticity at large stretching.

The appropriate example of diffusion law satisfying (\ref{alim}),
that is possessing power-law tails, is presented by
\begin{equation}
\Psi(z) =  \frac {(3/2 +\eta) !}{(2\pi \eta)^{3/2} \, \eta !} \,
  \left(1 + \frac z{2\eta} \right)^{-5/2 - \eta}  \,   \label{psi}
\end{equation}
with free parameter \,$\eta > 0$\,
(factorial \,$x!$\, is standard synonym of the gamma-function
\,$\Gamma(x+1)$\,).  At that, obviously,
\,$A_{max} = (5/2+\eta )\, T$\,. The condition \,$\eta > 0$\,
is necessary for finiteness of the mean diffusivity in (\ref{dif}).

Such distribution, with \,$\eta =1$\,, for the first time was obtained in \cite{p1}
from consideration of Brownian motion (``self-diffusion'' \cite{i1})
of test,  or ``marked'', atom of a gas.
Similar distribution was found for molecular Brownian motion
in a liquid \cite{p0803,tmf}).
Though, strictly speaking, this is approximation of formally
more exact but more complicated expressions taking into account,
among other factors, violation of the scale invariance.

Formula  (\ref{psi})  turned out to be a reasonable approximations
also for BP whose mass \,$M$\, differs from mass \,$m$\,
of medium (gas) atoms. At that,  various mathematical approaches
\cite{p1007,p1209,p1311} to the BBGKY equations lead to
identical estimate of the parameter \,$\eta$\, as a function of mass ratio,
\,$\eta = M/m$\,.

Hence, investigation of complete (infinitely-many-dimensional)
Liouville equation qualitatively justifiers results of our
semi-heuristic analysis of initial terms of this equation.

We may further lower formal rigor and try to visually
``by fingers'' interpret mathematical connections between
statistics of Brownian motion and its microscopic mechanism.
For instance, namely, let \,$\Pi$\, be internal pressure
of the medium  (gas)
and the quantity \,$A_{max} = A(\infty)$\, be identified with
\,$3T/2 + \Pi \Omega$\,, where \,$\Omega$\,  is gas volume
forced out by far walking BP from its vicinity, and
\,$\Pi \Omega$\, is related forcing work.
 In essence  \,$\Omega$\,  represents deficiency of BP's collisions
 with gas atoms facilitating its far going away.
At that, since position of center of mass of the system stays fixed,
an effective decrease of gas mass near BP,
\,$m \Omega n  $\,, - with \,$n$\, being mean concentration of gas
atoms, - just compensate local mass excess  \,$M+m$\, accompanying
current BP-atom collision.
From here we have  \,$\Omega = (M/m +1)/n$\, and,
taking \,$\Pi/n = T$\, for not too dense gas,
\,$A_{max} = (5/2+M/m)\, T$\,.

Such reasonings, of course, by themselves are rather unsafe,
but they can be supported by exact results on pair
many-particle non-equilibrium statistical correlations
\cite{p0803,tmf,ufn1,p1209}.
In particular, that is a theorem stating that short-range character
of one-time spatial pair BP-atom correlation  (boundedness of
``correlation volume'' \,$\Omega$\,)  implies long-range
(``long-living'')  behavior of many-time self-correlations
in BP's motion and thus invalidity of Gaussian diffusion law
(and, reciprocally, validity of the latter requires non-locality
of BP-gas correlations in space)  \cite{p0803,tmf}.

  \subsection{Uncertainty and flicker fluctuations of diffusivity}

The expansion  (\ref{nip}) of non-Gaussian diffusion law (\ref{psi}) over
Gaussian ones yields  for related probability distribution
of   \,$\zeta =\Delta/D$\, expression

\begin{equation}
 U(\zeta,0) = \frac 1{\eta ! \, \zeta} \, \left(\frac \eta\zeta \right)^{\eta +1} \,
\exp{\left( -\frac \eta\zeta \right)} \,  .   \label{nid}
\end{equation}
According to the FDR \cite{ufn,bk2,ufn1}  this distribution
transmits onto BP's mobility (at least ``low-field'' one) and
therefore can be observed in measurements of diversity of
 ``time-of-flight'' (time of drift) values of Brownian particles
 under influence of external force (for example,
injected electrons or holes in semiconductors) \cite{p1008}.

Effects of non-Gaussian statistics were observed also directly in equilibrium,
by measuring ``fourth cumulants'' (irreducible fourth-order correlations)
of electric  current or voltage noise \cite{ufn}.
At that, low-frequency, at frequencies \,$f \sim 1/t$\,, fluctuations
of power spectral density of thermal white noise were under
investigation, i.e. in essence uncertainty and fluctuations of rate
of charge transfer and rates (coefficients)  of diffusion
of charge carriers.

In our example (\ref{psi})-(\ref{nid}), for  \,$\eta>1$\,, it is not hard to obtain
\begin{equation}
 \frac {\langle \, (R^2)^2 \, \rangle}{\langle \, R^2 \,\rangle^2} - 3 =
3 \, \left( \frac {\langle \, \Delta^2 \, \rangle}{\langle \, \Delta \, \rangle^2}
- 1 \right)  = \frac 3{\eta -1} \,  \nonumber
\end{equation}
with \,$\langle  \Delta  \rangle = D$\,
and  \,$\langle R^2  \rangle = 6Dt $\,.
This formula shows not only degree of uncertainty
of diffusion rate but also defects of approximation of pure
scale invariance:\, divergence of variance of diffusion rate
at \,$\eta \leq 1$\,  and  pure independence of the variance
on duration of observations.
The latter makes  the  rate fluctuations effectively quasi-static,
with spectrum (power spectral density)
\,$S_D(f) \, \propto \, D^2 \, \delta(f)$\, concentrated at zero frequency.
This is usual result of a simplest (though non-trivial)   approach
to 1/f-noise from microscopic theory \cite{kmg,p1302}.

Undoubtedly, in a more precise theory beyond ideal scale invariance
\cite{p1,p1007}  the slow tails of diffusion law are somehow
``cut off'', at least at
\,$R^2\gtrsim V_0^2 t^2$\, (\,$\Delta\gtrsim D/\xi \sim V_0^2 t$\,
in (\ref{nip})),
so that the  \,$\Delta$\,'s variance, as well as all the higher-order
statistical moments of \,$R$\, and \,$\Delta$\,, definitely are finite
and hardly exceed values corresponding to free BP's flight:\,
\,\,$\langle (R^2)^k \rangle \lesssim (2k+1)!! \, (V_0^2t^2)^k$\, and
\,$\langle \Delta^k \rangle \lesssim (2k+1)!! \, (V_0^2t/2)^k$\,.
What is for the delta-function \,$\delta(f)$\,, it in a definite way
``spreads'', with keeping dimensionality and singularity at zero,
 into \,$\sim 1/f$\,, where \,$\sim$\, replaces some function of \,$\ln{(\tau f)}$\,.

The first of these corrections is easy describable
by replacing \,$U(\zeta,0)$\,, - for instance, in (\ref{nid}), -
by approximate expression
\,$  %
 U(\zeta, \xi) \approx U(\zeta, 0) \, \Xi(\zeta\xi)  \, ,
$\,  %
in which \,$\Xi(0)=1$\, and  \,$\Xi(\cdot) $\, in sufficiently fast way
tends to zero at infinity. Then instead of (\ref{psi})  one obtains
\,$  %
 \Psi(z, \xi) \approx \Psi(z) \, \Theta (z \xi)  \, ,
$\,  %
 where scale-invariant factor \,$\Psi(z)$\, is the same as before, -
for instance, in (\ref{psi}), - and also \,$\Theta(0)=1$\,
and \,$\Theta(\cdot) $\,  fast decreases to zero at infinity,
thus cutting off the \,$\Psi(z)$\,'s tail.
As the result, the quadratic cumulant of \,$\Delta$\, becomes finite
even at \,$\eta \leq 1$\,. At once it acquires  time dependence,
so that the fourth cumulant of BP's displacement increases with time
\,$\propto t^{3- \eta}$\,, if \,$\eta < 1$\,, and \,$\propto t^2 \ln{(t/\tau)}$\,
at \,$\eta = 1$\,.
Correspondingly, the quasi-static spectrum  \,$\propto \delta(f)$\,
transforms to ``flicker''  spectrum,

\begin{equation}
 S_D(f) \sim  \frac {D^2}{\pi f} \, \left[ \frac 1{\tau f} \right] ^{1-\eta}  \, ,  \,  \label{sp0}
\end{equation}
at \,$\tau f \ll 1$\,, in particular, to  \,$\propto 1/f$\,, when \,$\eta = 1$\,.

However, at \,$\eta > 1$\,  such correction is insufficient for full
``spreading'' of frequency delta-function,
which says that scale invariance violation in this case
has some another or more complex character.
  A notion of how else it may look we can obtain, for instance,
if consider \cite{ufn,pr157,bk1,bk2,i2,pr195} diffusion law
possessing property of infinite divisibility in the sense of the probability
theory  \cite{fel} but asymptotically only for \,$\xi =\tau/t \rightarrow 0$\,,
 since no real transport process can be physically divided into
infinitely small independent pieces.
The respective kernel in the expansion (\ref{nip})  simplistically
is describable by formula

\begin{equation}
 U(\zeta,\xi) \approx \frac { \alpha(\xi) \,
\exp_+{[- (\zeta -\zeta_0(\xi))/c \, ]} }
{[\, \zeta -\zeta_0(\xi) + \alpha(\xi)]^{\, 2} } \,  , \,  \label{qg}
\end{equation}
where  \,\,$\alpha(\xi) = 1/\ln{(1/\xi)} =[\ln{(t/\tau)}]^{-1}$\,,
function \,$\exp_+(x)=$\,$\exp(x)$\, at \,$x>0$\, and
\,$\exp_+(x)=0$\, at \,$x<0$\,, function \,$\zeta_0(\xi) $\,
is determined by condition (\ref{dif}),
i.e. \,$\int \zeta \, U(\zeta ,\xi) \, d\zeta =1$\,, while
\,$c = r_0^2/D\tau_0$\, with \,$r_0$\, and  \,$\tau_0$\, being
minimal space and time scales down to which
 the ``infinite divisibility'' of random walk is physically meaningful
 ((\ref{qg}) presumes for simplicity that the constant  \,$c$\, is not too small,
\,$c \gg \alpha(\xi)$\,). As it is seen from here, at  \,$\xi  \rightarrow 0$\, expression (\ref{qg})
turns to \,$U(\zeta,0) = \delta(\zeta -1)$\,. Thus, the scale-invariant
``seed'' of such the diffusion law is purely Gaussian, which motivates to
name it ``quasi-Gaussian'' \cite{i2}. In \cite{pr195} it was considered in detail,
including its generalizations and comparison with experiments \cite{p1008}.

For tails of the quasi-Gaussian law at \,$z \gg 1$\, from (\ref{qg})  and
(\ref{nip}) one can find

\[
 \frac {\Psi(z,\xi)}{\Psi(0,\xi)}  \, \sim  \,  \alpha(\xi) \,  \frac {2 c \sqrt{\pi}}{ z^{3/2}} \,
 \exp{\left(-\sqrt{\frac {2z}c} \right)}  \, , \,
\]
that is tails satisfy the boundedness requirement (\ref{flim}),
although lie on boundary of set of diffusion laws permitted by (\ref{flim}).
And for spectrum of flicker fluctuations of diffusion rate, - or,
generally, rate of a transport process, - the kernel (\ref{qg})  yields

\begin{equation}
 S_D(f) \, \approx \, \frac {D^2 c}f \, \left[ \ln \, \frac 1{\tau f} \right]^{\gamma}  \, , \,  \label{sp}
\end{equation}
where \,$\gamma = -2$\,.

Spectrum (\ref{sp}) results mainly from difference between most probable
and average (ensemble-averaged) values of the rate, namely,
\,$\zeta_0(\xi)$\,  and  \,$1$\, in (\ref{qg}) in relative units.
More precisely, (\ref{sp}) reflects logarithmically slow decay of this
difference with observation time:
\,$1-\zeta_0(\xi) \approx \alpha(\xi) \, \ln{(c/\alpha(\xi))}$\,.

The kernel  (\ref{nid})  by its shape is quite similar to  (\ref{qg})  (both consist
of more or less sharp ``wall'' on the left and comparatively gentle slope on the right),
but analogous difference in   (\ref{nid})  is fixed.
 Imparting a time dependence to it may be one more, parallel,
 scenario of spreading of spectrum \,$\propto \delta(f)$\,
 under improved analytical approximations of solutions to the BBGKY
 equations. From our point of view, this is practically important
 problem of of statistical mechanics.

 Though, even presently available approximations of microscopic theory
 are  able to realistic quantitative estimates of 1/f-noise amplitude.
 %
 Estimates obtained in  \cite{i1} and in  \cite{p1,p1008}  in different
 approximations ((\ref{sp}) with \,$\gamma =1$\, and (\ref{sp0})  with
 \,$\eta =1$\, or (\ref{sp}) with \,$\gamma =0$\,, respectively),
 although differing one from another by factor $\ln {[1/(\tau f)]}$\,,
 nevertheless, both are in satisfactory agreement with experimental
 data on liquids and gases \cite{ufn,p1008}, with taking into account
 diversity of these data.

 On the other hand, the scheme of quasi-Gaussian random walk rather well
 predicts or explains level of electric  1/f-noise in various
 systems \cite{ufn,pr157,bk1,i2,p1008}. Since transported 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 essentially long-range, it is not surprising that
 transport statistics there is non-Gaussian  in essentially other manner
 than in case of molecular Brownian motion.

 Analysis of relation of this statistics to a quantum many-particle
 Liouville equation or equivalent ``quantum BBGKY hierarchy'', -
 for e.g. standard electron-phonon Hamiltonians, -
 also is actually important problem \cite{p1207}.

At today's stage of development of statistical mechanics it is useful to state
that unprejudiced treatment of this science inevitably discovers flicker fluctuations
of rates of transport processes, even diffusivity  (rate of random walk)
of particle in ideal gas \cite{p0803,tmf,p1209,p1311}
and, moreover, even in the formal Boltzmann-Grad limit
 (under vanishingly small gas parameter) \cite{p1411}.

This fact excellently highlights inconsistency of attempts to reduce
1/f-noise and related long-living statistical correlations and
dependences to some very long memory or relaxation times.
And thus it highlights inconsistency of the underlying opinion
 that any statistical correlations  between random phenomena
gives up some literal or at least indirect physical correlations between them.

In the next Section  by means of elementary logics only we shall show that
in reality in many-particle systems just physical
disconnectedness of inter-particle collisions
leads to uncertainty and 1/f-noise of relative frequency of collisions
and rate of wandering of each particle.
Thus we from a new viewpoint shall justify both the general logics of Introduction
and the following elementary mathematical analysis of molecular random walk.    %

\section{Myths and reality of random walks}

\subsection{Gaussian probability law and two meanings of
independence of random events}

First, recall why the Gaussian law have appeared and appears
in various theoretical models.  This is because it naturally comes
from assumption of statistical independence of BP' displacements
(increments of random walk) at non-intersecting time intervals.
And, most importantly, because physicists have gotten accustomed to identify
statistical independence of random events in the sense of the probability theory
with their independence in the sense of their non-influencing one on another.

Both these circumstances have more than three hundred years history.
A history of the Gaussian law had taken beginning from the celebrated
``law of large numbers''  \cite{jb} discovered by J.\,Bernoulli  who
investigated statistics of sequences of observations on vicissitudes of life
or, for instance, coin tossing or playing dice, under assumption that
unpredictable outcomes of successive  ``random trials'' are mutually
independent. To be more precise, that their probabilities are independent,
that is joint probability of several random events decomposes
 (factorizes) into product of their individual probabilities.

Exactly in such the way the (statistical) independence is introduced
in modern probability theory \cite{kol}.
But there it is nothing but formal mathematical definition,
and therefore, - as A.\,Kolmogorov warned in  \cite{kol}, -
deduction of this probability property from seeming independence
of physical phenomena as such is possible only as a hypothesis
to be verified by experiments.

In other words, any evidences of independence of physical random events
at every concrete their realization, - in the sense, for instance, of absence
of cause-and-consequence connections between them, -
as such can not be sufficient  ground for declaring statistical independence of these
events in a set (statistical ensemble) of realizations (observations).

Logically inverting this thesis, we obtain that even when statistical
experiments reveal statistical dependence in an ensemble of realizations
of random events, this observation does not necessarily mean
existence of  some real interaction of the events.

Just such situations do occur when one meets 1/f-noise.

Hence, identifying of the two meanings of ``independence''
is nothing but fallacy. Unfortunately, it traditionally governs relations
of physicists to randomness, even despite its careful
 disclosure, - from viewpoint of fundamental statistical mechanics, -
by N.\,Krylov more than sixty years ago  \cite{kr}.

\subsection{Collisions, chaos and noise in system of hard balls}

Mathematicians know N.\,Krylov as one of pioneers of modern theory
of dynamical chaos.
 According to it ей \cite{los,gz}, for instance,
motion of  \,$N\geq 3$\, elastic hard balls in a box or disks on torus,
obeying deterministic laws of mechanics, is indistinguishable
from a random process \cite{gz,aa}.
For us here, it is important that statistical characteristics of this
process are crucially sensible to ratio of its observation time
and total number of balls participating in it.

More precisely, let us consider role of parameter
 \,$t /\tau  N$\,, where  \,$\tau $\, is mean free path time of
a given ball  and thus characteristic time of relaxation
of its velocity because of collisions with other balls \cite{p1,tmf}. %
At

\[
t /\tau  \, \gg \, N \,,
\]
clearly, number  \,$\propto t /\tau $\, of quantities
describing trajectory of any particular ball is much greater than
number \,$\propto N$\,  of quantities establishing initial state
of the whole system, so that each particular trajectory contains
one and the same exhaustive information on the system.
Moreover, this information is contained even in any small
part of the particular trajectory with duration \,$\sim N\tau \ll t$\,.

Due to this circumstance,  fluctuations in numbers of collisions
of given ball from any time sub-interval \,$\sim N\tau$\,  to next
one behave like statistically independent random values,
or ``white noise'' (which is well understandable:  presence of some
relationship or correlation between them would be recognition
of some system's initial state specificity yet non-realized on
shorter time intervals, in contradiction to the condition \,$t\gg N\tau$\,).
 %
Correspondingly,  relative frequency of the ball's collisions
time-averaged over whole observation time is almost non-random,
that is one and the same for all balls and  all initial conditions
(at fixed full system's energy, of course), while statistics of
fluctuations in number and rate of collisions  (of given ball)
at intervals \,$t\gg N\tau$\, obeys the law of large numbers,
i.e. is asymptotically Gaussian.

\subsection{Paradox of independence}

Such the picture of chaos of collisions like usual noise is
quite pleasant for physicists. But we should not forget that it had
required the condition   \,$t /\tau \gg N$\, establishing
rigid (detetrministic)  non-local in time and space
(non-vanishing at \,$t /\tau \rightarrow \infty$\,
and spanning all the balls) physical (cause-and-consequence)
inter-dependence between collisions.
Just at the expense of this dependence, - paradoxically! -
 the statistical independence of time-distant and space-distant events
 (collisions) was ensured.

In other words, interestingly, creation of ideal disorder,  - with which
statistical independence is usually associated,  -
needs vigilant underlying control of it
and, in this sense, global strict order.
At this point we involuntarily remind how Dront
in the B.\,Zakhoder's Russian translation of the L.\,Carroll's    %
 ``Alice's adventures in wonderland'' agitated other personages
to ``fit into strict disorder'', or ``stand up strictly anyhow''.
Along with these laughable words, comparisons suggest themselves     %
with the mysterious ``quantum non-locality'' and
``entangled quantum states''.

\subsection{Uncertainty and 1/f\,-noise of relative frequency of collisions
and rate of diffusion}

However, in the real world it is not simple to mark off
temporal disorder of random events in so strict way as to
subordinate it to the law of large numbers.

It is not simple by those simple reason that  real many-particle
systems are characterized by just opposite ratio of duration of
observations (practically achievable in experiments)
and number of particles in the system:\,

\[
 t /\tau  \, \ll \, N \,  .
\]
Therefore, the appeal to arbitrary large averaging times,
so much beloved in mathematical physics,
has no factual grounds  \cite{aa}.

The above inverse inequality is satisfied even for rather small
volumes of solids and fluids isolated from the rest of the world \cite{tmf}.

All the more, this inequality is true if one takes into account physical
impossibility of complete isolation  and hence necessity to include
to \,$N$\,  particles (and generally degrees of freedom)
of all huge surroundings of a system under interest.
And definitely this inequality covers objects of the Gibbs statistical
mechanics, in which  number of particles \,$N$\, is not limited,
and which was under N.\,Krylov's critical analysis \cite{kr}.

Now, number \,$\propto t /\tau $\, of quantities sufficient for description
of observed trajectory of one or another particle (ball)
all the time stays small as compared with number \,$\propto N$\,
of independent  causes, i.e.  variables of  system`s (initial) state,
determining the trajectory.

But averaging over relatively few number of consequences
determined by much larger number of causes definitely is unable to produce a certain result,
since the result remains dependent on many unknown
free parameters  and does not represent all possible
variants of course of events, all the more can not represent them
under some certain proportion.
Therefore, time averaging of observations of particle's motion
 in any particular experiment (at each realization of system's phase
 trajectory) inevitably brings unpredictably new value of
 relative frequency of the particle's collisions, all the more, new
 distribution (histogram) of collisions (or more complex events)
in respect to their inner characteristics.
In other words, an experimenter meets 1/f-noise (see Introduction).

From here we see that,  instead of fabrication of hypotheses on
relative frequencies or ``probabilities'' and  ``independences''
of events  constituting random walks  it would be better for all that
to follow Newton \cite{new} and devote ourselves to investigation of
equations of (statistical) mechanics.

\subsection{Game of independences and problems of statistical mechanics}

 Just said is just to what N,\,Krylov called in his book \cite{kr} clarifying
 falseness of the widespread prejudices (citation \cite{fn})
 ``... as if a probability law exists regardless of theoretical scheme
and full experiment''
and ``... as if   ``obviously independent'' phenomena should have   %
independent probability distributions''.

The ``full experiment'' here means concrete realization of system's
phase trajectory considered as a single whole, - as an origin
of practical observations, - without its artificial division into
``independent'' time fragments (thus, we in Sections 2 and 3 above
have analyzed just a full experiment).

As far as,  - at \,$N \gg t/\tau $\,, - time-smoothed relative frequency,
or rate,  of a given sort of random phenomena or events (collisions of
a given particle with others) varies  from one experiment
to another, demonstrating non-self-averaging, we can not
(have no grounds to)  introduce
for such event a separately definite (individual)  a priori ``probability''.
This means that all the events occur seeming commonly statistically
dependent, since, figuratively speaking, all equally are responsible
for resulting, each time new, rate of their appearance
(a posteriori probability).
This is so  in spite of that physically all the events are independent,
since at \,$N \gg t/\tau $\,  are determined by interactions
with different groups from total set of \,$N$\, particles.
Consequently,  we come to crash (inapplicability) of the Bernoulli's law of
large numbers based on postulate of statistical independence.

 Here, we clearly see another side of ``paradox of independence'':\,
 a true full-value chaos implies infinitely long statistical
dependences and correlations.

It is clear also why the molecular Brownian motion,
being conjugated with such full-valued chaos,
does not want to go into ``Procrustean bed'' of  Gaussian statistics
and, all the more, Boltzmann's kinetics.

\section{Conclusion}

 Unfortunately, the above underlined popular careless ideas
of independences and probabilities of random phenomena
(once again citing \cite{kr})
``... are so much habitual that even a person who had agreed
with our argumentation then usually automatically returns to them    %
 as soon as he faces with a new question.   %
The origin of stableness of these ideas is in that they are based
on common intuitive  notion about statistical laws, and therefore they
would be permissible and advisable if the talk concerned
learning of phenomena of empirical reality.
However, such ideas turn out to be quite unsatisfactory as a bench-mark
for substantiation of probability laws when the talk is about connections
between statistical laws to principles of the micro-mechanics''.

Fortunately, at present we have understanding of errors of
replacing micro-mechanics by speculative probabilistic constructions,
let beautiful in themselves and likely.
Besides, as we noted above, there is already an experience of
consecutive investigation of equations of statistical mechanics
in application to transport processes.  It clearly shows that mechanics   %
of systems of very many interacting particles, or degrees of freedom,
in now way  prescribes for the interactions to keep definite rates
of changing system's micro-state (transition probabilities), even when
molecular chaos  takes form of a macroscopic order
(let even thermodynamic equilibrium).

The point is that any realization  of  ``elementary'' act
 of interactions in fact is a product of full  (initial) micro-state  of the system,
 so that number of causes of visible randomness always highly exceeds
 number of its manifestations under time averaging even in
 most long realistic experiments.
As the consequence, any particular experiment presents to researcher's
eyes its own unique assortment of relative frequencies
(``probabilities''), or time rates, of random events composing
a process under observations.
That is just the 1/f\,-noise.

Hence,  being surprised at 1/f\,-noise
is not more reasonable than being surprised at noise in general.
The Nature needs 1/f\,-noise as expression
of all inexhaustible resources of the Nature's randomness
in  any particular ``irreversible'' processes
as well as in originality  of the wholly observed realization
of our  Universe's evolution at all its time scales.
A purely stochastic world, without 1/f\,-noise,   in which anything
can be easily time-averaged, would be too tedious
(and even, possibly, would repress a free will \cite{str}).

Unfortunately, as we have seen above,
1/f\,-noise  involves a ``bad'' statistics  absolutely alien to
 the law of large numbers and resembling one what sometimes
enforces its observers, - for example, in \cite{sh}, -  to suspect action of
mysterious ``cosmic factors''. This fact significantly   complicates
theoretical tasks.

Fortunately,  although an influence from cosmos never is
undoubtedly excluded, a source of randomness  quite sufficient
for 1/f-noise creation is contained, - as we noted above, -
already in so simple system as molecular Brownian particle
interacting with ideal gas.
And, generally, - as we have demonstrated above, -  a source
of 1/f-noise definitely exists in any medium which allows
Brownian motion.
 Hence, one has every prospect of success in building  and
 experimental verification  of theory of  1/f\,-noise
and accompanying statistical anomalies starting from
very usual Hamiltonians.

We believe that the presented notes will induce somebody
of interested readers to work in this intriguing area
of statistical physics.

\,\,\,

redpajama_set_name: RedPajamaArXiv


Verizon Wireless reported this week that, in the month of June 2007, their network sent and received more than 10 billion text messages - a record in the American SMS market. In addition Verizon Wireless customers also sent and received over 200 million photo- and video-based multimedia messages in the same month.
Use of text messaging with inexpensive unlimited plans and text friendly phones - available from every provider - has been close to exponential in growth over the past year. As a comparison, during the fourth quarter of 2006 (this is a three month period), Verizon Wireless customers sent and received 17.7 billion text messages and over 353 million picture and video messages.


text:
Sobering cross over stat...
system February 10, 2020, 4:29am #1
For those fans in the West looking for an open back door into the playoffs and hopefully into the Grey Cup here is a sobering stat I came across:
The crossover rule has been in effect seven times.[since 1997]. The following list tells which team crossed over
BC	1997	crossed to East and Lost Semi-Final 35-45
SSK 2002 crossed to East and Lost Semi-Final 12-24
BC	2003	crossed to East and Lost Semi-Final 7-28
EDM 2008 crossed to East and lost Final 26-36
BC	2009 crossed to East and Lost Final 18-56
EDM2012 crossed to East and Lost Semi-Final 26-42
For you eastern fans whose team finishes up in 1st or 2nd place it might be an opportunity to do a little revelling. What I find interesting is that none of those games were even close.
Bottom Line? The west crossing over to the east is not free passage to the Grey Cup.
dcmoses February 10, 2020, 4:29am #2
"beaglehound":
1997 - 8-10 BC lost to 13-5 Montreal
2002 - 8-10 Saskatchewan lost to 8-10 Toronto
2003 - 11-7 BC lost to 9-9 Toronto
2005 - 9-9 Saskatchewan lost to 10-8 Montreal
2008 - 10-8 Edmonton beat 8-10 Winnipeg and lost to 11-7 Montreal
2009 - 8-10 BC beat 9-9 Hamilton and lost to 15-3 Montreal
2012 - 7-11 Edmonton lost to 9-9 Toronto
So let's put this "sobering stat into perspective. Out of 9 games played out of the 7 crossover seasons only ONCE did a team with a better record lose to a team with an inferior regular season record. When applied to the current season, the crossover team from the west will very likely have a better record than BOTH eastern playoff teams.
Bottom Line? The west crossing over to the east this season is free passage to the Grey Cup if history has anything to say about it.
"dcmoses":
how many of those teams won the cup?
"FootbalYouBet":
As usual you completely miss the point. In all but one game the team with the better record won and the team with the better record was always the 2nd place or first place eastern team. Not going to be the case this year. The team in the east playoffs with the best record will be the western crossover team.
Why do I feel I wasted my time explaining this to you? :roll:
my point was that it doesn't matter the record going into the playoffs, the crossover team has little chance winning the cup, and not much more of a chance to even get there
HfxTC February 10, 2020, 4:29am #6
Who says the crossover team will have a better record than the top team in the East ? Your getting way, way ahead of yourself
"HfxTC":
I said very likely (in my post) not will (your words), but I'll play along. Can we at least agree that the crossover is going to happen despite eastern apologists claims the division is catching up. That crossover will be either BC or Saskatchewan.
If BC, they finish with Bombers, Eskies and Stamps. Even a 1-2 finish makes them 9-9 and one of the east teams must go 4-0 down the stretch to top that. Do you really see that happening? I don't.
If the Riders crossover they finish with Edmonton twice and Calgary. I'll play devil's advocate and say the Riders go 0 for 3 there. They also finish 9-9 so first in the east still needs to go 4-0 to have a better record. Still don't see it happening.
So explain to me how the crossover team having a better record than first in the east is so unlikely that the mere suggestion is getting "way, way ahead of" myself.
TravelPatB February 10, 2020, 4:29am #8
Whoever the crossover team is from the west - if there is one - which is more likely than not - will IMO be underdogs in any game they play in the East as bettors will take into account the better play from Eastern teams - which has been the stronger division since mid-September.
don't forge that third place in the east only has to tie the west team. Every body has 9 wins, no crossover
if it is possible that all 3 east teams finish at 9-9, then hard for BC to cross over as they would have to win at least one of their games against the league two best teams, by standings anyhow.
Do you do any research? It's all divisional play for the last four weeks of the schedule. It is IMPOSSIBLE for 3 east teams to win 3 of 4 to finish 9-9. Not virtually impossible, physically impossible. As in it can not happen. :roll:
HfxTC February 10, 2020, 4:29am #11
That one of the three teams runs the table is possible but unlikely, but even if they don't and finish with identical record to BC for example 9-9 that still does not support your theory that the Western team will win because it has a [u]better record. As far as I know identical records is not better.
Your making an educated assumption with a month left of CFL football to play IMO. But you don't stop there, you then add that this western will walk through their two opponents to get to the GC. That's a whole lot of W.A.G.
FootbalYouBet February 10, 2020, 4:29am #12
actually, I said IF because I did not feel like doing the math. So I did it. It is possible for there to be a 3-3-2 finish in the east, so then all bc has to do is win 1. IF BC does win 1, then I see it coming down to the last week between ham and mont for the east second spot with loser odd team out. However, standings are deceptive as all 3 eastern teams have been on fire lately. I think BC would be the underdog irrelevant of standings.
system February 10, 2020, 4:29am #13
dcmoses, you said… "Bottom Line? The west crossing over to the east this season is free passage to the Grey Cup if history has anything to say about it."
On the contrary. FYB has it right. History has proven that 7 straight times not a single Western cross over team made it into the Grey Cup let alone win it.
History since 1997 has proven that in order for a Western team to get into the Grey Cup it has to beat both the 2nd and 1st place teams in the east. History has plenty to say about it. History tells me that not a single western cross over team has succeeded in getting past both eastern playoff teams. I would not call that getting free passage to the Grey Cup. If you have made a point, I'm sorry but I've missed it.
Maybe a western team will cross over and win the Grey Cup this year. Maybe not.
I agree all you have to look is how well BC played in the East this year even prior to these teams getting their act together... Saskatchewan didn't look too hot in Montreal on the weekend either
I was showing a best case scenario for the east. Only a 4-0 record for one team even gives that team any hope of having a better record than the crossover team. The reason I point it out is, as I showed, only once in a game involving a crossover team has the team with the better record lost. Yes, identical records of 9-9 are a tie, but again, not likely to happen. That would see one east team go 3-1 and either BC goes 1-2 or the Riders go 0-3. Go ahead and bet your house on that. I wouldn't bet a nickel on that possibility even if it was FYB's nickel.
Eastern apologists hurry up, you only have 2-3 more weeks to back up the east "juggernaut" (I would have laughed without the quotation marks :lol: ) before two teams are eliminated and a crossover is awarded. And that crossover team will (I'll bet on this) have a better record than first in the east.
I trust that your point is…
" In all but one game the team with the better record won and the team with the better record was always the 2nd place or first place eastern team. Not going to be the case this year. The team in the east playoffs with the best record will be the western crossover team. "
You raise a valid point dcmoses. However, given the way the eastern teams have come on lately this year and the injury issues both the Lions and SSK face I would not put my money on the team with the greater number of wins going to the Grey Cup. Even the coaches will say that when playoffs begin it is a whole new season.
Also the wins vs losses that translate into a team's overall standing is not necessarily indicative of how strong that team may be heading into the playoffs.
EvilDoctor February 10, 2020, 4:29am #17
You are just embarrassing yourself. For one thing, it's highly UNLIKELY that a Western team will have a BETTER record than all the Eastern teams. At least one team, more likely two teams will go 3-1. It's highly unlikely that Saskatchewan will win another game this year. The only conceivable game that B.C. can win is against Winnipeg. And you are forgetting one major thing. Eastern teams have now won 9 of the last 11 from Western opponents....every Western opponent. Those two losses? Both by Ottawa.
thirdperson February 10, 2020, 4:29am #18
Winning the grey cup is hard for all teams. Improbable but not impossible for crossover team to advance to grey cup. For crossover team, east opponents are still more attractive than facing tough Calgary and Edmonton. In recent decades, east division is overrated because of weaker competition. Without Ottawa, top 3 east teams would have worser records.
"EvilDoctor":
It's you that's embarassing yourself. Read the OP, then read my posts and you may understand why I posted what I did. However, based on your keen desire to insult I think the point will sail way over your head.
Also the wins vs losses that translate into a team's overall standing is not necessarily indicative of how strong that team may be heading into the playoffs.
Your stats were misleading beagle, which is why I posted. Yes the east has been lucky in the last 4-6 weeks, but I believe their records are a true reflection of the teams and come playoffs the crossover will win. You used crossover history to show the odds were not in the crossover teams favour and I used the same history to show it was in the crossovers favour.
In the end it is just a historical pattern that has no impact on this season....but it's fun to speculate. 8)