J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Key Ideas for Heightening the Informativeness of Plasma
Physical Theorizing
Vasily I. EROFEEVLaboratory of Nonlinear Physics, Institute of Automation & Electrometry of the Siberian Branch of Russian
Academy of Sciences, Novosibirsk 630090, Russian Federation(Received: 26 August 2008 / Accepted: 19 December 2008)

It is pointed that approaches of traditional plasma theory cannot provide sucient reliability of naltheoretical deductions: The recipes of the theory yield equally rigorous justications to incompatible
versions of the same physical phenomenon. This general property stems from two mutually intertwined
reasons. First, the asymptotic convergence of intermediate iterative calculations: It supposes thatvariations in the leading order of perturbation expansion yield diversity of derivable scenarios of theplasma physical evolution and also imply some restrictions on appropriateness of specic computing
methods in intermediate kinetic calculations. (Thus, intermediate transition from time originals to
Fourier (Laplace) transforms is fraught with substantial loss in the informativeness of nal results.)
Second, common substitution of real plasmas by plasma ensembles: The laws of evolution of the ensemble
statistics cannot be independent on the ensemble content. For gaining the informativeness of plasma
kinetic considerations with account for these reasons, a specic correlation analysis is created. It can becharacterized as some technique of reducing full plasma description to more simplied kinetic one. The
current results are discussed of its application to studies of a homogeneous weakly turbulent plasma.
Some comments are formulated regarding possible other informative plasma research.
Keywords: informativeness of plasma theoretical description, correlation analysis of plasma kinetics,
weakly turbulent collisionless plasma, Coulomb collisions in a quiet weakly nonideal plasma

1. Introduction

physicists rendered the collapse as a real physicalphenomenon that provides back transfer of wave
energy to short-wavelength region of the spectrum.
In view of this, a large number of reports have beencollected in history of plasma natural experiments
about observations of various evidences for Langmuir
wave collapses. The most recognized of them weredeveloped in series of beam-plasma experiments by
Wong and his colleagues (University of California,
Los Angeles) [58]. They observed formation of asolitary spatially localized eld structure and its
subsequent evolution that resembled the Langmuir
wave collapse. The collapse theory was discussed
also in connection with beam-plasma experiments bygroup of Vyacheslavov (BudkerINP) [911], and by
Benford et al. [1214]. We have mentioned here only
the most important of respective plasma observations:
More expanded review of the collapse-related activity
was given by Robinson [15].
Having posed the laboratory plasma observationsof above three groups to an extra scrutiny, we haveshown that they evidence more explicitly against
the Langmuir wave collapse rather then in its
favor [1618]. (Particularly, in experiments by Wonget al. the characteristic time of observed wave
collapse constitutes about 20 inverse growth ratesof original Zakharov's instability with respect toshort-wavelength modulations in plasma density [4].
The growth rate of the given instability depicted

One of basic branches of plasma physics
is the plasma turbulence theory. Factually, fromthe very moment when Langmuir and Tonks
observed electronic oscillations in a fully ionized gas
and proposed term plasma [1], physicists began
theoretical studies of turbulent plasma phenomena.
We shall omit discussion of the story of theory
development: plasma physicists do know it rather
well. We note only that activities of 6070-th inlapsed XX-th century resulted in appear of the weakplasma turbulence theory. One of most importantof its beginnings was the idea of conservation ofLangmuir wave quanta in nonlinear processes [2] thatcomplied with former semi-empiric deductions [3].
This property yielded wave transfer to low wave
numbers through the wave scattering by plasma
particles. With this, a high level of turbulence wasbelieved to build up at the long-wavelength regionof the spectrum, due to the ineciency of wave
dissipation there in comparison with the wave energy
supply. The corresponding piece of the turbulence
spectrum was called a Langmuir condensate. The
puzzle of its dissipation had stimulated studies thatcan be characterized as the creation of strong plasma
turbulence theory. The most important of them was
the development of concept of Langmuir wave collapse[4]. Since the appear of the latter, many of plasma
author's e-mail: yerofeyev@iae.nsk.su

59©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

V.I. Erofeev, Key Ideas for Heightening the Informativeness of Plasma Physical Theorizing

strongly depend on the ensemble content1 . Studying
some plasma ensemble, one never can be surethat evolution of the ensemble statistics repeats
macroscopic evolution of the given single plasma.
Naturally, for each of single plasmas one can compose
plenty of ensembles that follow some period of time themacroscopic behavior of a given plasma in dynamics
of their statistics, but many others demonstrate quitedierent evolution of their statistics from the very
beginning. To have a possibility to select a proper
ensemble, one should perform preliminary study ofmacroscopic evolution of the single plasma itself,
because otherwise he cannot judge whether certain
ensemble adequately models behavior of his plasma
or not. From the viewpoint of common sense, after
learning the physics of a single plasma the search forthe proper ensemble becomes a senseless enterprise2 .
Thus, plasma experimental realities have helped
us to arm an absolute truth of an inappropriatenessof plasma ensemble studies that does not even needin any experimental checking: Our argumentation
against the ensemble method is cogent in itself.
An important comment on inappropriateness ofplasma ensemble substitutions can be advanced fromthe viewpoint of the information theory. Note thatnone of constructive plasma descriptions can give a
picture of a certain plasma evolution that does notdiverge over time from the real plasma macrophysicalevolution. The reason is an incompleteness of dataon positions and momentums of individual plasma
particles. (On the one side, they are never known
to the full extent. On the other, their full -scaleaccount is technically unfeasible.) With this, thegeneral problem of a theorist is to reduce thenonconstructive full plasma description 3 to a more
simplied one that models as better as possible
the current plasma macrophysical evolution. Thisimplies an extremely careful separation of the theory
informational basis from the full plasma information.
The ensemble substitution has no correlation with thegiven principle. It can be said that generally with
implementing the ensemble averaging the researcher
manually lessens the informativeness of nal plasma
macrophysical scenario and thus looses lots in thescientic soundness of nal conclusions.
In usual plasma considerations the ensemble

the granted rate of the most slow initial stage ofLangmuir collapse: With extra reasons of the collapse
initiation the very collapse would have developedon a shorter time scales.). We stress that in all
three series of experiments the necessary conditions
for the collapse were well satised (see details in
Refs. [1618]). Note that it is only via a suciently
intense decay of Langmuir wave quanta that thedevelopment of collapse scenarios might have been
precluded there. We supposed that the reason ofwave dissipation is the transfer of wave energy toplasma electrons during their nonresonant diusion
in momentums. We discovered that considerationof this phenomenon within tradition of plasma
theory implies equally rigorous substantiations bothfor the commonly accepted concept of Langmuir
wave quanta conservation and one of their intense
energetic decay. The incompatibility of these two
conclusions demonstrates that the plasma turbulence
theory fails in its main destination of modelling
realities of plasma physical evolution. To put it morecorrectly, the commonly accepted principles of plasma
theoretical studies cannot yield appropriate reliabilityof respective nal deductions.
The above conclusion has helped us to uncoverbasic false cornerstone of plasma theory: The
substitution of real plasmas by plasma ensembles.
After the Gibbs's success with erecting the statistical
thermodynamics [19], the practice of ensemble
averaging was widely rendered as the only appropriate
one in many of specic physical theories. Particularly,
the idea of the ensemble method had exerted a
key inuence on formation of the most basic ofplasma physical theoretical notions and approaches.
The method underlies the plasma kinetics after
Bogoliubov [20]  Born  Green [21]  Kirkwood [22]
 Yvon [23] (that is collectively known as the BBGKY
kinetics), it is hidden behind any hydrodynamic
plasma modelling, behind the wave phase averaging,etc. In short, plasma theorists did always freely
substituted real plasmas by plasma ensembles, either
deliberately or unintentionally, whereas the ensemble
averaging generally veils the picture of the plasma
physical evolution. We have thus stated that thenonequilibrium statistical mechanics is useless forphysical studies of evolving plasmas. More generally,
our results have disclosed that the existing common
belief in virtues of the given theoretical discipline has
no scientically sound foundation.
An expanded discussion of history of ensemble
studies and their trend in modern physics was given
in paper [18]. For illustration, we expose here only themost evident visualization of senselessness of plasma
ensemble considerations.
Following the most easy common sense, lawsof evolution of the ensemble statistics should

1 We have pointed above that traditional theory permit
equally rigorous substantiations to conclusions of Langmuir
wave quanta conservation and that of their intense energetic
decay. We have commented in Ref. [18] that respective
reasonings oriented on diering plasma ensembles.
2 The reader is reminded that substitution of physical system
by system ensemble was proposed by J.W. Gibbs after L.
Boltsmann [24] exclusively for simplication of the system
physical study.
3 For a classical ionized plasma, the full description is given
by simultaneous Klimontovich [25]Dupree [26] and Maxwell
equations.

60

V.I. Erofeev, Key Ideas for Heightening the Informativeness of Plasma Physical Theorizing

substitution is strongly intertwined with second
reason of losses in theory informativeness: The ignoreof asymptotic nature of the theory convergence.
This thesis can be developed as follows. Whentheorist reduces full plasma description to a simplied
one, he(she) inevitably generates some nonlinear
successive perturbations. Regardless the essence ofcorresponding iterations, they will converge at best
asymptotically only: While heightening the expansion
order one gets a factorial growth in number ofterms to be taken account of. We stress that theasymptotic convergence of the theory suppose absenceof any unique limit even for converging ordersof the expansion: The conditional limit for thoseorders depends on the choice of the lowest orderapproximation. The diering conditional limits of thetheory imply the varying scenarios of the plasma
evolution. Hence, nal results of kinetic calculation
depend essentially on the choice of the lowest orderapproximation in respective perturbation expansion.
Correspondingly, a problem arises of a rational choiceof this approximation: its variations predetermine
the physical content of basic notions of the theory.
This position can be exemplied by the notion wave :above mentioned deductions of Langmuir wave quantaconservation and of their intense dissipation stemmed
from delta-functional and Lorentzian approximationsof wave frequency line, respectively (see details in
Refs. [16, 17]). Undoubtedly, the problem of thoroughchoice of basic theoretical notions is important for
many of branches of plasma theory and, respectively,
for many activities in plasma studies4 .
The asymptotic nature of perturbation
convergence impose also some restrictions on
appropriateness of specic computational techniques
for intermediate kinetic calculations. For instance,
the ordinary transition from time originals of particle
distributions and electromagnetic eld to theirfrequency Fourie (Laplace) transforms leads tolessening of the theory informativeness5 .
It should be stressed that the above two reasonsof theory non-informativeness cannot be separated:
one implies another and vice versa. Really, had thepicture of ensemble evolution not depended on theensemble content, one may have substantiated byensemble variations the diversity of lowest orderapproximations. Equally, variations of lowest orderapproximation within the practice of ensemble studies
suppose appeals to diering ensembles; absence ofdependence of plasma evolution picture on thetheory leading order would have meant then the

independence of the picture of the ensemble evolution
on the ensemble content.
In view of above stated, it is necessary to gain
existing practice of plasma physical theorizing, inorder of maximal heightening the informativeness ofnal theoretical results, by creating new approacheswith both refraining from traditional plasma
ensemble substitutions and proper accounting for theasymptotic nature of successive iterations. Below we
exhibit basic principles of such an approach for thecase of a weakly turbulent homogeneous plasma.
The narration is constructed as follows. The
beginnings of our plasma kinetic approach are given
in Section 2. The current results on the highinformative kinetics of a homogeneous plasma witha weak Langmuir turbulence are listed in Section 3.
Section 4 contain comments on informative studies foralternative plasma physical situations. We comment
here also the top level of informativeness of respective
plasma macrophysical scenario.

2. Logics of informative kinetic study
for a homogeneous turbulent plasma
The most logical reasoning in plasma kinetic
studies was developed through our former researches,
and its diverse aspects were enlightened rather well in
papers [31, 32, 34] (see also Refs. [16, 17, 33, 35]). We
formulate below only its milestones.
The basis of full plasma description is the
Klimontovich -Dupree equation:
µ
¶
∂Nα
∂Nα
+ v·
+
∂t
∂r
¶
µ
∂Nα
1
= 0.
(1)
eα E + [v × B] ·c∂p
It represents in a cumulative form the motion
equations of all the individual charged plasma
particles: The trajectories of charged particles are thecharacteristics of this partial dierential equation.
For the nonmagnetized plasma with Langmuir
turbulence, the terms with magnetic elds can beomitted. Then the electric eld E is governed byequation ∂E/∂t = −4πj , with current density j(r, t)
being the standard integral of microdistributions
P 3
Nα =δ (r − ri (t))δ 3 (p − pi (t)).
i

The microdistribution cannot be rendered asa constructive notion of the theory: it depends
substantially on the positions and momenta of allplasma particles that are never known to a full extent.
Besides, the integration of full plasma description
is technically infeasible, because of large numbersof the particles. For eliminating dependencies on
positions and momenta of individual particles, thenecessary statistic of the microdistribution, known
as a distribution function fα = hNα (r, p, t)i, was

4 Particularly, the reconsideration of theory fundamentals

might have been useful for diverse items of nuclear fusion
research, e.g. for studies on the magnetized target fusion [27,28]
and for studies on the inertial connement fusion [29, 30].
5 See details in Ref. [17].

61V.I. Erofeev, Key Ideas for Heightening the Informativeness of Plasma Physical Theorizing

and of nal expressing the RHS of Eq. (2) and ∂nk /∂t
in terms of wave spectral density and distributionfunctions. The choice of the leading order of two-timecorrelation function is the key problem here. Let us
comment it a bit.
As we have formerly stated, the only consistentgoal of any plasma calculation is to approximate
as better as possible the current tendency ofplasma macrophysical evolution. The corresponding
computing should use predominantly the featuresof the plasma state at the current moment andat the relatively recent past. (Really, the further
back in time the plasma state one considers, theless denite is the inuence of that state on thecurrent plasma evolution. This is a consequence ofthe lack of detailed information on positions andmomenta of individual plasma particles.) Meanwhile,the customary frequency harmonics of electromagnetic
eld and distribution functions suppose weighting ofparticle distributions and electromagnetic eld at alltime axis −∞ < t < ∞ (semi-axis 0 < t − t0 < ∞in the case of Laplace transformation) without any
discrimination of the time remote states. Note alsothat the period of the plasma existence is rather oftena nite one, especially in laboratory installations, andthen the appeals to Fourier transforms imply the
supplementing of real plasma history by nonexistentplasma states. Figuratively speaking, the researcher
manually introduces extra entropy into the plasma
description when he operates by temporal Fourier(Laplace) transforms of electromagnetic eld andparticle distributions.
Thus, it is more natural to develop the lowest
b k (t, t0 )order approximation of correlation function Φ
stemming from the original two-time representationof this function rather than from its frequency
spectrum. (Note that it is both convenient andwell appropriate to perform the transition to spatial
Fourier transformof the two-time correlation function,Rb t0 ) = Φ(R,b
Φ(t,t, t0 ) exp(−ik · R)d3 R. In view ofnal correlation radius of the turbulent wave eld, thisfunction is dened unambiguously at any moment.)
In the case of stationary plasma the representationb k (t, t0 )of certain plasma natural oscillation in Φ
should obey harmonic oscillations following the wave
dispersion law ωk (a nonlinear one generally):Xb (0) (t, t0 ) ≡ Φ
b (0) (t − t0 ) =esk ⊗ E
e∗ ×
Φ
E
sk
k
k

traditionally developed by the plasma ensemble
averaging. With refraining from the latter, one
is forced to substitute it by a contextuallyoriented averaging in phase space : An appropriatearrangement of the averaging depends essentially on
the plasma physical problem under consideration.
Thus, for a homogeneous plasma with Langmuir
turbulence one can average over 6-dimensional
parallelepipeds with extended spatial dimensions: At
the expense of them one can reach appropriate
small momentum gradations of a statistically reliable
particle distribution fα (p, t).
Function fα evolves according to equation

∂
∂fα
= −eα
hE(r, t)Nα (r, p, t)i .
∂t
∂p

(2)

That is, the distrubution fα is advanced in time byfunction [N E]α (R, p, t, t0 ) = hNα (r, p, t)E(r + R, t0 )ithat we call a two-point correlation function.
The latter is advanced in time by a three-point
correlation function [N EE]α (R, R0 , p, t, t0 , t00 ) =hNα (r, p, t) E(r + R, t0 ) ⊗ E(r + R0 , t00 )i, etc. (Sign⊗ symbolizes the direct tensor product.) The
corresponding chain of evolution equations is innite.
It can be truncated in the case of a weakly turbulent
plasma at any reasonable order, and then one reduces
the problem to a study of coordinated evolution of the
distributions fe,i (p, t) and the two-time correlationbfunction Φ(R,t, t0 ) = hE(r, t) ⊗ E(r + R, t0 )i. Notethat the latter is conceptually the solution to Maxwell
equations whereat the charges and charge currents
are respective integrals of the two-point correlation
functions. In our case of potential Langmuir wave eld,

b
∂Φ
=
∂t
X Z
−4π
eα vα (p) ⊗ [N E]α (R, p, t, t0 )d3 p (3)α(here vα (p) is the particle nonrelativistic velocity).
We comment that within above truncated plasma
description, the integrand of two-point correlation
function can be iteratively expressed in termsof the two-time correlation function itself. The
corresponding calculations were properly described inRef. [31] and afterwards reconsidered in Ref. [34] forgaining the accuracy of plasma consideration. We notethat the calculations generate equations that resemble
the canonical redaction of Wyld diagram technique[36] in its two-time reading.
The remaining path to informative description of
the plasma evolution consists of selecting the leading
order of the two-time correlation function, of theiterative expressing the two-time correlation function
through its cumulant of a wave spectral density nk(a characteristic of the leading order of the function),s=±× exp(−isωsk (t − t0 ) − γsk (t − t0 )).
ek is the amplitude of the wave, and γk is its(Here E
damping rate.) A natural generalization of above form
for the common case of evolving plasma isX
b (0) (t, t0 ) = k ⊗ k×Φkk 2 s=±

62

V.I. Erofeev, Key Ideas for Heightening the Informativeness of Plasma Physical Theorizing

µ
¶

Rt
Rt

0

n
(t
)
exp
−is
ω
(τ
)dτ
−
γ
(τ
)dτ
,

sk
sk
sk


t0
t0


 t > t0 ,
Ã 0
!
t
t0
R
R


 nsk (t) exp is ωsk (τ )dτ − γsk (τ )dτ ,



t
t


t < t0 .

¯¯¯ 2(k · v) 1 + ²i (k2 , ω 00 ) ¯2
¯
¯
+
¯ ωk
²(k2 , ω 00 ) ¯

The notation nk (t) stands for the wave spectral
density. Note that structure (4) of the solution isb k (t, t0 ), and it
dictated by the evolution equation of Φ
complies with the causality principle. In more details
we discussed this in Ref. [31].

The beginnings of above approach were rstexemplied on a problem of three-wave interactions
for a drift wave turbulence [31, 32]. In corresponding
research, the usual understanding of three-wave
phenomena was conrmed [32].
Papers [1618] unroll the physics of Langmuir
wave dissipation in a turbulent plasma due to thenonresonant electron diusion in momentums, anduncover the most important consequences of thisphenomenon. Let us specify this in a more expanded
manner. In Ref. [16], we have shown that respective
wave dissipation is intense enough for precluding theformation of Langmuir wave condensate, especially at
intense pumping of weakly turbulent short-wavelength
Langmuir turbulence. Ref. [18] was devoted todemonstration of impossibility of Vedenov-Rudakov's
long-wavelength modulational instability of a plasma
with intense Langmuir turbulence [37]. Besides,
paper [18] supplements the study of Ref. [17] inshowing impossibility of Zakharov's short-wavelength
plasma modulational instability in a turbulent plasma.
Together, these two papers state the impossibility ofLangmuir wave collapses in a turbulent plasma andthus explain the non-occurrence of Langmuir collapses
in above cited beam-plasma experiments of Wong's,
Vyacheslavov's and Benford's teams.
In paper [33], we reconsidered the physics
of Langmuir wave scattering induced by plasma
electrons. We shall correct a bit the nal result of thestudy. In terms of our nk , the rate of wave scatter is
¯
Z
¯
2πωnk
4πe4
∂nk
¯
=
d3 k1 ×∂t
Re ∂(ω²(k, ω))/∂ω ¯ω=ωk m3e½00

δ(ω − (k2 · v))

(5)

(6)

(see, e.g., Refs. [38, 39]).
Regarding the very picture of the phenomenon
as compared to its traditional analog, we shall say
the following. The fair quantitative coincidence oftraditional and new images of the scatter takes
place for long waves (λ À (mi /me )2/5 rD ). Further,intensities of two scatters for short Langmuir waves(rD ¿ λ ¿ rD (mi /me )1/3 ) have the same scaling
in the case of maxwellian isotropic plasma electron
distribution. But new scatter of short Langmuir waves,
unlike to former one, yields here the Langmuir quanta
decay: In some elementary scatters both the scatteringand the scattered wave quanta help each other todeliver a piece of their energy to bulk plasma electrons.
Finally, the intensity of traditional scatter of above
specied short Langmuir waves in a plasma with afairly nonisotropic electron distribution constitutes apiece of the order rD /λ of new (real) wave scatter
intensity. That is, at this case traditional approaches
veil the bulk of the short wave scatter.
In paper [35], the physics was considered ofelectron redistribution in momentums during related
scatter of Langmuir waves. We have shown therethat traditional theory undervalues rate of change ofelectron distribution in all possible situations.

3. Final deductions developed withnovel correlation analysis of plasma
kinetics(k · k1 )2 nk1
k 2 k12 ωk4

ω =ωk −ωk1

µ
¶
∂fe
k2 ·
.
∂v

Note that the traditional analog of this equationdiers with the given one in content of braces: Former
images of the phenomenon assumed it to be(4)

d3 k2 δ 3 (k − k1 − k2 )

¯
¯
¯
¯ 00

2 ¾¯

(k · v)10
ωk2

4. General comments on informativeplasma theoretical studies
Narration of Sections 2,3 was oriented on theeasiest problem of an evolving homogeneous weakly
turbulent plasma. Meanwhile, the very homogeneous
plasma turbulence is nothing but an abstractionthat hardly can be evoked in real plasmas byany external inuence. Say, when one tries to heata plasma by an electron beam, he(she) observes
spatially inhomogeneous beam-plasma interaction.
Besides, the latter process depends essentially on
dynamics of the beam injection. On the other side,
the weakly turbulent plasma phenomena are not the
only ones that yield obtaining of any informativenal theoretical conclusions. In general, the revealingof plasma contexts that possess this property, andthe developing of theoretical means for inferringrespective nal deductions, should constitute an
extremely important component of plasma research.
We stress that there are no any universal calculation

Z
d3 v×
2

(k · v)|1 + εi (k2 , ω 00 )|+4+|ε(k2 , ω 00 )|2ωk63

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[22] J.G. Kirkwood Journ. Chem. Phys. 14, 180 (1946).
[23] J. Yvon, La theorie des uids et l'equation d'etat:
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S. Brush (Univ. California Press, Berkeley, 1964).
[25] Yu.L. Klimontovich, Zh. Exp. Teor. Fiz. 33, 982(1957); The Statistical Theory of Nonequilibrium
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technique : each of plasma problems dictate its own
logics of obtaining informative nal conclusions.
The easiest illustration to this thesis constitute theproblem of thermodynamic relaxation of a weakly
nonideal nonequilibrium homogeneous plasma. In sucha plasma, the particles exercise Coulomb collisions,which lead to the plasma thermalization. Similarly tophenomena in a collisionless weakly turbulent plasma,
this process permits developing of its somewhat
informative scenario. At the same time, the above
concept of plasma natural oscillation (the leading
solution (4)) and respective fashion of its treating
are useless here. The consideration of Coulomb
collisions requires diering lowest order approximationbof Φ(R,t, t0 ), which supposes respective change infurther calculation organization. One more illustrationis the plasma leakage from a tokamak: Toroidal
geometry of plasma volume dictates respective fashionof the phase space averaging.
Finally, the possibility of developing informative
conclusions depends essentially on the order of
expansion parameter in the perturbation expansion:
No informative nal deductions can be developed
without smallness of that parameter. (Really,
successive iterations diverge then from the very
beginning.) With expansion parameter ε, the most
optimal order of the expansion is about 1/ε. Up
to this top level, the adding of extra orders leads
to enlarging the time interval of reliability of therespective plasma scenario. In the plasma turbulencecase, the expansion parameter is the ratio of nonlinear
damping rate γ nl to the turbulence dispersion innatural frequencies ∆ω , then conclusions on current
plasma evolution up to n-th order are reliable up
to time delays ∼ (∆ω n−1 )/(γ nl )n . Similarly, inother cases the parameter ε constitutes a ratio of
two characteristic inverse times, with analogous
estimation of the period of the scenario reliability.
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