A decay of electromagnetic A decay of electromagnetic wave quanta in a turbulent wave quanta in a turbulent plasma during their interaction plasma during their interaction with Langmuir waveswith Langmuir waves

EROFEEV VasilyMESHCHERIAKOV Dmitry

a) Laboratory of nonlinear physicsa) Laboratory of nonlinear physicsInstitute of Automation and Electometry Institute of Automation and Electometry

Siberian Branch of Russian Academy Siberian Branch of Russian Academy of Sciences of Sciences

b) Novosibirsk State Universityb) Novosibirsk State UniversityPhysical DepartmentPhysical Department

c) Novosibirsk State Universityc) Novosibirsk State UniversityDepartment of Information TechnologiesDepartment of Information Technologies

IA&E SB RASIA&E SB RAS

a,ba,ba,ca,c

A key objective of developing natural sciences consists in heightening the information content of conclusions regarding phenomena in surrounding world. The progress of theoretical physics in XX-th century has complied with this objective insufficiently well. An impressive illustration to this statement has been worked out by the plasma studies:

Traditional concepts of nonlinear plasma Traditional concepts of nonlinear plasma theory cannot provide appropriate level of theory cannot provide appropriate level of reliability of final predictionsreliability of final predictions.

(An InformativenessInformativeness of plasma physical of plasma physical scenarios scenarios is unacceptable lowis unacceptable low.)

The problem of wave quanta decay Traditional viewpoint: Nonlinear phenomena in

weakly turbulent plasmas conserve the total number of Langmuir wave quanta% A.S Kompaneyets, Zh. Eksp. Teor. Fiz. 31, 876 (1956).– L. M. Gorbunov, V. P. Silin, Sov. Phys. JETP 20, 135

(1965).– V.N. Tsytovich, Sov. Phys. Uspekhi 15, 632 (1973)

Alternative opinion: Langmuir waves quickly transmit their energy to bulk plasma electrons during a stochastic electron acceleration– F.G. Bass, Ya.B. Fainberg and V.D. Shapiro, Sov. Phys.

JETP 22, 230 (1966).

First reason of theory non-informativeness: The ENSEMBLE METHOD (substitution of real plasmas by plasma ensembles)““Incarnations” of the ensemble method in

plasma theory:

BBGKY plasma kinetics Hydrodynamic modelling of nonlinear plasma

phenomena Wave phase averaging Hamiltonian approaches to description of

phenomena in a turbulent plasma … …

The picture of ensemble evolution strongly depends on the ensemble content: Differing ensembles exhibit diverging interference of their evolving statistic.

A general practice in physical theorizing was to regard particular deductions on the interference of the ensemble statistics as a genuine laws of the system physical evolution.

Information-theoretical aspect of plasma description:

Impossibility to predict plasma behavior during infinite time period

The main goal: to develop reliable scenario of plasma evolution for as a longer period as possible

Absence of full data on particle positions and momentums

Careful separation of informational basis of the theory from full (never known!) plasma information

Noncompliance of plasma ensemble substitutions with above principle

Second reason of theory noninformativeness: AN ASYMPTOTIC CONVERGENCE OF SUCCESSIVE ITERATIONS

Dependence of final deductions regarding the physical laws of the plasma evolution on the lowest order approximation of the perturbation theory. Necessity of selecting the most rational choice of the lowest order approximation:

% First successive iterations of nonlinear perturbation expansion converge to conditional limit that depends on the theory leading order%Differing conditional limits stands for varying scenarios of the plasma evolution

Second reason of theory noninformativeness: AN ASYMPTOTIC CONVERGENCE OF SUCCESSIVE ITERATIONS (slide 2)

Restrictions on appropriateness of certain mathematical procedures in intermediate calculations: Fourier and Laplace transformations are fraught with deviation of final theoretical deductions from objective physics of plasma evolution:

% In computing scenario of plasma evolution, one should use predominantly the data on current plasma state and on its relatively recent past

% Temporal Forier and Laplace transforms do not discriminate data on plasma states at remote periods of time

Two earlier reasons of theory non-informativeness cannot be separated:%Had the picture of ensemble evolution not

depended on the ensemble content, one may have substantiated by ensemble variations the diversity of lowest order approximations.

%Variations of lowest order approximation within the practice of ensemble studies suppose appeals to differing ensembles; absence of dependence of plasma evolution picture on the theory leading order would have meant then the independence of the picture of the ensemble evolution on the ensemble content.

• It is necessary to gain existing practice of physical theorizing by creating new approaches that both refrain from traditional plasma ensemble substitutions and take proper account of the asymptotic nature of successive iterations.

• An approach of this type, The correlation analysis of plasma kinetics, is created for studies of turbulent plasma phenomena.

• The revealing of other plasma contexts yielding any informative final theoretical conclusions and the developing of theoretical means for inferring respective conclusions should constitute an extremely important component of further plasma research.

Principles of getting high-informativeplasma kinetic scenarios: A REFRAIN FROM THE PLASMA ENSEMBLE SUBSTITUTION

With refraining from the the plasma ensemble averaging, one is forced to substitute the latter by a contextually oriented averaging in phase space of plasma particles. Particularly, the statistic of distribution function is defined as a density of particleswithin voluminous areas of -space:

),,( tNf pr

Principles of getting high-informativeplasma kinetic scenarios: A DIRECT TIME INTEGRATION OF INTERMEDIATE EVOLUTION EQUATIONS

The direct time integration discriminates theindeterminate data on time remote plasma

statesvia the “phase mixing” within correspondingnonlinear integrals

Comments on top informativeness of final theoretical deductions

% The possibility of developing informative conclusions depends essentially on the theory expansion parameter. With expansion parameter , the most optimal order of the expansion is about . Up to this top level, the adding of extra orders leads to enlarging the time interval of reliability of respective plasma scenario. In the plasma turbulence case, the expansion parameter is

the ratio of typical wave damping rate to the width of turbulence spectrum in natural frequencies , then

conclusions on current plasma evolution up to -th order are reliable up to time delays of the order of

% Presumably, in other cases the expansion parameter should also constitute a ratio of two characteristic inverse times, with analogous estimation of the period of the scenario reliability.

/1

./)( 1 nn

n

Thermalization of electromagnetic wave quanta in a turbulent plasma

The rate of waveenergy dissipation:

Here is the quanta density of electromagnetic waves with polarization

, is the natural frequency of the wave, and are the

components of wave collision integral : , is the nonlinear wave damping rate.

Comment: Total wave energy in the unit of plasma volume is given by

integral

2

3~

2

kkkk

k penl

N NtS

Nk

tS N~

k

k Nnl kk

NtSSttN nl

NN

kkkk

k 2~

StNk

nlk

Nd kkk 33

)2(

Beam-plasma experiments Wong et al.:

% A.Y. Wong and P.Y. Cheung, Phys. Rev. Lett. 52, 1222 (1984).% P.Y. Cheung and A.Y. Wong, Phys. Rev. Lett. 55, 1880 (1985).% M.D. McFarland and A.Y. Wong, Phys. Plasmas 4, 945 (1997).% M.D. McFarland and A.Y. Wong, Phys. Rev. Lett. 84, 666 (2000).

Vyacheslavov et al:% L.N. Vyacheslavov et al.,% Proc. of the IV-th International Workshop “Strong Microwaves in

Plasmas,” Nizhny Novgorod, Inst. of Applied Physics, 1999, ed. A.G. Litvak (Nauka, Moscow, 2000) Vol. 2, p. 405

% L.N. Vyacheslavov et al., JETP Letters 75, 41 (2002).% L.N. Vyacheslavov et al., PPCF 44, B279 (2002).

Benford et al:– D. Levron, G. Benford and D. Tzach, Phys. Rev. Lett. 58, 1336

(1987).– G. Benford, X. Zhai and D. Levron, Phys. Fluids B 3, 560 (1991).– G. Benford and X.L. Zhai, Phys. Fluids B 5, 1914 (1993).

Reduction of full plasma descriptionFull plasma description = Klimontovich-

Dupree equation + Maxwell equationsMicrodistribution (Klimontovich):

The microdistribution cannot be rendered as a constructive notion of the theory: it depends essentially on the positions and momenta of all plasma particles.

))(())((),,( 33 tttN ii

i pprrpr

Distribution function:

An infinite hierarchy of evolution equations for multipoint correlation functions:

% Distribution function is advanced in time by the two-point correlation function

% The two-point correlation function is advanced in time by the three-point correlation function

% …% After the hierarchy truncation at a reasonable order, the system can be

reduced to simultaneous evolution equations of distribution function and two-point correlation function

),,( tNf pr

f

,),(),,(),,,( ttNttEN RrEprpR

),,,,,( tttNEE pRR

.),(),()( tttN RrERrE

The case of a plasma with weak Langmuir turbulence: logics of obtaining simplified kinetic description% The characteristic time of plasma and spectrum evolution,

, is great compared to inverse spectrum width in natural frequences .

% In collision integral of plasma particles, the effect of two-point correlation function can be expressed in terms of the two-time correlation function

% The two-point correlation function drives via Maxwell equations

the two-time correlation function

% Within domain , the evolution equation of two-time correlation function can be directly integrated: the function can be expressed in terms of wave spectral density .

% The resulting expression (of the two-time correlation function) can be used for obtaining time derivatives of distribution functions and spectral density

1

1T

),(),( tt RrErE

11 tt

)(tnk