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VOLUME 88, NUMBER 25

P H YS IC AL R E VI EW LETTE R S

24 JUNE 2002

Fluctuation-Dissipation Relations for a Nonlocal Plasma
V. V. Belyi
IZMIRA N, Troitsk, Moscow Region, 142190 Russia and International Solvay Institutes for Physics and Chemistry, ULB-CP 231, 1050 Brussels, Belgium (Received 19 March 2002; published 6 June 2002) A generalized version and time is derived. In a of the electrostatic field asymmetry with respect of the Callen-Welton fluctuation-dissipation formula that is nonlocal in space nonuniform plasma there appear significant differences between the fluctuations and those of the electron density, and the spatial inhomogeneities lead to an to the sign inversion of the frequency.
PACS numbers: 52.25.Dg, 05.40. ­ a, 05.10.Gg, 52.25.Gj

DOI: 10.1103 / PhysRevLett.88.255001

The study of fluctuations attracts a great deal of attention. Besides being of interest from a fundamental point of view, there are situations for which nonequilibrium fluctuations play an important role, namely, in the neighborhood of bifurcations where the system has to choose a branch [1]. Moreover fluctuations find applications in diagnostic procedures. Indeed, plasma parameters such as temperature, mean velocity, density, and their respective profiles can be determined from incoherent (Thomson) scattering diagnostics [2], i.e., by the proper interpretation of the data obtained from the scattering of a given electromagnetic field interacting with the system. The key point for this interpretation is the knowledge of the intensity of the dielectric function fluctuations or equivalently of the electron form factor. The electron form factor and the electrostatic field fluctuations have been the object of active investigation since the early 1960s [2]. In thermodynamic equilibrium, the electrostatic field fluctuations satisfy the famous Callen-Welton fluctuation-dissipation theorem [3], linking their intensity to the imaginary part of the dielectric function and to the temperature. The matter becomes more delicate in the local-equilibrium case. We have indeed shown that in the collisional regime the Callen-Welton formula should be revised [4]. There then appear new terms explicitly displaying dissipative nonequilibrium contributions and containing the interparticle collision frequency, the differences in the temperatures and the velocities, and also functions of the real parts of the dielectric susceptibilities. However, it is not evident that the plasma parameters can be kept constant in both space and time. Inhomogeneities in space and time of these quantities will certainly also contribute to the fluctuations. In this Letter, using the Langevin approach and the time-space multiscale technique, we show that not only the imaginary part but also the derivatives of the real part of the dielectric susceptibility determine the amplitude and the width of the spectral lines of the electrostatic field fluctuations, as well as the form factor. As a result of the inhomogeneity, these properties become asymmetric with respect to the inversion of the sign of the frequency. In the kinetic regime the form factor is more sensitive to space gradients than the spectral function of the electrostatic field fluctuations. Note that for 255001-1 0031 - 9007 02 88(25) 255001(4) $20.00

simple fluids and gases a general theory of hydrodynamic fluctuations for nonequilibrium stationary inhomogeneous states has been developed in a series of publications [5]. In particular, it has been found that there exists an asymmetry of the spectrum for Brillouin scattering from a fluid in a shear flow or in a temperature gradient. The situation for the plasma problem we are considering is, however, quite different. To derive nonlocal expressions for the spectral function of the electrostatic field fluctuation and for the electron form factor, we use the Langevin approach to describe kinetic fluctuations [6]. The starting point of our procedure is the same as in [4]. A kinetic equation for the fluctuation d fa of the one-particle distribution function (DF) with respect to the reference state fa is considered. In the general case the reference state is a nonequilibrium DF which varies in space and time both on the kinetic scale 2 (mean-free path lei and interparticle collision time nei 1 ) and on the larger hydrodynamic scales. These scales are much larger than the characteristic fluctuation time v 21 . In the nonequilibrium case we can therefore introduce a small parameter m nei v , which allows us to describe fluctuations on the basis of a multiple space and time scale analysis. Obviously, the fluctuations vary on both the "fast" r , t and the "slow" mr , mt time and space d fa x, t , mt , mr and fa x, t scales: d fa x, t fa p , mt , mr . Here x stands for the phase-space coordinates r , p . The Langevin kinetic equation for d fa has the form [4,6] b La
xt S dfa x, t 2 d fa x, t

2ea d E r , t ?

fa x, t , p (1)

where ea is the charge of the particle of specie a, d E b is the electrostatic field fluctuation, and the operator Laxt b a x,t ; Ga x, t b b a xt is defined by L t 1 v ? r 1 G ba , and dIa is the linearized collision operab ea E ? p 2 dI S tor. The Langevin source d fa in Eq. (1) is determined by S b the following equation [4,6]: Laxt d fa x, t d fb x0 , t 0 dab d t 2 t 0 d x 2 x0 fb x0 , t 0 . The solution of Eq. (1) © 2002 The American Physical Society 255001-1


VOLUME 88, NUMBER 25 has the form d fa x, t
S d fa x, t 2

P H Y S IC AL R E VIEW LETTER S

24 JUNE 2002

XZ
b

d x0

Z

t

dt
2`

0

fb x0 , t 0 , p 0 (2) 00 b where the Green function Gab x, t , x ,t of the operator Laxt 0 b axt Gab x, t , x0 , t 0 is determined by L dab d x 2 x 3 d t 2 t 0 with the causality condition: Gab x, t , x0 ,t 0 0, S when t , t 0 . Thus, d fa x, t d fb x0 , t 0 and Gab x, t , x0 ,t 0 S are connected by the relation: d fa x, t d fb x0 , t 0 Gab x, t , x0 , t 0 fb x0 , t 0 , t . t 0 . For stationary and spab tially uniform systems the DF fa and the operator Ga 3 Gab x, t , x0 ,t 0 eb d E r 0 , t 0 ?

do not depend on time and space. In this case, the dependence on time and space of the Green function Gab x, t , x0 ,t 0 appears only through the difference t 2 t 0 and r 2 r 0 . However, when the DF fa p , mr , mt and b Ga p , mr , mt are slowly varying quantities in time and space, and when nonlocal effects are considered, the time and space dependence of Gab x, t , x0 ,t 0 is more subtle: Gab x, t , x0 , t
0

G

ab

p , p 0 , r 2 r 0 , t 2t 0 , mr 0 ,mt 0 .

d fa x, t

(3) For the homogeneous case this nontrivial result was obtained for the first time in our previous work [7]. Recently, this result was extended to inhomogeneous systems [8]. Here we want to stress that the nonlocal effects appear due to the slow time and space dependences mr 0 and mt 0 . At first order, the expansion with respect to m, Eq. (2) leads to µ Z` XZ S 0 2 mr ? d fa x, t 2 eb d p d r d t 1 2 mt mt mr 0 b
ab

3G with r r 2 r 0 and t t 2 t0. From the Poisson equation

r , t , p , p 0 , mt , mr d E r 2 r , t 2 t ?

fb p 0 , mt , mr , p 0

(4)

dE r, t

2

1 X Z d fb x0 , t d x0 , eb r b jr 2 r 0 j

(5) R` R

and performing the Fourier-Laplace transformation for the fast variables d E k, v 0 dt d r d E r , t exp 2Dt 1 i v t 2 i k ? r , we have µ Z X k b 21 fa p , r , t 2 s d E k, v , t , r d E k, v 1 2i ? . L 4p iea d p 1 1 i d E k, v , r , t ? v t k r k 2 av k p a (6) Here and in the following for simplicity we omit m, keepP ´ v, k 1 1 a xa v , k , ing in mind that derivatives over coordinates and time are µ taken with respect to the slowly varying variables. The 21 e xa v , k 2i ? xa v , k, t , r , 11i b resolvent Lav k in Eq. (6) is determined by the following v t r k R R` 21 b d r 0 d t exp 2Dt 1 relation: Lav kdab d p 2 p 0 (8) i vt 2 i k ? r Gab r , t , p , p 0 , mt , mr . One should bear 4p ie2 R 21 a b av kk ? 3 dp L 2 k2 and where xa v , k, t , r p in mind that the derivatives v and k in Eq. (6) act only fa p , t , r is the susceptibility for a collisional plasma. k b 21 on the operator k 2 Lavk . The approximation in which In the same approximation the spectral function of the Eq. (6) was derived corresponds to the geometric optics S Langevin source d Ed E v ,k takes the form approximation [9]. At first order and after some manipuZ X lations, one obtains from Eq. (6) the transport equation in S 2 32p 2 ea Re d p d Ed E v ,k the geometric optics approximation, which is not consida µ ered in the Letter, and the equation for the spectral function of the electrostatic field fluctuations: 2i ? 3 11i v t k r 1 S 1 b 21 Re´ v , k d Ed E v ,k 2 d Ed E v ,k 0, 3 2 Lav kfa p , r , t . (9) j´ v , k j2 e k (7) If Re´ v , k fi 0, it follows from Eqs. (7) and (9) that P e where we introduced ´ v , k e 1 1 a xa v , k , the spectral function of the nonequilibrium electrostatic field fluctuations is determined by the expression P2R 1 b 21 2 32p a ea Re d p 1 1 i v t 2 i k ? r k 2 Lav k fa p , r , t d Ed E v ,k . (10) j´ v , k j2 e 255001-2 255001-2


VOLUME 88, NUMBER 25

P H Y S IC AL R E VIEW LETTER S 11i
v t

24 JUNE 2002 ?
r 1 k2

X 8pQa e d Ed E v ,k Imxa v , k . (12) ? Re´ v , k, t , r (11) e va j´ v , k j2 a k r determines the width of the spectral lines near the resoIn this case the small parameter m is determined on the nance. Note that when expanding the Green function in slower hydrodynamic scale. For the case of equal temperaEq. (4) in terms of the small parameter m, there appear 0, one obtains a generalized expression for tures and Va additional terms at first order. It is important to note that the Callen-Welton formula: the imaginary part of the dielectric susceptibility is now replaced by the real part, which is greater than the imagie 8pQ Im´ v , k . (13) d Ed E v ,k nary part by the factor m21 . Therefore, the second and v j ´ v , k j2 e third terms in Eq. (11) in the kinetic regime have an effect comparable to that of the first term. At second order in the To calculate explicitly d Ed E v ,k we will restrict our expansion in m the corrections appear only in the imagianalysis to the vicinity of the resonance, i.e., v 6v0 , nary part of the susceptibility, and they can reasonably be where Re´ v0 , k 0. We can develop ´ v , k e v2 neglected. It is therefore sufficient to retain the first order 2 Re´ v0 sgnv v bv v0 sgnv 1 i Im´ 1 v t 2 k ? r 3 corrections to solve the problem. g e For the local equilibrium case where the reference R 3 Re´ bv v0 sgnv . Thus d Ed E v ,k v2v0 sgnv 2 1g 2 e state fa is Maxwellian, we have the identity, d p 3 8p T v Re´ v b¡ v0 , where v 2 Re´ e g Im´ 1 Re´ 2 ? Re´ (14) v t k r v v v0 sgnv is the effective damping decrement. For the case where the 2 2 vL nei 4p ne 2 Q kD 2 system parameters are homogeneous in space but vary in Im´ v 2 v , vL m m , and time, the correction is still symmetric with respect to the ¡ vL n 2 n change of sign of v , but the intensities and broadening are e 16 2 k? sgnv g nei 1 2. (15) different, and the intensity integrated over the frequencies n t r nkD remains the same as in the stationary case. However, when 2 n 6v n the plasma parameters are space dependent this symmetry On the hydrodynamic scale j n t j, j nk 2L k ? r j , nei , and D is lost. In the same manner as for simple fluids and gases e g . 0. [5] the spectral asymmetry is related to the appearance of For the spatially homogeneous case there is no differspace anisotropy in inhomogeneous systems. The real part ence between the spectral properties of the longitudinal of the susceptibility Re´ is an even function of v . This electric field and of the electron density. They are conproperty implies that the contribution of the third term nected by the Poisson equation. This statement is no longer to the expression of the damping decrement (14) is an valid when considering an inhomogeneous plasma. Indeed odd function of v . Moreover this term gives rise to an the longitudinal electric field is linked to the particle denanisotropy in k space. Let us estimate this correction for sity by the nonlocal Poisson relation (5). In the latter case, 2 2 kQ v vL Re´ 1 2 vL 1 1 3 mv 2 , the plasma mode v0 2 an analysis similar to that made above can also be performed for the particle density. From Eq. (2) there follows S d na k, v , r , t d na k, v , r , t µ X 4p i keb ea Z b 21 fa p , r , t dp 1 1 i 2i ? . (16) 1 Lav kd nb k, v , r , t ? 2 k v t k r p b b 21 One should remember that now the derivatives v and k in Eq. (16) act only on the operator Lav k. At the first order approximation and after some manipulations, one obtains the following expression for the electron form factor for a two-component (a e, i ) plasma: ã ã ã ã z}|{ z}|{ z}|{ 2ne k 2 1 1 xi v , k 2 z}|{ xe v , k 2 Qi 2ne k 2 Im xe v , k 1 z}|{ (17) d ne d ne v ,k z}|{ 2 2 Im xi v , k , ve kD ´ v, k ´ v , k Qe vi kD where we used for local equilibrium the following exz}|{ 1 Qa k 2 S 1 1 i v t 2 i k 2 ri kj ki kj xa v , k, t , r . xa v , k dab va 2p ea 3 pression for the "source" d na d nb v ,k 2 P z}|{ z}|{ z}|{ z}|{ Im xa v , k , and ´ v, k 1 1 a xa v , k : As above we can expand ´ v , k near the plasma 2 255001-3 255001-3

e The effective dielectric function ´ v , k in the denominator of Eq. (10) determines the spectral properties of the electrostatic field fluctuations and its imaginary part e Re´ v , k, t , r Im´ v , k Im´ v , k 1 v t

2i

e xa v , k (va v 2 kV a , and Q fa p , t , r 2 is the temperature in energy units), and Eq. (10) takes the form

k i Qa 2 va 4p ea

b 21 Lav kfa p , t , r

i v

R
a

dp 3


VOLUME 88, NUMBER 25

P H Y S IC AL R E VIEW LETTER S

24 JUNE 2002

between the vectors of induction and the electric field. This phase shift results from the finite time needed to set the polarization in the plasma with dispersion [9]. Such a phase shift in the plasma with space dispersion appears due to the medium inhomogeneity. These results are important for the understanding and the classification of the various phenomena that may be observed in applications; in particular, the asymmetry of lines can be used as a diagnostic tool to measure local gradients in the plasma. Fruitful discussions with M. Tlidi are gratefully acknowledged. This research was supported by the Russian Foundation for Basic Research (Grant No. 00-02-17139).

FIG. 1. The electron form factor d ne d ne v ,k (solid line) and the spectral function of electrostatic field fluctuations d Ed E v,k (dashed line) as a function of frequency. 2 nei nkD kD n 6. k ? r 54vL ; k

vL . Thus, for the electron line, dne 3 z}|{ g 2ne k 2 d ne v , k 2 z}|{ v kD Re´ v bv vL , where 21 2 v2sgnv L g z}|{ 1 2 Re´ 22 kj g Im´ 1 t v k ri ¡ Re´ 3 kj Re´ (18) ki v v vL sgnv is the effective damping decrement for the electron form factor. At this stage of calculation, let us note that the damping decrements for the electrostatic field fluctuations [Eq. (14)] and for the electron density fluctuations [Eq. (18)] are not the same. The origin of this difference is that the Green function for electrostatic field fluctuation and density particle fluctuations are not the same. This property holds only in the inhomogeneous situation. An estimation for theplasma mode is then z}|{ vL 2 n 1 k g nei 1 n t nk 2 ¡ µ 6k 2 n 2. (19) 1 1 2 sgnv ? r kD From this equation we see that the inhomogeneous correction in Eq. (19) is greater than the one in Eq. (15) by 2 the factor 1 1 kD 6k 2 . For the same inhomogeneity, i.e., the same gradient of the density, we plot the form factor d ne d ne v ,k together with the d Ed E v ,k as functions of frequency (Fig. 1). This figure shows that the asymmetry of the spectral lines is present both for d ne d ne v ,k and for d Ed E v ,k . However, this effect is more pronounced in d ne d ne v ,k than in d Ed E v ,k. We have shown that the amplitude and width of the spectral lines of the electrostatic field fluctuations and form factor are affected by new nonlocal dispersive terms. They are not related to Joule dissipation and appear because of an additional phase shift resonance v 255001-4

[1] P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stabilit y and Fluctuations (Wiley, New York, 1971); G. Nicolis and I. Prigogine, Nonequilibrium Systems. From Dissipative Structure to Order Through Fluctuations (Wiley, New York, 1979); W. Horsthemeke and R. Lefever, Noise-Induced Transitions (Springer-Verlag, Berlin, 1984). [2] J. P. Dougherty and D. T. Farley, Proc. R. Soc. London A 259, 79 (1960); W. Thompson and J. Hubbard, Rev. Mod. Phys. 32, 716 (1960); J. Sheffield, Plasma Scattering of Electromagnetic Radiation (Academic, New York, 1975); A. Akhiezer, I. Akhiezer, R. Polovin, A. Sitenko, and K. Stepanov, Plasma Electrodynamics, Linear Theory (Pergamon, Oxford, 1975), Vol. 1. [3] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951). [4] V. V. Belyi and I. Paiva-Veretennicoff, J. Plasma Phys. 43, 1 (1990). [5] I. Procaccia, D. Ronis, M. A. Collins, J. Ross, and I. Oppenheim, Phys. Rev. A 19, 1290 (1979); D. Ronis, I. Procaccia, and I. Oppenheim, Phys. Rev. A 19, 1307 (1979); Phys. Rev. A 19, 1324 (1979); I. Procaccia, D. Ronis, and I. Oppenheim, Phys. Rev. A 20, 2533 (1979); J. Machta, I. Oppenheim, and I. Procaccia, Phys. Rev. A 22, 2809 (1980); A.-M. S. Tremblay, M. Arai, and E. D. Siggia, Phys. Rev. A 23, 1451 (1981); T. R. Kirkpatrick, E. G. D. Cohen, and J. R. Dorfman, Phys. Rev. A 26, 972 (1982); 26, 97 (1982); V. V. Belyi, Theor. Math. Phys. 58, 421 (1984). [6] B. B. Kadomtsev, Sov. Phys. JETP 32, 934 (1957); S. V. Gantsevich, V. L. Gurevich, and R. Katilus, Sov. Phys. JETP 57, 503 (1969); Sh. M. Kogan and A. Ya. Shulman, Sov. Phys. JETP 56, 862 (1969); Sh. M. Kogan and A. Ya. Shulman, Sov. Phys. JETP 57, 2112 (1969); S. V. Gantsevich, V. L. Gurevich, and R. Katilus, Sov. Phys. JETP 59, 533 (1970); S. V. Gantsevich, V. L. Gurevich, and R. Katilus, Riv. Nuovo Cimento 2, 1 (1979); Yu. L. Klimontovich, Kinetic Theory for Nonideal Gases and Nonideal Plasma (Academic, New York, New York). [7] V. V. Belyi, Yu. A. Kukharenko, and J. Wallenborn, Phys. Rev. Lett. 76, 3554 (1996). [8] V. V. Belyi, Yu. A. Kukharenko, and J. Wallenborn, Contrib. Plasma Phys. 42, 3 (2002). [9] Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, Berlin, 1990); M. Bornatici and Yu. A. Kravtsov, Plasma Phys. Controlled Fusion 42, 255 (2000).

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