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Kinetic equation for a dense classical plasma
Volume 40A, number 3
PHYSICS LETTERS
17 July 1972
KINETIC EQUATION FOR A DENSE CLASSICAL PLASMA M. BAUS *
Faculty of Sciences, Universityof Brussels, Belgium Received 25 April 1972 The Balescu-Guernsey-Lenard equation for an homogeneous classical dilute electron gas is extended to higher densities.
In the traditional derivation of the Balescu-Guernsey-Lenard equation [ 1] two difficulties are immediately apparent: 1) one neglects the triple correlations by considering their equilibrium magnitude and 2) one eliminates a small-distance divergence using Landau's argument [2] by introducing a short-distance cut-off evaluated in equilibrium. In non-equilibrium situations these approximations clearly become unsatisfactory. Moreover, neglecting the triple correlations amounts to neglecting part of the collective effects which are responsible for the system's intrinsic large and small distance cut-offs. Indeed, part of the triple correlation contribution corresponds to a screening between the colliding particles produced by the nearby field particles just as the screening produced by the distant field particles is taken into account through the Vlassov terms appearing in the second member of the B.B.G.K.Y. hierarchy. This triple correlation effect is important for distant colliding particles for which it will introduce a cut-off at the mean free path, but also for colliding particles which are close because they now move in the force field of nearby field particles. Further, triple correlations can certainly not be ne, glected for those charged systems for which the number of particles in a Debye sphere is not large such as dense systems, or systems which have strong correlations such as turbulent plasmas and the electron liquids encountered in solid statephysics [3]. We now report some results Obtained using as guide the recent progress made in dense neutral gas theory by Pomeau [4]. The system of equations formed by the first three members of the B.B.G.K.Y. hierarchy for the correlation functions [5] is solved under the conditions we can 1) neglect four particle correlations * Charg6 de recherches du Fonds National Beige de la Recherche Scientifique.
because we do not want to retain correlations between field particles, 2) neglect the term nonlinear in the binary correlations in order to be able to solve the system of equations, 3) use the adiabatic hypothesis, 4) neglect interactions between the colliding particles but not between a colliding and a field particle because we assume the particles to be weakly coupled to each other but not to the medium as a whole, i.e., collective effects dominate, 5) take the long time limit for a stable system. The resulting kinetic equation for a homogeneous electron gas reads then:
af(p;t)_ f f bt n dp _~
d6ofdk 2rr a87r3
IVkl2
k" 3
le(k,w)l 2 0 - p
× {Re Pk(vr,co) Im Fk(o, 6o)-RePk(o,co) Im F.k(o t ,co)} . (1) In eq. (1) Re and Im denote respectively the real and imaginary part and moreover:
ek(O ,CO) =[ik "O--iog+Ck(O,60)]-1 f(p: t)
(2a)
Fk(o,w) = [ik.o-iw+ Ck(o,o~)]-I ik.-~ f(p;t) (2b) e(k, w) = 1 + n f dp Vk Fk(O, oo)
(2c)
where Ck(o, oo) is a finite frequency collision operator for an inhomogeneous system but linearized around f(p; t). The remaining notations are standard: n is the number desity, p = mu, Vk is the Fourier transformed Coulomb potential, etc. The kinetic eq. (1) conserves the total number, momentum and kinetic energy of the particles. An H-theorem can be proven for (1) each time the hermitian part of Ck(~, co) is negative definite, i.e. when the system is stable, and moreover the H-quan213
Volume 40A, number 3
PHYSICS LETTERS
tity becomes stationary for a maxwellian. Eq. (1) clearly reduces to the Balescu-Guernsey-Lenard equation when we substitute a priori +0 for Ck(O,w) in (2). However for Ck(o , w) 4 : 0 the behaviour of the functions defined in (2) will be very different from their Balescu-Guernsey-Lenard counterpart. Indeed, the effective potential (see (2c)) is now no longer evaluated on the basis of a reversible Vlassov equation but contains now correlational effects (compare with ref. [3]) whereas ( 2 a - b ) indicate the sharp wave-particle resonance of the Balescu-Guernsey-Lenard equation is broadened. Moreover individual particle and transport modes at finite k and w can now be excited in the system. Needless to say that in order for eq. (1) to be closed we have to evaluate (2) which can be done explicitly only by model considerations. To get some insight into (1) we have used a zero frequency, infinite wavelength, Fokker-Planck model [6] for Ctc(o,co) in which case we can show explicitly that eq. (1) conserves the non-negative character o f f ( p ; t) whereas the k-integral now converges for large Ik [ proving the conjecture, recently made by the author [7], that collisions with field particles for intermediate range ira-
214
17 July 1972
pact parameters will modify completely the binary collision analysis of the small distance behaviour of the Balescu-Guernsey-Lenard equation. Finally, with this model the dependence of (1) on the plasma expansion parameter becomes non-analytic. Details about the statements made here will be published elsewhere. We thank R. Balescu, A. Grecos and P. R6sibois for their comments.
References [ 1] D. Montgomery and D. Tidman, Plasma kinetic theory (Mc. Graw-Hill, N.Y. 1964). [2] L.D. Landau, Phys. Z. Soviet Un. 10 (1936) 154. [3] K.F. Berggren, Phys. Rev. A1 (1970) 1783; S. Ichimaru, Phys. Rev. A2 (1970)494; Phys. Fluids 13 (1970) 1560. [4] Y. Pomenau, Phys. Rev. A3 (1971) 1174, Phys. Lett. 27A (1968) 601. [5] E.A. Frieman and R. Goldman, J. Math. Phys. 8 (1967) 1410. [6] J.P. Dougherty, Phys. Fluids 7 (1964) 1788. [7] M. Baus, Ann. Phys. 62 (1971) 135.
PHYSICS LETTERS
17 July 1972
KINETIC EQUATION FOR A DENSE CLASSICAL PLASMA M. BAUS *
Faculty of Sciences, Universityof Brussels, Belgium Received 25 April 1972 The Balescu-Guernsey-Lenard equation for an homogeneous classical dilute electron gas is extended to higher densities.
In the traditional derivation of the Balescu-Guernsey-Lenard equation [ 1] two difficulties are immediately apparent: 1) one neglects the triple correlations by considering their equilibrium magnitude and 2) one eliminates a small-distance divergence using Landau's argument [2] by introducing a short-distance cut-off evaluated in equilibrium. In non-equilibrium situations these approximations clearly become unsatisfactory. Moreover, neglecting the triple correlations amounts to neglecting part of the collective effects which are responsible for the system's intrinsic large and small distance cut-offs. Indeed, part of the triple correlation contribution corresponds to a screening between the colliding particles produced by the nearby field particles just as the screening produced by the distant field particles is taken into account through the Vlassov terms appearing in the second member of the B.B.G.K.Y. hierarchy. This triple correlation effect is important for distant colliding particles for which it will introduce a cut-off at the mean free path, but also for colliding particles which are close because they now move in the force field of nearby field particles. Further, triple correlations can certainly not be ne, glected for those charged systems for which the number of particles in a Debye sphere is not large such as dense systems, or systems which have strong correlations such as turbulent plasmas and the electron liquids encountered in solid statephysics [3]. We now report some results Obtained using as guide the recent progress made in dense neutral gas theory by Pomeau [4]. The system of equations formed by the first three members of the B.B.G.K.Y. hierarchy for the correlation functions [5] is solved under the conditions we can 1) neglect four particle correlations * Charg6 de recherches du Fonds National Beige de la Recherche Scientifique.
because we do not want to retain correlations between field particles, 2) neglect the term nonlinear in the binary correlations in order to be able to solve the system of equations, 3) use the adiabatic hypothesis, 4) neglect interactions between the colliding particles but not between a colliding and a field particle because we assume the particles to be weakly coupled to each other but not to the medium as a whole, i.e., collective effects dominate, 5) take the long time limit for a stable system. The resulting kinetic equation for a homogeneous electron gas reads then:
af(p;t)_ f f bt n dp _~
d6ofdk 2rr a87r3
IVkl2
k" 3
le(k,w)l 2 0 - p
× {Re Pk(vr,co) Im Fk(o, 6o)-RePk(o,co) Im F.k(o t ,co)} . (1) In eq. (1) Re and Im denote respectively the real and imaginary part and moreover:
ek(O ,CO) =[ik "O--iog+Ck(O,60)]-1 f(p: t)
(2a)
Fk(o,w) = [ik.o-iw+ Ck(o,o~)]-I ik.-~ f(p;t) (2b) e(k, w) = 1 + n f dp Vk Fk(O, oo)
(2c)
where Ck(o, oo) is a finite frequency collision operator for an inhomogeneous system but linearized around f(p; t). The remaining notations are standard: n is the number desity, p = mu, Vk is the Fourier transformed Coulomb potential, etc. The kinetic eq. (1) conserves the total number, momentum and kinetic energy of the particles. An H-theorem can be proven for (1) each time the hermitian part of Ck(~, co) is negative definite, i.e. when the system is stable, and moreover the H-quan213
Volume 40A, number 3
PHYSICS LETTERS
tity becomes stationary for a maxwellian. Eq. (1) clearly reduces to the Balescu-Guernsey-Lenard equation when we substitute a priori +0 for Ck(O,w) in (2). However for Ck(o , w) 4 : 0 the behaviour of the functions defined in (2) will be very different from their Balescu-Guernsey-Lenard counterpart. Indeed, the effective potential (see (2c)) is now no longer evaluated on the basis of a reversible Vlassov equation but contains now correlational effects (compare with ref. [3]) whereas ( 2 a - b ) indicate the sharp wave-particle resonance of the Balescu-Guernsey-Lenard equation is broadened. Moreover individual particle and transport modes at finite k and w can now be excited in the system. Needless to say that in order for eq. (1) to be closed we have to evaluate (2) which can be done explicitly only by model considerations. To get some insight into (1) we have used a zero frequency, infinite wavelength, Fokker-Planck model [6] for Ctc(o,co) in which case we can show explicitly that eq. (1) conserves the non-negative character o f f ( p ; t) whereas the k-integral now converges for large Ik [ proving the conjecture, recently made by the author [7], that collisions with field particles for intermediate range ira-
214
17 July 1972
pact parameters will modify completely the binary collision analysis of the small distance behaviour of the Balescu-Guernsey-Lenard equation. Finally, with this model the dependence of (1) on the plasma expansion parameter becomes non-analytic. Details about the statements made here will be published elsewhere. We thank R. Balescu, A. Grecos and P. R6sibois for their comments.
References [ 1] D. Montgomery and D. Tidman, Plasma kinetic theory (Mc. Graw-Hill, N.Y. 1964). [2] L.D. Landau, Phys. Z. Soviet Un. 10 (1936) 154. [3] K.F. Berggren, Phys. Rev. A1 (1970) 1783; S. Ichimaru, Phys. Rev. A2 (1970)494; Phys. Fluids 13 (1970) 1560. [4] Y. Pomenau, Phys. Rev. A3 (1971) 1174, Phys. Lett. 27A (1968) 601. [5] E.A. Frieman and R. Goldman, J. Math. Phys. 8 (1967) 1410. [6] J.P. Dougherty, Phys. Fluids 7 (1964) 1788. [7] M. Baus, Ann. Phys. 62 (1971) 135.