# Kinetic theory of spatially homogeneous systems with long-range interactions:

III. Application to power-law potentials, plasmas, stellar systems,

and to the HMF model

###### Abstract

We apply the general results of the kinetic theory of systems with long-range interactions to particular systems of physical interest. We consider repulsive and attractive power-law potentials of interaction with in a space of dimension . For , strong collisions must be taken into account and the evolution of the system is governed by the Boltzmann equation or by a modified Landau equation; for , strong collisions are negligible and the evolution of the system is governed by the Lenard-Balescu equation. In the marginal case , we can use the Landau equation (with appropriately justified cut-offs) as a relevant approximation of the Boltzmann and Lenard-Balescu equations. The divergence at small scales that appears in the original Landau equation is regularized by the effect of strong collisions. In the case of repulsive interactions with a neutralizing background (e.g. plasmas), the divergence at large scales that appears in the original Landau equation is regularized by collective effects accounting for Debye shielding. In the case of attractive interactions (e.g. gravity), it is regularized by the spatial inhomogeneity of the system and its finite extent. We provide explicit analytical expressions of the diffusion and friction coefficients, and of the relaxation time, depending on the value of the exponent and on the dimension of space . We treat in a unified framework the case of Coulombian plasmas and stellar systems in various dimensions of space, and the case of the attractive and repulsive HMF models.

## I Introduction

The dynamics and thermodynamics of systems with long-range interactions is currently a topic of active research in physics houches ; assise ; oxford ; cdr ; proceedingdenmark ; bgm . A system is said to be long-ranged if the potential of interaction decays at large distances as with . In that case, the potential energy diverges at large scales implying that all the particles interact with each other and that the system displays a collective behavior. For such systems, the mean field approximation becomes exact in a proper thermodynamic limit cdr . In paper1 ; paper2 (Papers I and II), we have developed a general framework to tackle the kinetic theory of systems with long-range interactions and we have presented the basic kinetic equations (Vlasov, Landau, Lenard-Balescu, Fokker-Planck). These equations were initially introduced in plasma physics ichimaru ; pitaevskii ; nicholson ; balescubook and stellar dynamics spitzerbook ; bt ; hut but they are actually valid for a much larger class of systems with long-range interactions. In this paper, we consider specific applications of this general formalism. We provide explicit analytical expressions of the diffusion and friction coefficients, and of the relaxation time, for plasmas and stellar systems in various dimensions of space, and for the attractive and repulsive Hamiltonian Mean Field (HMF) models. Our approach provides a unified framework to treat these different systems. In this paper, we restrict ourselves to systems that are spatially homogeneous or for which a local approximation may be implemented. The case of spatially inhomogeneous systems is more complicated and must be treated with angle-action variables as in angleaction ; kindetail ; heyvaerts ; newangleaction ; aanew .

We first consider repulsive and attractive power-law potentials of interaction with in a space of dimension , generalizing the traditional Coulombian and Newtonian potentials (corresponding to ). For , where is the effective number
of particles^{1}^{1}1For attractive interactions, represents the number of particles
in the Jeans sphere, which is of the order of the total number of particles . For repulsive interactions
with a neutralizing background, represents the number of particles in the Debye sphere., the evolution of the system is dominated by weak collisions and collective effects. We can therefore make a weak coupling approximation and expand the equations of the BBGKY hierarchy in
powers of (see Appendix A). For , we get the Vlasov equation. At the order , the evolution of the distribution function is governed by the Lenard-Balescu equation. However, for singular potentials, the Lenard-Balescu equation may present divergences at small scales reflecting the importance of strong collisions that have been neglected in the weak coupling approximation. For power-law potentials,
there exist a critical index (for Coulombian or Newtonian interactions, it corresponds to a critical dimension ). For (i.e. for Coulombian or Newtonian potentials), strong collisions must be taken into account while collective effects may be neglected. In that case, the evolution of the system is described
by the Boltzmann equation or by a modified Landau equation (see Appendix B). The relaxation time^{2}^{2}2Here, we consider the relaxation time of a test particle in a thermal bath. For , it coincides with the relaxation time of the system as a whole. However, for spatially homogeneous systems in , the collision term vanishes and the relaxation time of the system as a whole is larger than (see discussion in Paper II). scales as , where is the dynamical time. For (i.e. for Coulombian or Newtonian potentials), strong collisions are negligible while collective effects are important. In that case, the Lenard-Balescu equation is rigorously valid. The relaxation time scales as . For (i.e. for Coulombian or Newtonian potentials), strong collisions and collective effects must be taken into account but their importance is weak. The Boltzmann equation is marginally valid provided that a large-scale cut-off is introduced at the Debye length. Similarly, the Lenard-Balescu equation is marginally valid provided that a small-scale cut-off is introduced at the Landau length. In that marginal situation, we can use the Landau equation (with appropriately justified cut-offs) as a good approximation of the Boltzmann and Lenard-Balescu equations. The relaxation
time scales as . These scalings
agree with those obtained by Gabrielli et al. gjm based on an extension of the Chandrasekhar
binary collision theory. The binary collision theory is usually adapted
to systems with short-range interactions. It can be used when and it is marginally
valid when (corresponding to 3D Coulombian or 3D
Newtonian potentials). For it produces a divergence at large scales. By contrast,
the Lenard-Balescu theory is adapted to systems with long-range interactions. It can be used when
and it is marginally valid when . For it produces
a divergence at small scales. Therefore, these two theories are
complementary to each other.

We discuss in detail the physical regularization of the divergences that occur in the Landau equation. The small-scale divergence is regularized by the effect of strong collisions. They are taken into account in the Boltzmann equation or in the modified Landau equation. For repulsive potentials with a neutralizing background (as in plasma physics), the large-scale divergence is regularized by collective effects accounting for Debye shielding. In a plasma, since the Coulomb force between like-sign (resp. opposite-sign) charges is repulsive (resp. attractive), the sign of the polarization cloud surrounding a test charge is opposite to that of the test charge. As a result, the interaction is screened on a distance of the order of the Debye length dh . Collective effects are accounted for in the Lenard-Balescu equation. This equation does not present any divergence at large scales, contrary to the Landau equation, and the Debye length appears naturally. For attractive potentials (as in stellar dynamics), the large-scale divergence is regularized by the spatial inhomogeneity of the system and its finite extent. The Jeans length jeansbook , which is of the order of the system’s size, represents a natural large-scale cut-off. Spatial inhomogeneity is accounted for in the Landau and Lenard-Balescu equations written in angle-action variables angleaction ; kindetail ; heyvaerts ; newangleaction ; aanew . These equations do not present any divergence at large scales, contrary to the case where we make a local approximation. In the gravitational case, collective effects lead to a form of anti-shielding. The test star draws neighboring stars into its vicinity and these add their gravitational force to that of the test star itself. The “bare” gravitational force of the test star is thus augmented rather than shielded. The polarization acts to increase the effective gravitational mass of a test star. As a result, collective effects tend to increase the value of the diffusion coefficient and reduce the relaxation time of the system gilbert ; weinberg ; aanew .

For 3D plasmas, the Lenard-Balescu equation is marginally valid provided that a small-scale cut-off is introduced at the Landau length in order to take into account the effect of strong collisions. The relaxation time scales as where is the number of electrons in the Debye sphere. The Landau equation with a small-scale cut-off at the Landau length and a large-scale cut-off at the Debye length provides an excellent approximation of the Lenard-Balescu equation. For 2D plasmas, the Lenard-Balescu equation is rigorously valid. The relaxation time scales as . The Landau equation with a cut-off at the Debye length is a poor approximation of the Lenard-Balescu equation but the discrepancy is not dramatic. In particular, the velocity dependence of the diffusion coefficient is the same, up to a numerical factor. Therefore, we may use the Landau equation by “adapting” the value of the large scale cut-off (it is a fraction of the Debye length). For 1D plasmas, the Lenard-Balescu equation is rigorously valid but it trivially reduces to zero. The relaxation time of the system as a whole scales as while the relaxation time of a test particle in a thermal bath scales as . The Landau equation leads to wrong results. It predicts that the diffusion coefficient of a test particle in a thermal bath decays like a Gaussian while in reality, when collective effects are properly accounted for, it decays as .

For 3D stellar systems, the Lenard-Balescu equation and the Landau equation written with angle-action variables are marginally valid provided that a small-scale cut-off is introduced at the Landau length in order to take into account the effect of strong collisions. The relaxation time scales as where is the number of stars in the cluster. The Landau equation based on a local approximation with a small-scale cut-off at the Landau length and a large-scale cut-off at the Jeans length is relatively accurate. For 2D stellar systems, the Lenard-Balescu equation and the Landau equation written with angle-action variables are rigorously valid. The relaxation time scales as . The Landau equation based on a local approximation produces a linear divergence at large scales but it may be used provided that a large scale cut-off is introduced and properly “adapted” (it is a fraction of the Jeans length). For 1D stellar systems, the Lenard-Balescu equation and the Landau equation written with angle-action variables are rigorously valid. The relaxation time scales as . The Landau equation based on a local approximation leads to wrong results. In particular, it predicts a relaxation time of the system as a whole scaling as while the right scaling is when spatial inhomogeneity is accounted for.

For the HMF model, explicit expressions of the diffusion and friction coefficients, and of the relaxation time, can be obtained in the homogeneous phase. The relaxation time of the system as a whole is larger than (scaling presumably as ) while the relaxation time of a test particle in a bath scales as . The diffusion coefficient depends on the temperature. For the attractive HMF model, there is a critical point ( and the diffusion coefficient increases close to the critical point implying a decrease of the relaxation time.

Some results of kinetic theory have been obtained previously in particular cases, using different approaches, and we give the corresponding references. However, the originality of our approach is to develop a general framework to treat the kinetic theory of systems with long-range interactions (Papers I and II) and, from this framework, consider particular systems. This provides a more unified description of systems with long-range interactions.

The paper is organized as follows. In Sec. II, we consider power-law potentials of interactions and introduce appropriate notations. In Sec. III, we provide an estimate of the relaxation time for systems interacting with power-law potentials and we discuss how it depends on the value of the exponent . In Sec. IV and V, we treat the case of Coulombian plasmas and self-gravitating systems in dimensions. In Sec. VI, we consider the attractive and repulsive HMF models. The Appendices gather additional results of kinetic theory that complete the main discussion of the paper.

## Ii Power-law potentials of interaction

In this section, we consider power-law potentials of interaction of the form with in a space of dimension . We introduce appropriate notations that generalize those commonly used in the study of plasmas and self-gravitating systems.

### ii.1 Repulsive interactions: plasmas

The potential of interaction of a one-component Coulombian plasma with a neutralizing background in -dimensions is the solution of the Poisson equation where is the surface of a unit sphere, is the charge of the electron, and is its mass. Its Fourier transform is .

More generally, we consider a repulsive power-law potential whose Fourier transform may be written as . The potential of interaction in physical space is of the form , where is a constant depending on and (its precise expression is not needed since only the Fourier transform of the potential of interaction matters in the kinetic theory). The exponent of power-law decay is . The Coulombian potential corresponds to , i.e. . For (i.e. ), is the logarithmic potential.

We introduce the parameter where is a typical velocity of the system (for a system at statistical equilibrium, is the inverse temperature). We define the Debye wavenumber and the plasma pulsation by the relations and where is the mass density. They are related to each other by . We also introduce the dynamical time . It represents the typical time needed by a particle to travel the Debye length with the velocity . Finally, we introduce the parameter that gives the number of electrons in the Debye sphere. The normalized potential introduced in Paper II may be written as .

The number of charges in the Debye sphere scales like . For the Coulombian potential (), we get . The number of charges in the Debye sphere is always an increasing function of the temperature. By contrast, its dependence with the density depends on the dimension of space. In , decreases with the density. In , does not depend on the density. In , increases with the density.

Remark 1: in the previous formulae, represent the charge of the electron only when . However, when , it is always possible to write the potential of interaction in the above form (where may be regarded as a coupling constant) so as to facilitate the connection with Coulombian plasmas when .

### ii.2 Attractive interactions: self-gravitating systems

The potential of interaction of a self-gravitating system in -dimensions is the solution of the Poisson equation where is the gravitational constant. Its Fourier transform is .

More generally, we consider an attractive power-law potential whose Fourier transform may be written as . The potential of interaction in physical space is . The exponent of power-law decay is . The Newtonian potential corresponds to , i.e. . For (i.e. ), is the logarithmic potential.

We introduce the parameter where is a typical velocity of the system (for a system at statistical equilibrium, is the inverse temperature). We define the Jeans wavenumber and the gravitational pulsation by the relations and where is the mass density. They are related to each other by . We also introduce the dynamical time . It represents the typical time needed by a particle to travel the Jeans length with the velocity . Finally, we introduce the parameter that gives the number of particles in the Jeans sphere. The normalized potential may be written as .

For attractive power-law potentials in dimensions with , the virial theorem reads where is the kinetic energy and the potential energy (see Appendix I of n2 ). The kinetic energy may be estimated by and the potential energy by where is the system’s size. Since the virial theorem implies , we find from the foregoing expressions that . When , the virial theorem reads leading to the exact result (the velocity dispersion can take a unique value in a steady state) n2 . At statistical equilibrium , so we get . Using , we find in all cases that . Therefore, the Jeans length represents the typical size of the system. Accordingly, the parameter gives the total number of particles in the system.

Remark 2: in the previous formulae, represent the gravitational constant only when . However, when , it is always possible to write the potential of interaction in the above form (where may be regarded as a coupling constant) so as to facilitate the connection with self-gravitating systems when . The statistical mechanics of systems with attractive power-law interactions has been studied in ic .

### ii.3 Characteristic lengths

There are three characteristic lengths in the problem:

The length , where is the numerical density, gives the typical distance between particles.

The Debye length for repulsive potentials gives the effective range of the interaction due to the screening by opposite charges, and the Jeans length for attractive potentials gives the typical size of the system. These expressions rely on a mean field approximation so they are valid only for long-range interactions ( i.e. ). We shall denote commonly the Debye and Jeans lengths by and define .

The Landau length is the distance at which binary collisions become “strong”. This corresponds to an impact parameter yielding a deflexion at . This happens when the energy of interaction or between two particles becomes comparable to their kinetic energy . Equating these two quantities, we get and . These definitions make sense only for decaying potentials ( i.e. ). Introducing the notations defined in the preceding sections, the Landau wavenumber may be rewritten as . For attractive interactions, using and , we get .

Combining the previous expressions, we obtain

(1) |

For (i.e. ) and , we find that .

The coupling parameter is equal to the ratio between the potential energy or of two particles separated by the average inter-particle distance and the typical kinetic energy of a particle . This yields or . Using the previous relations, we find that . The weak coupling approximation corresponds to . For long-range interactions ( i.e. ), the conditions and are equivalent. We may also define the coupling parameter where or is the potential energy of two particles separated by the Debye length or by the Jeans length. In that case, we find . The parameter is called the plasma parameter or the graininess parameter. It is equal to the reciprocal of the number of particles in the Debye or Jeans sphere.

## Iii Estimate of relaxation time for systems with power-law potentials

The normalized Fourier transform of the power-law potentials defined previously may be written as

(2) |

where for attractive interactions and for repulsive interactions (in the repulsive case, we assume that there exist a neutralizing background that maintains the spatial homogeneity of the system). The Coulombian potential (plasmas) corresponds to , and (Debye wavenumber) and the Newtonian potential (gravity) corresponds to , and (Jeans wavenumber).

To estimate the relaxation time, we consider the Landau equation (II-5)^{3}^{3}3Here and in the following, (II-n) refers to Eq. (n) of Paper II. in which strong collisions and
collective effects are neglected (see Appendix A for more details about the approximations
made in this section). Therefore, we only consider weak collisions. We also assume that the system
is spatially homogeneous or that a local approximation may be implemented. In the thermal bath approach, the
diffusion coefficient is given by Eq. (II-45) with

(3) |

The relaxation time may be estimated by (see Paper II). Defining the dynamical time by and performing the change of variables , we obtain

(4) |

where . For repulsive interactions represents the number of particles in the Debye sphere and for attractive interactions represents the total number of particles in the system. The relaxation time scales like except if the integral in Eq. (3) diverges. In that case, it must be regularized at small or large scales and the regularized integral may depend on . We must distinguish different cases according to whether (or ) is larger, smaller, or equal to the critical index

(5) |

For a Coulombian or a Newtonian potential (, ), this critical index corresponds to the critical dimension .

### iii.1 The case

If (i.e. or for a Coulombian or a Newtonian potential), the integral (3) converges for (large scales) implying that collective effects (and spatial inhomogeneity effects for attractive interactions) are weak. On the other hand, it diverges algebraically for (small scales). This divergence is regularized by taking strong collisions into account. Heuristically, we can introduce a small-scale cut-off at the Landau length at which collisions become strong. Performing the integrals in Eqs. (3) and (4) with , we obtain

(6) |

We now develop a more precise description (see Appendix A). In the limit , the evolution of the system is dominated by weak collisions and collective effects (and spatial inhomogeneity for attractive interactions). Three-body collisions are negligible. In principle, strong collisions should also be negligible but the divergence of the diffusion coefficient (3) shows that they are important at small scales. Since the divergence is strong (algebraic), the Lenard-Balescu equation is not applicable. On the other hand, collective effects and spatial inhomogeneity do not seem to be crucial since the diffusion coefficient (3) converges at large scales. If we only take weak and strong collisions into account, we obtain the Boltzmann equation. This equation does not present any divergence. Since weak collisions dominate over strong collisions when , we can expand the Boltzmann equation for small deflexions (or directly use the Fokker-Planck equation). In that case, we obtain the modified Landau equation (63) of Appendix B. This equation does not diverge at small scales since strong collisions have been accounted for. We note, however, that a more rigorous kinetic equation should take into account collective effects (and spatial inhomogeneity for attractive interactions) even if their presence is not required to make the integral converge at large scales.

### iii.2 The case

If (i.e. or for a Coulombian or a Newtonian potential), the integral (3) converges for (small scales) implying that strong collisions are negligible. On the other hand, it diverges algebraically for (large scales). In the case of repulsive interactions (e.g. plasmas), the divergence is regularized by collective effects which account for Debye shielding. Heuristically, we can introduce a cut-off at the Debye length and take . In the case of attractive interactions (e.g. gravity), the divergence is regularized by the finite extent of the system. Heuristically, we can introduce a cut-off at the Jeans length and take . Performing the integrals in Eqs. (3) and (4) with , we obtain

(7) |

We now develop a more precise description (see Appendix A). For repulsive interactions (e.g. plasmas), in the limit the system is dominated by weak collisions and collective effects (strong collisions and three-body collisions can be neglected). Therefore, its evolution is rigorously described by the Lenard-Balescu equation (II-3). This equation does not present any divergence. In the thermal bath approximation, the diffusion tensor is given by Eq. (II-39) with

(8) |

We note that the large-scale divergence that occurs in the diffusion coefficient (3) is regularized by Debye shielding. If we make the Debye-Hückel approximation, or consider small velocities , the diffusion tensor
is given by Eq. (II-45) with^{4}^{4}4The value of the integral is

(10) |

If we neglect collective effects and introduce a large-scale cut-off at the Debye length, we get the Landau equation (II-5). In the thermal bath approach, the diffusion tensor is given by Eq. (II-45) with Eq. (7). However, the Landau equation with a cut-off at the Debye length is not quantitatively correct since the results depend strongly (algebraically) on the precise value of the large-scale cut-off. It may, however, provide a reasonable approximation of the Lenard-Balescu equation provided that the cut-off is suitably adapted to the situation.

For attractive interactions (e.g. gravity), in the limit the system is dominated by weak collisions, spatial inhomogeneity, and collective effects (strong collisions and three-body collisions can be neglected). Therefore, its evolution is rigorously described by the Lenard-Balescu equation written with angle-action variables angleaction ; kindetail ; heyvaerts ; newangleaction ; aanew . If we neglect collective effects, it reduces to the Landau equation written with angle-action variables. These equations do not display any divergence. The large-scale divergence that occurs in the diffusion coefficient (3) is regularized by the finite extent of the system. If we neglect collective effects, make a local approximation, and introduce a large-scale cut-off at the Jeans length, we get the Vlasov-Landau equation (II-16). In the thermal bath approach, the diffusion tensor is given by Eq. (II-45) with Eq. (7). However, the Vlasov-Landau equation with a cut-off at the Jeans length is not quantitatively correct since the results depend strongly (algebraically) on the precise value of the large-scale cut-off. It may, however, provide a reasonable approximation of the Landau equation written with angle-action variables provided that the cut-off is suitably adapted to the situation. Finally, we note that collective effects tend to reduce the relaxation time (7-b) as explained in Appendix C.

### iii.3 The marginal case

If (i.e. or for a Coulombian or a Newtonian potential), the integral (3) diverges logarithmically at small and large scales. This implies that both strong collisions and collective effects (or spatial inhomogeneity for attractive interactions) must be taken into account. However, their influence is weak since the divergence is only logarithmic. We are therefore in a marginal situation. Heuristically, we can introduce a small-scale cut-off at the Landau length^{5}^{5}5We assume because the Landau length
is ill-defined in when . and a large-scale cut-off at the Debye length (for repulsive interactions) or at the Jeans length (for attractive interactions). Performing the integrals in Eqs. (3) and (4) with and , we obtain

(11) |

We now develop a more precise description (see Appendix A). For repulsive interactions (e.g. plasmas), in the limit the evolution of the system is dominated by weak collisions and collective effects. Three-body collisions are negligible. In principle, strong collisions should also be negligible but the divergence of the diffusion coefficient (3) shows that they are important at small scales. Therefore, the evolution of the system is rigorously described by Eqs. (61-a) and (61-b) with . These equations do not present any divergence. However, it is difficult to make them more explicit. The usual strategy is to consider successively the contribution of collisions with small and large impact parameters, and then connect these two limits. If we ignore collective effects and introduce a large-scale cut-off at the Debye length , we get the Boltzmann equation. This equation does not diverge at small scales since strong collisions are taken into account. This equation is marginally valid since the divergence at large scales is weak (logarithmic). For , weak collisions dominate over strong collisions and we can expand the Boltzmann equation for small deflexions (or directly use the Fokker-Planck equation). In the dominant approximation , we get the Landau equation (II-5) with the diffusion coefficient (11) with in which the Landau length appears naturally (see Appendix D). If we ignore strong collisions and introduce a small-scale cut-off at the Landau length , we get the Lenard-Balescu equation. This equation is marginally valid since the divergence at small scales is weak (logarithmic). In the thermal bath approach, the diffusion tensor is given by Eq. (II-39) with

(12) |

We note that the large-scale divergence that occurs in the Landau diffusion coefficient (3) is
regularized by Debye shielding. If we make the Debye-Hückel approximation, or consider small
velocities , the diffusion coefficient is given by Eq. (II-45) with^{6}^{6}6The
value of the integral is

(14) |

In the dominant approximation , the Lenard-Balescu diffusion coefficient and the Debye-Hückel diffusion coefficient can be approximated by Eq. (11) with in which the Debye length appears naturally. In conclusion, in the dominant approximation , the evolution of the system is rigorously described by the Landau equation with a small scale cut-off at the Landau length and a large-scale cut-off at the Debye length. These cut-offs are not ad hoc but they are justified by the above arguments.

For attractive interactions (e.g. gravity), in the limit the evolution of the system is dominated by weak collisions, collective effects, and spatial inhomogeneity. Three-body collisions are negligible. In principle, strong collisions should also be negligible but the divergence of the diffusion coefficient (3) shows that they are important at small scales. Therefore, the evolution of the system is rigorously described by Eqs. (62-a) and (62-b) with . These equations do not present any divergence. However, it is difficult to make them more explicit. Again, the usual strategy is to consider successively the contribution of collisions with small and large impact parameters, and then connect these two limits. If we ignore collective effects and introduce a large-scale cut-off at the Jeans length , we get the Boltzmann equation. This equation does not diverge at small scales since strong collisions are taken into account. This equation is marginally valid since the divergence at large scales is weak (logarithmic). For , weak collisions dominate over strong collisions and we can expand the Boltzmann equation for small deflexions (or directly use the Fokker-Planck equation). In the dominant approximation , we get the Vlasov-Landau equation (II-16) with the diffusion coefficient (11) with in which the Landau length appears naturally (see Appendix D). If we ignore strong collisions and introduce a small-scale cut-off at the Landau length , we get the Lenard-Balescu equation written with angle-action variables angleaction ; kindetail ; heyvaerts ; newangleaction ; aanew . This equation is marginally valid since the divergence at small scales is weak (logarithmic). If we neglect collective effects, it reduces to the Landau equation written with angle-action variables. These equations do not diverge at large scales since they take into account the finite extent of the system. If we neglect collective effects and make a local approximation (which is justified by the fact that the divergence of the diffusion coefficient at large scales is weak), we get the Vlasov-Landau equation (II-16). This equation presents a logarithmic divergence at large scales (). Since this divergence is cured by the finite extent of the system, it is natural to introduce a large-scale cut-off at the Jeans scale. Since this procedure is essentially heuristic, the precise value of the large-scale cut-off is not known. However, in the dominant approximation , the precise value of the numerical factor is not important since it appears in a logarithmic factor and we finally obtain the Vlasov-Landau equation (II-16) with the diffusion coefficient (11) with . In conclusion, in the dominant approximation , the evolution of the system is reasonably well described by the Vlasov-Landau equation (II-16) with a small scale cut-off at the Landau length and a large-scale cut-off at the Jeans length. This equation is not exact (the rigorous equation is the Lenard-Balescu equation written in angle-action variables with a small-scale cut-off at the Landau length justified by the previous arguments) but it may be used as a simplified kinetic equation. Finally, we note that collective effects tend to reduce the relaxation time (11-b) as explained in Appendix C.

### iii.4 Existence of quasi stationary states

The previous scalings of the relaxation time agree with those obtained by Gabrielli et al. gjm
from the Chandrasekhar binary collision theory^{7}^{7}7Their approach assumes that
collective effects and spatial inhomogeneity can be neglected.
Our approach is more general since it can take these effects
into account as explained previously.. For , the Lenard-Balescu equation is rigorously valid and the relaxation
time scales as . For , the Lenard-Balescu equation is marginally valid and the relaxation time scales as . This corresponds to the usual situation encountered in plasma physics and stellar dynamics () since the dimension of space is critical for Coulombian and Newtonian interactions. For , the Lenard-Balescu equation is not valid and the relaxation time scales as .

For (corresponding to ), the relaxation time diverges when (for , the relaxation time increases linearly with and, for , it increases less rapidly than ). In that case, the Vlasov regime becomes infinite in the thermodynamic limit and quasi stationary states (QSS) may form as a result of violent relaxation lb . On the other hand, for (corresponding to ), the relaxation time tends to zero when . In that case, as emphasized by Gabrielli et al. gjm , no QSS can form (except if the potential includes a sufficiently large soft core). In the intermediate case , the relaxation time is independent on .

Remark 3: If we define the mean free path by , we find that for , for , and for . For , we find that when so that the mean free path is much larger than the Debye length or the Jeans length. This is another manner to understand why the Vlasov equation becomes exact when .

Remark 4: For specific power-law potentials, the dynamics may be particular. For (harmonic potential) there is no (violent or slow) relaxation and the system oscillates permanently as discovered by Newton newton ; chandranewton ; lblb .

## Iv Coulombian plasmas

For Coulombian plasmas, the Fourier transform of the potential of interaction is

(15) |

and the Fourier transform of the Debye-Hückel potential is

(16) |

In physical space, we have and in , and in , and in .

### iv.1 3D Coulombian plasmas

If we neglect strong collisions and collective effects, the evolution of the system is described by the Landau equation (II-5). In the thermal bath approach, the diffusion tensor is given by Eq. (II-45) with

(17) |

The integral diverges logarithmically at small and large scales. This implies that both strong collisions and collective effects (Debye shielding) must be taken into account. However, their effect is weak since the divergence is logarithmic (we are in the marginal case of Sec. III.3). If we introduce a small-scale cut-off at the Landau length at which binary collisions become strong and a large-scale cut-off at the Debye length at which the interaction is shielded, we get

(18) |

where represents the number of electrons in the Debye sphere. In plasma physics, is called the Coulomb logarithm.

Strong collisions are taken into account in the Boltzmann equation. This corresponds to the two-body encounters (or impact) theory. In this equation, the integral over the impact parameter does not diverge at small scales. Therefore, the correct treatment of strong collisions regularizes the logarithmic divergence that appears at small scales in the Landau equation. However, for a Coulombian potential in , the integral over the impact parameter diverges logarithmically at large scales. Therefore, the Boltzmann equation is marginally valid provided that a large-scale cut-off is introduced at the Debye length (see the next argument). If we expand the Boltzmann equation for small deflexions (or directly use the Fokker-Planck equation), and consider the dominant approximation , we obtain the Landau equation (II-5) with the diffusion coefficient (18) with in which the Landau length appears naturally (see Appendix D).

Collective effects are taken into account in the Lenard-Balescu equation (II-3). This corresponds to the wave theory. In this equation, the integral over the wavenumber does not diverge at large scales. Therefore, the correct treatment of collective effects regularizes the logarithmic divergence that appears at large scales in the Landau equation. However, for a Coulombian potential in , the integral over the wavenumber diverges logarithmically at small scales. Therefore, the Lenard-Balescu equation is marginally valid provided that a small-scale cut-off is introduced
at the Landau length (see the previous argument). In the thermal bath approach, the diffusion tensor is given by Eq. (II-39) with^{8}^{8}8The value of the integral is

(20) |

This integral converges at large scales () due to Debye shielding. For small velocities , the diffusion coefficient is given by (II-45) with^{9}^{9}9The value of the integral is

(22) |

This corresponds to the Debye-Hückel approximation. In the dominant approximation ,
the Lenard-Balescu diffusion coefficient (20) and Debye-Hückel diffusion coefficient (22)
reduce to the Landau diffusion coefficient (18) with in which the Debye length appears naturally. This shows that the Landau equation with a large-scale cut-off at the Debye length provides a very good approximation of the Lenard-Balescu equation in . This implies that dynamical screening is
negligible^{10}^{10}10This is true for the Maxwell distribution. This is not true anymore
for distributions for which the large velocity population plays a crucial role as compared to the bulk of
the distribution. in . Indeed, in the dominant approximation, we can neglect the velocity effects encapsulated in the dielectric function and use the Debye-Hückel potential accounting for static screening.

For a Coulombian plasma in (marginal case), the impact theory and the wave theory are two approximate theories that are complementary to each other. Collective interactions between charges are not included in the impact theory implying that the integrand in the Boltzmann equation is valid only for impact parameters sufficiently smaller than the Debye length (). On the other hand, the curvature of orbits at small impact parameters is not included in the wave theory implying that the integrand in the Lenard-Balescu equation is valid only for wavelengths sufficiently larger than the Landau length (). For , the range of validity of these two theories greatly overlaps in the region corresponding to the domain of validity of the Landau equation. A possibility to unify these approaches and determine precisely the numerical factor in the Coulomb logarithm was first recognized by Hubbard hubbard2 and subsequently developed by Kihara and Aono ka and Gould and DeWitt gould among others. Roughly speaking, the idea is to sum the Boltzmann () and Lenard-Balescu () equations and subtract the Landau equation (). This leads to a fully convergent kinetic equation. It has the form of the Landau equation in which the Coulomb logarithm is exactly given by where is Euler’s constant. If we neglect the constant term as compared to (dominant approximation), this procedure justifies using the Landau equation with a small-scale cut-off at the Landau length and a large-scale cut-off at the Debye length.

Conclusion: The evolution of the system as a whole is described by the Lenard-Balescu equation (II-3) with a cut-off at the Landau length (this cut-off is not ad hoc but is justified by the correct treatment of strong collisions). The relaxation of a test particle in a thermal bath is described by the Fokker-Planck equation (II-36) with the diffusion tensor given by Eqs. (II-39) and (20). In the dominant approximation , the Lenard-Balescu equation can be replaced by the Landau equation (II-5) with a cut-off at the Debye length (this cut-off is not ad hoc but is justified by the correct treatment of collective effects). In that case, the evolution of the system as a whole is described by the Landau equation (II-5) or (II-9) with . The relaxation of a test particle in a thermal bath in described by the Fokker-Planck equation (II-36) with the diffusion tensor given by Eqs. (II-47) and (II-48) where given by Eq. (18). Using the results of Sec. III, we find that the relaxation time is given by

(23) |

where is the dynamical time. A short historic of the early development of the kinetic theory of 3D Coulombian plasmas with many references is given in Paper II.

### iv.2 2D Coulombian plasmas

If we neglect strong collisions and collective effects, the evolution of the system is described by the Landau equation (II-5). In the thermal bath approach, the diffusion tensor is given by Eq. (II-45) with

(24) |

This integral converges at small scales implying that strong collisions are negligible. However, it diverges rapidly (linearly) at large scales implying that collective effects (Debye shielding) are important. If we introduce a large-scale cut-off at the Debye length, we obtain

(25) |

where represents the number of electrons in the Debye disk.

Collective effects are taken into account in the Lenard-Balescu equation (II-3). This equation is rigorously valid for a Coulombian potential in (see Sec. III.2). The large-scale divergence that occurs in the Landau equation is regularized by Debye shielding. In the thermal bath approach, the diffusion tensor is given by Eq. (II-39) with

(26) |

This function is perfectly well-defined. For small velocities , the diffusion coefficient is given by Eq. (II-45) with

(27) |

This corresponds to the Debye-Hückel approximation. For , we can use the asymptotic results given in Sec. IV.D of Paper I. The diffusion coefficient is given by Eq. (II-53-d) with replaced by and the diffusion coefficient is given by Eq. (II-53-c) with replaced by where

(28) |

We note that the Debye-Hückel approximation is not correct at large velocities (compare Eqs. (27) and (28)). Therefore, dynamical screening is important for 2D plasmas, contrary to 3D plasmas in the dominant approximation. This makes the dimension particularly interesting. On the other hand, the Lenard-Balescu diffusion coefficient (26) and the Debye-Hückel diffusion coefficient (27) are different from the Landau diffusion coefficient (25). Therefore, the Lenard-Balescu equation cannot be rigorously approximated by the Landau equation with a large-scale cut-off at the Debye length. However, comparing Eqs. (25), (27) and (28), we see that the discrepancy is not dramatic since the velocity dependence of the diffusion coefficient is the same (at least asymptotically) and the value of the prefactor is only slightly different ( and instead of ). Therefore, the Landau equation may be used as a rough approximation of the Lenard-Balescu equation by “adapting” the value of the large scale cut-off.

Conclusion: The evolution of the system as a whole is described by the Lenard-Balescu equation (II-3). The relaxation of a test particle in a thermal bath in described by the Fokker-Planck equation (II-36) with the diffusion tensor given by Eqs. (II-39) and (26). The Landau equation (II-5) with a cut-off at the Debye length is not rigorously equivalent to the Lenard-Balescu equation in but the discrepancy is not dramatic. If we use this approximate theory, the evolution of the system as a whole is given by the Landau equation (II-5) or (II-9) with , or in the Debye-Hückel approximation. The evolution of a test particle in a thermal bath is described by the Fokker-Planck equation (II-36) with the diffusion tensor given by Eqs. (II-51) and (II-52) where given by Eq. (25), or by Eq. (27) in the Debye-Hückel approximation. Using the results of Sec. III, we find that the relaxation time is given by

(29) |

where is the dynamical time.

A kinetic theory of 2D Coulombian plasmas has been developed by Benedetti et al. benedetti for the cut-off Coulombian potential and by Chavanis landaud for the true Coulombian potential. However, collective effects are treated in an approximate manner in Appendix C of that paper. The present treatment is more satisfactory.

### iv.3 1D Coulombian plasmas

If we neglect strong collisions and collective effects, the evolution of the system is described by the Landau equation (II-5). In the thermal bath approach, the diffusion coefficient is given by Eqs. (II-46) and (II-54). This leads to

(30) |

This integral converges at small scales implying that strong collisions are negligible. However, it diverges rapidly (quadratically) at large scales implying that collective effects (Debye shielding) are important. If we introduce a large-scale cut-off at the Debye length, we obtain

(31) |

where represents the number of electrons in the Debye segment.

Collective effects are described by the Lenard-Balescu equation (II-3). This equation is rigorously valid for a Coulombian potential in (see Sec. III.2). The large-scale divergence that occurs in the Landau equation is regularized by Debye shielding. In the thermal bath approach, the diffusion coefficient is given by Eqs. (II-43) and (II-44) leading to