Preface
The well-known equations for fully ionized plasma are Vlasov equation, Landau equation in kinetic regime and two-fluid equation, Magnetohydrodynamics equation(MHD) in hydrodynamic regime. MHD can be viewed as some average of kinetic equation, so it makes someone feel safety to use them since they are built on a sounder fundation. But how about kinetic equations? How to justify them? Actually, the popular kinetic equation like Vlasov equation is some truncated form of a more rigorious form called Liouville equation. If we want to feel safer when use Vlasov equation, it is better to know the derivation of it from Liouville equation.
The whole subject of plasma physics can be reduced to the understanding of the motion of a set of charged particles in an electromagnetic field. However, it is easier saying than doing, as soon as the plasma is expressed on a quantitative basis, one realizes the overwhelming complexity of the problem. Since we cannot track the motion of all charged species in the system, we shall rather ask questions about the global properties of the plasma, which are observable on a macroscopic scale (e.g. the motion of the fluid rather of its consituent particles, the transport of energy from one point to another, the propagation of a wave through the medium, etc.) In doing so, the complexity of the system turns into an advantage. Indeed, the large number of particles and the extreme irregularity of their motion lead to the picture of a quasi-chaotic system, to which some of the concepts of the probability theory become applicable. We thus go over from ordinary mechanics into the realm of statistical mechanics and of its daughter kinetic theory.
This article mainly refers to R.Balescu, Transport Processes In Plasmas, Vol1.
Lie algebra
The special importance of the Lie bracket for the dynamics comes from the following postulate:
The evolution in time of any dynamical function, due to the motion of the system, is determined by the equation:
\[ \dot{a} = [a, H] \]
This is the most general form of the equation of motion in the Hamiltonian formalism. From classic mechanics to quantum mechanics, physics language evolves from Newtons' formalism to Langrange (Hamilton) formalism. We treat the physical system as force and geometry to energy and functionals. The Lie algebra is a effective language to describe more general physics for post-Newton's era. Some extra reading is required here to derive this dynamic equation.Reasons for developing Lie algebra are firstly put forward, then we restart from the begining to learn details.
Any theory of a system evolving in time must be constructed upon two concepts: a definition of the state of the system, and a definition of the law of evolution. In classical Hamiltonian mechanics, the state of a system of \(f\) degrees of freedom requires the specification of \(2 f\) variables which are taken (in a first stage) as the generalized coordinates \(q_i\) and momenta \(p_i\), \[ \left(q_i, p_i\right), \quad i=1,2, \ldots, f . \] The law of motion is governed by a specified function of the state variables: the Hamiltonian \(H(q, p)\). Here \((q, p)\) is an abbreviation for the set \(\left(q_1, \ldots, q_f, p_1, \ldots, p_f\right)\). Here, and in all forthcoming sections up to section 1.9, we assume the Hamiltonian to be independent of the time, \(t\). The system is then called an autonomous system. The motion, i.e., the change in time of the positions and of the momenta, is determined by the Hamilton equation \[ \dot{q}_i(t)=\frac{\partial H(q, p)}{\partial p_i}, \quad \dot{p}_i(t)=-\frac{\partial H(q, p)}{\partial q_i} \]
Besides \(q_i\) and \(p_i\), we are interested in many other quantities which take definite values in each state of the system. The Hamiltonian, as well as other physical quantities, such as the total momentum, the angular momentum, etc., are obvious physical examples. A subtler example is provided by the solution of eqs. (2.2). The coordinate \(q_i(t)\) at time \(t\) is a function of the initial coordinates and momenta \(q_i, p_i\) and of the time. We decide from here on to take the set \((q, p)\), representing the coordinates and the momenta at aiven (initial) time, as the variables spanning a (fixed) phase space of the dynamical system. All the quantities mentioned above are then defined (in classical mechanics) as functions of the phase space coordinates \((q, p)\), possibly depending on some "external parameters" \(\alpha\); they will be called dynamical functions and will be denoted by \(a, b, \ldots\) : \[ a \equiv a(q, p ; \alpha) . \] Typical examples of external parameters are: the time \(t\), parameters characterizing the system (such as the mass \(m\), or the charge \(e\) ), or characterizing the environment (such as the magnetic field \(\boldsymbol{B}\) ), or universal constants (such as the speed of light \(c\) ), or simply numerical constants.
We now consider the set \(\mathscr{D}\) of all the dynamical functions and define the permissible operations on its members; in other words, we endow the set with an algebraic structure. We postulate that, for any member \(a, b, \ldots \in \mathscr{D}\), the result of the following operations is also a member of the set \(\mathscr{D}\) : Multiplication by an external parameter: \[ \alpha a=c, \quad c \in \mathscr{D} . \] Linear combination *: \[ \alpha a+\beta b=d, \quad d \in \mathscr{D} . \] Multiplication: \[ a \cdot b=e, \quad e \in \mathscr{D} . \] Inversion: \[ a^{-1}=f, \quad f \in \mathscr{D} . \] These operations have their usual algebraic meaning and properties. Clearly, aking \(q\) and \(p\) as "building blocks", we can construct by means of these operations very wide classes of dynamical functions (e.g. polynomials, analytical functions, meromorphic functions, Fourier series, etc.). But the truly characteristic feature of the set \(\mathscr{D}\) is obtained by defining an additional operation, the Lie bracket, \[ [a, b]=g, \quad g \in \mathscr{D}, \] with the following properties. It is an antisymmetric operation: \[ [a, b]=-[b, a] . \] It is a non-associative operation, governed by the Jacobi relation: \[ [[a, b], c]+[[b, c], a]+[[c, a], b]=0 . \] It is related as follows to the three basic operations (2.4)-(2.6): \[ \begin{aligned} & {[\alpha a, b]=\alpha[a, b],} \\ & {[\alpha a+\beta b, c]=\alpha[a, c]+\beta[b, c],} \\ & {[a b, c]=a[b, c]+b[a, c] .} \end{aligned} \] The set \(\mathscr{D}\) of dynamical functions, endowed with the operations (2.4)-(2.8), is called a Lie algebra.
The three rules (2.11)-(2.13) imply that the Lie bracket has properties analogous to a first order differential operator. They imply the following important, easily derived relation \[ [a(q, p), b]=\sum_{i=1}^f\left(\frac{\partial a}{\partial q_i}\left[q_i, b\right]+\frac{\partial a}{\partial p_i}\left[p_i, b\right]\right) . \] Iterating this relation, we find \[ \begin{aligned} & {[a(q, p), b(q, p)]} \\ & \qquad \sum_{i=1}^f \sum_{j=1}^f\left(\frac{\partial a}{\partial q_i} \frac{\partial b}{\partial q_j}\left[q_i, q_j\right]+\frac{\partial a}{\partial q_i} \frac{\partial b}{\partial p_j}\left[q_i, p_j\right]\right. \\ & \left.\quad+\frac{\partial a}{\partial p_i} \frac{\partial b}{\partial q_j}\left[p_i, q_j\right]+\frac{\partial a}{\partial p_i} \frac{\partial b}{\partial p_j}\left[p_i, p_j\right]\right) . \end{aligned} \]
This very important relation enables us to calculate explicitly the Lie bracket of any pair of functions of the phase space coordinates, provided we know the fundamental Lie brackets of the "building stones" \(q_i, p_i\). At this stage in the Hamiltonian formalism, we postulate the following value of the fundamental Lie brackets: \[ \begin{aligned} & {\left[q_i, q_j\right]=0,} \\ & {\left[p_i, p_j\right]=0,} \\ & {\left[q_i, p_j\right]=\delta_{i j}, \quad i, j=1,2, \ldots, f .} \end{aligned} \] For completeness, the following relations \[ \begin{aligned} & {\left[\alpha, q_i\right]=0,} \\ & {\left[\alpha, p_i\right]=0, \quad i=1,2, \ldots, f} \end{aligned} \] must be added for any external parameter \(\alpha\). Actually, eqs. (2.17) can be considered as a general definition of an external parameter. Equations (2.16) can be called the Lie multiplication table of the Lie algebra \(\mathscr{D}\) in the variables \(\left(q_i, p_i\right)\). Any set of variables satisfying (2.16) is called a set of canonical variables. It is also said that \(q_i\) and \(p_i\) are canonically conjugate.
Combining (2.15) and (2.16), we are in a position of calculating explicitly the Lie bracket of any pair of dynamical functions. The former equation reduces to \[ [a(q, p), \quad b(q, p)]=\sum_{i=1}^N\left(\frac{\partial a}{\partial q_i} \frac{\partial b}{\partial p_i}-\frac{\partial a}{\partial p_i} \frac{\partial b}{\partial q_i}\right) . \] This is the familiar expression of the Poisson bracket. It appears as a specific realization of the Lie bracket, valid when the fundamental variables \(q_i, p_i\) are canonical.
Liouville equation
Derivation of the Liouville equation
It is well known that the statistical state of a system is determined by a phase-space distribution function \(F(q, p, t)\) :a non-negative function of the \(6 \mathrm{~N}\) phase-space coordinates and of the time, which is normalized to one, as
\[ \int d^{3 N} q d^{3 N} p F(q, p ; t)=1 \]
Here, \(F(q, p, t)\) can be interpreted as the probability density for finding the system at time \(t\) at the point \((q, p)\) in phase space. The fact that \(F\) is normalized to one at all times is a non-trivial consequence Liouville's theorem, which follows the fact that the volume of any region of phase space is invariant when its constituent points move according to Hamilton's equations, therefore we have
\[ \frac{\mathrm{d} F(q, p ; t)}{\mathrm{d} t} \equiv \frac{\partial F}{\partial t}+\sum_{j=1}^N\left(\dot{\boldsymbol{q}}_j \cdot \frac{\partial F}{\partial \boldsymbol{q}_j}+\dot{\boldsymbol{p}}_j \cdot \frac{\partial F}{\partial \boldsymbol{p}_j}\right)=0 \]
Combining this results general evolution formula of a dynamical function \(\dot{a}=[a, H]\), we have
\[ \frac{\partial F}{\partial t}+\sum_{j=1}^N\left(\left[\boldsymbol{q}_j, H\right] \cdot \frac{\partial F}{\partial \boldsymbol{q}_j}+\left[\boldsymbol{p}_j, H\right] \cdot \frac{\partial F}{\partial \boldsymbol{p}_j}\right)=0 \]
Using Lie bracket property \([a(q, p), b]=\sum_{i=1}^N\left(\frac{\partial a}{\partial q_i}\left[q_i, b\right]+\frac{\partial a}{\partial p_i}\left[p_i, b\right]\right)\), the above equation can be written as \[ \frac{\partial F}{\partial t}+[F, H]=0 \]
The expression on the R.H.S can also be written as a linear operator, \[ \partial_t F=L F \] where, \(L\) is defined as the differential operator \[ L \equiv \sum_{j=1}^N\left(\left[\boldsymbol{q}_j, H\right] \cdot \frac{\partial}{\partial \boldsymbol{q}_j}+\left[\boldsymbol{p}_j, H\right] \cdot \frac{\partial}{\partial \boldsymbol{p}_j}\right) \]
Reduced distribution function
The distribution contains all the information about the system, but it is high-dimensional and hence hard to deal with. Considering that some information is irrelevant to our purpose, we have compress the information, namely, reduce the distribution too fewer quantities. This is called reduced distribution functions ( \(\mathrm{rdf}\) ) \(\hat{f}_s^{\alpha_1 \cdots \alpha_s}\left(\boldsymbol{q}_1, \ldots, \boldsymbol{q}_s, \boldsymbol{p}_1, \ldots, \boldsymbol{p}_s ; t\right)\), note that here we use s-tuple distribution \(f_s\), this is not the meaning of probability, but rather represents the probable number of s-tuples of particles such that they occupy specific state \({ }^{[1]}\). In a two-component system, it is rather complicated to write down complete rdf since the combinatorial factor is so complex. Here the explicit definition for the one- and two-particle distribution functions are given:
\[ \begin{aligned} & \hat{f}_1^\alpha\left(\boldsymbol{q}_1, \boldsymbol{p}_1 ; t\right)=N_\alpha \int \mathrm{d} \boldsymbol{q}_2 \mathrm{~d} \boldsymbol{p}_2 \cdots \mathrm{d} \boldsymbol{q}_N \mathrm{~d} \boldsymbol{p}_N F(q, p; t) \\ & \hat{f}_2^{\alpha \alpha}\left(\boldsymbol{q}_1, \boldsymbol{p}_1, \boldsymbol{q}_2, \boldsymbol{p}_2 ; t\right)=N_\alpha\left(N_\alpha-1\right) \int \mathrm{d} \boldsymbol{q}_3 \mathrm{~d} \boldsymbol{p}_3 \cdots \mathrm{d} \boldsymbol{q}_N \mathrm{~d} \boldsymbol{p}_N F(q, p ; t) \\ & \hat{f}_2^{\alpha \beta}\left(\boldsymbol{q}_1, \boldsymbol{p}_1, \boldsymbol{q}_2, \boldsymbol{p}_2 ; t\right)=N_\alpha N_\beta \int \mathrm{d} \boldsymbol{q}_3 \mathrm{~d} \boldsymbol{p}_3 \cdots \mathrm{d} \boldsymbol{q}_N \mathrm{~d} \boldsymbol{p}_N F(q, p ; t) \end{aligned} \]
For the first one-particle distribution, it describes the distribution function for a particle in state \((\boldsymbol{q}, \boldsymbol{p})\), all other particles' states have been integrated out. Let's briefly review some basic knowledge of probability. Now let's discuss the weather condition. Let \(A\) be a random variable denoting tomorrow's weather, 0 being sunny, 1 being rainy. Let \(B\) denote the day after tomorrow's weather, so now we have a probability distribution for two random variables.
\[ \begin{array}{|l|l|l|l|l|} \hline \mathrm{P}(\mathrm{A}=\mathrm{x}, \mathrm{B}=\mathrm{y}) & (1,0) & (1,1) & (0,0) & (0,1) \\ \hline val & 0.1 & 0.3 & 0.2 & 0.4 \\ \hline \end{array} \]
Now if I only want to know \(P(A)\), then I should integrate out \(B\). This is equivalent to get one-particle distribution.
The reason why there is a \(N_\alpha\) is that it gaurantees if we integrate out \(\left(\boldsymbol{q}_1, \boldsymbol{p}_1\right)\) there will be \(N_\alpha\) particles.
\[ \begin{aligned} & \int \mathrm{d} q_1 \mathrm{~d} p_1 \hat{f}_1^\alpha\left(q_1, p_1 ; t\right)=N_\alpha, \\ & \int \mathrm{d} \boldsymbol{q}_1 \mathrm{~d} p_1 \mathrm{~d} \boldsymbol{q}_2 \mathrm{~d} \boldsymbol{p}_2 \hat{f}_2^{\alpha \beta}\left(\boldsymbol{q}_1, p_1, \boldsymbol{q}_2, p_2 ; t\right)=N_\alpha N_\beta . \end{aligned} \]
Actaully, according to the definition of \(f_s\), it means that there are \(N_\alpha\) number of 1-tuple of particles such that the particle is in the state \(\left(\boldsymbol{q}_1, \boldsymbol{p}_1\right)\). It behaves like that I try to put 1 balls into \(N_\alpha\) holes, so there will be \(N_\alpha\) possible arrangements. Therefore, we know the relation,
\[ f_s=A_{N_\alpha}^s F_s \]
For two-particle distribution, it behaves like that I try to put 2 balls into \(N_\alpha\) holes, so there will be \(N_\alpha\left(N_\alpha-1\right)\) possible arrangements. And after understanding this, you can understand the above formula and also know why it is complicate to write down rdf for stuple.
And, btw, the one-particle distribution doesn't mean that there is only one particle in the system, but rather, it compress \(6 \mathrm{~N}\) dimensional distribution to 6 dimenisinal distiribution. (Just like when when there is little variation in \(z\) direction, we can reduce 3D problem into 2D problem). If all other \(6(\mathrm{~N}-1)\) dimensions have little influence on the one-particle distribution, this is a good compression. It is called "one-particle" distribution because we only consider state for one particle instead of considering state for one particle along with exact state for another particle mean time. The importance of the reduced distribution function comes from the fact that all import quantities of the transport theory are defined as averages of dynamical quantities \(b^\alpha(\boldsymbol{q}, \boldsymbol{p})\) weighted with \(\hat{f}_1^\alpha(\boldsymbol{q}, \boldsymbol{p} ; t)\) (since these dynamical quantities don't have "manyparticle" dependence)
\[ \langle b\rangle=\sum_\alpha \int \mathrm{d} \boldsymbol{q} \mathrm{d} \boldsymbol{p} b^\alpha(\boldsymbol{q}, \boldsymbol{p}) \hat{f}_1^\alpha(\boldsymbol{q}, \boldsymbol{p} ; t) \]
The main goal of kinetic theory is the derivation of a closed equation of evolution for the reduced distribution function \(\hat{f}_1^\alpha(\boldsymbol{q}, \boldsymbol{p} ; t)\). Up to now, there is no error introduced, but if we want to derive a equation for \(r d f\), there will be some approximation.
Liouville equation for independent particles
To put it simple, we start by a fictitious system of non-interacting particles. This is useful even in some quite important descriptions of the interacting plasmas. The Hamiltonian of this ideal system is then,
\[ H(q, p)=\sum_{j=1}^N\left(\frac{1}{2 m_{\alpha_j}}\left|p_j-c^{-1} e_{\alpha_j} A\left(q_j\right)\right|^2+e_{\alpha_j} \Phi_0\left(q_j\right)\right) = \sum_{j=1}^NH_{0}^{\alpha_j}, \]
where the index \(\alpha_j\) labels the species of particles \(j\). \(\boldsymbol{A}(x)\) is the magnetic vector potential, while \(\Psi(x)\) is electric potential, both are external stationary fields. \(H_{0j}^{\alpha_j}\) is single particle \(j\)'s Hamiltonian. This additive structure is characteristic of system of non-interacting particles.
Recall that for Lie bracket, we have such linear relation:
\[ \begin{aligned} & {[\alpha a, b]=\alpha[a, b],} \\ & {[\alpha a+\beta b, c]=\alpha[a, c]+\beta[b, c],} \\ & {[a b, c]=a[b, c]+b[a, c] .} \end{aligned} \]
Therefore, for Liouville operator
\[ LF = [H, F] = [\sum_{j=1}^NH_{0}^{\alpha_j}, F] = \sum_{j=1}^N[H_{0}^{\alpha_j},F]= [\sum_{j=1}^N L_j^{\alpha_j}] F \]
Then the equation for the distribution function is
\[ \partial_t F = LF = [\sum_{j=1}^N L_j^{\alpha_j}] F = \sum_{j=1}^N[H_{0}^{\alpha_j},F] \]
Multiply both sides by \(N_\alpha\) and integrate over the positions and momenta of all the particles, except those of particle 1 of species \(\alpha\), we have
\[ \partial_t \hat{f}_1^\alpha\left(\boldsymbol{q}_1, \boldsymbol{p}_1 ; \boldsymbol{t}\right)=L_1^\alpha \hat{f}_1^\alpha\left(\boldsymbol{q}_1, \boldsymbol{p}_1 ; \boldsymbol{t}\right) \]
because the contribution of all the one-particle Liouville other than \(L_1^{\alpha}\) vanish by partial integration. But if Hamiltonian doesn't have such good linear property as to be written as separate single particle's Hamiltonian, then we won't have such single equation for one-particle distribution.
We now adopt a more compact notation: we drop the index 1 from \(f_1^\alpha\), it being understood that \(\hat{f}^\alpha\) denotes a one-particle reduced distribution function. Moreover, in the text, we shall refer to \(\hat{f}^\alpha\) simply as the distribution function (of species \(\alpha\) ). The index 1 is also superfluous on \(L_1^\alpha\) and on the variables \(\boldsymbol{q}_1\), \(\boldsymbol{p}_1\). Thus, we simply write \[ \partial_t \hat{f}^\alpha=L^\alpha \hat{f}^\alpha(\boldsymbol{q}, \boldsymbol{p} ; t), \] or, equivalently, \[ \partial_t \hat{f}^\alpha=\left[H_0^\alpha, \hat{f}^\alpha\right], \] or, even more explicitly, \[ \partial_t \hat{f}^\alpha=\left[H_0^\alpha, \boldsymbol{q}\right] \cdot \frac{\partial \hat{f}^\alpha}{\partial \boldsymbol{q}}+\left[H_0^\alpha, \boldsymbol{p}\right] \cdot \frac{\partial \hat{f}^\alpha}{\partial \boldsymbol{q}}, \] and \[ H_0^\alpha=\frac{1}{2 m_\alpha}\left|\boldsymbol{p}-\frac{e_\alpha}{c} \boldsymbol{A}(\boldsymbol{q})\right|^2+e_\alpha \Phi_0(q) . \] The important feature of this result is the following. For a non-interacting system, the reduced one-particle distribution function obeys an equation of evolution which has the same form as the Liouville equation in the reduced phase space defined by the position and the momentum of a single particle.
Summary
In this article, we briefly introduce the Liouville equation and conceopt of reduced distribution function. In next article, we will explain how to construct a closed equation for rdf and introduce BBGKY equations.
The BBGKY equations
Hamiltonian of interacting system
We now go one step further in the complexity of the description of a plasma, by taking account of the interactions among the charged particles. Each particle in the plasma feels, besides the external field, an electromagnetic field which is wildly varying, as it depends on the instantaneous coordinates and velocities of all the other particles in the system. This is the essence of the interaction effects. The simplest way of accounting for the interactions in plasmas is by postulating the following form of the Hamiltonian:
\[ \begin{aligned} H(q, v) & =\sum_{j=1}^N\left[\frac{1}{2} m_{\alpha_j} v_j^2+e_{\alpha_j} \Phi_0\left(\boldsymbol{q}_j\right)\right]+\sum_{j<n=1}^N V_{j n}^{\alpha_j \alpha_n}\left(\boldsymbol{q}_j, \boldsymbol{q}_n\right) \\ & \equiv \sum_{j=1}^N H_{0_j}^{\alpha_j}+\sum_{j<n=1}^N V_{j n}^{\alpha_j \alpha_n} . \end{aligned} \]
Here the first term is the kinetic and potential energy of one particle, \(\Phi_0(\boldsymbol{x})\) is the potential field for the external electric field. No magnetic field appears in the Hamiltonian since magnetic force line is always perpendicular to the velocity of a particle, so it will do no work. If we don't consider the second term, then the Hamiltonian describes non-interacting system with an additive structure.
The new feature appearing in the second term is a sum, over all distinct pairs of particles, of functions depending non-additiviely on the positions of two particles, \(j\) and \(n\) (of species \(\alpha_j\), \(\alpha_n\)). We assume that the interaction potential describes the Coulomb interaction of two charged point particles:
\[ V_{j n}^{\boldsymbol{\alpha}_j \boldsymbol{\alpha}_n} \equiv V^{\boldsymbol{\alpha}_j \boldsymbol{\alpha}_n}\left(\boldsymbol{q}_j, \boldsymbol{q}_n\right)=e_{\alpha_j} e_{\boldsymbol{\alpha}_n} \frac{1}{\left|\boldsymbol{q}_j-\boldsymbol{q}_n\right|} \]
This is a non-relativistic fully ionized interacting system. In sipte of these limitations, this model covers a large spectrum of interesting problems.
Let's try to apply the Liouville operator,
\[ \begin{align*} LF &= [H, F] = [ \sum_{j=1}^N H_{0_j}^{\alpha_j}+\sum_{j<n=1}^N V_{j n}^{\alpha_j \alpha_n} , F] \\ &= [ \sum_{j=1}^N H_{0_j}^{\alpha_j}, F] + [\sum_{j<n=1}^N V_{j n}^{\alpha_j \alpha_n} , F] \\ &= \sum_{j=1}^N [H_{0_j}^{\alpha_j}, F] + \sum_{j<n=1}^N[ V_{j n}^{\alpha_j \alpha_n} , F] \end{align*} \]
As shown above, there exsits a correlation part. Coresspondingly, the Liouville equation an equation for reduced distribution functions is
\[ \begin{aligned} & \partial_t f^\alpha\left(\boldsymbol{q}_1, \boldsymbol{v}_1 ; t\right) =L_1^\alpha f^\alpha\left(\boldsymbol{q}_1, \boldsymbol{v} ; t\right)+\sum_{\beta=\mathrm{e}, \mathrm{i}} \int \mathrm{d} \boldsymbol{q}_2 \mathrm{~d} \boldsymbol{v}_2 L_{12}^{'\alpha \beta} f^{\alpha \beta}\left(\boldsymbol{q}_1, \boldsymbol{v}_1, \boldsymbol{q}_2, \boldsymbol{v}_2 ; t\right) . \end{aligned} \]
where, \(L_1^{\alpha} = [H_{0_1}^{\alpha}, F]\), and the operator \(L_{12}^{'\alpha \beta}\) is the interaction Liouvillian, describing the interaction of particle 1 (of species \(\alpha\)) and 2 (of species \(\beta\)):
\[L_{12}^{'\alpha \beta} = [V_{12}^{\alpha \beta}, F]\]
for any function \(F\) of the coordinates of particles 1 and 2. Compare to non-interacting case, here the interacting potential make the distribution function equation become more complicated and non-closed. To solve one-particle distribution, one has to know two-particle distribution. However, when one tries to solve two-particle distribution, the solution requires three-particle distribution fucntion and so on. Thus, the exact solution requires the solution the \(N\) coupled equations for all the rdf: the well-known BBGKY hierarchy.
Weak coupling approximation
In order to define this approximation, we first introduce the concept of a binary correlation function \(g^{\alpha \beta}(\boldsymbol{q}_1, \boldsymbol{v}_1,\boldsymbol{q}_2, \boldsymbol{v}_2;t)\),
\[ g^{\alpha \beta}\left(\boldsymbol{q}_1, \boldsymbol{v}_1, \boldsymbol{q}_2, \boldsymbol{v}_2 ; t\right) =f^{\alpha \beta}\left(\boldsymbol{q}_1, \boldsymbol{v}_1, \boldsymbol{q}_2, \boldsymbol{v}_2 ; t\right)-f^\alpha\left(\boldsymbol{q}_1, \boldsymbol{v}_1 ; t\right) f^\beta\left(\boldsymbol{q}_2, \boldsymbol{v}_2 ; t\right) \]
The correlation function measures the deviation from statistical independence of the two particles. It is expected, on physical grounds, that \(g^{\alpha,\beta}\rightarrow 0\) as the relative distance between the particles increases. One may usually define a quantity \(r_c\), called the range of the correlations, noting that
\[ g^{\alpha \beta}\left(q_1, v_1, q_2, v_2 ; t\right) \approx 0 \text { for }\left|q_1-q_2\right|>r_{\mathrm{c}} \]
The following normalization property exists:
\[ \int \mathrm{d} q_1 \mathrm{~d} v_1 \mathrm{~d} q_2 \mathrm{~d} v_2 g^{\alpha \beta}\left(q_1, v_1, q_2, v_2 ; t\right)=0 . \]
Since
\[ \int \mathrm{d} q_1 \mathrm{~d} v_1 \mathrm{~d} q_2 \mathrm{~d} v_2 f^{\alpha \beta}\left(q_1, v_1, q_2, v_2 ; t\right)= \int \mathrm{d} q_1 \mathrm{~d} v_1 \mathrm{~d} q_2 \mathrm{~d} v_2 f^\alpha\left(\boldsymbol{q}_1, \boldsymbol{v}_1 ; t\right) f^\beta\left(\boldsymbol{q}_2, \boldsymbol{v}_2 ; t\right) = N_\alpha N_\beta \]
The physical origin of the correlation is the existence of interactions, which cause the particles to feel their mutual influence.
It is well known that, in a plasma, the relative importance of the Coulomb interactions, compared to the mean kinetic energy (i.e. the thermal energy) is measured by the characteristic dimensionless parameter \(\mu_{\mathrm{p}}\), involving the charge \(e_\alpha\), the mean particle density \(n_\alpha\) and the temperature \(T_\alpha\) : \[ \mu_{\mathrm{P}}=\left(\frac{4}{3} \pi \lambda_{\mathrm{D}}^3 n_{\mathrm{i}}\right)^{-2 / 3}=(36 \pi)^{1 / 3} \frac{e^2 n_{\mathrm{i}}^{-1 / 3}\left(Z T_{\mathrm{e}}+T_{\mathrm{i}}\right)}{T_{\mathrm{e}} T_{\mathrm{i}}(1+Z)}, \] where \(\lambda_D\) is the Debye length (Trubnikov 1965), \[ \lambda_{\mathrm{D}}=\left(\frac{4 \pi Z e^2\left(n_{\mathrm{e}} T_{\mathrm{e}}+n_{\mathrm{i}} T_{\mathrm{i}}\right)}{T_{\mathrm{e}} T_{\mathrm{i}}(1+Z)}\right)^{-1 / 2} . \] The plasma parameter is defined by various authors in slightly different ways, differing by a numerical factor, or by the exponent in eq. (4.7). The definition adopted here has a very simple physical interpretation. Up to the factor \((36 \pi)^{1 / 3} \simeq 4.836\), the right-hand side of eq. (4.7) represents the ratio of the Coulomb energy, evaluated at the mean distance of the ions, \(n_{\mathrm{i}}^{-1 / 3}\), to the mean thermal energy, defined as \[ T^{-1}=\frac{Z T_{\mathrm{e}}+T_{\mathrm{i}}}{(1+Z) T_{\mathrm{e}} T_{\mathrm{i}}}=\left(\frac{1}{n_{\mathrm{i}} T_{\mathrm{i}}}+\frac{1}{n_{\mathrm{e}} T_{\mathrm{e}}}\right) /\left(\frac{1}{n_{\mathrm{i}}}+\frac{1}{n_{\mathrm{e}}}\right), \] where we used the electroneutrality condition \(n_{\mathrm{e}}=Z n_{\mathrm{i}}\) and where the temperature is expressed in energy units (see section 3.2). Other authors prefer to use the parameter \(N_{\mathrm{D}}\), \[ N_{\mathrm{D}}=\frac{4}{3} \pi \lambda_{\mathrm{D}}^3 n_{\mathrm{i}}=\mu_{\mathrm{P}}^{-3 / 2} . \] The parameter \(N_{\mathrm{D}}\) represents the number of particles in a sphere of radius equal to the Debye length \(\lambda_{\mathrm{D}}\).
We define a plasma as being weakly coupled when the following relation is satisfied:
\[ \mu_p \ll 1 \]
Clearly, this condition can be met in two ways. For a given temperature, the condition is satisfied for sufficiently low density; alternatively, for a given density, it holds for sufficiently high temperatures. Thus in a weak coupling regime, the potential interaction energy of the particles is, on the average, very small compared to their mean kinetic energy. A nautral truncation of the BBGKY hierarchy is obtained by negelecting all correlation functions of more than two particles.
The equation for \(f^\alpha\) is obtained, \[ \begin{aligned} \partial_t f^\alpha(1 ; t)= & L_1^\alpha f^\alpha(1 ; t)+\sum_\beta \int \mathrm{d} 2 L_{12}^{\prime \alpha \beta} f^\alpha(1 ; t) f^\beta(2 ; t) \\ & +\sum_\beta \int \mathrm{d} 2 L_{12}^{\prime \alpha \beta} g^{\alpha \beta}(1,2 ; t) . \quad\text{(1)} \end{aligned} \] where \(f(1;t) \equiv f(\boldsymbol{q}_1, \boldsymbol{p}_1; t)\) (using the obvious abbreviation \(j\) for the \(\operatorname{set}\left(q_j, v_j\right)\) ).
This equation is coupled to the equation for the correlation function (see Balescu 1975), in which the three-body correlation is set equal to zero: \[ \begin{aligned} \partial_t g^{\alpha \beta}(1,2 ; t) &=\left(L_1^\alpha+L_2^\beta\right) g^{\alpha \beta}(1,2 ; t)+L_{12}^{\prime \alpha \beta} g^{\alpha \beta}(1,2 ; t) \\ &\quad+\sum_\gamma \int \mathrm{d} 3\left[L_{13}^{\prime \alpha \gamma} f^\alpha(1 ; t) g^{\beta \gamma}(2,3 ; t)+L_{23}^{\prime \beta \gamma} f^\beta(2 ; t) g^{\alpha \gamma}(1,3 ; t)\right. \\ &\left.\quad+\left(L_{13}^{\prime \alpha \gamma}+L_{23}^{\prime \beta \gamma}\right) f^\gamma(3 ; t) g^{\alpha \beta}(1,2 ; t)\right]+L_{12}^{\prime \alpha \beta} f^\alpha(1 ; t) f^\beta(2 ; t) \end{aligned} \] Even this set of two coupled equations for \(f^\alpha\) and \(g^{\alpha \beta}\) is too complicated for explicit calculations. By using arguments which will not be described in detail for lack of space, it can be shown that the following two simpler versions are appropriate for describing a plasma. In a first stage, we retain \[ \begin{aligned} & \partial_t g^{\alpha \beta}(1,2 ; t)-\left(L_1^\alpha+L_2^\beta\right) g^{\alpha \beta}(1,2 ; t) \\ & =\sum_\gamma \int \mathrm{d} 3\left[L_{13}^{\prime \alpha \gamma} f^\alpha(1 ; t) g^{\beta \gamma}(2,3 ; t)+L_{23}^{\prime \beta \gamma} f^\beta(2 ; t) g^{\alpha \gamma}(1,3 ; t)\right] \\ & \quad+L_{12}^{\prime \alpha \beta} f^\alpha(1 ; t) f^\beta(2 ; t) . \quad[B L] \end{aligned} \]
The second, simpler version is \[ \partial_t g^{\alpha \beta}(1,2 ; t)-\left(L_1^\alpha+L_2^\beta\right) g^{\alpha \beta}(1,2 ; t)= L_{12}^{\prime \alpha \beta} f^\alpha(1 ; t) f^\beta(2 ; t) . \quad[L] \]
The main difference between the two approximations can be readily recognized. In second version, only the two particles 1 and 2, appearing in the unknown correlation function, are involved. One therefore expects this approximation to describe an evolution driven by binary collisions of the particles (in the weak coupling approximation). The end result of this approximation will be the Landau kinetic equation. On the contrary, in first version, we also retain interactions with a third particle taken from the "bath". This implies that we admit a certain class of many-body interactions in the approximate description of the evolution. The product of this procedure is the Balescu-Lenard equation, which contains a dynamic description of the screening or polarization in the plasma, a typical collective effect.
The last stage in reaching a true kinetic equation is the elimination of the correlation function. This is a crucial and non-trivial step, which will be discussed later. Let us say here that if the formal solution of correlation function is substituted into dynamic equaiton, the resulting equation is quite untractable, because of its mathematical complexity. A careful study reveals, however, the possible emergence of a regime in which the correlation function at time \(t\) becomes a functional of the one-particle distribution functions at the same time \(t\) (Bogoliubov 1962, Balescu 1975):
\[ g^{\alpha \beta}(t)=g^{\alpha \beta}\{f(t)\} . \]
When this kinetic regime is established, the substitution of this correlation function into dynamic equation results in
\[ \partial_t f^\alpha(1 ; t)=L_1^\alpha f^\alpha(1 ; t)+\sum_\beta \int \mathrm{d} 2 L_{12}^{\alpha \beta} f^\alpha(1 ; t) f^\beta(2 ; t)+\mathscr{K}^\alpha\{f(t)\} \] where \(\mathscr{K}^\alpha\{f(t)\}\) represents term \(\sum_\beta \int \mathrm{d} 2 L_{12}^{\prime \alpha \beta} g^{\alpha \beta}(1,2 ; t)\) is the so-called collision term.
This is the typical, generic form of a kinetic equation. This concept is defined as a closed, Markovian, generally non-linear equation of evolution for the one-particle distribution function \(f^\alpha(1 ; t)\). The word "Markovian" means (broadly speaking) that the kinetic equation only involves the value of the unknown distribution function at time \(t\). In particular, the rate of change \(\partial_t f^\alpha(1 ; t)\) is not influenced by the values taken by the distribution function in the past, i.e. \(f^\alpha(1 ; t-\tau), \tau>0\).
We may already characterize the role of the various terms in the above equation in driving the change in time of the distribution function. \(L_1^\alpha\) describes the effect of the free motion (kinetic energy) as well as the effect of the external electric and magnetic fields. The interactions affect the motion in two ways. The second term \(\sum_\beta \int \mathrm{d} 2 L_{12}^{\alpha \beta} f^\alpha(1 ; t) f^\beta(2 ; t)\) on the right-hand side will be shown to represent the effect of an average self-consistent field acting on particle 1 and produced by all the other particles in the system. Besides this "smooth" background action, the interactions also act more "violently" upon a sufficiently close approach of two (or more) particles. This important effect is contained in the collision term \(\mathscr{K}^\alpha\). Its specific form depends widely on the nature and the state of the system. For instance, it is quite different when only binary collisions are considered or when collective effects are included; its form depends on the density and the temperature, being quite different for a fusion plasma or for a degenerate quantum plasma; it can be affected by the presence of a (very strong) electric or magnetic field, etc.
The most important property of the above equation is the following. When the kinetic regime is established, the Hamiltonian (or Liouvillian) structure of the evolution process is lost. The detailed reason for this "symmetry breaking" will appear later. Let us say here that this feature opens the gate towards the description of a dissipative and irreversible evolution. This property was absent in the initial Hamiltonian description, but is indispensable for understanding macroscopic physics.
The Vlasov kinetic equation
Note that the interaction Liouvillian operator has the form
\[ \begin{aligned} L_{12}^{\prime \alpha \beta} &= [V_{12}^{\alpha \beta}, F]\\ & =\left[\nabla_{12} V^{\alpha \beta}\left(\boldsymbol{q}_1-\boldsymbol{q}_2\right)\right] \cdot\left(\partial_1-\partial_2\right)F \\ & =e_\alpha e_\beta\left(\nabla_{12} \frac{1}{\left|\boldsymbol{q}_1-\boldsymbol{q}_2\right|}\right) \cdot\left(\partial_1-\partial_2\right)F, \end{aligned} \] where the following abbreviations were used: \[ \nabla_{12} \equiv (\frac{\partial}{\partial q_1}, \frac{\partial}{\partial q_2} ), \quad \partial_1 \equiv \frac{\partial}{\partial p_1} \]
The second term on the right-hand side of the dynamic equation can be transformed as \[ \begin{aligned} & \sum_\beta \int \mathrm{d} 2 L_{12}^{\prime \alpha \beta} f^\alpha(1 ; t) f^\beta(2 ; t) \\ & =\sum_\beta \int \mathrm{d} 2\left(\nabla_{12} V_{12}^{\alpha \beta}\right) \cdot\left(m_\alpha^{-1} \partial_1-m_\beta^{-1} \partial_2\right) f^\alpha(1 ; t) f^\beta(2 ; t) \\ & =\left(\nabla_{1} \sum_\beta \int \mathrm{d} 2 V_{12}^{\alpha \beta} f^\beta(2 ; t)\right) \cdot m_\alpha^{-1} \partial_1 f^\alpha(1 ; t) \\ & \equiv-\frac{e_\alpha}{m_\alpha} \bar{E}\left(\boldsymbol{q}_1 ; t\right) \cdot \partial_1 f^\alpha(1 ; t), \end{aligned} \] where \[ \bar{E}\left(q_1 ; t\right) \equiv-\nabla_1 \bar{\Phi}\left(q_1 ; t\right) \] and: \[ \overline{\boldsymbol{\Phi}}\left(\boldsymbol{q}_1 ; t\right)=\sum_\beta e_\beta \int \mathrm{d} \boldsymbol{q}_2 \mathrm{~d} \boldsymbol{v}_2 \frac{1}{\left|\boldsymbol{q}_1-\boldsymbol{q}_2\right|} f^\beta\left(\boldsymbol{q}_2, \boldsymbol{v}_2 ; t\right) . \]
These results have a very simple physical interpretation. We recognize in the above equality \(\sum_\beta \int \mathrm{d} 2 L_{12}^{\prime \alpha \beta} f^\alpha(1 ; t) f^\beta(2 ; t)\) the gradient of the average interaction potential which, in turn, can be expressed in the form of an electric field \(\overline{\boldsymbol{E}}\) deriving from a scalar potential \(\overline{\boldsymbol{\Phi}}\), defined as integral over coulomb potential times particle 2's distribution function. The latter has exactly the form of the potential produced by a macroscopic continuous charge distribution \(\sigma\),
\[ \sigma\left(\boldsymbol{q}_2 ; t\right)=\sum_\beta e_\beta \int \mathrm{d} \boldsymbol{v}_2 f^\beta\left(\boldsymbol{q}_2, \boldsymbol{v}_2 ; t\right) \]
More precisely, the potential \(\bar{\Phi}\left(\boldsymbol{q}_1 ; t\right)\) is a solution of the Poisson equation
\[ \nabla^2 \bar{\Phi}\left(q_1 ; t\right)=-4 \pi \sigma\left(q_1 ; t\right) . \] This effect of the interactions can be described as follows. Each particle (1) feels the action of an average electric field produced by all the other particles in the system; these particles act like a "smeared-out" continuous charge distribution \(\sigma\left(q_1 ; t\right)\) : they lost their discrete, point-like character in the process of averaging. This electric field is self-consistent in the following sense: \(\overline{\boldsymbol{E}}\) acts on the distribution function \(f^\alpha\) and makes it change in time; but as \(f^\alpha\) changes, it modifies, in turn, the potential through.
We now note that the vlasov equation has been reduced to the same form as the electric field term in the one-particle Liouville equation. The latter field \(\boldsymbol{E}_0\) is, by definition, an external field, produced by sources outside the plasma: the corresponding potential \(\Phi_0\) therefore obeys the homogeneous Laplace equation, i.e. with \(\sigma=0\). We now define the total electric field in the plasma, \(\boldsymbol{E}(\boldsymbol{q} ; t)\), as \[ \boldsymbol{E}(\boldsymbol{q} ; t)=\boldsymbol{E}_0(\boldsymbol{q} ; t)+\overline{\boldsymbol{E}}(\boldsymbol{q} ; t) . \] It derives from the potential \(\Phi=\Phi_0+\bar{\Phi}\), which obeys the Poisson equation with source inside the plasma (with the same source term \(\sigma\) ), with appropriate boundary conditions.
We may thus write the kinetic equation, omitting the collision term \(\mathscr{K}^\alpha\), in the explicit form
\[ \partial_t f^\alpha(1 ; t)=-v_1 \cdot \nabla_1 f^\alpha(1 ; t)-\frac{e_\alpha}{m_\alpha}\left(\frac{1}{c}\left(v_1 \times \boldsymbol{B}\right)+\boldsymbol{E}\right) \cdot \partial_1 f^\alpha(1 ; t), \]
which is coupled to the Poisson equation
\[ \nabla \cdot \boldsymbol{E}(\boldsymbol{q} ; t)=4 \pi \sum_\beta e_\beta \int \mathrm{d} \boldsymbol{v} f^\beta(\boldsymbol{q}, \boldsymbol{v} ; t), \]
together with
\[ \nabla \times \boldsymbol{E}(\boldsymbol{q} ; t)=0 . \]
This is the celebrated Vlasov equation (Vlasov 1938) which plays a fundamental role in plasma physics. It is also called the "collisionless kinetic equation" (or, very improperly, the "collisionless Boltzmann equation"). One feature is immediately striking: it has the same form as the one-particle Liouville equation. But one should never forget the essential difference between these equations. In the Liouville equation, the electric field \(\boldsymbol{E}_0\) is external, i.e., it is a prescribed function of \(\boldsymbol{q}\) and \(t\). As a result, the Liouville equation is linear in \(f^\alpha\). On the contrary, the "linear" form of the Vlasov equation is only apparent, because here the electric field is to be considered as a second unknown function, along with \(f^a\); the Vlasov equation is coupled to the Poisson equation. Hence, the Vlasov equation is non-linear. This property is a necessary consequence of the interactions.
Now we only consider the Coulombic effects, so the self-consistent effects only reveal on the electric field. We haven't consider the magnetic interaction between particles, that requires a relativistic theory, which is not an easy matter. A simple treatment, sufficient for the derivation of the Vlasov equaion, is given by Klimontovich(1964). The average velocity of the particles builds up an electric current, which acts as a source of a self-consistent magnetic field \(\boldsymbol{B}(\boldsymbol{q};t)\) that must be added to the external field in order to produce the total magnetic field.
As a result, the Vlasov equation is still valid as it stands, but it is coupled to the full set of Maxwell equations, \[ \begin{aligned} & \nabla \cdot \boldsymbol{E}(\boldsymbol{q} ; t)=4 \pi \sum_\beta e_\beta \int \mathrm{d} v f^\alpha(\boldsymbol{q}, \boldsymbol{v} ; t), \\ & \nabla \times \boldsymbol{E}(\boldsymbol{q} ; t)=-c^{-1} \partial_t \boldsymbol{B}(\boldsymbol{q} ; t), \\ & \nabla \cdot \boldsymbol{B}(\boldsymbol{q} ; t)=0, \\ & \nabla \times \boldsymbol{B}(\boldsymbol{q} ; t)=c^{-1} \partial_t \boldsymbol{E}(\boldsymbol{q} ; t)+(4 \pi / c) \sum_\beta e_\beta \int \mathrm{d} \boldsymbol{v} v f^\alpha(\boldsymbol{q}, \boldsymbol{v} ; t) . \end{aligned} \] Clearly, if we let \(c \rightarrow \infty\), we come back to the previous picture.
The Landau kinetic equation
The Vlasov equation ignore the collision term, but now we try to derive the collision term. Our goal is the elimination of the correlation function \(g^{\alpha, \beta}(1,2;t)\) from the last term in Eq.(1). For this purpose, we must solve one of the truncated versions of the second BBGKY equation: we choose here the simplest case which only involve two particles. This linear, inhomogeneous equation for \(g^{\alpha \beta}(1,2 ; t)\) is formally solved in terms of a propagator \(U_{12}^{\alpha \beta}(t)\), which obeys \[ \partial_t U_{12}^{\alpha \beta}(t)-\left(L_1^\alpha+L_2^\beta\right) U_{12}^{\alpha \beta}(t)=0, \quad U_{12}^{\alpha \beta}(0)=I, \] where \(I\) is the identity operator. The propagator is used to propagate information along some characteristic line. The solution of the initial value problem for Eq.(L) is then \[ \begin{aligned} g^{\alpha \beta}(1,2 ; t)= & \int_0^t \mathrm{~d} \tau U_{12}^{\alpha \beta}(\tau) L_{12}^{\prime \alpha \beta} f^\alpha(1 ; t-\tau) f^\beta(2 ; t-\tau) \\ & +U_{12}^{\alpha \beta}(t) g^{\alpha \beta}(1,2 ; 0), \end{aligned} \] where \(g^{\alpha \beta}(1,2 ; 0)\) is the (given) initial condition for the correlation function. Here the information deciding \(g^{\alpha\beta}\) at time \(t\) is divided into two part: the first part is correlation part, it is an integral over some characteristic line and varies all the way. The second part is initial state which propagates without any distraction. This is very similar to integral solution of the BGK equation.
Substituting this result into the last term of Eq.(1), we obtain \[ \begin{aligned} \mathscr{K}^\alpha= & \sum_\beta \int \mathrm{d} 2 \int_0^t \mathrm{~d} \tau L_{12}^{\alpha \beta} U_{12}^{\alpha \beta}(\tau) L_{12}^{\prime \alpha \beta} f^\alpha(1 ; t-\tau) f^\beta(2 ; t-\tau) \\ & +\sum_\beta \int \mathrm{d} 2 L_{12}^{\alpha \beta} U_{12}^{\alpha \beta}(t) g^{\alpha \beta}(1,2 ; 0) . \end{aligned} \] This equation is formally exact (within the weak coupling approximation) and yields a closed equation for \(f^\alpha(1 ; t)\). It is called the master equation and was originally derived (in a more general context) by Prigogine and Résibois (1961). This equation has some quite unfamiliar properties. A striking feature is its non-Markovian character. It is an integro-differential equation in time; hence, the rate of change \(\partial_t f^\alpha(1 ; t)\) at time \(t\) depends on the integral of \(f^\alpha(1 ; t-\tau)\) over the whole past history. And Markovian character means that the current state only depends on the former time step, such as random walk.
Such a structure cannot be expected to lead to the familiar macroscopic equations of hydrodynamics or electrodynamics, which are all Markovian. (For macroscopic equation, if you discretize the equation with a timestep \(\Delta t\), the state of \(t^{n+1}\) only depends on state of \(t^{n}\). That's why there will be constraint of timestep.) Moreover, the second term is very strange: it implies that the rate of change at time \(t\) of \(f^\alpha(1 ; t)\) depends on the initial value of the correlations, i.e. on the initial preparation of the system. In order to get out of these difficulties, we must exploit more thoroughly the physics of the problem and disentangle the relevant features from the less relevant ones.
We first note that the two-body propagator \(U_{12}^{\alpha \beta}(t)\) relates to the unperturbed motion of independent particles. It is well known (or easily checked) that this quantity factorizes as \[ U_{12}^{\alpha \beta}(t)=U_1^\alpha(t) U_2^\beta(t) \] where the one-particle propagator satisfies the simpler Liouville equation \[ \partial_t U_1^\alpha(t)=L_1^\alpha U_1^\alpha(t) \] Next, we note that the determination of the propagator is equivalent to the solution of the equations of motion of a single particle. Indeed, the Liouville theorem asserts that the phase space density at time \(t\), at the point \((q, v)\), is the same as the density at time 0 , at the point where the particle was at time \((-t)\) (the phase space fluid element density remains unchanged): \[ U_1^\alpha(t) f^\alpha\left(q_1, v_1 ; 0\right)=f^\alpha\left(q_1, v_1 ; t\right)=f^\alpha\left(q_1(-t), v_1(-t) ; 0\right), \] where \(q(-t)\) obeys the "backward" equation of motion, \[ \dot{q}(-t)=-[q(-t), H]=[H, q(-t)] \] with the initial condition \[ \boldsymbol{q}(0)=\boldsymbol{q} \] and similar equations for \(v(-t)\). But we know that even this one-particle problem is, in general, very difficult in the presence of inhomogeneous external fields. The feature that saves us and leads to the derivation of an explicit kinetic equation is the fact that in a plasma there exist several characteristic time-and length-scales and that these scales may be widely separated, at least in some cases. In order to discuss these, we must formulate a certain number of intuitive, qualitative statements or anticipations, which must be checked a posteriori.
We have already mentioned that the correlation function introduces a characteristic length, the range of correlation \(r_{\mathrm{c} \alpha}\), defined by the fact that \(g^{\alpha \beta} \simeq 0\) whenever the relative distance \(\left|\boldsymbol{q}_1-\boldsymbol{q}_2\right|\) exceeds \(r_{\mathrm{c} \alpha}\). On the basis of elementary plasma physics (the Debye theory) it can be argued that \(r_{\mathrm{c} \alpha}\) is of the order of the Debye length of species \(\alpha\), \[ r_{\mathrm{c} \alpha}=\lambda_{\mathrm{D} \alpha}=\left(\frac{4 \pi e_\alpha^2 n_\alpha}{T_\alpha}\right)^{-1 / 2}, \]
where \(n_\alpha\) is the number density and \(T_\alpha\) is the temperature (expressed in energy units) of species \(\alpha\). To this characteristic length corresponds a characteristic time, the inverse plasma frequency of species \(\alpha\), \[ \tau_{\mathrm{c} \alpha}=\omega_{\mathrm{P} \alpha}^{-1}=\left(\frac{4 \pi e_\alpha^2 n_\alpha}{m_\alpha}\right)^{-1 / 2} \] Next, we consider the range of the interactions: this poses a problem for the Coulomb forces, which decrease very slowly with distance and can be considered, for many purposes, to have infinite range. Nevertheless, the elementary Debye theory tells us that the statistical correlations produce a screening of these forces for distances larger than \(\lambda_{\mathrm{D} \alpha}\). Hence, the effective range of the interactions can be estimated as being equal to \(r_{c \alpha}\) as well. It then follows that \(\tau_{\mathrm{c} \alpha}\) estimates the time spent by a typical particle in the sphere of interaction of another, i.e. the duration of a collision.
An independent characteristic time is provided by the collision term as a whole. Anticipating the fact that the collisions drive the system towards equilibrium, the collision term can be, very roughly speaking, represented as \[ \mathscr{K}^\alpha \approx \frac{f^\alpha-f_{\mathrm{eq}}^\alpha}{\tau_\alpha} . \] We thus introduce the relaxation time \(\tau_\alpha\) of species \(\alpha\), which may be defined as the time over which the distribution function \(f^\alpha\) undergoes a significant change under the effect of the collisions. It will be determined more precisely in chapter 4. Associated with it is the mean free path of species \(\alpha\), defined as \[ \lambda_{\mathrm{mfp} \alpha}=V_{\mathrm{T} \alpha} \tau_\alpha, \] where \(V_{\mathrm{T} \alpha}\) is the thermal velocity of species \(\alpha\), \[ V_{\mathrm{T} \alpha}=\left(\frac{2 T_\alpha}{m_\alpha}\right)^{1 / 2} . \] A third, independent characteristic length is associated with the spatial variation of the macroscopic quantities of the system (density, temperature,...), of the electric and magnetic fields, and also of the distribution function itself.
Assuming that all these lengths are of the same order, we define \(L_{\mathrm{H}}\), the gradient length scale, or hydrodynamic length, as \[ L_{\mathrm{H}}=\left(\frac{1}{A}\left|\frac{\partial A}{\partial \boldsymbol{q}}\right|\right)^{-1}, \] where \(A\) is any one of the above mentioned quantities. Here the gradient is normalised by the magnitude of the quantity itself. We also define a hydrodynamic time as \[ \tau_{\mathrm{H} \alpha}=L_{\mathrm{H}} / V_{\mathrm{T} \alpha} . \] Last, but not least, there is a time scale defined by the Larmor frequency, \[ \tau_{B \alpha}=\left|\Omega_\alpha\right|^{-1}=\frac{m_\alpha c}{\left|e_\alpha\right| B}, \] and, corresponding to it, a length scale defined by the Larmor radius: \[ \rho_{\mathrm{L} \alpha}=V_{\mathrm{T} \alpha} \tau_{B \alpha} \] The order relations between these scales determines various regimes of evolution. The most familiar one is characterized as follows. The gradient scale length \(L_{\mathrm{H}}\) is a macroscopic quantity, typically measured in meters, centimeters,...; it is produced by a macroscopic preparation of the system. At the other extreme, the range of the correlations or of the interactions, \(r_{c \alpha}\), is a purely microscopic quantity. For a neutral gas, it is of the order of the size of a molecule, i.e. a few ångstroms. For a plasma, the Debye length can be much larger, depending on the density and the temperature. For laboratory systems (such as a thermonuclear plasma) it does not exceed \(10^{-3} \mathrm{~cm}\). As for the mean free path, it is always much longer than the range of the interactions, provided the plasma is weakly coupled; but its relation to the hydrodynamic length is much more variable. In neutral, not too dilute gases, one has \(L_{\mathrm{H}} \gg \lambda_{\mathrm{mfp} \alpha}\). But in a plasma at high temperature, the mean free path may easily become as large as several meters.
In conclusion, we may assume the following relationships as representing typical situations in a weakly coupled plasma: \[ r_{\text {c } \alpha} \ll \lambda_{\text {mfp } \alpha} \leq L_{\mathrm{H}}, \] together with \[ \tau_{\mathrm{c} \alpha} \ll \tau_\alpha \lessgtr \tau_{\mathrm{H} \alpha} . \]
Alternatively, we may have \[ r_{\text {c } \alpha} \ll L_{\mathrm{H}}<\lambda_{\text {mfp } \alpha}, \] together with \[ \tau_{\mathrm{c} \alpha} \ll \tau_{\mathrm{H} \alpha}<\tau_\alpha \] Considering now the magnetic effects, we assume that the magnetic field is sufficiently small in order to ensure the conditions * \[ \left|\Omega_\alpha\right| \tau_{\mathrm{c} \alpha}<1, \quad \frac{r_{\mathrm{c} \alpha}}{\rho_{\mathrm{L} \alpha}}<1 . \] It may be useful to note that this condition can be satisfied in practice even with very strong magnetic fields, such that \[ \frac{L_{\mathrm{H}}}{\rho_{\mathrm{L} \alpha}} \gg 1, \quad \frac{\lambda_{\mathrm{mfp} \alpha}}{\rho_{\mathrm{L} \alpha}} \gg 1 . \] It should be realized that the first of these conditions precisely defines the applicability of the drift approximation. Forgetting now the details of the rough estimation of the various quantities, we only retain the basic five assumptions and make the best use of them in deriving an expression for the collision operator.
In kinetic theory, we are mainly interested in the approach to equilibrium. This process takes place on the characteristic time scale \(\tau_\alpha\), which is much longer than the duration of the collisions, \(\tau_{\mathrm{c} \alpha}\). We therefore decide to study the evolution by "smoothing out" the details of the motion that arise on the short time scale \(\tau_{c \alpha}\). The resulting "smooth", or "asymptotic" evolution law will be an appropriate description of the phenomena, whenever we are not interested in short-living transient processes. (Note that a similar philosophy underlies the description of the particle motion in the drift approximation, where we use average)
The result of this treatment is certainly familiar to many readers: it is the explicit form of the kinetic equation in the weak coupling approximation,
\[ \begin{aligned} \mathscr{K}^\alpha= & \sum_\beta 2 \pi e_\alpha^2 e_\beta^2 \ln \Lambda \int \mathrm{d} v_2 m_\alpha^{-1} \frac{\partial}{\partial v_{1 r}} G_{r s}(g) \\ & \times\left(m_\alpha^{-1} \frac{\partial}{\partial v_{1 s}}-m_\beta^{-1} \frac{\partial}{\partial v_{2 s}}\right) f^\alpha\left(\boldsymbol{q}_1, \boldsymbol{v}_1 ; t\right) f^\beta\left(\boldsymbol{q}_1, \boldsymbol{v}_2 ; t\right) .\quad\text{(2)} \end{aligned} \]
Here, we use the notation \(g\) for the relative velocity vector, \[ g=v_1-v_2 \] The important Landau tensor \(G_{r s}(a)\) is defined, for an arbitrary vector \(\boldsymbol{a}\), as \[ G_{r s}(a)=\frac{a^2 \delta_{r s}-a_r a_s}{a^3} . \] Finally, the Coulomb logarithm, \(\ln \Lambda\), is defined as \[ \ln \Lambda=\ln \frac{\frac{3}{2}\left(T_{\mathrm{e}}+T_{\mathrm{i}}\right) \lambda_{\mathrm{D}}}{Z e^2} . \] Eq.(2) represents the celebrated Landau collision term, that provides the basis of most of the existing works on plasma transport theory. It was first obtained by Landau (1936) from the Boltzmann equation of the kinetic theory of gases, combined with the weak coupling assumption. It has been later derived more rigourously by many authors (Bogoliubov 1946, Balescu 1963, 1975, Klimontovich 1964, 1982) and its properties are discussed in many textbooks on plasma physics (e.g. Montgomery and Tidman 1964, Ichimaru 1974, Krall and Trivelpiece 1986, Golant et al. 1980, Rosenbluth and Sagdeev 1983).
A fundamental result of this section is the fact that the collision operator \(\mathscr{K}^\alpha\) appears as an operator acting on \(f^\alpha(\boldsymbol{q}, \boldsymbol{v} ; t)\), i.e. a functional of the distribution function at time \(t\). We have thus proved the existence of a kinetic regime. It appears as an asymptotic regime, valid for times much longer than the duration of a collision(Here the collison time scale is \(\tau_{c\alpha}\)). A fuller discussion of the more fundamental aspects of this question can be found in the book by Balescu (1975). For our more restricted purpose, the main point is the existence of a closed kinetic equation for \(f^\alpha\).
For detailed derivation of the collision term, refer to Balescu Appendix 2A.1. One important notion here is that in plasma, collision time scale is different from the time scale for reaching equilibrium state. A particle may experience a set of collisions before come to equilibrium state.
