From conformal invariance towards dynamical symmetries of the collisionless Boltzmann equation

Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the $2D$ conformal algebra. In the case without external forces, the symmetry of the conformally invariant transport equation is first generalised by considering the particle momentum as an independent variables. This new conformal representation can be further extended to include an external force. The construction and possible physical applications are outlined.


Introduction
The Boltzmann transport equation (BTE) [1, 14,7,8] furnishes a semi-classical description of the effects of particle transport, including the influence of external forces, on the effective single-particle distribution function f = f (t, r, p) of a small cell in phase phase, centred at position r and momentum p. For a system with identical particles of mass m, the Boltzmann equation reads ∂f ∂t (1.1) Here, dN = f (t, r, p, )dr dp is the number of particles in a cell of phase volume dr dp, centred at position r and momentum p [8]. In addition, F = F (t, r) is the force field acting on the particles in the fluid. The term on the right-hand-side is added to describe the effect of collisions between particles. It is a statistical term and requires knowledge of the statistics the particles obey, like the Maxwell-Boltzmann, Fermi-Dirac or Bose-Einstein distributions. In his famous 'Stoßzahlansatz' (or hypothesis of molecular chaos), Boltzmann obtained an explicit form for it. In a modern notation, for example for an interacting Fermi gas, where a particle from a state with momentum p is scattered to a state with momentum p ′ , whereas a second particle is scattered from a momentum q to a momentum q ′ , the collision term reads where w({pq} → {p ′ q ′ }) is the normalised transition probability from the two-particle state with momenta {p, q} to the state labelled by {p ′ , q ′ }. Clearly, solving this widely studied equation is a very difficult task. It might be hoped that symmetries could be helpful. The equation without the collision term is known as the Vlasov equation [17]. Relationship with Landau damping and a physicists' derivation can be found in [18,5]. In this work, we shall explore a class of symmetries of the (collisionless) BTE.
Throughout, we shall restrict to d = 1 space dimension 1 . We start from a non-standard representation, isomorphic to the infinite-dimensional Lie algebra of conformal transformations in d = 2 dimensions. 2 This Lie algebra is spanned by the generators X n , Y n n∈Z and can be defined from the commutators [9,12] where µ is a parameter. An explicit realisation in terms of time-space transformation is [9,12]: 1 By analogy with other constructions of local scale symmetries, see [9,4,15,13] and especially [12] and refs. therein, we expect a straightforward extension of the results reported here to d > 1. Since we shall construct here a finite-dimensional Lie algebra of dynamical conformal symmetries of the 1D collisionless BTE, one should indeed expect that an extension to d > 1 exists. That symmetry algebra should contain three generators X ±1,0 , along with a vector of generators Y n and also spatial rotations. 2 For the sake of clarify, we shall adopt the following convention of terminology: the infinite-dimensional Lie algebra X n , Y n n∈Z will be called a (centreless) 'conformal Virasoro algebra'. Its maximal finite-dimensional sub-algebra X n , Y n n∈{−1,0,1} will be called a 'conformal algebra'.
such that µ −1 can be interpreted as a velocity ('speed of light/sound') and where x, γ are constants. 3 Writing X n = ℓ n +l n and Y n = µ −1l n , where the generators ℓ n ,l n n∈Z satisfy [ℓ n , ℓ m ] = (n − m)ℓ n+m , [l n ,l m ] = (n − m)l n+m , [ℓ n ,l m ] = 0, it can be seen that, provided µ = 0, the above Lie algebra (1.2) is isomorphic to a pair of Virasoro algebras vect(S 1 ) ⊕ vect(S 1 ) with a vanishing central charge. However, this isomorphism does not imply that physical systems described by two different representations of the conformal Virasoro algebra, or the conformal algebra, with commutators (1.2), were trivially related. For example, it is well-known that if one uses the generators of the standard representation of conformal invariance or else the non-standard representation (1.4) in order to find co-variant two-point functions, the resulting scaling forms are different [9]. Now, consider the maximal finite-dimensional sub-algebra X ±1,0 , Y ±1,0 , which for µ = 0 in turn is isomorphic to the direct sum sl(2, R) ⊕ sl(2, R). The explicit realisation follows from from (1.3) Here, the generators X −1 , Y −1 describe time-and space-translations, Y 0 is a (conformal) Galilei transformation, 4 X 0 gives the dynamical scaling t → λt of r → λr (with λ ∈ R) such that the so-called 'dynamical exponent' z = 1 since both time and space are re-scaled in the same way and finally X +1 , Y +1 give 'special' conformal transformations. In the context of statistical mechanics of conformally invariant phase transitions, one characterises co-variant quasi-primary scaling operators through the invariant parameters (x, µ, γ), where x is the scaling dimension. In order to return to the Boltzmann equation, we consider eq. (1.5) in the form where f = f (t, r, v) is interpreted as a single-particle distribution function and where we consider v as an additional variable. Eq. (1.7) is a simple Boltzmann (or Vlasov) equation, without an external force and without a collision term, and in one space dimension. From (1.6), with v fixed (and normalised to v = 1), its solution space is conformally invariant 6 . In section 2, we shall generalise the above representation of the conformal algebra to the situation with v as a further variable. In section 3, we shall further extend this to the case when an external force F = F (t, r, v), possibly depending on time, spatial position and velocity, is included. The aim of these calculations is to determine which situations of potential physical interest with a non-trivial conformal symmetry might be identified. This explorative study aims at identifying lines for further study, which might lead later to a more comprehensive understanding of the possible symmetries of Boltzmann equations. Taking into account the collision term is left for future work. We shall concentrate on d = 1 space dimension throughout. Conclusions and final comments are given in section 4.

Collisionless Boltzmann equation without external forces
In our construction of conformal dynamical symmetries of the 1D collisionless BTE, we shall often meet Lie algebras of a certain structure. These will be isomorphic to the two-dimensional conformal algebra.
Proposition 1: The Lie algebra X n , Y n n∈Z defined by the commutators where k, q are constants, is isomorphic to the pair of centreless Viraso algebras vect(S 1 ) ⊕ vect(S 1 ).
Proof: For either k = 0 or q = 0 this is either evident or else has already been seen in section 1. In the other case, consider the change of basis X n = ℓ n +l n and Y n = αℓ n − βl n where ℓ n ,l n are two families of commuting generators of vect(S 1 ) and α and β are constants such that α + β = 0. It then follows k = αβ and q = α − β. q.e.d.
This implies in particular the isomorphism of the maximal finite-dimensional sub-algebras, or 'conformal algebras' in the terminology chosen here. By definition, this 'conformal algebra' obeys the commutators (2.1), but with n, m ∈ {−1, 0, 1}.
Our construction of dynamical symmetries of the equation (1.7) follows the lines of the construction of local scale-invariance in time-dependent critical phenomena [9]. The physically motivated requirements are: First of all it is clear that the equation is invariant under timetranslations: Some kind of dynamical scaling must be present as well. Its most general form is Whenever, the dynamical exponent z = 1, we shall find an explicit dependence on v. In general, we look for a family of generators X n , for which we make the ansatz We shall find X n from the following three conditions (throughout, we use the notations 1. X n must be a symmetry for the equation (1.7), hence [L, X n ] = λ nL . This gives 2. The generator X 0 is assumed to be in the Cartan sub-algebra, hence [X n , X 0 ] = α n,0 X n . It follows 3. The action of X −1 is as a lowering operator, hence [X n , X −1 ] = α n,−1 X n−1 . It followṡ a n = α n,−1 t,ḃ n = α n,−1 r/z (2.10) c n = α n,−1 v(1 − z)/z,ḋ n = α n,−1 x/z.
These conditions, combined with the following initial conditions: must be sufficient for determination of all admissible forms of X n .
In the special case n = 1, we have α 1,0 = 1 and find the most general form of X 1 as a symmetry of (1.7) as follows: 7 and with a certain set of undetermined constants.
For conformal invariance, a family of generators Y n must also be found. Its construction is straightforward if the explicit form of Y −1 is known. Really X 1 must act as a raising operator, in both hierarchies, such that [9] [ However, the usual realization of Y −1 = −∂ r as space translations does not work, since if we set all undetermined constants in eq. (2.12) to zero, one It is better to work with the form as we shall do from now on.
We first consider the special case, when all the constants in the expression (2.12) for X 1 vanish: Proposition 2: The six generators Then the generators are modified as follows: we conclude that the cases A 12 = 0 and A 12 = 0 must be treated separately.
case B1: A 12 = 0. It follows that the constants in (2.12) are given by: Proposition 3: Let z = 1 and A 110 be arbitrary constants. Then the six generators span a representation of the conformal algebra. 8 These generators give more symmetries of the equation (1.7).
Proof: From the above, the commutator (1.2) are readily verified, with µ → −µ. For the dynamical symmetries, one checks the commutators which proves the assertion. q.e.d.
In contrast to the previous case A, the representation acting only on (t, r) but keeps v is a constant parameter, can no longer be obtained by simply setting z = 1. Rather, one must set A 110 = 0 first and only then the limit z → 1 is well-defined. It turns out that for A 12 = 0, the algebra also can be closed, but only if A 12 = µ and A 110 = 0 (then all others constants also vanish). Proof: The commutation relation are directly verified. The isomorphism with the conformal algebra follows from Proposition 1. The requirement to have an symmetry algebra of equation (1.7) implies a relation between the constants k, q (called α, β in Proposition 1) and µ, namely q = (k − µ 2 )/µ. In this case at hand, we have k = µ, q = 1 − µ. It is then verified that

close into a Lie algebra, with the following non-zero commutation relations
which proves that these are dynamical symmetries of (1.7). q.e.d.
We now ask whether the finite-dimensional representations (2.19, 2.24, 2.25), with µ = 0, acting on functions f = f (t, r, v), and having a dynamical exponent z = 1, can be extended to representations of an infinite-dimensional conformal Virasoro algebra. The answer turns out to be negative: Similar no-go results have been found before for variants of representations of the Schrödinger and conformal galilean algebras [11]. On the other hand, for µ = 0 extensions to a representation of a conformal Virasoro algebra with z = 1 exist [3].
Proof: Since for the finite-dimensional representations (2.19, 2.24, 2.25), we have [X n , X n ′ ] = (n − n ′ )X n+n ′ , [X n , Y m ] = (n − m)Y n+n ′ , n, n ′ , m = 0, ±1 we suppose that this must be valid for all admissible n, m ∈ Z. Now using the condition (2.10) for n = 2, a conformal Virasoro algebra should contain a new generator X 2 . Starting from the most general form, we find that the coefficients are obtained from: where a 21 (r, v), b 21 (r, v), c 21 (r, v), d 21 (r, v) are unknown functions of their arguments, but do no longer depend on the time t. We want to satisfy [X 2 , Y −1 ] = 3Y 1 . However when calculating we see that closure is not possible for z = 1. Indeed, although the dependence on r, v of the functions a 21 , b 21 , c 21 , d 21 can be chosen to satisfy the above closure condition, the t-dependence can not be absorbed into these functions. Hence our new representations (2.19, 2.24, 2.25) of the conformal algebra (2.1) are necessarily finite-dimensional. q.e.d.

Symmetry algebra of collisionless Boltzmann equation with an extra force term
We write the collisionless Boltzmann equation in the form We want to determine the admissible forms of an external force F (t, r, v) such that the equation (3.1) is invariant under a representation of the conformal algebra (2.1). The unknown representation must include the "force" term and in particular for F (t, r, v) = 0 it should coincide with the representations of conformal algebra obtained in previous section.
The idea of the construction is similar to the one used in section 2. First, we impose invariance under basic symmetries: where ϕ(u) is an arbitrary function, of the scaling variable u := r z−1 v.
It turns out that for the following calculations, it is more convenient to make a change of independent variables (t, r, v) → (t, r, u). In the new variables, the generator of dynamical scaling just reads: Next, in order to be specific, we make the following ansatz for the analogue of space translations 9 In the same coordinate system, the collisionless Boltzmann equation becomeŝ Here, some comments are in order. In the structure of Boltzmann equation (3.7), as well as in the form (3.6) of the modified space translations Y −1 , enters an unknown function Φ(t, r, u). 9 Indeed, we might also require to find Y −1 from the conditions to be (i) a symmetry of Boltzmann equation and (ii) to form a closed Lie algebra with the other basic symmetries X −1,0 . Such requirements lead to a system of differential equations and the ansatz (3.6) is a particular solution of this system, which has the special property that the Boltzmann operator can be linearly expressedB = −µX −1 − Y −1 by the generators. We believe this to be a natural auxiliary hypothesis.
Therefore, the form of X 1 cannot be found only from its commutator with the other generators X n , but the constraints form the entire conformal algebra must be used, as well as the requirement that X 1 and Y 0,1 are dynamical symmetries of eq. (3.7) (3.8) In fact, commuting the unknown generators X 1 , Y 0 , Y 1 with X −1 and X 0 , we can fix the t-and r-dependence of the yet undetermined functions which occur in them with the four functions In particular, looking for a representation of the analog of extended Galilei algebra X −1 , X 0 , Y −1 , Y 0 , we find that the unknown functions a 0 (u), b 0 (u), c 0 (u), d 0 (u) must satisfy the system Because of (3.14), one must distinguish two cases: 1. Φ(u) = 0, when d 0 (u) can be arbitrary 2. Φ(u) = 0, when d 0 (u) = d 0 = cste. is a constant.
The results derived here can be used as a starting point to derive forms of the transition rates w in the collision terms which would be compatible with the dynamical symmetries of the collision-free equations. This kind of approach would be analogous to the one used for finding dynamical symmetries of non-linear Schrödinger equations, see e.g. [2,16]. We hope to return to this elsewhere.