# From Conformal Invariance towards Dynamical Symmetries of the Collisionless Boltzmann Equation

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## Abstract

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## 1. Introduction

## 2. Collisionless Boltzmann Equation without External Forces

**Proposition 1.**The Lie algebra ${\langle {X}_{n},{Y}_{n}\rangle}_{n\in \mathbb{Z}}$ defined by the commutators:

**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 ${X}_{n}={\ell}_{n}+{\overline{\ell}}_{n}$ and ${Y}_{n}=\alpha {\ell}_{n}-\beta {\overline{\ell}}_{n}$, where ${\ell}_{n},{\overline{\ell}}_{n}$ are two families of commuting generators of $\mathfrak{vect}\left({S}^{1}\right)$ and α and β are constants, such that $\alpha +\beta \ne 0$. It then follows $k=\alpha \beta $ and $q=\alpha -\beta $. ☐

- ${X}_{n}$ must be a symmetry for the Equation (7); hence, $[\widehat{L},{X}_{n}]={\lambda}_{n}\widehat{L}$. This gives:$$\begin{array}{ccc}& & \mathrm{\mu}{\dot{a}}_{n}+v{a}_{n}^{\prime}+\mathrm{\mu}{\lambda}_{n}=0,\phantom{\rule{1.em}{0ex}}\mathrm{\mu}{\dot{b}}_{n}+v{b}_{n}^{\prime}-{c}_{n}+{\lambda}_{n}v=0\hfill \end{array}$$$$\begin{array}{ccc}& & \mathrm{\mu}{\dot{c}}_{n}+v{c}_{n}^{\prime}=0,\phantom{\rule{1.em}{0ex}}\mu {\dot{d}}_{n}+v{d}_{n}^{\prime}=0.\hfill \end{array}$$
- The generator ${X}_{0}$ is assumed to be in the Cartan sub-algebra; hence, $[{X}_{n},{X}_{0}]={\alpha}_{n,0}{X}_{n}$. It follows:$$\begin{array}{ccc}\hfill (1+{\alpha}_{n,0}){a}_{n}-t{\dot{a}}_{1}-\frac{r}{z}{a}_{n}^{\prime}-\frac{1-z}{z}v{\partial}_{v}{a}_{n}& =& 0\hfill \end{array}$$$$\begin{array}{ccc}\hfill (1/z+{\alpha}_{n,0}){b}_{n}-t{\dot{b}}_{n}-\frac{r}{z}{b}_{n}^{\prime}-\frac{1-z}{z}v{\partial}_{v}{b}_{n}& =& 0\hfill \end{array}$$$$\begin{array}{ccc}\hfill ((1-z)/z+{\alpha}_{n,0}){c}_{n}-t{\dot{c}}_{1}-\frac{r}{z}{c}_{n}^{\prime}-\frac{1-z}{z}v{\partial}_{v}{c}_{n}& =& 0\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\alpha}_{n,0}{d}_{n}-t{\dot{d}}_{n}-\frac{r}{z}{d}_{n}^{\prime}-\frac{1-z}{z}v{\partial}_{v}{d}_{n}& =& 0.\hfill \end{array}$$
- The action of ${X}_{-1}$ is as a lowering operator; hence, $[{X}_{n},{X}_{-1}]={\alpha}_{n,-1}{X}_{n-1}$. It follows:$$\begin{array}{ccc}\hfill {\dot{a}}_{n}& =& {\alpha}_{n,-1}t,\phantom{\rule{1.em}{0ex}}{\dot{b}}_{n}={\alpha}_{n,-1}r/z\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\dot{c}}_{n}& =& {\alpha}_{n,-1}v(1-z)/z,\phantom{\rule{1.em}{0ex}}{\dot{d}}_{n}={\alpha}_{n,-1}x/z.\hfill \end{array}$$

_{12}= A

_{110}= A

_{100}= B

_{110}= B

_{100}= D

_{0}= 0.

**Proposition 2.**The six generators:

**Proof.**It is readily checked that the generator Equation (26) satisfies the commutation relations (2), with $\mathrm{\mu}\mapsto -\mathrm{\mu}$. On the other hand, for any $f=f(t,r,v)$, one has:

_{12}≠ 0, A

_{110}≠ 0, A

_{100}≠ 0, B

_{110}≠ 0, B

_{100}≠ 0, D

_{0}≠ 0.

_{12}= 0. It follows that the constants in Equation (19) are given by:

**Proposition 3.**Let $z\ne 1$ and ${A}_{110}$ be arbitrary constants. Then, the six generators:

**Proof.**From the above, the commutator Equation (2) is readily verified, with $\mathrm{\mu}\mapsto -\mathrm{\mu}$. For the dynamical symmetries, one checks the commutators:

_{12}≠ 0, A

_{110}≠ 0, B

_{110}≠ 0, A

_{100}≠ 0, B

_{100}≠ 0, D

_{0}≠ 0.

**Proposition 4.**Let z be an arbitrary constant. Then, the generators $\u2329{\mathcal{X}}_{\pm 1,0},{\mathcal{Y}}_{\pm 1,0}\u232a$, where:

**Proof.**The commutation relation is directly verified. The isomorphism with the conformal algebra follows from Proposition 1. The requirement to have a symmetry algebra of Equation (7) implies a relation between the constants $k,q$ (called $\alpha ,\beta $ in Proposition 1) and μ, namely $q=(k-{\mathrm{\mu}}^{2})/\mathrm{\mu}$. In this case at hand, we have $k=\mathrm{\mu}$, $q=1-\mathrm{\mu}$. It is then verified that $[\widehat{L},{\mathcal{X}}_{-1}]=[\widehat{L},{\mathcal{Y}}_{-1}]=0$ and:

**Proposition 5.**The representation Equations (26), (30) and (31) of the finite-dimensional conformal algebra ${\u2329{X}_{n},{Y}_{n}\u232a}_{n\in \{\pm 1,0\}}$ with commutator Equation (8) cannot be extended to representations of an infinite-dimensional conformal Virasoro algebra with commutator Equation (8) when $z\ne 1$.

**Proof.**Since for the finite-dimensional representations Equations (26), (30) and (31), we have:

## 3. Symmetry Algebra of Collisionless Boltzmann Equation with an Extra Force Term

- From invariance under time translation ${X}_{-1}=-{\partial}_{t}$, it follows:$$[{X}_{-1},\widehat{B}]=-\dot{F}=0\to F=F(r,v)$$
- From invariance under dynamical scaling ${X}_{0}=-t{\partial}_{t}-\frac{r}{z}{\partial}_{r}-\frac{1-z}{z}v{\partial}_{v}-\frac{x}{z}$, we obtain that:$$[\widehat{B},{X}_{0}]=-\widehat{B},$$if $F(r,v)$ satisfies the equation $(r{\partial}_{r}+(1-z)v{\partial}_{v}-(1-2z))F(r,v)=0$, with solution:$$F(r,v)={r}^{1-2z}\phi \left({r}^{z-1}v\right),$$

- $\Phi \left(u\right)=0$, when ${d}_{0}\left(u\right)$ can be arbitrary
- $\Phi \left(u\right)\ne 0$, when ${d}_{0}\left(u\right)={d}_{0}=\mathrm{cste}.$ is a constant.

**Proposition 6.**The generator Equation (77) close into the following Lie algebra:

**Proof.**The commutation relation Equation (78) is directly checked. From the commutators $[\widehat{B},{X}_{-1}]=[\widehat{B},{Y}_{-1}]=0$ and:

**Proposition 7.**Let $\Phi \left(u\right)=(z-1){u}^{2}+\phi \left(u\right)$. Consider the generators:

**Proof.**The commutators are satisfied for $k=0$ and $q=-\mathrm{\mu}$ if condition Equations (83) and (84) are fulfilled. Under the same conditions, the symmetries are proven by the relations:

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Boltzmann, L. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Wien. Ber.
**1872**, 66, 275–370. [Google Scholar] - Haug, H. Statistische Physik; Vieweg: Braunschweig, Germany, 1997. [Google Scholar]
- Huang, K. Statistical Mechanics, 2nd ed.; Wiley: New York, NY, USA, 1987; p. 53ff. [Google Scholar]
- Kreuzer, H.J. Nonequilibrium Thermodynamics and Its Statistical Foundations; Oxford University Press: Oxford, UK, 1981; Chapter 7. [Google Scholar]
- Vlasov, A.A. On vibration properties of electron gas. JETP
**1938**, 8, 291–318. (in Russian). [Google Scholar] - Elskens, Y.; Escande, D.; Doveil, F. Vlasov equation and N-body dynamics. Eur. Phys. J.
**2014**, D68. [Google Scholar] [CrossRef] - Vilani, C. Particle systems and non-linear Landau damping. Phys. Plasmas
**2014**, 21, 030901:1–030901:9. [Google Scholar] - Duval, C.; Horváthy, P.A. Non-relativistic conformal symmetries and Newton-Cartan structures. J. Phys. A
**2009**, 42, 465206:1–465206:32. [Google Scholar] [CrossRef] - Henkel, M. Phenomenology of local scale-invariance: From conformal invariance to dynamical scaling. Nucl. Phys.
**2002**, B641, 405–486. [Google Scholar] [CrossRef] - Henkel, M.; Hosseiny, A.; Rouhani, S. Logarithmic exotic conformal Galilean algebras. Nucl. Phys.
**2014**, B879, 292–317. [Google Scholar] [CrossRef] - Martelli, D.; Tachikawa, Y. Comments on Galilean conformal field theories and their geometric realization. J. High Energy Phys.
**2010**, 1005, 091:1–091:31. [Google Scholar] [CrossRef] - Henkel, M.; Pleimling, M. Non-Equilibrium Phase Transitions Volume 2: Ageing and Dynamical Scaling Far from Equilibrium; Springer: Heidelberg, Germany, 2010. [Google Scholar]
- Henkel, M.; Unterberger, J. Schrödinger invariance and space-time symmetries. Nucl. Phys.
**2003**, B660, 407–435. [Google Scholar] [CrossRef] - Henkel, M.; Schott, R.; Stoimenov, S.; Unterberger, J. The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states. Confluentes Math.
**2012**, 4, 1250006:1–1250006:23. [Google Scholar] [CrossRef] - Cherniha, R.; Henkel, M. On nonlinear partial differential equations with an infinite-dimensional conditional symmetry. J. Math. Anal. Appl.
**2004**, 298, 487–500. [Google Scholar] [CrossRef] - Fushchych, W.I.; Shtelen, W.M.; Serov, M.I. Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics; Kluwer: Dordrecht, The Netherlands, 1993. [Google Scholar]
- Boyer, C.D.; Sharp, R.T.; Winternitz, P. Symmetry-breaking interactions for the time-dependent Schrödinger equation. J. Math. Phys.
**1976**, 17, 1439–1451. [Google Scholar] [CrossRef] - Stoimenov, S.; Henkel, M. Dynamical symmetries of semi-linear Schrödinger and diffusion equations. Nucl. Phys.
**2005**, B723, 205–233. [Google Scholar] [CrossRef]

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Stoimenov, S.; Henkel, M.
From Conformal Invariance towards Dynamical Symmetries of the Collisionless Boltzmann Equation. *Symmetry* **2015**, *7*, 1595-1612.
https://doi.org/10.3390/sym7031595

**AMA Style**

Stoimenov S, Henkel M.
From Conformal Invariance towards Dynamical Symmetries of the Collisionless Boltzmann Equation. *Symmetry*. 2015; 7(3):1595-1612.
https://doi.org/10.3390/sym7031595

**Chicago/Turabian Style**

Stoimenov, Stoimen, and Malte Henkel.
2015. "From Conformal Invariance towards Dynamical Symmetries of the Collisionless Boltzmann Equation" *Symmetry* 7, no. 3: 1595-1612.
https://doi.org/10.3390/sym7031595