Hydrodynamics and kinetics of Vlasov and Liouville equations V.V. - PowerPoint PPT Presentation

Hydrodynamics and kinetics of Vlasov and Liouville equations V.V. Vedenyapin, M.A. Negmatov, N.N. Fimin Keldysh Institute of Applied Mathematics of RAS

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary We describe the derivation of Vlasov-Maxwell equation from classical Lagrangian and a similar derivation of the Vlasov-Poisson-Poisson charged gravitating particles. The last term we use for combination of electrostatic and gravitational forces. By using an exact substitution we derive some versions of the equations of the electromagnetic hydrodynamics from Vlasov-Maxwell equations and present them to the Godunov’s double-divergence form. For them we get generalized Lagrange identity. The Lagrange identity is convenient here as a test to compare different forms of equations. We analyzes the steady-state solutions of the Vlasov-Poisson-Poisson equation: their types changes at a certain critical mass m 2 = e 2 /G having a clear physical meaning with different behavior of particles - recession or collapse trajectories.

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary Under Vlasov equation simply imply the following equation for an arbitrary K(x, y) pair interaction potential of particles ∂F � v, ∂F � � � K ( x, y ) F ( t, v, y ) dvdy, ∂F � ∂t + = 0 . − ∇ x ∂x ∂v Let us consider the substitution N � F ( t, v, x ) = ρ i δ ( v − V i ( t )) δ ( x − X i ( t )) . i =1 Substitution takes place if X i ( t ) and V i ( t ) satisfy N -body equations of motion ˙  X i = V i ,    N ˙ � V i = − ∇ 1 K ( X i , X j ) ρ j .    j =1

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary Generally, the Vlasov type equations are used with some prefix: Vlasov-Poisson Equation (for gravity, electrons, and plasma) Vlasov-Maxwell Equation (plasma, the galaxy) Vlasov-Einstein Equation Simplest derivation of the Vlasov-Maxwell equation from classical Lagrangian is highly desirable, it is provides us a firm basis: For the classification of equations with the same name; To assess their validity; The nature of the approximations made by various authors.

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary Derivation of equations of the Vlasov-Maxwell and Vlasov-Poisson-Poisson: We start with the usual action of the electromagnetic field, the action of Lorentz-Shwartzchield T � � � � g µν ˙ X µ α ( q, t ) ˙ X ν S L = S V M = − m α c α ( q, t ) dt + (1) α q 0 T e α � � � A µ ( X α ( q, t ) , t ) ˙ X µ + α ( q, t ) dt + c α q 0 1 � F µν F µν d 4 x = S p + S p-f + S f + 16 πc S p – is a particle action, S f – is a field action, S p-f – is a particles-fields action.

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary We have to seek a variation in a special way: first we obtain δ ( S p + S p-f ) = 0 than evolution of fields δ ( S p-f + S f ) = 0 . However, for particles, we proceed to distribution functions. � � 1 � x i δ ˙ ˙ x i � � d x i ˙ x i dt. δS p = mc 2 � 1 − v 2 /c 2 dt = m δ ˙ c 2 � dt � 1 − v 2 /c 2 q � d δS p-f = e � � c∂A 0 ∂x i δx i + ∂A i � � x i δx j − � δx i ∂x j ˙ dtA i dt. c q and from δ ( S p + S p-f ) = 0 we have: dp αi � − 1 ∂A i ∂t − ∂A 0 ∂x i − 1 � x j = e α c F ij ˙ , (2) α dt c p αi = ∂L p m α ˙ x αi α /c 2 , F ij = ∂A i ∂x j − ∂A j = ∂x i � ∂x i 1 − ˙ x 2 α

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary The equation for the distribution function is obtained as the equation of translation along the trajectories of the resulting dynamic system of charges in the field. It is seen that is convenient to take the distribution function of the momentum (instead of velocity). It should express velocity through momentums: m 2 v 2 mv i 1 − v 2 /c 2 ⇒ p 2 = p i = 1 − v 2 /c 2 . � v 2 p 2 p i γ − 2 = 1 + 1 − c 2 = γ − 2 , ( m 2 c 2 , v i = ( γm ) . Hence we can find the equation for the distribution function f α ( x, p, t ) : � ∂f α ∂f α � v α , ∂f α � + e α � − ∂A i ∂t − c∂A 0 ∂x − F ij v j ∂t + = 0 . (3) α ∂x c ∂p i

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary Fields equations. We use the distribution function instead of density: � e α � δA µ ( x ) v µ α f α ( x, p ) d 3 pd 4 x, δS p-f = c 2 1 � 1 � δF µν F µν d 4 x = δA µ ∂ µ F µν d 4 x. δS f = 16 πc 2 8 πc 2 If δ ( S p-f + S f ) = 0 then: ∂ µ F µν = − 4 π � � v µ α f α ( x, p ) d 3 p. e α (4) c α System of equations (3),(4) is Vlasov-Maxwell with some small adjustments: we have explicite expression of velocities over momentum. Similarly, we can derive the system of equations of the Vlasov-Poisson with gravitation in the nonrelativistic case.

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary The Lagrangian of electrostatics derived from the general Lagrangian, and gravitation part is derived by analogy with electrostatics. So, in the nonrelativistic case: � x 2 x 2 1 − ˙ c 2 ≈ 1 − ˙ α α 2 c 2 � m α ˙ x 2 m α c 2 + � α ( q,t ) Particles action: S p = − � � . α,q α,q 2 Particle-fields action (electrostatic): S e p-f = − � � α e α ϕ ( x, t ) f α ( x, p, t ) dxdpdt. Particle-fields action (gravity): S g � p-f = − � α m α U ( x, t ) f α ( x, p, t ) dxdpdt. ( ∇ ϕ ) 2 dxdt. Fields action (electrostatic): S e 1 � f = 8 π ( ∇ U ) 2 dxdt Fields action (gravity): S g 1 � f = − 8 πG

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary p-f + S g f + S g Lagrangian S = S p + S e p-f + S e f � m α ˙ x 2 α ( q, t ) � � � S = e α ϕ ( x, t ) f α ( x, p, t ) dxdpdt − − 2 α,q α � U ( x, t ) f α ( x, p, t ) dxdpdt + 1 � 1 � ( ∇ ϕ ) 2 dxdt − ( ∇ U ) 2 dxdt. � − m α 8 π 8 πG α Varying this expression as before we obtain a system of Vlasov-Poisson-Poisson plasma with gravitation: � p ∂f α , ∂f α � � ∂U ∂ϕ ∂x , ∂f α � ∂t + − m α ∂x + e α = 0 , m α ∂x ∂p i � � ∆ U = 4 πG m α f α ( x, p, t ) dp, α � � ∆ ϕ = − 4 π e α f α ( x, p, t ) dp. α

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary As shown, a complete system of Vlasov-Maxwell equations is obtained by varying the action of electro-magnetism with the transition to the distribution function: � ∂f α ∂f α � v α , ∂f α � � − 1 ∂A i ∂t − ∂A 0 ∂x − 1 c F ij v j ∂t + + e α = 0 , (5) α ∂x c ∂p i ∂F µν = − 4 π α f α ( x, p, t ) dp, F µν = ∂A µ − ∂A ν � � v µ e α , ( µ, ν : 1 , . . . , 4) , ∂x ν c ∂x ν ∂x µ α � E i = − 1 ∂A i ∂t − ∂A 0 p p 2 ∂x i , [ v α , H ] = − F ij v j α , v α = , γ α = 1 + α c 2 . m 2 c m α γ α Lagrange’s Identity is defined as the second time derivative of the moment of inertia through the kinetic and potential energy. Following to V. V. Kozlov, "The generalized Vlasov kinetic equation", Russian Math. Surveys, 63:4 (2008) we show that the Lagrange Identity can be extended to the case of Vlasov-Maxwell equations.

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary Let us introduce the moment of inertia of the particles respect to origin of coordinates: � � f α ( t, x, p ) x 2 d 3 pd 3 x, I ( t ) = α T ( t ) = 1 � � f α ( t, x, p ) v 2 α d 3 pd 3 x, 2 α e α � x, E + 1 � � � f α d 3 pd 3 x − Π = c [ v α , H ] γ α m α α e α � � α c 2 ( p, x )( p, E ) f α d 3 pd 3 x. − γ 3 α m 3 α Lagrange Identity is valid as: ¨ I = 4 T − 2Π . (6)

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary Prove: From (5) we have � � ∂f α � ¨ � ( v α , x ) d 3 pd 3 x − I = − 2 ∂x , v α α � ∂f α � E + 1 � � d 3 pd 3 x. − 2 ( v α , x ) e α c [ v α, H ] ∂p i α The first integral in this expression with integrating by parts can be transformed to : � � ( v α , v α ) f α d 3 pd 3 x = 4 T. 2 α The second integral can be transformed, if we count : � p 2 ∂v i p i where, v i = , γ α 1 + α c 2 , ∂p j m α γ α m 2 ∂v i δ ij p i ∂v j = − p j α c 2 ⇒ F ij = 0 γ 3 α m 3 ∂p j γ α m α ∂p i Then we can get: � ∂v αj � − 1 ∂A i ∂t − ∂A 0 ∂x i − 1 � � c F ij v j f α d 3 pd 3 x = − 2 x j e α α ∂p i c α

Introduction Derivation Lagrange’s identity VM and EMHD VPP Summary � � � � � δ ij p i p j − 1 ∂A i ∂t − ∂A 0 ∂x i − 1 � f α d 3 pd 3 x c F ij v j = − 2 x j e α − α γ α m α γ 3 α m 3 α c 2 c α e α � x, E + 1 � � � f α d 3 pd 3 x + = − 2 c [ v α, H ] γ α m α α p i p j � � α c 2 x j e α E i _ f α d 3 pd 3 x = − 2Π +2 γ 3 α m 3 α We use: p i F ij v i α = 0 . So finaly, the second term is transformed into: � e α � α c 2 ( p, x ) ( p, E ) f α d 3 pd 3 x. +2 γ 3 α m 3 α Lagrange’s Identity can be useful in studies of stability. Derivation shows that the second term in the functional Π is associated with relativism.