A Poisson map from kinetic theory to hydrodnamics with non-constant entropy
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A Poisson map from kinetic theory to hydrodynamics withnon-constant entropy
Ching Lok Chong
Abstract
Kinetic theory describes a dilute monatomic gas using a distributionfunction f ( q, p, t ), the expected phase-space density of particles at a given position q with a given momentum p . The distribution function evolves according to thecollisionless Boltzmann equation in the high Knudsen number limit. Fluid dynam-ics provides an alternative description of the gas using macroscopic hydrodynamicvariables that are functions of position and time only. The mass, momentum andentropy densities of the gas evolve according to the compressible Euler equationsin the limit of vanishing viscosity and thermal diffusivity. Both systems can beformulated as noncanonical Hamiltonian systems. Each configuration space is aninfinite-dimensional Poisson manifold, and the dynamics is the flow generated bya Hamiltonian functional via a Poisson bracket. We construct a map J from thespace of distribution functions to the space of hydrodynamic variables that respectsthe Poisson brackets on the two spaces. This map is therefore a Poisson map. Itmaps the p -integral of the Boltzmann entropy f log f to the hydrodynamic entropydensity. This map belongs to a family of Poisson maps to spaces that include gen-eralised entropy densities as additional hydrodynamic variables. The whole familycan be generated from the Taylor expansion of a further Poisson map that de-pends on a formal parameter. If the kinetic-theory Hamiltonian factors throughthe Poisson map J , an exact reduction of kinetic theory to fluid dynamics is pos-sible. However, this is not the case. Nonetheless, by ignoring the contribution tothe Hamiltonian from the entropy of the distribution function relative to its localMaxwellian, a distribution function defined by the p -moments R d n p (1 , p, | p | ) f ,we construct an approximate Hamiltonian that factors through the map. The re-sulting reduced Hamiltonian, which depends on the hydrodynamic variables only,generates the compressible Euler equations. We can thus derive the compressibleEuler equations as a Hamiltonian approximation to kinetic theory. We also givean analogous Hamiltonian derivation of the compressible Euler–Poisson equationswith non-constant entropy, starting from the Vlasov–Poisson equation. C.L. ChongOCIAM, Mathematical Institute, University of Oxford, Andrew Wiles Building,Radcliffe Observatory Quarter, Woodstock Road, Oxford,OX2 6GG, UKE-mail: [email protected] Ching Lok Chong
Contents C ∞ ( T ∗ M ) + to s ∗ A . . . . . . . . . . . . . . . . . . . . . 164.4 A one-parameter family of Poisson maps from C ∞ ( T ∗ M ) + to s ∗ . . . . . . . . 205 Fluid dynamics as a Hamiltonian approximation to kinetic theory . . . . . . . . . . . 255.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 The near global Maxwellian regime . . . . . . . . . . . . . . . . . . . . . . . . . 30 There are different levels of descriptions of a dilute monatomic gas. The first is the N -body description . The positions and momenta of each of the N gas particles evolveaccording to Hamilton’s equations generated by the full N -particle Hamiltonian H N ( q , p , . . . , q N , p N ). This is a system of 6 N ordinary differential equations. If weaverage over an ensemble of initial conditions, we arrive at the N -particle kinetic de-scription , where the joint probability distribution function f N ( q , p , . . . , q N , p N , t )for the positions and momenta of all N particles evolves according to the Liouvilleequation ∂f N ∂t + { f N , H N } = 0 , (1.1)where {· , ·} is the canonical Poisson bracket on R N . If we assume that the gas par-ticles interact only through a pairwise potential φ ( | q i − q j | ), where 1 ≤ i < j ≤ N ,the Liouville equation transforms into the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy [4, 7, 14, 24]. The dynamical variables in the BBGKY hi-erarchy are the s -particle distribution functions f s ( q , p , . . . , q s , p s , t ), each of whichgives the expected number of s -tuples of particles with prescribed positions andmomenta. The time evolution of f s depends on f s +1 . For example, the first twoequations of the BBGKY hierarchy are ∂f ∂t + p · ∂f ∂q = Z d q d p ∂φ ( | q − q | ) ∂q · ∂f ∂p , (1.2) ∂f ∂t + p · ∂f ∂q + p · ∂f ∂q − ∂φ ( | q − q | ) ∂q · (cid:18) ∂f ∂p − ∂f ∂p (cid:19) = Z d q d p ∂φ ( | q − q | ) ∂q · ∂f ∂p + ∂φ ( | q − q | ) ∂q · ∂f ∂p . (1.3)Here and henceforth we assume that the gas particles have unit mass. In the limitas N → ∞ , certain scaling hypotheses allow us to approximate the f term on the Poisson map from kinetic theory to hydrodynamics with non-constant entropy 3 right-hand side of (1.2) in terms of f , leading to 1 -particle kinetic theory . We brieflyreview how various 1-particle kinetic equations arise from the BBGKY hierarchy.There are three timescales in the BBGKY hierarchy [14, 24]. The first is the streaming time T , corresponding to the typical time that a particle takes to coversome typical macroscopic lengthscale L . The second is the collision time τ c , cor-responding to the typical duration of each pairwise interaction. The third is the mean free time τ mf , corresponding to the typical duration between two succes-sive pairwise interactions experienced by a particular particle. These timescalescorrespond to the sizes of the various terms appearing in the BBGKY hierarchyequations:1 T ∼ f s p ∂f s ∂q ∼ VL , (1.4)1 τ mf ∼ f s Z d qd p ∂φ∂q ∂f s +1 ∂p ∼ φ nd V , (1.5)1 τ c ∼ f s ∂φ∂q ∂f s ∂p ∼ φ V d , (1.6)where V is the typical velocity of the particles, n is the number density of theparticles, φ is the typical magnitude of the interaction potential, and d is theinteraction range. The total number N of particles scales as N ∼ nL . It is alsouseful to consider the mean free path λ mf = 1 / ( nd ), the typical distance a par-ticle travels between two successive pairwise encounters. Importantly, the termsinvolving φ on the left hand side of the BBGKY hierarchy equations scale as 1 /τ c ,and only appear in the f s -equation for s ≥
2. This distinguishes the f -equation(1.2) from the rest of the hierarchy.The Boltzmann–Grad limit describes the dynamics of a dilute gas of particlesinteracting through strong but short-range collisions, where the mean free path λ mf is finite compared to the macroscopic lengthscale L [4, 7]. In this limit, nd → nd L = L/λ mf ∼
1. The ratio Kn = λ mf /L is known as the Knudsen number .In a short-range binary collision where the angle of deflection is of order unity, thework done on each particle is of the same order as the kinetic energy of the particles,so we can estimate the magnitude of the potential as φ ∼ V . The asymptoticrelationship between the timescales is 1 /T ∼ /τ mf ≪ /τ c . The f -equation (1.3)under this scaling can then be used to simplify the right-hand side of (1.2), whichscales as 1 / (Kn T ). Boltzmann’s Stoßzahlansatz , which is the assumption that, inalmost all binary collisions, the colliding particles are uncorrelated pre-collision,allows a further simplification that leads to a closed equation for f known as the Boltzmann equation ∂f ∂t + p · ∂f ∂q = 1Kn C [ f ] , (1.7)where C [ f ] is a quadratic integral operator known as the Boltzmann collision opera-tor . We can consider further nested asymptotic limits within the Boltzmann–Gradlimit. In the high Knudsen number limit Kn → ∞ , one obtains the collisionlessBoltzmann equation , where one ignores the right-hand side of (1.7) entirely. Theopposite limit Kn → Ching Lok Chong
Another limit where reduction to 1-particle kinetic theory is possible is the
Vlasov limit , where the interactions are weak but long range [5, 21, 24]. The scalingsin this limit are d ∼ L → ∞ and φ ∼ V /N →
0, so the total potential energyper particle φN is finite. The asymptotic relationship between the timescales is1 /T ∼ /τ mf ≫ /τ c , which is opposite to that of the Boltzmann–Grad limit.The limiting system of equations is sometimes called the Vlasov hierarchy or the mean field hierarchy . In the mean field hierarchy, the terms involving φ on the lefthand side of the f s equations are dropped, thus admitting solutions of factorisedform f s ( q , p , . . . , q s , p s , t ) = f ( q , p , t ) . . . f ( q s , p s , t ) [5, 7, 21]. The f equation(1.2) is then closed. The physical picture for such factorised solutions is that eachparticle couples with others only through the collective effect of all N − φ is the Green’s function of Poisson’s equation, and (1.2) becomes thewell-known Vlasov–Poisson equation : ∂f ∂t + p · ∂f ∂q − ∂∂q (cid:18)Z d q d p e f ( q , p , t )4 π | q − q | (cid:19) · ∂f ( q , p , t ) ∂p = 0 , (1.8)where e is the suitably rescaled charge of the particles. In the following we restrictour attention to collisionless 1-particle kinetic theory, and the subscript 1 willhenceforth be dropped.An alternative theory that describes the gas is fluid dynamics , where we workwith macroscopic hydrodynamic variables that are functions of position and timeonly. It is convenient to take the momentum density m ( q, t ), mass density ρ ( q, t ), andthe entropy density s ( q, t ) as the hydrodynamic variables. In the limit of vanishingviscosity and thermal diffusivity, the time evolution of the hydrodynamic variables m, ρ, s are governed by the compressible Euler equations ∂ρ∂t + ∂∂q · m = 0 , (1.9) ∂m∂t + ∂∂q · (cid:18) m mρ (cid:19) = − ∂P ( ρ, s ) ∂q , (1.10) ∂s∂t + ∂∂q · (cid:18) mρ s (cid:19) = 0 , (1.11)where P ( ρ, s ) is the pressure as determined by the equation of state of a monatomicideal gas.The N -body description, the BBGKY hierarchy, the collisionless 1-particle ki-netic descriptions, and the hydrodynamic description are all conservative systems ,which means that their smooth solutions are time-reversible and possess a numberof conservation laws. This can be attributed to the fact that they are all noncanon-ical Hamiltonian systems [20, 41, 48]. The time evolution of any functional F [ · ] ofthe dynamical variables in each theory is given by dFdt = { F, H } NC , (1.12)where H a the Hamiltonian functional , and {· , ·} NC is an abstract Poisson bracket or noncanonical Poisson bracket that is required to satisfy a number of familiaridentities motivated from the canonical Poisson bracket in classical mechanics. The Poisson map from kinetic theory to hydrodynamics with non-constant entropy 5 respective configuration spaces for 1-particle kinetic theory and fluid dynamics canbe treated as infinite-dimensional
Poisson manifolds , and the time evolution on eachspace is the flow of generated by a chosen Hamiltonian functional H [20, 34, 37, 41].The passage along the hierarchy of kinetic descriptions, from N -body to Liou-ville to BBGKY to collisionless 1-particle kinetic theory, is manifestly Hamiltonian[33, 44]. It is then natural to ask whether the hydrodynamic description can be de-rived from the 1-particle kinetic description in a way that respects the Hamiltonianstructure. We seek a map J taking f to the hydrodynamic variables m [ f ] , ρ [ f ] , s [ f ]that respects the Poisson brackets on the two spaces. Mathematically, such a J iscalled a Poisson map . Since the 1-particle distribution function f ( q, p, t ) describesnumber density of particles in phase space, the mass and momentum densitiesshould be related to f as ρ [ f ] = R d n pf and m [ f ] = R d n p pf . Given the Poisson map J , suppose that the kinetic-theory Hamiltonian H KT factors through the Poissonmap J , in the sense that there is some other functional H red of the hydrodynamicvariables such that H KT = H red ◦ J . The image of the dynamics of kinetic the-ory under J is then exactly reproduced by the noncanonical Hamiltonian systemgoverned by the hydrodynamic Poisson bracket and the reduced Hamiltonian func-tional H red . This is the programme of reduction of noncanonical Hamiltonian systems that we will pursue in this paper, which we explain in [19, 34, 35, 36, 37]. Thereare also other ways to obtain reduced Hamiltonian systems from larger ones, whichwe discuss in section [6, 23, 31, 32, 37, 38, 40, 45, 50, 51, 52].We construct the Poisson map J in section . The map J maps the p -integral of the Boltzmann entropy f log f onto the hydrodynamic entropy den-sity. It belongs to a family of Poisson maps J A that include generalised entropies s a [ f ] = R d n p f (log f ) a for a = 0 , , , . . . , A as additional hydrodynamic variables(section ). These Poisson maps are non-linear generalisations of that obtainedin [19, 37], where only m [ f ] and ρ [ f ] are considered. The non-linearities appearingin these maps are closely related to the eigenfunctions of the Euler differential op-erator x ( d/dx ) (section ). Furthermore, the J A arises from the truncated Taylorexpansion of a Poisson map J that depends on a formal parameter ξ . This largermap J has the Tsallis entropy ρ ξ [ f ] = R d n pf ξ [53] as a hydrodynamic variable(section ).One then wonders how this could possibly circumvent the moment closure prob-lem , where a finite number of moments ( p -integrals of f ) cannot reproduce kinetictheory exactly. An elementary example is the raw moment hierarchy, where oneconsiders the raw moments µ l = R d n p p . . . pf ( l times), and find that the time evo-lution of µ l depends on µ l +1 . Sadly, our generalised entropies do not circumvent themoment closure problem. The Hamiltonian functional H KT [ f ] = R d n qd n p | p | f/ J A ,and in particular, not through J . This can be attributed to the difference in thedefinition of temperature in kinetic theory and fluid dynamics. In kinetic theory,the temperature θ [ f ]( q ) is defined by the variance in the distribution of momentumat position q . In fluid dynamics however, taking ρ, m and s as our hydrodynamicvariables, the temperature T ( ρ, s ) is obtained by inverting the thermodynamicequation of state s = s ( ρ, T ) for a monatomic ideal gas. In section we showthat the two definitions of temperature agree if and only if the distribution func-tion f is a local Maxwellian f m , a distribution function defined by the moments R d n p (1 , p, | p | ) f (see (5.8), section ). Since generic local Maxwellians do not stayas local Maxwellians under the collisionless Boltzmann equation or the Vlasov– Ching Lok Chong
Poisson equation, the Poisson map J never provides an exact reduction fromkinetic theory to fluid dynamics. Other approaches to moments and the momentclosure problem are discussed in section [15, 16, 45, 50, 51, 52].The Poisson map J defers the moment closure problem to the Hamiltonianfunctional instead of solving it. We can make progress by taking the perspec-tive of approximate reduction , where one makes physically motivated but otherwiseuncontrolled approximations to derive reduced equations. In our case, the differ-ence between θ [ f ] and T ( ρ [ f ] , s [ f ]) is controlled by the relative entropy r [ f | f m ] of f against its local Maxwellian f m [8, 46]. This difference can be thought of the“non-hydrodynamic energy” in the kinetic system, and we show that this termstays small if the initial distribution function is close to a global Maxwellian insection . By ignoring this relative entropy term in the Hamiltonian functional,the remaining term is precisely the pullback of the fluid Hamiltonian through J .We can thus derive the compressible Euler equations from 1-particle kinetic theoryin a manifestly Hamiltonian manner, such that the Hamiltonian structure of thehydrodynamic equations is inherited from 1-particle kinetic theory through thePoisson map J . This type of derivation of reduced conservative models throughuncontrolled approximations in the Hamiltonian functional, or equivalently inthe Lagrangian density of a variational principle, is not uncommon in fluid dy-namics and mathematical physics, and we give some examples in section [25, 26, 27, 30, 33, 39, 47, 55, 56, 57]. To set out our notation and prove a few basic identities, we begin with a basicexposition on symplectic and Poisson geometry, assuming some familiarity withdifferentiable manifolds. The main references are [6, 54]. Readers familiar withsymplectic and Poisson geometry can skip this section and only refer to it for thenotation used in the subsequent sections. We will assume that all manifolds andother objects are smooth.2.1 Basic definitionsA symplectic manifold S is a 2 n -dimensional manifold together with a symplectic -form ω on S that is closed and nondegenerate: dω = 0 and ω n | x = 0 for all x ∈ S .Given a function f on S , its Hamiltonian vector field X f is the unique solution tothe equation ι X f ω = df, (2.1)where ι denotes the interior product. The canonical Poisson bracket {· , ·} : C ∞ ( S ) × C ∞ ( S ) → C ∞ ( S ) on S is then defined as { f, g } = − L X f g = L X g f = ω ( X f , X g ) , (2.2)where f and g are functions on S , and L X is the Lie derivative along the vectorfield X . The canonical Poisson bracket satisfies a list of important identities: for Poisson map from kinetic theory to hydrodynamics with non-constant entropy 7 all α, β ∈ R and all f, g, h ∈ C ∞ ( S ), { αf + βg, h } = α { f, h } + β { g, h } , ( R – bilinearity ) (2.3) { f, g } = −{ g, f } , ( antisymmetry ) (2.4) { f, gh } = { f, g } h + { f, h } g, ( Leibniz identity ) (2.5)0 = {{ f, g } , h } + {{ g, h } , f } + {{ h, f } , g } . ( Jacobi identity ) (2.6)Conversely, a manifold P equipped with a binary operation {· , ·} NC : C ∞ ( P ) × C ∞ ( P ) → C ∞ ( P ) satisfying (2.3-2.6) is called a Poisson manifold , and the opera-tion {· , ·} NC is called a noncanonical Poisson bracket . All symplectic manifolds arePoisson manifolds, but not necessarily vice versa. To distinguish between Poissonbrackets on symplectic manifolds and non-symplectic Poisson manifolds, we willalways put subscripts on the latter and leave the former unsubscripted. The localtheory of finite-dimensional Poisson manifolds is extensively studied in [54]. On afinite-dimensional Poisson manifold P , the Poisson bracket can be written in termsof an antisymmetric contravariant rank-2 tensor Π , called the Poisson tensor , suchthat { f, g } NC = Π ( df, dg ) for all functions f, g on P . Now let y : R → R be a realfunction. If f : P → R is a function on a finite-dimensional Poisson manifold P ,then y ( f ) = y ◦ f is also a function on P . We have { y ( f ) , g } NC = y ′ ( f ) { f, g } NC . (2.7)While the Leibniz identity directly implies (2.7) for polynomial y , the general casefor smooth functions relies on the identification of Poisson brackets with Poissontensors. The identity (2.7) will become important in sections and .2.2 Poisson brackets and affine functions on a cotangent bundleLet M be an n -dimensional manifold. Its cotangent bundle T ∗ M is a symplecticmanifold, whose Poisson bracket has some special properties that we will makeextensive use of later. The points in T ∗ M are pairs ( q, p ), where q is a point on M , and p ∈ T ∗ q M is a cotangent vector at q . The cotangent vector spaces T ∗ q M are called the fibres of T ∗ M . On a coordinate neighbourhood U of q , there areinduced coordinates ( q , . . . , q n , p , . . . , p n ) on T ∗ U , where q i are the coordinatesof the basepoint q and p i = h p, ∂/∂q i i are the components of the cotangent vector p . These are often called canonical coordinates in physics. Here and henceforth wewill use the summation convention on the Latin indices i, j = 1 , . . . n .The symplectic form on T ∗ M can be written as ω = dq i ∧ dp i in canonicalcoordinates. The Poisson bracket {· , ·} associated with ω corresponds precisely tothe canonical Poisson bracket in classical mechanics: { f, g } = ∂f∂q i ∂g∂p i − ∂g∂q i ∂f∂p i , (2.8)for functions f, g on T ∗ M . Some natural geometric constructions on M can beformulated using the canonical Poisson bracket on T ∗ M . Let u be a vector field on M . We can associate u to a function h u, p i : ( q, p )
7→ h u ( q ) , p i q on T ∗ M , where h· , ·i q : T q M × T ∗ q M → R denotes the dual pairing between the tangent and cotangentspaces at q ∈ M . In local coordinates where u = u i ( q )( ∂/∂q i ), this function is Ching Lok Chong h u, p i = u i ( q ) p i . Any function on T ∗ M that is linear in p arises from the dualpairing of p with a vector field on M . We call these functions fibrewise linear .A function g on M defines a function π ∗ g = g ◦ π on T ∗ M by pullback, where π : T ∗ M → M is the projection. In canonical coordinates, these functions areprecisely those that depend on q only: g ( q, p ) = g ( q ). We call these function fibrewiseconstant . By analogy with affine functions defined on a vector space, we call afunction on T ∗ M fibrewise affine if it is a linear combination of fibrewise linear andfibrewise constant functions.Fibrewise affine functions on the cotangent bundle form a Lie subalgebra ofthe Lie algebra ( C ∞ ( T ∗ M ) , {· , ·} ). This construction is first described in [19, 37]in the context of cotangent lifts and moment(um) maps. Let u, v be vector fieldson M and g, h be functions on M . Then we have {h u, p i , h v, p i} = − h [ u, v ] , p i , (2.9) (cid:8) h u, p i , π ∗ g (cid:9) = − π ∗ ( L u g ) , (2.10) (cid:8) π ∗ g, π ∗ h (cid:9) = 0 , (2.11)where [ u, v ] = L u v is the Lie bracket for vector fields on M , given in coordinatesas [ u, v ] i = u j ∂v i ∂q j − v j ∂u i ∂q j . (2.12)While symmetric contravariant (upper index) tensors T i ...i k ( q ) on M induce fiber-wise polynomial functions T i ...i k ( q ) p i . . . p i k on T ∗ M , the quadratic and higherdegree polynomials generate a subalgebra that necessarily includes polynomialsof arbitrarily large degree. The fibrewise affine functions constitute a maximal“small” subalgebra of the fibrewise polynomials. This is somewhat analogous tothe situation on a symplectic vector space R n , where the polynomials of degreeat most 2 constitute a maximal finite-dimensional subalgebra of the Lie algebraPol( R n ) of polynomials under the canonical Poisson bracket.The Lie algebra of fibrewise affine functions on the cotangent bundle can alsobe constructed as a semidirect product Lie algebra without reference to the canon-ical Poisson bracket. The space of vector fields on M , denoted Vect( M ), is a Liealgebra under the Lie bracket for vector fields (2.12). The action of vector fieldson functions by Lie differentiation defines a representation of Vect( M ) on C ∞ ( M ).Thus we can construct the semidirect product Lie algebra s = Vect( M ) ⋉ C ∞ ( M ).As a vector space, s consists of pairs ( u, g ) where u is a vector field on M and g is a function on M . The Lie bracket on s is[( u, g ) , ( v, h )] s = ([ u, v ] , L u h − L v g ) . (2.13)Now consider the linear map s → C ∞ ( T ∗ M ) defined by( u, g )
7→ h u, p i + π ∗ g. (2.14)This map is injective, and its image is precisely the space of fibrewise affine func-tions on T ∗ M . The identities (2.9-2.11) shows that this map is an anti-homomorphismof Lie algebras, namely that (cid:8) h u, p i + π ∗ g, h v, p i + π ∗ h (cid:9) = − (cid:0) h [ u, v ] , p i + π ∗ ( L u h − L v g ) (cid:1) . (2.15)Thus s can be considered as the Lie subalgebra of fibrewise affine functions in( C ∞ ( T ∗ M ) , {· , ·} ), up to a sign. Poisson map from kinetic theory to hydrodynamics with non-constant entropy 9 n -dimensional symplectic manifold S has a standard non-vanishing 2 n -form dV = ω n /n ! called the symplectic volume form . (Despite the notation, dV is notnecessarily an exact form.) For the cotangent bundle T ∗ M of an n -dimensionalmanifold M , the local expression for dV in canonical coordinates is dV = dq ∧ . . . ∧ dq n ∧ dp ∧ . . . ∧ dp n . (2.16)The volume form induces a weakly non-degenerate symmetric bilinear form on thespace of functions on S by integration:( f, g ) = Z S dV fg, (2.17)where f, g are functions on S that satisfy sufficient decay conditions. The condi-tion “weakly non-degenerate” means that if ( f, g ) = 0 for all sufficiently decayingfunctions g , then f = 0. Moreover, since { f, g } ω n is an exact 2 n -form, we have Z S dV { f, g } = 0 , Z S dV { f, g } h = Z S dV f { g, h } (2.18)for any f, g, h ∈ C ∞ ( S ), subject to decay conditions on the integrands. Theseidentities for S = T ∗ M will be used extensively in section .One way to make the decay conditions precise is to take an exhaustion of S by compact submanifolds – take a sequence S k of compact submanifolds withboundary such that ∪ S k = S and S k ⊂ Int( S k +1 ), where Int denotes the interiorof a manifold with boundary. The decay conditions on f can then be phrasedin terms of the decay rates of the numerical sequence sup x ∈ S k +1 \ S k ( | f ( x ) | ), andsimilarly for its derivatives. For example, if S = R n is the classical phase space fora particle in R n , we can take S k to be the ball of radius k ; if S = T ∗ M where M isa compact Riemannian manifold, we can take S k to be S q B k ( q ), where B k ( q ) isthe ball of radius k on T ∗ q M under the norm induced by the Riemannian metric.We will henceforth assume that all functions satisfy sufficient decay conditions.2.4 The Lie–Poisson bracket on the dual of a Lie algebraThe dual space to a Lie algebra is an important example of a non-symplecticPoisson manifold. Let g be a Lie algebra with Lie bracket [ · , · ], which we assumeto be finite-dimensional for now, and let g ∗ be its dual vector space. There is anatural Poisson bracket on g ∗ , called the Lie–Poisson bracket , that turns g ∗ into aPoisson manifold. For functions f ( µ ) , g ( µ ) on g ∗ , it is given by { f, g } g ( µ ) = ± (cid:28) µ, (cid:20) ∂f∂µ , ∂g∂µ (cid:21)(cid:29) g , (2.19)where the angle bracket denotes the dual pairing between g ∗ and g , and we in-terpret ∂f /∂µ to be taking values in g instead of g ∗∗ . The bracket {· , ·} g satisfies (2.3-2.6)), so it is a Poisson bracket. Geometrically, if g is the Lie algebra of aLie group G , we can also obtain the Lie–Poisson structure on g ∗ reducing thesymplectic structure on T ∗ G by the natural left-(or right-) G action [35, 36, 37].In applications to fluid dynamics and kinetic theory, the relevant Lie algebrasare typically infinite-dimensional, and the functions on these infinite-dimensionalLie algebras become functionals , so the ordinary derivative on these spaces has tobe replaced with the functional derivative . We proceed formally by taking “smoothdual spaces” [28] and restricting our attention to sufficiently regular functionals.A smooth dual space is a subspace of the topological dual space that has a weaklynon-degenerate pairing with the original space. The choice of a smooth dual is non-unique and is usually motivated by context. For example, the space of functionson a symplectic manifold C ∞ ( S ) is an infinite-dimensional Lie algebra under thecanonical Poisson bracket {· , ·} . The smooth dual of C ∞ ( S ) can be taken to be thespace of smooth volume forms Ω n ( S ), perhaps with sufficient decay conditions,with the dual pairing given by integration. If we restrict our attention to sufficientlyregular functionals on Ω n ( S ) so that their functional derivatives exist and canbe considered to take values in C ∞ ( S ), we can formally consider Ω n ( S ) to bean “infinite-dimensional Poisson manifold” equipped with the Lie–Poisson bracketfrom C ∞ ( S ). Rigorous analytical treatments of Lie–Poisson dynamics on infinite-dimensional spaces can be found in [11, 13, 28]. In this section we briefly review noncanonical Hamiltonian mechanics and outline the reduction of noncanonical Hamiltonian systems that we will pursue in the subsequentsections. Detailed expositions on noncanonical Hamiltonian mechanics with anemphasis on fluid dynamics and kinetic theory can be found in [41, 48, 52].Consider a space M of dynamical variables for a set of configurations. Usually, M is a smooth manifold of possibly infinite dimension. For physical applications, M is often a space of functions on a finite-dimensional manifold M , or more generallya space of sections of a given vector bundle over M , in which case M is often calleda space of field variables . Associated to the space of configurations M is a space F ( M ) of sufficiently regular functionals M → R , which can be thought of as acollection of observable quantities on the field variables in M .Suppose now that the configuration space M is a Poisson manifold, with Pois-son bracket {· , ·} M . Define the noncanonical Hamiltonian system on M generatedby a Hamiltonian functional H ∈ F ( M ) to be the dynamical system where anysufficiently regular functional F ∈ F ( M ) evolves in time as dFdt = { F, H } M . (3.1)If M is a space of fields, letting F [ z ] = h ψ, z i M for arbitrary test functions ψ recovers the equation of motion for the state z ∈ M in a weak sense.In many applications, the configuration space M is often the smooth dual spaceto an infinite-dimensional Lie algebra (see section ). For example, the kinetictheory of dilute gases or electrostatic plasmas can be formulated on the dualof C ∞ ( T ∗ M ), where the Lie algebra structure is given by the canonical Poissonbracket [20, 37]. Ideal compressible hydrodynamics and magnetohydrodynamics Poisson map from kinetic theory to hydrodynamics with non-constant entropy 11 can be formulated on the duals of appropriate semidirect product Lie algebras ofVect( M ) [20, 34, 35, 42]. The Hamiltonian functional is the total energy of thesystem in each case.3.1 Poisson maps and reductionGiven a noncanonical Hamiltonian system ( M , {· , ·} , ˜ H ), we can determine thetime evolution of all functionals ˜ F ∈ F ( M ). However, M may have unimportantdegrees of freedom. We would like to reformulate the noncanonical Hamiltoniansystem on a reduced configuration space M that contains only the relevant degreesof freedom. We formalise this notion using Poisson manifolds and Poisson maps.Given Poisson manifolds ( M , {· , ·} ) and ( M , {· , ·} ), we say that a map J : M → M is a Poisson map , if for all functionals
F, G ∈ F ( M ), { F ◦ J , G ◦ J } = { F, G } ◦ J , (3.2)where F ◦ J and G ◦ J are now functionals on M by composition. The pullbackmap F F ◦ J is commonly denoted as J ∗ : F ( M ) → F ( M ). Returning to thenoncanonical Hamiltonian system ( M , {· , ·} , ˜ H ), suppose we are interested in thetime evolution of functionals ˜ F ∈ F ( M ) that depend on z ∈ M only through itsimage J [ z ] ∈ M , i.e. ˜ F = F ◦ J for some F ∈ F ( M ). (3.3)If the Hamiltonian functional also factors through J i.e. ˜ H ∈ F ( M ) is of the form˜ H = H ◦ J for some H ∈ F ( M ), (3.4)then, since J is a Poisson map, d ˜ Fdt = (cid:8) ˜ F , ˜ H (cid:9) = { F, H } ◦ J , (3.5)so we can replace the evolution equation for ˜ F with an evolution equation ˙ F = { F, H } for F , and forget about M altogether. A Hamiltonian functional thatfactors through a Poisson map is called collective in [19].The adjoint linear map of a Lie algebra homomorphism is an important exam-ple of a Poisson map. Let g and h be Lie algebras, and φ : g → h be a Lie algebra(anti)-homomorphism. The adjoint linear map φ ∗ : h ∗ → g ∗ is a Poisson map ifwe equip the duals spaces g ∗ and h ∗ with the Lie–Poisson bracket of the same(opposite) sign. In fact, the adjoint of a linear map φ : g → h is a Poisson map ifand only if φ is a Lie algebra homomorphism ([37], Lemma 8.2). (This result stillholds formally if the Lie algebras are infinite-dimensional.) The Poisson map from1-particle kinetic theory to fluids with constant entropy is precisely the adjointlinear map to the inclusion of s = Vect( M ) ⋉ C ∞ ( M ) into C ∞ ( T ∗ M ) as the Liesubalgebra of fibrewise affine functions met in section [19, 37].To carry out our programme of reduction, we construct a Poisson map from theconfiguration space for 1-particle kinetic theory to that of ideal compressible fluidswith non-constant entropy (section ). However, the kinetic-theory Hamiltonian isnot the pullback of an effective Hamiltonian in the hydrodynamic variables, so wemake an “approximation” to the kinetic-theory Hamiltonian to make it so (section ). The rest of this paper is devoted to carry out this approximate reduction . . We will not pursuethese methods any further in sections and .The classic example is Marsden–Weinstein reduction of symplectic or Poissonmanifolds [6, 31, 32, 37, 38, 40]. Let M be a symplectic or Poisson manifold, G be a Lie group and g be the Lie algebra of G . Suppose we have a Poissonmap J : M → g ∗ , often called a moment(um) map . This induces a Hamiltonian G -action on M . Under suitable conditions, the Marsden–Weinstein quotient J − (0) /G is correspondingly a symplectic or Poisson manifold. A G -invariant Hamiltonianfunctional on M induces a reduced Hamiltonian system on J − (0) /G . Marsden–Weinstein reduction achieves a similar goal to the reduction illustrated in section , where unimportant degrees of freedom are forgotten to form a reduced Hamil-tonian system, but it uses the Poisson map J in a different way.More recently, various Hamiltonian “reduced fluid models” for drift kineticsare derived by constructing a new but related Poisson bracket on the space ofreduced variables in [50, 51, 52]. A similar construction is used to find reduceddescriptions for the one-dimensional Vlasov–Amp´ere system in [45]. We give a briefsummary of the techniques involved. Let ( M , {· , ·} ) be the Poisson manifold ofthe more primitive variables, M be the space of reduced variables, and π : M → M be the map from primitive to reduced variables. In these applications, giventwo functionals F, G on M , the Poisson bracket of their pullbacks { π ∗ F, π ∗ G } cannot be expressed as the pullback of some other functional on M . There areexcess variables not described by the image π ( M ) that appear in the expression of { π ∗ F, π ∗ G } . These excess variables are eliminated by imposing a closure relation ,which is an additional functional relationship between the excess and reducedvariables. We can describe this as an embedding σ : M → M , whose imagecorresponds to the locus of the closure relation on M . We require that π ◦ σ = id is the identity map on M . Define a bracket {· , ·} on M by { F, G } = σ ∗ (cid:8) π ∗ F, π ∗ G (cid:9) , (3.6)where F and G are arbitrary functionals on M . The bracket {· , ·} always satisfies(2.3 - 2.5). If the closure relation σ is so chosen that {· , ·} satisfies the Jacobiidentity (2.6), ( M , {· , ·} ) becomes a Poisson manifold. (The Jacobi identity for {· , ·} is checked manually in [45, 50, 51].) While σ ∗ π ∗ = id ∗ , the composition π ∗ σ ∗ is not the identity for functionals on M . The relation (3.6) is not the same as(3.2), and in particular π is not a Poisson map.In the applications described in [45, 50, 51], the primitive Hamiltonian ˜ H gen-erating the dynamics on M is the pullback of some effective Hamiltonian H on M through π , that is, ˜ H = π ∗ H . The reduced dynamics is then given by dFdt = { F, H } = σ ∗ (cid:8) π ∗ F, ˜ H (cid:9) , (3.7)so the functional F evolves according to the value of (cid:8) π ∗ F, ˜ H (cid:9) restricted on σ ( M ). This framework provides a manifestly Hamiltonian way to impose certain“good” closure relations to obtain reduced evolution equations, which is often an Poisson map from kinetic theory to hydrodynamics with non-constant entropy 13 improvement over imposing the closure relations directly in the primitive evolutionequations.In a separate class of applications, one seeks physically relevant Poisson sub-manifolds of the Poisson manifold of primitive variables. The Hamiltonian func-tional restricted to the submanifold then generates a reduced Hamiltonian system.For example, the extended magnetohydrodynamics (MHD) system, which is an ex-tension to ordinary MHD that incorporates the Hall effect [49], is known to be anoncanonical Hamiltonian system [1]. In [23], a Poisson submanifold correspondingto extended MHD configurations that are translationally invariant along the z -axisis found, and the corresponding reduced Hamiltonian system is studied in detail.This type of reduction facilitates a more compact description of a system underspecial conditions, instead of removing unimportant degrees of freedom from thesystem. In this section we describe the formulation of kinetic theory and fluid dynamics asnoncanonical Hamiltonian systems [20, 27, 34, 37, 42], and obtain a Poisson mapbetween the underlying Poisson manifolds.In the following, M is an oriented n -dimensional manifold, and T ∗ M is itscotangent bundle. The assumption on orientability is inessential and can be re-moved by replacing all top-degree differential forms with smooth densities. Varioussmoothness and decay conditions are implicitly assumed. For physical applications,it suffices to take M = T n (or R n ) and T ∗ M = T n × R n (respectively R n ), where M is endowed with the standard metric δ ij and standard volume element d n q .4.1 The Lie–Poisson bracket for kinetic theoryConsider the space of functions C ∞ ( T ∗ M ) on the cotangent bundle T ∗ M , whichis a Lie algebra under the canonical Poisson bracket {· , ·} on T ∗ M . Its dual spaceis the space of volume forms on T ∗ M , which we denote as Ω n ( T ∗ M ). We use thesymplectic volume element dV = ω n /n ! to identify C ∞ ( T ∗ M ) with Ω n ( T ∗ M ) bythe one-to-one correspondence f fdV . The dual pairing h fdV, g i C ∞ ( T ∗ M ) = Z T ∗ M dV fg = ( f, g ) (4.1)gives a weakly non-degenerate symmetric invariant bilinear form on the Lie algebra C ∞ ( T ∗ M ) (see section ). We interpret the functions f ∈ C ∞ ( T ∗ M ) as the1-particle distribution functions on phase space in kinetic theory. The (+)-Lie–Poisson bracket is { F, G } KT [ f ] = Z T ∗ M dV f (cid:26) δFδf , δGδf (cid:27) , (4.2)where F [ f ] , G [ f ] are functionals of f . Given a Hamiltonian functional H [ f ], theevolution equation for f is the familiar Liouville equation: ∂f∂t = (cid:26) δHδf , f (cid:27) . (4.3) For M = T n or R n , choosing the Hamiltonian functional H KT [ f ] = Z d n qd n p | p | f (4.4)recovers the collisionless Boltzmann equation for a dilute gas. The Vlasov–Poissonequation for an electrostatic plasma can be obtained from the Vlasov–PoissonHamiltonian [20, 37, 58]: H V P [ f ] = Z d n qd n p | p | f ( q, p ) + e Z d n qd n pd n q ′ d n p ′ f ( q, p ) f ( q ′ , p ′ ) G ( q, q ′ ) , (4.5)where e is the nondimensionalised charge of the particles. We write the potentialenergy using the Green’s function G ( q, q ′ ) for the Laplacian −∇ on T n or R n .This avoids having to introduce an explicit equation for the electrostatic potential ϕ ( q ) = e Z d n q ′ d n p ′ f ( q ′ , p ′ ) G ( q, q ′ ) . (4.6)For a plasma on T n , we assume that there is a uniformly charged inert backgroundto keep the plasma neutral overall.The Jeans equation for a self-gravitating stellar system in galactic dynamicscan be obtain from a slightly different Hamiltonian [3, 22] H SG [ f ] = Z d n qd n p | p | f ( q, p ) − G2 Z d n qd n pd n q ′ d n p ′ f ( q, p ) f ( q ′ , p ′ ) G ( q, q ′ ) , (4.7)where G is the dimensionless gravitational constant. The Jeans equation is iden-tical to the Vlasov–Poisson equation, except that e has been replaced with − G.The sign difference leads to qualitative differences between the respective solutionsof the Vlasov–Poisson and Jeans equations [17]. The self-gravitating Hamiltonian H SG [ f ] is not bounded from below unlike H KT [ f ] and H V P [ f ]. This differencebecomes important in section .Since the distribution function f ( q, p ) represents the density of particles inphase space, the appropriate configuration space for distribution functions inkinetic theory should be taken as the space of positive distribution functions C ∞ ( T ∗ M ) + , where C ∞ ( T ∗ M ) + = (cid:8) f ∈ C ∞ ( T ∗ M ) : f ( q, p ) > q, p ) ∈ T ∗ M (cid:9) . (4.8)The strict positivity of f also means that we can make sense of expressions suchas f log f and its f -derivatives without having to define them as limits as f → C ∞ ( T ∗ M ) + is a Poisson submanifold because the Liouville equation (4.3) preservesthe condition f ( q, p ) > q, p ) for any choice of Hamiltonian functional. Poisson map from kinetic theory to hydrodynamics with non-constant entropy 15 s A = Vect( M ) ⋉ C ∞ ( M, R A +1 ), where thevector fields u ∈ Vect( M ) act on the R A +1 -valued functions ( g ( q ) , . . . , g A ( q )) ∈ C ∞ ( M, R A +1 ) by componentwise Lie differentiation. We will not use the summa-tion convention on R A +1 and instead display the indices a = 0 , . . . , A explicitly.We will work implicitly with a standard Euclidean basis on R A +1 throughout. Thesemidirect product Lie bracket on s A is[( u, ( g , . . . g A )) , ( v, ( h , . . . h A ))] s A = ([ u, v ] , ( L u h − L v g , . . . , L u h A − L v g A )) . (4.9)The dual space s ∗ A of the Lie algebra s A is Vect ∗ ( M ) × Ω n ( M, R A +1 ), whereVect ∗ ( M ) = Ω ( M ) ⊗ C ∞ ( M ) Ω n ( M ) is the space of 1-form densities on M , and Ω n ( M, R A +1 ) is the space of R A +1 -valued volume forms on M , which we canalso think of as the space of A + 1 real-valued volume forms by taking compo-nents. A typical element of s ∗ A can be written as ( m i d n q, ( s d n q, . . . , s A d n q )), where m = m i d n q is a 1-form density, and s a d n q are volume forms for a = 0 , . . . , A . Thedual pairing of s ∗ A with s A is given by integration: (cid:10) ( m i d n q, ( s d n q, . . . , s A d n q )) , ( u, ( g , . . . , g A )) (cid:11) s A = Z M d n q m i u i + A X a =0 s a g a ! . (4.10)The ( − )-Lie–Poisson bracket on s ∗ A is given, for functionals F, G ∈ F ( s ∗ A ), by { F, G } s A = − (cid:28) m, (cid:20) δFδm , δGδm (cid:21)(cid:29) Vect( M ) − A X a =0 (cid:28) s a , L δFδm δGδs a − L δGδm δFδs a (cid:29) C ∞ ( M ) , (4.11)where the angle brackets denote the pairing between the vector space indicated bythe subscript and its dual. For A = 1, we can interpret m as the momentum density of a fluid, s = ρ as the mass density of a fluid, and s = s as the entropy density of a fluid. For M = T n or R n , the Lie–Poisson bracket (4.11) has the explicit form { F, G } s = − Z d n q m i (cid:18) δFδm j ∂∂q j δGδm i − δGδm j ∂∂q j δFδm i (cid:19) + s (cid:18) δFδm j ∂∂q j δGδρ − δGδm j ∂∂q j δFδρ (cid:19) + s (cid:18) δFδm j ∂∂q j δGδs − δGδm j ∂∂q j δFδs (cid:19) , which coincides with the bracket for ideal fluids with non-constant entropy [42].The Hamiltonian functional that generates the compressible Euler equations is H fluids [ m, ρ, s ] = Z d n q (cid:18) | m | ρ + ρU ( ρ, s ) (cid:19) , (4.12) where U is the internal energy density of the fluid as a local function of ρ and s ,given by some equation of state.We have chosen to work with a general A instead of fixing A = 1. This becomesuseful in section , where we show that that there is a family of Poisson maps J A from C ∞ ( T ∗ M ) + to s ∗ A for A = 0 , , , . . . , with the property that J A can beobtained by truncating J B for some B ≥ A . The hydrodynamic mass and entropydensities can thus be thought of as the first two members of a series of generalisedentropies , as we shall explain in section .4.3 The Poisson map from C ∞ ( T ∗ M ) + to s ∗ A Now we construct the map J A : C ∞ ( T ∗ M ) + → s ∗ A , which we will soon prove tobe a Poisson map. It is defined by the dual pairing h J A [ f ] , ( u, ( g , . . . , g A )) i s A = Z T ∗ M dV h u, p i f + A X a =0 f (log f ) a g a ! . (4.13)In components, we have J A [ f ] = ( m [ f ] , ( s [ f ] , . . . , s A [ f ])), where m i [ f ] d n q = d n q Z d n p p i f, s a [ f ] d n q = d n q Z d n p f (log f ) a (4.14)are the momentum density and the generalised entropy densities of the distributionfunction, respectively, obtained by integrating along the cotangent fibres i.e. in-tegrating over p . In particular, s [ f ] is the mass density and s [ f ] is the spatialdensity of the Boltzmann entropy f log f . Here and henceforth we use the signconvention that the entropy density s [ f ] is a convex function of f – the oppositesign convention is more common in physics.Now we state the main result of this paper: Proposition 1.
The map J A : C ∞ ( T ∗ M ) + → s ∗ A (4.13) is a Poisson map from thespace of positive -particle distribution functions C ∞ ( T ∗ M ) + (as defined in section ) to the space s ∗ A of hydrodynamic variables with generalised entropies (as definedin section ). The case for A = 0 is known in [19, 37] and can be immediately deduced fromthe fact that Lie algebra s = Vect( M ) ⋉ C ∞ ( M ) acts on T ∗ M by Hamiltonianvector fields generated by the associated fibrewise affine function. The novelty hereis that the Poisson map can be extended to account for the non-constant entropybracket [42], and that J A maps the p -integral of the Boltzmann entropy onto thehydrodynamic entropy density. We also find that there is family of spaces withhydrodynamic variables, all of which arises naturally from kinetic theory, that in-cludes fluids with constant entropy as its first member and fluids with non-constantentropy as its second member. Taking the p -integral of the Boltzmann entropy toobtain a hydrodynamic entropy density is not a new idea, even within the frame-work of noncanonical Hamiltonian systems [18]. In Porpotision 1, we settle thestatus of s [ f ] as a component of a Poisson map. This offers a partial passage fromkinetic theory to fluid dynamics by offering a Poisson map between the respec-tive configuration spaces. However, as we shall see in section , the reduction of Poisson map from kinetic theory to hydrodynamics with non-constant entropy 17 noncanonical Hamiltonian systems described in section cannot be carried out ex-actly, because the kinetic-theory Hamiltonian does not factor through the Poissonmap. Proof.
Define the functions y a ( x ) : R + → R for a = 0 , , , . . . as y a ( x ) = x (log x ) a . (4.15)We set (log x ) = 1 by convention so that y ( x ) = x , and also set y a = 0 for a < x = 0 for a ≥
1. Each generalisedentropy density s a [ f ] can be written in terms of the y a as s a [ f ] = R d n p y a ( f ).The functions y a ( x ) satisfy the recurrence relations x dy a dx − y a = ay a − , (4.16) x d y a dx = a dy a − dx . (4.17)The recurrence relation (4.16) determines the functions y a ( x ) uniquely up to anoverall multiplicative constant. Let V b be the ( b + 1)-dimensional vector spacespanned by the functions y ( x ) , . . . , y b ( x ) for each b = 0 , . . . , A . Then the Eulerdifferential operator x ( d/dx ) maps each V b into itself. The operator x ( d/dx ) is inJordan normal form using the basis { y , . . . , b ! y b } .We can put this in the language of the generalised eigenvector-eigenvalue problem ( L − λI ) k v = 0 [2]. The space V b can also be defined as the generalised eigenspace of x ( d/dx ) of rank b + 1 with eigenvalue 1 i.e. the kernel of ( x ( d/dx ) − b +1 , sincefor any b (cid:18) x ddx − (cid:19) b +1 y b ( x ) = 0 but (cid:18) x ddx − (cid:19) b y b ( x ) = 0 . (4.18)The natural sequence of inclusions V ⊂ V ⊂ . . . ⊂ V A is called a complete flag .The ordered basis y ( x ) , . . . , y A ( x ) for V A is said to be adapted to the completeflag, since the first b + 1 elements form a basis for V b .Now suppose we have functionals ˜ F , ˜ G ∈ F ( C ∞ ( T ∗ M ) + ) such that ˜ F = F ◦ J A and ˜ G = G ◦ J A for some F, G ∈ F ( s ∗ A ). To prove that (cid:8) ˜ F , ˜ G (cid:9) KT = { F, G } s A ◦ J A , (4.19)we need to compute the functional derivative of ˜ F = F ◦ J A . The functional chainrule gives (cid:28) δ ˜ Fδf , δf (cid:29) C ∞ ( T ∗ M ) = (cid:28) δFδm , δm [ f ] (cid:29) Vect( M ) + A X a =0 (cid:28) δFδs a , δs a [ f ] (cid:29) C ∞ ( M ) , (4.20)and δm [ f ] = d n q Z d n p pδf, δs a [ f ] = d n q Z d n p y ′ a ( f ) δf. (4.21) Comparing both expressions gives δ ˜ Fδf = (cid:28) δFδm , p (cid:29) + A X a =0 y ′ a ( f ) δFδs a . (4.22)We have suppressed the pullback π ∗ of the cotangent bundle projection for nota-tional simplicity. Here δF /δm is a vector field on M , and δF /δs a is a function on M for each a , so they all depend on q only. Now we compute (cid:8) ˜ F , ˜ G (cid:9) KT = Z d n qd n p f (cid:26) δ ˜ Fδf , δ ˜ Gδf (cid:27) = Z d n qd n p f (cid:26)(cid:28) δFδm , p (cid:29) , (cid:28) δGδm , p (cid:29)(cid:27) + A X a =1 f (cid:26) y ′ a ( f ) δFδs a , (cid:28) δGδm , p (cid:29)(cid:27) + A X b =1 f (cid:26)(cid:28) δFδm , p (cid:29) , y ′ b ( f ) δGδs b (cid:27) + A X a,b =1 f (cid:26) y ′ a ( f ) δFδs a , y ′ b ( f ) δGδs b (cid:27) . (4.23)Using (2.9), the term on the first line of (4.23) is Z d n qd n p f (cid:26)(cid:28) δFδm , p (cid:29) , (cid:28) δGδm , p (cid:29)(cid:27) = − Z d n qd n p f (cid:28)(cid:20) δFδm , δGδm (cid:21) , p (cid:29) , = − (cid:28) m [ f ] , (cid:20) δFδm , δGδm (cid:21)(cid:29) Vect( M ) . (4.24)The second line of (4.23) contains two similar groups of terms. We compute oneof them as follows: Z d n qd n p f (cid:26) y ′ a ( f ) δFδs a , (cid:28) δGδm , p (cid:29)(cid:27) = Z d n qd n p (cid:18) fy ′ a ( f ) (cid:26) δFδs a , (cid:28) δGδm , p (cid:29)(cid:27) + fy ′′ a ( f ) δFδs a (cid:26) f, (cid:28) δGδm , p (cid:29)(cid:27)(cid:19) , = Z d n qd n p (cid:18) ( y a ( f ) + ay a − ( f )) (cid:26) δFδs a , (cid:28) δGδm , p (cid:29)(cid:27) + δFδs a (cid:26) ay a − ( f ) , (cid:28) δGδm , p (cid:29)(cid:27)(cid:19) , = Z d n qd n p (cid:18) ( y a ( f ) + ay a − ( f )) (cid:26) δFδs a , (cid:28) δGδm , p (cid:29)(cid:27) − ay a − ( f ) (cid:26) δFδs a , (cid:28) δGδm , p (cid:29)(cid:27)(cid:19) , = Z d n qd n p y a ( f ) (cid:18) L δGδm δFδs a (cid:19) = (cid:28) s a [ f ] , L δGδm δFδs a (cid:29) C ∞ ( M ) . (4.25)We have used the identity (2.7) as well as the recurrence relations (4.16,4.17) of thefunctions y a ( x ) = x (log x ) a to manipulate terms involving y a ( f ) and its derivativesin and out of the canonical Poisson bracket. The terms on the last line of (4.23) Poisson map from kinetic theory to hydrodynamics with non-constant entropy 19 vanish, because Z d n qd n p f (cid:26) y ′ a ( f ) δFδs a , y ′ b ( f ) δGδs b (cid:27) = Z d n qd n p f y ′ a ( f ) y ′ b ( f ) (cid:26) δFδs a , δGδs b (cid:27) + y ′ a ( f ) δGδs b (cid:26) δFδs a , y ′ b ( f ) (cid:27) + δFδs a y ′ b ( f ) (cid:26) y ′ a ( f ) , δGδs b (cid:27) + δFδs a δGδs b (cid:8) y ′ a ( f ) , y ′ b ( f ) (cid:9) ! = Z d n qd n p f (cid:18) y ′ a ( f ) δGδs b (cid:26) δFδs a , y ′ b ( f ) (cid:27) + δFδs a y ′ b ( f ) (cid:26) y ′ a ( f ) , δGδs b (cid:27)(cid:19) = Z d n qd n p (cid:18) fy ′ a ( f ) y ′′ b ( f ) δGδs b (cid:26) δFδs a , f (cid:27) + fy ′′ a ( f ) y ′ b ( f ) δFδs a (cid:26) f, δGδs b (cid:27)(cid:19) = Z d n qd n p (cid:18) δGδs b (cid:26) δFδs a , I ab ( f ) (cid:27) + δFδs a (cid:26) I ba ( f ) , δGδs b (cid:27)(cid:19) = − Z d n qd n p ( I ab ( f ) + I ba ( f )) (cid:26) δFδs a , δGδs b (cid:27) = 0 . (4.26)We have { δF /δs a , δG/δs b } = 0 because both arguments are functions of q only.The function I ab ( x ) that we have introduced on the second-to-last line of (4.26) isan antiderivative to xy ′ a ( x ) y ′′ b ( x ): I ab ( x ) = Z x dw wy ′ a ( w ) y ′′ b ( w ) . (4.27)We can express xy ′ a ( x ) y ′′ b ( x ) as a polynomial in log x , so the integral (4.27) con-verges and I ab ( x ) is a linear combination of x (log x ) c for suitable integers c .Putting everything together, we have (cid:8) ˜ F , ˜ G (cid:9) KT = − (cid:28) m [ f ] , (cid:20) δFδm , δGδm (cid:21)(cid:29) Vect( M ) − A X a =0 (cid:28) s a [ f ] , L δFδm δGδs a − L δGδm δFδs a (cid:29) C ∞ ( M ) , (4.28)which is precisely the ( − )-Lie–Poisson bracket on s ∗ A (4.11), evaluated at the imageof the Poisson map J A [ f ] = ( m [ f ] , ( s [ f ] , . . . , s A [ f ])).The Poisson manifolds C ∞ ( T ∗ M ) + and s ∗ A are linear, in the sense that they areopen subsets of vector spaces and that the Poisson brackets are linear. However, thePoisson map J A is genuinely non-linear. It is not induced by a Lie algebra actionof s A on T ∗ M by Hamiltonian vector fields for A ≥