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5 1 0 Reciprocal Relations 2 in Dissipationless Hydrodynamics n a J Lev A.Melnikovsky∗ 7 2 P.L.Kapitza Institute for Physical Problems ] h Russian Academy of Sciences, Moscow, Russia c e m - t Abstract a t Hiddensymmetryindissipationlesstermsofarbitraryhydrodynamics s . equations is recognized. Wedemonstrate that all fluxesare generated by t a a single function and deriveconventional Eulerequations using proposed m formalism. - d I am enjoying the privilege to learnfrom Alexander FedorovichAndreev for n about 20 years. One of the doors he opened for me gives access to the power o of hydrodynamics. Universal derivation procedure for its equations (seemingly c pioneered by Landau) is based purely on the conservation laws and symmetry [ considerations. In present paper I would like to paraphrase this procedure in a 1 formal way without specifying particular system. v Generic equations of hydrodynamics are contained [1] in local conservation 2 laws 8 ∂jk 1 y˙ + α =0. (1) 7 α ∂xk 0 HereandbelowLatinsuperscriptsareusedfor3DspacecoordinatesandGreek . 1 subscripts enumerate the integrals of motion; summation over repeated indices 0 is assumed. Densities of these conserved quantities y form a complete set of α 5 proper thermodynamic variables and unambiguously specify all local equilib- 1 rium properties. : v In dissipationless approximationan additional conservation law exists: i X ∂fk r σ˙ + =0, (2) a ∂xk whereσ is the entropydensity S = σdV . The entropydensity is afunction of state (cid:0) R (cid:1) ∂σ dσ = dy . (3) α ∂y α ∗E-mail: [email protected] 1 Equations(1)haveinternalsymmetrywhichisrevealedifexpressedthrough the thermodynamically conjugate quantities (see [2]) Y = −∂σ/∂y . The α α transformationfromy toY isinvertibleandonemayuseeithersetofvariables α α to characterize the state. The fluxes jk and fk generally depend on Y (or, equally, on y ) and their α α α spatial derivatives. In dissipationless approximation the dependence on spatial derivatives is neglected. We may therefore substitute (3) in (2) and transform as follows ∂σ ∂fk ∂jk ∂fk ∂Y 0= y˙ + = Y α + β. ∂y α ∂xk (cid:18) α∂Y ∂Y (cid:19) ∂xk α β β This equation holds for arbitrary∂Y /∂xk and the expressionin parentheses is β identically zero: ∂j ∂f ∂ 0=Y α + = (Y j +f)−j . α α α β ∂Y ∂Y ∂Y β β β In other words, all fluxes j are generated by a single function g α ∂g j = (4) α ∂Y α and the entropy flux f is its Legendre transform f =g−Y j . (5) α α Thiscompletestheproof,thatthematrixofderivatives∂jk/∂Y issymmet- α β ric ∂j ∂j α β = (6) ∂Y ∂Y β α 1 in agreement with Onsager principle [3]. The proof does not rely upon any specific property of the entropy itself. In fact,equation(2)isjustoneoflocalconservationlawsand“thermodynamically conjugate quantities” Y could be defined with respect to any other integral α of motion. In practice it might be more convenient to use energy rather than entropy to this end. To illustrate this formalism, consider a classicalideal fluid. Energyper unit volume E is a function of other conserved quantities: 2 v dE =T dσ+ µ− dρ+vdj, (cid:18) 2 (cid:19) where µ is chemical potential, ρ is mass density, v is fluid velocity, and j is momentumdensity. ConjugatevariablesarethereforeY =−T,Y =v2/2−µ, σ ρ 1Macroscopic reversibility implies that jα and Yα (or yα) behave oppositely under time reversal. 2 Y =−v. Due to the fluid isotropy,the generating function is g=vh(T,Y ,v), j ρ where the scalar h can be obtained from identity j =j=ρv: ρ ∂g ∂h ρv= =−v . (cid:18)∂Y (cid:19) (cid:18)∂µ(cid:19) ρ T,v T,v Recalling the expression for the pressure differential dp = σdT +ρdµ, we get h=−p. Onecaneasilyverifythatthefunctiong=−pvgeneratesconventional Euler fluxes: ∂g j = =σv, σ (cid:18)∂Y (cid:19) σ Yρ,Yj ∂gi Πik =ji = =pδik+ρvivk. jk (cid:18)∂Y (cid:19) jk Yσ,Yρ References [1] L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1987). [2] L.D. Landau, E.M. Lifshitz, Statistical Physics, part 1 (Pergamon Press, Oxford, 1980). [3] L. Onsager, Phys.Rev. 37, 405 (1931). 3