Entropy and structure of the thermodynamical systems

28/10/2015
Auteurs : Géry de Saxcé
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14338

Résumé

With respect to the concept of affine tensor, we analyse in this work the underlying geometric structure of the theories of Lie group statistical mechanics and relativistic thermodynamics of continua, formulated by Souriau independently one of each other. We reveal the link between these ones in the classical Galilean context. These geometric structures of the thermodynamics are rich and we think they might be source of inspiration for the geometric theory of information based on the concept of entropy.

Entropy and structure of the thermodynamical systems

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With respect to the concept of affine tensor, we analyse in this work the underlying geometric structure of the theories of Lie group statistical mechanics and relativistic thermodynamics of continua, formulated by Souriau independently one of each other. We reveal the link between these ones in the classical Galilean context. These geometric structures of the thermodynamics are rich and we think they might be source of inspiration for the geometric theory of information based on the concept of entropy.

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Entropy and structure of the thermodynamical systems G´ery de Saxc´e LML UMR CNRS 8107 Universit´e Lille 1 GSI 2015 G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 1 / 27 A fragance of symplectic geometry A fragance of symplectic geometry G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 2 / 27 A fragance of symplectic geometry A fragance of symplectic geometry The phase space N, set of η =   t x v   is equipped with a skew-symmetric tensor field ω(δη, dη), the symplectic form (or Lagrange’s brackets) It allows to recover the canonical equations (equations of motion) as ∀δη, ω(δη, dη) = 0 G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 3 / 27 A fragance of symplectic geometry A fragance of symplectic geometry In Physics, a powerful tool is the symmetry group, a Lie group G acting on N by a → a · η We denote g∗ the dual of its Lie algebra g η → µ = ψ(η) ∈ g∗ is a momentum map (Souriau) if ω(Z · η, dη) = −d(ψ(η) Z) It allows to recover integrals of the motion (modern version of Noether’s theorem) G naturally acts on g by the adjoint representation Ad(a) Z = a Z a−1 and on g∗ by the induced action Ad∗ (coadjoint representation) G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 4 / 27 A fragance of symplectic geometry A fragance of symplectic geometry Theorem (Souriau) There exists θ : G → g∗ called a symplectic cocycle such that ψ(a · η) − Ad∗(a) ψ(η) = θ(a) modulo a coboundary, it defines a class of symplectic cohomology [θ] ∈ H1(G; g∗), generally null. A noticeable exception is Galileo’s group, the symmetry group of the classical mechanics G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 5 / 27 Momentum as affine tensor Momentum as affine tensor G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 6 / 27 Momentum as affine tensor Momentum as affine tensor Let M be a n-dimension manifold (in the sequel, the space-time, set of events represented by X = (t, x)), G be a subgroup of Aff (n) (for us, Galileo’s group) We denote ATXM the tangent space to M at X perceived as affine We denote A∗TXM the vector space of affine forms Ψ on ATXM We call momentum tensor a bilinear map µ : TXM × A∗TXM → R It is a mixed 1-covariant and 1-contravariant affine tensor It is represented in an affine frame f by µ( −→ V , Ψ) = (χ Fβ + ΦαLα β) V β where Fβ and Lα β are the components of µ in f G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 7 / 27 Momentum as affine tensor Momentum as affine tensor The transformation law of the components µ = (F, L) of µ is given by the induced action of Aff (n) F′ = F P−1 L′ = (P L + C F) P−1 Let a = (C, P) ∈ Aff (n) be dX → C + P dX and Z ∈ g be da = (dC, dP) If the space of µ is identified to g∗ thanks to the dual pairing µ Z = µ da = (F, L) (dC, dP) = F dC + Tr(L dP) the transformation law of µ is nothing else the coadjoint representation ! G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 8 / 27 Momentum as affine tensor Momentum as affine tensor This mathematical construction is not relevant for classical mechanics and we extend it by considering a generalized transformation law µ = a · µ′ = Ad∗ (a) µ′ + θ(a) which is an affine representation of G in g∗ (because we wish the momentum to be an affine tensor) Comparing to the Formula of the Theorem on symplectic cocycles, restated as ψ(η) = Ad∗ (a) ψ′ (η) + θ(a) where ψ → ψ′ = a · ψ is the induced action, it turns out that the values of the momentum map are just momentum tensors ! G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 9 / 27 Lie group statistical mechanics Lie group statistical mechanics G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 10 / 27 Lie group statistical mechanics Lie group statistical mechanics To discover the underlying geometric structure of the statistical mechanics, we are interested by the affine maps Θ(µ) represented by on Θ : g∗ → R Θ(µ) = z + µ Z , where z and Z ∈ g are the affine components of Θ According to the induced action, their transformation law is z = z′ − θ(a) Ad(a) Z′ , Z = Ad(a) Z′ G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 11 / 27 Lie group statistical mechanics Lie group statistical mechanics In Structure des syst`emes dynamiques (1970), Souriau proposed a statistical mechanics model using geometric tools Let dλ be a measure on the orbit orb (µ), identified to µ, and a Gibbs probability measure p dλ with p = e−Θ(µ) = e−(z+µ Z) The normalization condition orb(µ) p dλ = 1 links the components by z(Z) = ln orb(µ) e−µ Z dλ The corresponding entropy and mean momenta are : s = − orb(µ) p ln p dλ = z + M Z, M = orb(µ) µ p dλ = − ∂z ∂Z This construction is formal and, for reasons of integrability, the integrals will be performed only on a subset by an heuristic way G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 12 / 27 Relativistic thermodynamics of continua Relativistic thermodynamics of continua G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 13 / 27 Relativistic thermodynamics of continua Relativistic thermodynamics of continua Independently of his statistical mechanics, Souriau proposed in Lect. Notes in Math. 676 (1976) a thermodynamics of continua compatible with general relativity. It is organized around some key fields : the 4-velocity −→ U, the 4-flux of mass −→ N = ρ −→ U where ρ is the density, the 4-flux of entropy −→ S = ρ s −→ U = s −→ N where s is the specific entropy the absolute temperature θ = kB T or its inverse β = 1 / θ Planck’s temperature vector −→ W = β −→ U its gradient f = ∇ −→ W called friction tensor the momentum tensor T of a continuum a linear map from TXM into itself G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 14 / 27 Relativistic thermodynamics of continua Relativistic thermodynamics of continua Theorem (Galilean version) [de Saxc´e & Vall´ee IJES 2012] If (Massieu’s or) Planck’s potential ζ smoothly depends on s′, W and F = ∂x/∂s′ through right Cauchy strains C = FT F , then T = U Π + 0 0 −σv σ with Π = −ρ ∂ζ ∂W σ = −2ρ β F ∂ζ ∂C FT , represents a momentum tensor T and (∇ζ) N = −Tr (T f ) Combining with the geometric version of the 1st principle ∇ · T = 0, ∇ · −→ N = 0 the 4-flux of specific entropy −→ S = T −→ W + ζ −→ N is divergence free and the specific entropy s is an integral of the motion G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 15 / 27 Bridging the gap between both theories Bridging the gap between both theories G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 16 / 27 Bridging the gap between both theories Bridging the gap between both theories (1/5) Step 1 : parameterizing the orbit. Galileo’s group is the set of affine transformations t = t′ + τ0, x = R x′ + u t′ + k where u is the Galilean boost The infinitesimal action Z · X is δt = δτ0, δx = δ̟ × x + δu t + δk The dualing pairing is µ Z = l · d̟ − q · du + p · dk − e dτ0 where l is the angular momentum, q the passage, p the linear momentum and e the energy In the dual space g∗ of dimension 10, the generic orbits are submanifolds parameterized by (q, p, n) ∈ R3 × R3 × S2 where n = l / l G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 17 / 27 Bridging the gap between both theories Bridging the gap between both theories (2/5) Step 2 : modelling the deformation. We consider N identical spinless particles in a box of volume V representing the elementary volume of the continuum thermodynamics For a coordinate change t = t′, x = ϕ(t′, s′), the Jacobian matrix is ∂X ∂X′ = P = 1 0 v F If the box of initial volume V0 is at rest (v = 0) and the deformation gradient F is uniform in the box, dλ is preserved Replacing the orbit by the subset V0 × R3 × S2 and integrating gives z = 1 2 ln(det(C)) − 3 2 ln β + Cte G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 18 / 27 Bridging the gap between both theories Bridging the gap between both theories (3/5 and 4/5) Step 3 : boost method. A new coordinate system ¯X in which the box has the velocity v can be deduced from X = P ¯X + C by applying a boost u = −v. Leaving out the bars, we have z = 1 2 ln(det(C)) − 3 2 ln β + m 2 β w 2 +Cte . Step 4 : identification. Theorem The transformation law of the temperature vector W = (β, w) and Planck’s potential ζ is the same as the one of the components z, Z of the affine map Θ through the identification Z = (−W , 0), z = m ζ G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 19 / 27 Bridging the gap between both theories Bridging the gap between both theories (5/5) Step 5 : from Planck’s potential ζ = z m = 1 2 m ln(det(C)) − 3 2 m ln β + 1 2 β w 2 +Cte we deduce the linear 4-momentum Π = (H, −pT ) and Cauchy’s stresses H = ρ 3 2 kB T m + 1 2 v 2 , p = ρv, σ = −q 1R3 where we recover the ideal gas law q = ρ m kB T = N V kB T Book in preparation with Claude Vall´ee : Galilean Mechanics and Thermodynamics of Continua G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 20 / 27 Epilogue : link with Information Geometry Epilogue link with Information Geometry G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 21 / 27 Epilogue : link with Information Geometry Epilogue : link with Information Geometry There is a puzzling analogy with Koszul-Vinberg characteristic function [Barbaresco Entropy 2014] ψΩ(Z) = Ω∗ e−µ Z dλ where Ω is a sharp open convex cone and Ω∗ is the set of linear strictly positive forms on ¯Ω − {0} For Galileo’s group, the cone Ω∗ + of future directed timelike vectors (β > 0) is preserved by Galilean transformations The orbits are contained in Ω∗ + but the integral does not converge neither on the orbits nor on Ω∗ + G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 22 / 27 Epilogue : link with Information Geometry Epilogue : link with Information Geometry As function of W through Z, the Hessian matrix I of −z, is positive definite [Souriau SSD 1970] It is Fisher metric of the Information Geometry On this ground, we can construct a thermodynamic length of a path t → X(t) [Crooks PRL 2009] L = t1 t0 (δW (t))T I(t) δW (t)dt where δW (t) is tangent to the space-time at X(t) We can also define Jensen-Shannon divergence of the path J = (t1 − t0) t1 t0 (δW (t))T I(t) δW (t)dt G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 23 / 27 Epilogue : link with Information Geometry Thank you ! G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 24 / 27 Outline 1 A fragance of symplectic geometry 2 Momentum as affine tensor 3 Lie group statistical mechanics 4 Relativistic thermodynamics of continua 5 Bridging the gap between both theories 6 Epilogue : link with Information Geometry G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 25 / 27 Outline Putting momentum tensors and connections into duality µ = (F, L) components of the momentum tensor µ in the affine frame f ˜Γ = (ΓA, Γ) affine connection [´Elie Cartan 1923] dual pairing µ, ˜Γ = F ΓA + 1 2 Tr(L Γ) Factorization of the symplectic 2-form ω = 1 2 dµ ˜Γ G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 26 / 27 Outline Theorem Let be G-principal bundle : π : F → M of affine frames f πG : g∗ × F → (g∗ × F)/G of couples η = (µ, f ) ˜Γ a field of connection 1-forms on F ω = 1 2 dµ ˜Γ a field of 2-forms Then : the restriction to the orbit ψµ of the projection (µ, f ) → µ is a submersion on each orbit, ω is the pull-back by ψµ of the symplectic form of Kirillov-Kostant-Souriau theorem and invariant ψµ is a momentum map and ψµ ◦ La = Ad∗(a) ψµ + θ(a) Equations of motion : µ = orb (µ, f ) is parallel-transporded ˜∇µ = d µ + ad∗ (˜Γ)µ − Dθ(e)(˜Γ) = 0 (generalization of Euler-Poincar´e equation) G´ery de Saxc´e (LML UMR CNRS 8107 Universit´e Lille 1)Entropy and structure of the thermodynamical systems GSI 2015 27 / 27