Dirac structures in nonequilbrium thermodynamics

07/11/2017
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In this paper, we show that the evolution equations for nonequilibrium thermodynamics can be formulated in terms of Dirac structures on the Pontryagin bundle P = TQ  T*Q, where Q = Q x R denotes the thermodynamic con guration manifold. In particular, we extend the use of Dirac structures from the case of linear nonholonomic constraints to the case of nonlinear nonholonomic constraints. Such a nonlinear constraint comes from the entropy production associated with irreversible processes in nonequilibrium thermodynamics. We also develop the induced Dirac structure on N = T*Q x R and the associated Lagrange-Dirac and Hamilton-Dirac dynamical formulations.

Dirac structures in nonequilbrium thermodynamics

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application/pdf Dirac structures in nonequilbrium thermodynamics Hiroaki Yoshimura, François Gay-Balmaz
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In this paper, we show that the evolution equations for nonequilibrium thermodynamics can be formulated in terms of Dirac structures on the Pontryagin bundle P = TQ  T*Q, where Q = Q x R denotes the thermodynamic con guration manifold. In particular, we extend the use of Dirac structures from the case of linear nonholonomic constraints to the case of nonlinear nonholonomic constraints. Such a nonlinear constraint comes from the entropy production associated with irreversible processes in nonequilibrium thermodynamics. We also develop the induced Dirac structure on N = T*Q x R and the associated Lagrange-Dirac and Hamilton-Dirac dynamical formulations.
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Dirac structures in nonequilbrium thermodynamics Hiroaki Yoshimura1 and François Gay-Balmaz2 1 Waseda University, School of Science and Engineering, Okubo, Shinjuku, Tokyo 169-8555, Japan yoshimura@waseda.jp 2 CNRS & Ecole Normale Supérieure, LMD, IPSL, 24 Rue Lhomond 75005 Paris gaybalma@lmd.ens.fr Abstract. In this paper, we show that the evolution equations for non- equilibrium thermodynamics can be formulated in terms of Dirac struc- tures on the Pontryagin bundle P = TQ ⊕ T∗ Q, where Q = Q × R denotes the thermodynamic configuration manifold. In particular, we extend the use of Dirac structures from the case of linear nonholonomic constraints to the case of nonlinear nonholonomic constraints. Such a nonlinear constraint comes from the entropy production associated with irreversible processes in nonequilibrium thermodynamics. We also de- velop the induced Dirac structure on N = T∗ Q × R and the associated Lagrange-Dirac and Hamilton-Dirac dynamical formulations. Keywords: Nonequilibrium thermodynamics, Dirac structures, nonlin- ear constraints, irreversible processes, implicit systems 1 Dirac structures in thermodynamics Dirac structures are known as a geometric object that generalizes both (almost) Poisson structures and (pre)symplectic structures on manifolds (see, [2, 4]). They were named after Dirac’s theory of constraints [3], and various physical systems with constraints such as electric circuits and nonholonomic mechanical systems are shown to be represented in the context of Dirac structures and the associated implicit Hamiltonian systems [11, 12, 1]. On the Lagrangian side, it was shown by [13, 14] that the notion of implicit Lagrangian systems can be developed in the context of induced Dirac structures, together with its associated variational structure given by the Lagrange-d’Alembert-Pontryagin principle. 1.1 Fundamental setting for thermodynamics Simple discrete systems. A simple thermodynamical system3 is a macroscopic system for which one (scalar) thermal variable and a finite set of mechanical 3 In [9] they are called élément de système (French). We choose to use the English terminology simple system instead of system element. variables are sufficient to describe entirely the state of the system. From the second law of thermodynamics, we can always choose such a thermal variable as entropy S (see [9]). In this paper, we focus on the particular case of simple adiabatically closed systems; namely, we assume that there is no exchange of matter and heat with the exterior of the system Constraints for the thermodynamics of simple systems. Let us consider a simple thermodynamic system with a Lagrangian L = L(q, v, S) : TQ×R → R and a friction force Ffr : TQ×R → T∗ Q, where Q is a configuration manifold of the mechanical variables q of the system, and R denotes the space of the ther- modynamic variable S. We introduce the thermodynamic configuration manifold Q := Q × R. Following [5], we define the variational constraint as CV =  (q, S, v, W, δq, δS) ∈ TQ ×Q TQ ∂L ∂S (q, v, S)δS = Ffr (q, v, S), δq  , where (q, S) ∈ Q, (v, W) ∈ T(q,S)Q, and (δq, δS) ∈ T(q,S)Q. Since ∂L ∂S (q, v, S) 6= 0 (temperature is always positive), we obtain that CV is a submanifold of TQ ×Q TQ of codimension one. For each fixed (q, S, v, W) ∈ TQ, the annihilator of CV (q, S, v, W) is given by CV (q, S, v, W)◦ =  (q, S, α, T ) ∈ T∗ (q,S)Q α ∂L ∂S (q, v, S) = −T Ffr (q, v, S)  . The kinematic constraint CK ⊂ TQ is defined from CV as CK =  (q, S, v, W) ∈ TQ ∂L ∂S (q, v, S)W = Ffr (q, v, S), v  . 1.2 Dirac dynamical systems on the Pontryagin bundle. Let P = TQ ⊕ T∗ Q be the Pontryagin bundle over Q. We shall use the notation x = (q, S, v, W, p, Λ) for an element of the Pontryagin bundle P. A distribu- tion ∆P on P may be induced from CV using the projection π(P,Q) : P → Q, (q, S, v, W, p, Λ) 7→ (q, S) as ∆P(x) := (Txπ(P,Q))−1 (CV (q, S, v, W)). It is locally given by ∆P(x) :=  (x, δx) ∈ TP ∂L ∂S (q, v, S)δS = Ffr (q, v, S), δq  . Let ΩT ∗Q be the canonical symplectic structure on T∗ Q. We define the in- duced Dirac structure on P from the distribution ∆P and the presymplectic form ωP = π∗ (P,T ∗Q)ΩT ∗Q on P by D∆P (x) : =  (vx, αx) ∈ TxP × T∗ x P | vx ∈ ∆P(x) and hαx, wxi = ωP(x)(vx, wx) for all wx ∈ ∆P(x) . Using the local expressions ẋ = (q̇, Ṡ, v̇, Ẇ, ṗ, Λ̇) ∈ TxP, and ζ = (α, T , β, Υ, u, Ψ) ∈ T∗ x P, the condition (x, ẋ), (x, ζ)  ∈ D∆P (x) is equivalently given by              (ṗ + α) ∂L ∂S (q, v, S) = −(Λ̇ + T )Ffr (q, v, S), ∂L ∂S (q, v, S)Ṡ = Ffr (q, v, S), q̇ , β = 0, Υ = 0, u = q̇, Ψ = Ṡ. (1) Dirac dynamical formulation on P = T∗ Q ⊕ T∗ Q. Let E : P → R be the generalized energy given by E(q, S, v, W, p, Λ) = hp, vi + ΛW − L(q, v, S). Using dE(q, S, v, W, p, Λ) = q, S, v, W, p, Λ, −∂L ∂q , −∂L ∂S , p − ∂L ∂v , Λ, v, W  and the condition (1), the Dirac dynamical system (x, ẋ), dE(x)  ∈ D∆P (x) yields the evolution equations of the thermodynamics of simple systems:                 ṗ − ∂L ∂q (q, v, S)  ∂L ∂S (q, v, S) = −  Λ̇ − ∂L ∂S (q, v, S)  Ffr (q, v, S), ∂L ∂S (q, v, S)Ṡ = Ffr (q, v, S), q̇ , p = ∂L ∂v , Λ = 0, v = q̇, W = Ṡ, which are finally written as        d dt ∂L ∂q̇ (q(t), q̇(t), S(t)) − ∂L ∂q (q(t), q̇(t), S(t)) = Ffr (q(t), q̇(t), S(t)), ∂L ∂S (q(t), q̇(t), S(t))Ṡ(t) = Ffr (q(t), q̇(t), S(t)), q̇(t) . (2) These are the evolution equations for the nonequilibrium thermodynamics of simple closed systems, see [5–7]. In the above, the temperature is defined by minus the partial derivative of the Lagrangian with respect to the entropy, namely, T = −∂L ∂S , which is assumed to be positive. The friction force Ffr is dissipative, that is Ffr (q, q̇, S), q̇ ≤ 0, for all (q, q̇, S) ∈ TQ × R. For the case in which the force is linear in velocity, we have Ffr (q, q̇, S) = −λ(q, S)q̇, where λ(q, S) ≥ 0 is the phenomenological coefficient, determined experimentally. The internal entropy production of the simple system is given by I(t) = − 1 T Ffr (q, q̇, S), q̇ = 1 T λ(q, S)q̇2 . 2 The Lagrange-Dirac formulation 2.1 Induced Dirac structures on N = T ∗ Q × R. Here we present the thermodynamic analogue of the Lagrange-Dirac formulation in [13] for nonholonomic mechanics. Namely, we develop the Lagrange-Dirac formulation on N = T∗ Q × R associated with the induced Dirac structure on N from the variational constraint CV ⊂ TQ ×Q TQ and the canonical symplectic form on T∗ Q. Constraints. We assume that the Lagrangian L = L(q, v, S) : TQ × R → R of a simple thermodynamic system is hyperregular with respect to the mechanical variables (q, v), namely, the map FLS : TQ → T∗ Q, (q, v) 7→  q, ∂L ∂v (q, v, S)  is a diffeomorphism for each fixed S ∈ R. Given the variational constraint CV , we can define the constraint CV ⊂ T∗ Q ×Q TQ as CV (q, S, p, Λ) := CV (q, S, v, W), which can be explicitly described as CV (q, S, p, Λ) =  (q, S, δq, δS) | −T(q, p, S)δS = Ffr (q, p, S), δq . In the above, T(q, p, S) := −∂L ∂S (q, v, S) and Ffr (q, p, S) := Ffr (q, v, S), in which v is uniquely determined from the condition ∂L ∂v (q, v, S) = p. Since CV does not depend on W, we can define from CV the constraint CV (q, S, p, Λ) ∈ T(q,S)Q and it induces the following distribution on N: ∆N (q, S, p) := T(q,S,p)π(N,Q) −1 CV (q, S, p, Λ)  , locally given as ∆N (q, S, p) =  (q, S, p, δq, δS, δp) ∈ TN −T(q, S, p)δS = Ffr (q, S, p), δq . Using the distribution ∆N (q, S, p) and the presymplectic form ωN = π∗ (N,T ∗Q)ΩT ∗Q, where ΩT ∗Q is the canonical symplectic structure, the Dirac structure on N is defined by, for each n = (q, S, p) ∈ N, D∆N (n) : =  (vn, ζn) ∈ TnN × T∗ nN | vn ∈ ∆N (n) and hζn, wni = ωN (n)(vn, wn) for all wn ∈ ∆N (n) . Writing locally (n, ṅ) ∈ TN and (n, ζ) ∈ T∗ N, where ṅ = (q̇, Ṡ, ṗ), and ζ = (α, T , u), the condition (n, ṅ), (n, ζ)  ∈ D∆N (n) is equivalent to, for each n = (q, S, p),        (ṗ + α)T(q, S, p) = T Ffr (q, S, p), T(q, S, p)Ṡ = − Ffr (q, S, p), q̇ , u = q̇. (3) 2.2 The Lagrange-Dirac systems Recall from [13] that the Dirac differential for a Lagrangian L : TQ → R is defined by using the symplectic diffeomorphism γQ : T∗ TQ → T∗ T∗ Q, locally given by γQ(q, v, α, p) = (q, p, −α, v), as introduced in [10]. For the case of ther- modynamics, we introduce the symplectic diffeomorphism b γQ : T∗ (TQ × R) → T∗ (T∗ Q × R), (q, S, v, α, Λ, p) 7→ (q, S, p, −α, −Λ, v). Then we define the associated Dirac differential of L as b dDL(q, S, v) := (b γQ ◦ dL) (q, S, v) =  q, S, ∂L ∂v , − ∂L ∂q , − ∂L ∂S , v  . By this definition and the relations (3), it follows that we have (q, S, p, q̇, Ṡ, ṗ), b dDL(q, S, v)  ∈ D∆N (q, S, p), if and only if               ṗ − ∂L ∂q (q, v, S)  T(q, p, S) = − ∂L ∂S (q, v, S)Ffr (q, p, S), T(q, S, p)Ṡ = − Ffr (q, p, S), q̇ , v = q̇, p = ∂L ∂v (q, v, S). The last equality comes from the fact that (q, S, p, q̇, Ṡ, ṗ) and b dDL(q, S, v) both belong to the fibers at (q, S, p) ∈ T∗ Q × R and hence we have the following the- orem concerning the Lagrange-Dirac formulation for thermodynamics of simple systems. Theorem 1. Consider a simple system with a Lagrangian L = L(q, v, S) : TQ × R → R and a friction force Ffr : TQ × R → T∗ Q. Assume that L is hyperregular with respect to the mechanical variables (q, v) and define T(q, p, S) and Ffr (q, p, S) as before. Then the following statements are equivalent: – The curve (q(t), S(t), v(t), p(t)) ∈ M satisfies the equations                         ṗ(t) − ∂L ∂q (q(t), v(t), S(t))  T(q(t), p(t), S(t)) = − ∂L ∂S (q(t), v(t), S(t))Ffr (q(t), p(t), S(t)), T(q(t), v(t), S(t))Ṡ(t) = − Ffr (q(t), p(t), S(t)), q̇(t) , v(t) = q̇(t), p(t) = ∂L ∂v (q(t), v(t), S(t)). (4) – The curve (q(t), S(t), v(t), p(t)) ∈ M satisfies the Lagrange-Dirac system of the simple thermodynamic system (q, S, p, q̇, Ṡ, ṗ), b dDL(q, S, v)  ∈ D∆N (q, S, p). Moreover, the system (4) is an implicit version of the system of evolution equa- tions (2) for the thermodynamics of simple systems. 3 The Hamilton-Dirac formulation 3.1 Hamilton-Dirac systems on N = T ∗ Q × R Since we assume that L : TQ × R → R is hyperregular with respect to the mechanical variables (see (2.1)), we can define the Hamiltonian function H : N = T∗ Q × R → R by H(q, p, S) = hp, q̇i − L(q, q̇, S), where q̇ is uniquely determined from (q, p, S) by the condition ∂L ∂q̇ (q, q̇, S) = p. We shall make use of the same distribution and the same Dirac structure of §2. In (3) we can directly write the constraint in the Hamiltonian setting in view of T(q, S, p) = ∂H ∂S (q, S, p). Then, it follows that the Hamilton-Dirac system (q, S, p, q̇, Ṡ, ṗ), dH(q, S, p)  ∈ D∆N (q, S, p) is equivalent to           ṗ + ∂H ∂q (q, S, p)  ∂H ∂S (q, S, p) = ∂H ∂S (q, S, p)Ffr (q, S, p), ∂H ∂p = q̇, ∂H ∂S (q, S, p)Ṡ = − Ffr (q, S, p), q̇ . We obtain the following theorem. Theorem 2. Consider a simple system with a Lagrangian L = L(q, v, S) : TQ× R → R and a friction force Ffr : TQ × R → T∗ Q. Assume that the Lagrangian is hyperregular with respect to the mechanical variables, consider the associated Hamiltonian H : T∗ Q × R → R and define Ffr (q, p, S) as before. Then the following statements are equivalent: – The curve (q(t), S(t), p(t)) ∈ N satisfies the equations                           ṗ(t) + ∂H ∂q (q(t), p(t), S(t))  ∂H ∂S (q(t), p(t), S(t)) = ∂H ∂S (q(t), p(t), S(t))Ffr (q(t), p(t), S(t)), − ∂H ∂S (q(t), p(t), S(t))Ṡ(t) = Ffr (q(t), p(t), S(t)), q̇(t) , ∂H ∂p (q(t), p(t), S(t)) = q̇(t). (5) – The curve (q(t), S(t), p(t)) ∈ N satisfies the Hamilton-Dirac system (q, S, p, q̇, Ṡ, ṗ), dH(q, S, p)  ∈ D∆N (q, S, p). Moreover the system (5), equivalently written as                ṗ(t) = − ∂H ∂q (q(t), p(t), S(t)) + Ffr (q(t), p(t), S(t)), q̇(t) = ∂H ∂p (q(t), p(t), S(t)), ∂H ∂S (q(t), p(t), S(t))Ṡ(t) = − Ffr (q(t), p(t), S(t)), q̇(t) (6) is the Hamiltonian description of the system of evolution equations (2) for the thermodynamics of simple systems. The Hamilton-d’Alembert principle. To the Hamilton-Dirac formulation is naturally associated a variational structure. In our case, the variational formu- lation on N = T∗ Q × R is δ Z t2 t1 h hp, q̇i − H(q, S, p) i dt = 0 (7) for all variations (δq(t), δS(t), δp(t)) for the curve (q(t), S(t), p(t)) ∈ N that satisfy − ∂H ∂S (q, p, S)δS = Ffr (q, p, S), δq (8) with δq(t1) = δq(t2) = 0, and the curve is subject to the phenomenological constraint − ∂H ∂S (q, p, S)Ṡ = Ffr (q, p, S), q̇ . (9) The principle (7)–(9) is called the Hamilton-d’Alembert principle. From this prin- ciple one immediately obtains the system (6). We refer to [8] for a thorough treatment of Dirac structures in nonequilibrium thermodynamics. Acknowledgements. F.G.B. is partially supported by the ANR project GE- OMFLUID, ANR-14-CE23-0002-01; H.Y. is partially supported by JSPS Grant- in-Aid for Scientific Research (26400408, 16KT0024, 24224004), Waseda Uni- versity Grant for Special Research Project (2017K-167), and the MEXT “Top Global University Project”. References 1. Bloch, A. M., Crouch, P. E.: Representations of Dirac structures on vector spaces and nonlinear L–C circuits. In: Differential Geometry and Control (Boulder, CO, 1997). Vol. 64. 103–117. Amer. Math. Soc. Providence, RI (1997) 2. Courant, T., Weinstein, A.: Beyond Poisson structures. In: Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986). 27. 39–49, Hermann. Paris (1989) 3. Dirac, P. A. M.: Generalized Hamiltonian dynamics, Canad. J. Math. 2, 129–148 (1950) 4. Dorfman, I.: Dirac Structures and Integrability of Nonlinear Evolution Equations. 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