Abstract
We present the Dirac structures and the associated Dirac system formulations for non-simple thermodynamic systems by focusing upon the cases that include irreversible processes due to friction and heat conduction. These systems are called non-simple since they involve several entropy variables. We review the variational formulation of the evolution equations of such non-simple systems. Then, based on this, we clarify that there exists a Dirac structure on the Pontryagin bundle over a thermodynamic configuration space and we develop the Dirac dynamical formulation of such non-simple systems. The approach is illustrated with the example of an adiabatic piston.
H.Yoshimura is partially supported by JSPS Grant-in-Aid for Scientific Research (17H01097), JST CREST Grant Number JPMJCR1914, the MEXT Top Global University Project, Waseda University (SR 2021C-134) and the Organization for University Research Initiatives (Evolution and application of energy conversion theory in collaboration with modern mathematics).
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1 Variational Formulation of Non-simple Systems
Before exploring Dirac structures underlying the thermodynamics of non-simple systems, we review the variational setting of such non-simple systems by focusing on the internal irreversible processes associated with friction and heat conduction.
1.1 Setting for Thermodynamics of Non-simple Systems
Non-simple Systems with Friction and Heat Conduction. Consider an adiabatically closed system \(\mathbf {\Sigma }=\cup _{A=1}^P\mathbf {\Sigma }_A\) which consists of P simple thermodynamic systems \(\mathbf {\Sigma }_A\), in which we include the irreversible processes due to friction and heat conduction between subsystems. Here a simple thermodynamic system denotes a system that has only one variable to represent the thermodynamic state, usually denoted by entropy. Since \(\mathbf {\Sigma }\) is an interconnected system of simple subsystems \(\mathbf {\Sigma }_1,...,\mathbf {\Sigma }_P\), it becomes a “non-simple” system that has several entropy (or temperature) variables (see [7]) and we note that all the irreversible processes are internal. For each simple subsystem \(\mathbf {\Sigma }_A,\,A=1,...,P\), \(S_A \in \mathbb {R}\) indicates its entropy variable. Here, we assume that the mechanical configuration of \(\mathbf {\Sigma }\) is given by independent mechanical variables \(q=(q^{1},...,q^{n}) \in Q\), where Q is the mechanical configuration manifold of \(\mathbf {\Sigma }\).
Friction, Heat Conduction and External Forces. Let \(F^{\mathrm{ext}\rightarrow A}:T^*Q \times \mathbb {R}^P \rightarrow T^*Q\) be an external force that acts on \(\mathbf {\Sigma }_A\) and hence the total exterior force is \(F^\mathrm{ext}=\sum _{A=1}^P F^{\mathrm{ext}\rightarrow A}\). Let \(F^{\mathrm{fr} (A)}:T^*Q \times \mathbb {R}^P \rightarrow T^*Q\) be the friction forces associated with the irreversible processes of each subsystem \(\mathbf {\Sigma }_A\), which yield an entropy production for subsystem \(\mathbf {\Sigma }_A\). Associated with the heat exchange between \(\mathbf {\Sigma }_A\) and \(\mathbf {\Sigma }_B\), let \(J_{AB}\) be the fluxes such that for \(A\ne B\), \(J_{AB}=J_{BA}\) and for \(A=B\), \( J_{AA}:=- \sum _{B\ne A}J_{AB}, \) where \(\sum _{A=1}^PJ_{AB}=0\) for all B.
Thermodynamic Displacements. In our formulation, we introduce the concept of thermodynamic displacements, see [3, 4]. For the case of heat exchange, we define the thermal displacements \(\varGamma ^A\), \(A=1,...,P\) such that its time rate \(\dot{\varGamma }^A\) becomes the temperature of \(\mathbf {\Sigma }_A\). We also introduce a new variable \(\varSigma _A\) associated with the internal entropy production.
1.2 Variational Formulation of Non-simple Systems
The Lagrange-d’Alembert Principle for Non-simple Systems. Now we consider a variational formulation of Lagrange-d’Alembert type for non-simple systems with friction and heat conduction, which is a natural extension of Hamilton’s principle in mechanics (see [4]).
Given a Lagrangian \(L: TQ\ \times \mathbb {R}^P \rightarrow \mathbb {R}\) and an external force \(F^\mathrm{ext }: TQ\ \times \mathbb {R}^P \rightarrow T^{*}Q\), find the curves q(t), \(S_A(t)\), \(\varGamma ^A(t)\), \(\varSigma _A(t)\) which are critical for the variational condition
subject to the phenomenological constraint
and for variations subject to the variational constraint
with \(\delta q(t_1)=\delta q(t_2)=0\) and \( \delta \varGamma ^A(t_1)=\delta \varGamma ^A(t_2)=0\), \(A=1,...,P\).
By direct computations, we obtain the following evolution equations:
From the second equation in (3), the temperature of the subsystem \(\mathbf {\Sigma }_A\), i.e., \(T^A\) can be obtained as \( \dot{\varGamma }^A= -\frac{\partial L}{\partial S_A}=:T^A. \) Because \(\sum _{A=1}^PJ_{AB}=0\) for all B, the last equation in (3) yields \(\dot{S}_A = \dot{\varSigma }_A\). Hence, together with (1), we obtain the following Lagrange-d’Alembert equations for the curves q(t) and \(S_A(t)\):
The First Law of Energy Balance. For the total energy \(E: TQ\ \times \mathbb {R}^P \rightarrow \mathbb {R}\) given by \( E\left( q, v_{q}, S_A\right) = \left\langle \frac{\partial L}{\partial v_{q}}\left( q, v_{q}, S_A\right) , v_{q} \right\rangle -L\left( q, v_{q}, S_A\right) , \) we have \(\frac{d}{dt}E= \left\langle F^\mathrm{ext}, \dot{q} \right\rangle = P_W^\mathrm{ext}\) along the solution curve of (4). If the Lagrangian is given by \( L(q, v, S_1,...,S_P)= \sum _{A=1}^P L_A(q,v, S_A), \) the evolution equations for \(\boldsymbol{\Sigma }_A\) are
where \(F^{B\rightarrow A}\) is the internal force exerted by \(\mathbf {\Sigma }_B\) on \(\mathbf {\Sigma }_A\). From Newton’s third law, we have \(F^{B\rightarrow A}=-F^{A\rightarrow B}\). Denoting by \(E_A\) the total energy of \(\mathbf {\Sigma }_A\), we have
where \(P_W^{\mathrm{ext}\rightarrow A}= \left\langle F^{\mathrm{ext}\rightarrow A}, \dot{q} \right\rangle \) is the mechanical power that flows from the exterior into \(\mathbf {\Sigma }_A\), \(P^{B \rightarrow A}_W=\sum _{B=1}^P \left\langle F^{B \rightarrow A}, \dot{q} \right\rangle \) is the internal mechanical power that flows from \(\mathbf {\Sigma }_B\) into \(\mathbf {\Sigma }_A\), and \(P_H^{B \rightarrow A}=\sum _{B=1}^P J_{AB}\left( \frac{\partial L}{\partial S_B} - \frac{\partial L}{\partial S_A}\right) \) is the internal heat power from \(\mathbf {\Sigma }_B\) to \(\boldsymbol{\Sigma }_A\). It follows that the power exchange can be written as \( P_H^{B \rightarrow A}= J_{AB}(T^A-T^B). \) The Second Law and Internal Entropy Production. The total entropy of the system is \(S=\sum _{A=1}^PS_A\). Therefore, it follows from (4) that the rate of total entropy production of the system is given by
which becomes always positive because of the second law. This is consistent with the phenomenological relations of the form
In the above, \(\lambda ^A_{ij}\) and \(\mathscr {L}_{AB}\) are functions of the state variables, where the symmetric part of \(\lambda ^A_{ij}\) are positive semi-definite and with \(\mathscr {L}_{AB}\ge 0\) for all A, B. From the second relation, we get \(J_{AB}= -\mathscr {L}_{AB}T^AT^B=-\kappa _{AB}\), with \(\kappa _{AB}=\kappa _{AB}(q, S_A, S_B)\) the heat conduction coefficients between \(\mathbf {\Sigma }_A\) and \(\mathbf {\Sigma }_B\).
2 Dirac Formulation of Non-simple Systems
In this section, we develop the Dirac formulation for the dynamics of non-simple systems by means of an induced Dirac structure the Pontryagin bundle; for the details, see [1, 5, 6].
2.1 Dirac Structures in Thermodynamics
Thermodynamic Configuration Space. For our class of non-simple systems, let \( \mathscr {Q}=Q \times V\) be a thermodynamic configuration space, where Q denotes the mechanical configuration space with mechanical variables \(q \in Q\) as before and \(V=\mathbb {R}^{P} \times \mathbb {R}^{P} \times \mathbb {R}^{P}\) is the thermodynamic space with thermodynamic variables \((S_{A}, \varGamma ^{A}, \varSigma _{A}) \in V\). We denote by \(x=(q, S_{A}, \varGamma ^{A}, \varSigma _{A})\) an element of \(\mathscr {Q}\), by (x, v) an element in the tangent bundle \(T\mathscr {Q}\) where \(v=(v_{q}, v_{S_{A}}, v_{\varGamma ^{A}}, v_{\varSigma _{A}}) \in T_{x}\mathscr {Q}\), and by (x, p) an element of the cotangent bundle \(T^{*}\mathscr {Q}\), where \(p=(p_{q}, p_{S_{A}}, p_{\varGamma ^{A}}, p_{\varSigma _{A}}) \in T^{*}_{x}\mathscr {Q}\).
Nonlinear Constraints of Thermodynamic Type. Let \(C_{V} \subset T \mathscr {Q} \times _ \mathscr {Q} T \mathscr {Q} \) be the variational constraint locally given as
For every \((x,v) \in T \mathscr {Q} \), we consider the subspace of \(T_x \mathscr {D} \) given by
The kinematic constraint associated to \(C_V\) is defined by
which is given locally as
Variational and kinematic constraints \(C_V\) and \(C_K\) are called nonlinear constraints of thermodynamic type if they are related as in (8).
Note also that the annihilator of \(C_V(x,v) \subset T_x \mathscr {Q}\), defined by
is given, in coordinates \(\zeta =(\zeta _{q}, \zeta _{S_{A}}, \zeta _{\varGamma ^{A}}, \zeta _{\varSigma _{A}}) \in T ^{*}_{x}\mathscr {Q}\), by
Dirac Structures on the Pontryagin Bundle. \(\mathscr {P} =\) T\(\mathscr {Q} \oplus \) T\({}^{ *}\!\mathscr {Q}\). The Pontryagin bundle \( \mathscr {P} \) is defined as the Whitney sum bundle of \( \mathscr {P} =T \mathscr {Q} \oplus T^* \mathscr {Q}\), with vector bundle projection, \(\pi _{( \mathscr {P} , \mathscr {Q} )}: \mathscr {P} =T \mathscr {Q} \oplus T^* \mathscr {Q} \rightarrow \mathscr {Q}\), \(\mathrm {x}=(x,v,p) \mapsto x\). Given a variational constraint \(C_V \subset T \mathscr {Q} \times _ \mathscr {Q} T \mathscr {Q} \) as in (7), we define the induced distribution \(\varDelta _ \mathscr {P}\) on \(\mathscr {P}\) by
for each \((x,v,p) \in \mathscr {P}\). Locally, this distribution reads
Further, the presymplectic form on \(\mathscr {P} \) is defined from the canonical symplectic form \( \varOmega _{T^*\mathscr {Q}}\) on \(T^*\mathscr {Q}\) as \( \varOmega _\mathscr {P} := \pi _{(\mathscr {P} ,T^{*}\mathscr {Q})} ^*\varOmega _{T^*\mathscr {Q}} \), which is locally given by using local coordinates \((x,v,p)=(q, S_{A}, \varGamma ^{A}, \varSigma _{A},v_{q}, v_{S_{A}}, v_{\varGamma ^{A}}, v_{\varSigma _{A}},p_{q}, p_{S_{A}}, p_{\varGamma ^{A}}, p_{\varSigma _{A}})\) for each \(\mathrm {x}=(x,v,p) \in \mathscr {P}\) as
Definition 1
The Dirac structure \(D_{\varDelta _ \mathscr {P} }\) induced on \(\mathscr {P} \) from \(\varDelta _\mathscr {P} \) and \(\omega _\mathscr {P} \) is defined by, for each \(\mathrm {x} \in \mathscr {P}\),
Proposition 1
The local expression of the Dirac condition, for each \(\mathrm {x}=(x,v,p)\),
is equivalent to
In coordinates \((\alpha , \beta , \gamma )=(\alpha _{q}, \alpha _{S_{A}}, \alpha _{\varGamma ^{A}}, \alpha _{\varSigma _{A}},\beta _{q}, \beta _{S_{A}},\beta _{\varGamma ^{A}}, \beta _{\varSigma _{A}},\) \(\gamma _{q}, \gamma _{S_{A}}, \gamma _{\varGamma ^{A}}, \gamma _{\varSigma _{A}})\), this condition reads as
2.2 Dirac Formulation for Thermodynamics of Non-simple Systems
Dirac Dynamical Systems on \(\mathscr {P} = \) T\(\mathscr {Q} \oplus \) T\({}^{ *}\!\mathscr {Q}\). For a given Lagrangian \(L\left( q, v_{q}, S_A\right) \) on \(TQ \times \mathbb {R}^{P}\), we introduce an augmented Lagrangian by \(\mathscr {L}(q,S_{A}, \varGamma ^{A}, \varSigma _{A}, v_{q}, v_{\varGamma ^{A}}):=L\left( q, v_{q}, S_A\right) + v_{\varGamma ^{A}}(S_{A}-\varSigma _{A}).\)
In the above, note that the augmented Lagrangian may be regarded as a (degenerate) Lagrangian function \(\mathscr {L}(x,v)\) on \(T\mathscr {Q}\). Further, define the generalized energy \(\mathscr {E}\) on \(\mathscr {P} = T\mathscr {Q} \oplus T^*\mathscr {Q}\) as
Given an external force \(F^\mathrm{ext}(q, v_{q}, S_A)\), which may be regarded as a map \(F^\mathrm{ext}: T\mathscr {Q} \rightarrow T^*\mathscr {Q}\), a horizontal one-form \(\widetilde{F}^\mathrm{ext} : \mathscr {P} \rightarrow T^{*}\mathscr {P}\) is induced by
Along the solution curve \((x(t),v(t),p(t)) \in \mathscr {P} \) of the Dirac dynamical system in (10), the energy balance equation holds as
2.3 Example of the Adiabatic Piston
The Adiabatic Piston. Now we consider a piston-cylinder system that is consisted of two cylinders connected by a rod, each of which contains a fluid (or an ideal gas) and is separated by a movable piston, as in Fig. 1 (see [2]).
The system \(\mathbf {\Sigma } \) is an interconnected system that is composed of three simple systems; the two pistons \(\mathbf {\Sigma } _1 , \mathbf {\Sigma } _2 \) with mass \( m_1 , m _2 \) and the connecting rod \(\mathbf {\Sigma } _3 \) with mass \( m _3 \). As in Fig. 1, q and \(r=D-\ell -q\) denote the distances between the bottom and the top in each piston where \(D=\text {const}\). Choose the state variables \((q, v_{q} , S_1, S_2)\) (the entropy of \( \mathbf {\Sigma } _3 \) is constant), and the Lagrangian is
where \(M:= m_1 + m _2 + m _3\), \(U_1(q,S_1):=\mathsf {U}_1(S_1, V_1= \alpha _1 q, N_1)\), and \(U_2(q, S_2):= \mathsf {U}_2(S_2, V_2= \alpha _2 r, N_2)\), with \(\mathsf {U}_i(S_i, V_i, N_i)\) the internal energies of the fluids, \(N_i\) the constant numbers of moles, and \(\alpha _i\) the constant areas of the cylinders, \(i=1,2\). As in (6), we have \(F^{\mathrm{fr} (A)}(q, \dot{q}, S_A )= - \lambda ^A \dot{q}\), with \(\lambda ^A=\lambda ^A(q,S^A)\ge 0\), \(A=1,2\) and \(J_{AB}=-\kappa _{AB}=:-\kappa \), where \(\kappa =\kappa (S_1,S_2,q)\ge 0\) is the heat conductivity of the connecting rod.
From the Dirac system formulation (11), we obtain the evolution equations as
where \(T^i(q,S_i)=\frac{\partial U_i}{\partial S_i}(q,S_i)\), \(\frac{\partial U_1 }{\partial q}=-\varPi _1 (q, S_1)\alpha _1 \), and \( \frac{\partial U_2}{\partial q}=\varPi _2(q, S_2) \alpha _2 \).
Since the system is isolated, we recover the first law \( \frac{d}{dt} E=0\), where \(E= \frac{1}{2} M\dot{q}^2 +U_1(q,S_1)+U(q, S_2 )\). The second law is also recovered as
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Yoshimura, H., Gay-Balmaz, F. (2021). Dirac Structures in Thermodynamics of Non-simple Systems. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_98
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