Dirac Structures in Thermodynamics of Non-simple Systems

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).

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

$$\begin{aligned} \delta \int _{t _1 }^{ t _2} \!\Big [ L\left( q, \dot{q}, S_A\right) + \dot{\varGamma }^A( S_A- \varSigma _A)\Big ] \mathrm{d}t + \int _{t_1}^{t_2} \left\langle F^\mathrm{ext }, \delta q \right\rangle \,\mathrm{dt} =0, \end{aligned}$$

subject to the phenomenological constraint

$$\begin{aligned} \frac{\partial L}{\partial S_A}\dot{\varSigma }_A = \left\langle F^{\mathrm{fr}(A)}, \dot{q} \right\rangle + J_{AB}\dot{\varGamma }^B, \quad \text {for A=1,...,P}, \end{aligned}$$

(1)

and for variations subject to the variational constraint

$$\begin{aligned} \frac{\partial L}{\partial S_A}\delta \varSigma _A = \left\langle F^{\mathrm{fr}(A)}, \delta q \right\rangle + J_{AB}\delta \varGamma ^B, \quad \text {for A=1,...,P}, \end{aligned}$$

(2)

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:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}=\frac{\partial L}{\partial q} -\sum _{A=1}^P \frac{\dot{\varGamma }^A}{ \frac{\partial L}{\partial S_A} }F^{\mathrm{fr}(A)} + F^\mathrm{ext },\\[5mm] \displaystyle \frac{\partial L}{\partial S_A}+ \dot{\varGamma }^A=0, \quad A=1,...,P,\\[5mm] \displaystyle \dot{S}_A - \dot{\varSigma }_A +\sum _{B=1}^P \frac{\dot{\varGamma }^A}{ \frac{\partial L}{\partial S_A} }J_{BA}=0,\quad A=1,...,P. \end{array} \right. \end{aligned}$$

(3)

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 $S_A(t)$:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}- \frac{\partial L}{\partial q}= \sum _{A=1}^P F^{\mathrm{fr}(A)} +F^\mathrm{ext} ,\\ \displaystyle \frac{\partial L}{\partial S_A}\dot{S}_A= \left\langle F^{\mathrm{fr}(A)},\dot{q} \right\rangle - \sum _{B=1}^P J_{AB}\left( \frac{\partial L}{\partial S_B} - \frac{\partial L}{\partial S_A}\right) ,\quad A=1,...,P . \end{array} \right. \end{aligned}$$

(4)

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

$$ \frac{d}{dt}\frac{\partial L_A}{\partial \dot{q}}- \frac{\partial L_A}{\partial q}= F^{\mathrm{fr}(A)} +F^{\mathrm{ext}\rightarrow A}+\sum _{B=1}^PF^{B \rightarrow A}, \quad A=1,...,P, $$

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

$$\begin{aligned} \begin{aligned} \frac{d}{dt}E_A&= P_W^{\mathrm{ext}\rightarrow A} +\sum _{B=1}^P P^{B \rightarrow A}_W + \sum _{B=1}^P P_H^{B \rightarrow A}, \end{aligned} \end{aligned}$$

(5)

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

$$\begin{aligned} \dot{S}=-\sum _{A=1}^P\frac{1}{T^A} \left\langle F^{\mathrm{fr}(A)}, \dot{q} \right\rangle + \sum _{A<B}^{P}J_{AB}\left( \frac{1}{T^B}-\frac{1}{T^A} \right) (T^B-T^A), \end{aligned}$$

which becomes always positive because of the second law. This is consistent with the phenomenological relations of the form

$$\begin{aligned} F^{\mathrm{fr}(A)}_{i}=-\lambda ^A_{ij} \dot{q}^{j}\qquad \text {and}\qquad J_{AB}\frac{T^A-T^B}{T^AT^B}=\mathscr {L}_{AB}(T^B-T^A). \end{aligned}$$

(6)

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

$$\begin{aligned} \begin{aligned} C_{V}=\Bigg \{(x, v, \delta {x}) \in T \mathscr {Q}\times _ \mathscr {Q} T \mathscr {Q}\; \Biggr | \;&\frac{\partial L}{\partial S_A}\delta \varSigma _A = \left\langle F^{\mathrm{fr}(A)}, \delta q \right\rangle \\[-2mm]&+ \sum _{B=1}^PJ_{AB}\delta \varGamma ^B, A=1,...,P \Bigg \}. \end{aligned} \end{aligned}$$

(7)

For every $(x,v) \in T \mathscr {Q}$, we consider the subspace of $T_x \mathscr {D}$ given by

$$ C_V(x,v):=C_V \cap \big ( \{(x,v)\} \times T_x \mathscr {Q} \big ) \subset T_x \mathscr {Q}. $$

The kinematic constraint associated to $C_V$ is defined by

$$\begin{aligned} C_K=\{(x,v) \in T \mathscr {Q} \mid (x,v) \in C_V(x,v)\}, \end{aligned}$$

(8)

which is given locally as

$$\begin{aligned} \begin{aligned} C_{K}\!=\!\Bigg \{(x, v) \in T \mathscr {Q}\, \Biggr | \,\frac{\partial L}{\partial S_A}v_{\varSigma _A} \!\!=\! \left\langle F^{\mathrm{fr}(A)}, v_{q} \right\rangle \!+\! \sum _{B=1}^P J_{AB}v_{\varGamma ^B}, A=1,...,P \Bigg \}. \end{aligned} \end{aligned}$$

(9)

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

$$\begin{aligned} \begin{aligned} C_{V}(x,v)^{\circ }=\left\{ (x, \zeta ) \in T ^{*}_{x}\mathscr {Q}\; \bigr | \; \left\langle \zeta , \delta {x} \right\rangle =0, \;\; \forall \delta {x} \in C_{V}(x,v) \right\} \subset T ^{*}_{x}\mathscr {Q}, \end{aligned} \end{aligned}$$

is given, in coordinates $\zeta =(\zeta _{q}, \zeta _{S_{A}}, \zeta _{\varGamma ^{A}}, \zeta _{\varSigma _{A}}) \in T ^{*}_{x}\mathscr {Q}$, by

$$\begin{aligned} \begin{aligned} C_{V}(x,v)^{\circ }=\Bigg \{(x, \zeta ) \in T ^{*}_{x}\mathscr {Q}\; \Biggr | \;&\zeta _{q}+\frac{\zeta _{\varSigma _{A}}}{ \frac{\partial L}{\partial S_{A}}}F^{\mathrm{fr}(A)}=0, \quad \zeta _{S_{A}}=0,\\&\quad \zeta _{\varGamma _{A}}=\frac{\zeta _{\varSigma _{A}}}{ \frac{\partial L}{\partial S_{A}}}\sum _{B=1}^P J_{AB}, \;\; A=1,...,P \Bigg \}. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \varDelta _ \mathscr {P} (x,v,p):= \left( T_{(x,v,p)} \pi _{(\mathscr {P} ,\mathscr {Q})} \right) ^{-1} (C_V(x,v))\subset T_{(x,v,p)}\mathscr {P} , \end{aligned}$$

for each $(x,v,p) \in \mathscr {P}$. Locally, this distribution reads

$$ \varDelta _ \mathscr {P} (x,v,p)=\big \{(x,v,p, \delta {x}, \delta {v}, \delta {p}) \in T_{(x,v,p)}\mathscr {P} \mid (x, \delta {x}) \in C_V(x, v)\big \}. $$

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

$$ \varOmega _\mathscr {P}=dq \wedge dp_{q}+dS_{A} \wedge dp_{S_{A}}+d\varGamma _{A} \wedge dp_{\varGamma }+d\varSigma _{A} \wedge dp_{\varSigma _{A}}. $$

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 $\mathrm {x} \in \mathscr {P}$,

$$\begin{aligned} \begin{aligned} D_{\varDelta _ \mathscr {P} }(\mathrm {x}) :&=\big \{ (u_{\mathrm {x}}, \alpha _{\mathrm {x}}) \in T_{\mathrm {x}}\mathscr {P} \times T^{*}_{\mathrm {x}}\mathscr {P} \; \mid \; u_{\mathrm {x}} \in \varDelta _ \mathscr {P} (\mathrm {x}) \; \text {and} \\&\qquad \qquad \left\langle \alpha _{\mathrm {x}},w_{\mathrm {x}} \right\rangle =\varOmega _{\mathscr {P} }(\mathrm {x})(u_{\mathrm {x}},w_{\mathrm {x}}) \; \; \text{ for } \text{ all } \; \; w_{\mathrm {x}} \in \varDelta _ \mathscr {P} (\mathrm {x}) \big \}. \end{aligned} \end{aligned}$$

Proposition 1

The local expression of the Dirac condition, for each $\mathrm {x}=(x,v,p)$,

$$ \big ((x,v,p,\dot{x}, \dot{v}, \dot{p}),(x,v,p, \alpha , \beta , \gamma )\big ) \in D_{ \varDelta _ \mathscr {P} }(x,v,p) $$

is equivalent to

$$\begin{aligned} (x, \dot{x}) \in C_V(x,v), \quad \beta =0 , \quad \gamma =\dot{x}, \quad \dot{p} + \alpha \in C_V(x,v)^\circ . \end{aligned}$$

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

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \dot{p}_{q}+\alpha _{q}+(\dot{p}_{\varSigma _{A}}+\alpha _{\varSigma _{A}})\frac{1}{ \frac{\partial L}{\partial S_{A}}}F^{\mathrm{fr}(A)}=0,\;\;\dot{p}_{S_{A}}+\alpha _{S_{A}}=0,\\ \displaystyle \dot{p}_{\varGamma _{A}}+\alpha _{\varGamma _{A}}=\dot{p}_{\varSigma _{A}}+\alpha _{\varSigma _{A}}\frac{1}{ \frac{\partial L}{\partial S_{A}}}\sum _{B=1}^P J_{AB}, \\[6mm] \displaystyle \beta _{q}=0,\;\; \beta _{S_{A}}=0,\;\; \beta _{\varGamma ^{A}}=0,\;\; \beta _{\varSigma _{A}}=0,\\[3mm] \displaystyle \dot{q}=\gamma _{q},\; \dot{S}_{A}=\gamma _{S_{A}},\; \dot{\varGamma }_{A}=\gamma _{\varGamma ^{A}}, \; \dot{\varSigma }_{A}=\gamma _{\varSigma _{A}},\\[3mm] \displaystyle \frac{\partial L}{\partial S_A}\gamma _{\varSigma _{A}} \!\!=\! \left\langle F^{\mathrm{fr}(A)}, \gamma _{q} \right\rangle \!+\! \sum _{B=1}^P J_{AB}\gamma _{\varSigma _{B}}. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \mathscr {E} (x,v,p):= \left\langle p, v \right\rangle - \mathscr {L} (x,v). \end{aligned}$$

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

$$\begin{aligned} \left\langle \widetilde{F}^\mathrm{ext}(x,v,p) , u \right\rangle = \left\langle F^\mathrm{ext}(x,v), T\pi _{( \mathscr {P} , \mathscr {Q} )}(u) \right\rangle , \quad \text {for all}\;\; u \in T_{(x,v,p)}\mathscr {P}. \end{aligned}$$

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

$$ \frac{d}{dt}\mathscr {E} (x,v,p)= \left\langle F^\mathrm{ext}(x,v), \dot{x} \right\rangle . $$

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]).

Fig. 1.

The adiabatic piston problem

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

$$\begin{aligned} L(q, v_{q}, S _1, S _2 )= \frac{1}{2} M v_{q} ^2 -U_1 (q, S _1 )-U_2(q, S _2 ), \end{aligned}$$

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

$$ \left\{ \begin{array}{l} \displaystyle \dot{p}_{q}= \varPi _1 (q,S_1)\alpha _1 - \varPi _2(q,S_2) \alpha _2 - (\lambda ^1 + \lambda ^2 )\dot{q},\\[3mm] \displaystyle \dot{q}=v_{q}, \;\; p_{q}=Mv_{q},\\[3mm] \displaystyle T^1(q,S_1)\dot{S}_1= \lambda ^1 \dot{q}^2 +\kappa \left( T ^2(q,S_2)-T^1(q,S_1)\right) ,\\[3mm] \displaystyle T^2(q,S_2)\dot{S}_2= \lambda ^2\dot{q}^2 +\kappa \left( T^1(q,S_1)-T^2(q,S_2)\right) , \end{array} \right. $$

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

$$ \frac{d}{dt} S= \left( \frac{\lambda ^1}{T ^1} + \frac{\lambda ^2}{T ^2} \right) \dot{q}^2 + \kappa \frac{(T ^2 - T ^1 ) ^2 }{T ^1\,T ^2 } \ge 0. $$