Major in Complex Systems Science, Graduate School of Science and Engineering, Ibaraki University, Nakanarusawa-cho, Hitachi, 316-8511, Japan.

1 Introduction↩︎

1.1 Entropy in Open Systems↩︎

The notion of “entropy” in an open system is ambiguous. The thermodynamic entropy \(S\) of the system alone can exhibit nonmonotonic oscillations through exchange with a heat bath, and therefore does not directly serve as a Lyapunov function characterizing relaxation toward equilibrium. Nevertheless, we still retain the intuitive view that entropy-like quantity increases as the system approaches equilibrium. The quantity that supports this intuition must therefore be identified. It may be the thermodynamic entropy \(S\) itself, or it may be a distinct quantity derived from \(S\).

The thermodynamic entropy \(S\) is a central concept in thermodynamics and statistical mechanics, but its role is not unique. First, \(S\) is a thermodynamic state variable, on the same footing as internal energy and volume, and is connected to heat exchange through the first law,

\[dU=TdS-PdV.\]

Second, \(S\) appears as the central quantity in the second law: in an isolated system, it serves as an indicator of irreversibility, or equivalently as a Lyapunov function characterizing the approach to equilibrium. From the viewpoint of statistical mechanics, \(S\) is also related to the logarithm of the number of microscopic states, and hence to information, through Boltzmann’s relation \[S=k_{B}\mathrm{ln}W.\]

In equilibrium and isolated systems, these different roles are consistently unified in a single quantity, the thermodynamic entropy \(S\). In an open system that exchanges heat and work with external reservoirs, however, this unification is no longer automatic. Nevertheless, even in relaxation processes of open systems, the picture that a certain monotonic quantity increases as the system approaches equilibrium remains valid under appropriate conditions. This monotonic quantity is not necessarily the thermodynamic entropy \(S\) itself, but may be a different quantity constructed from it. Identifying this quantity is the central issue addressed below.

1.2 Challenges in Nonequilibrium Thermodynamics and the Description of Temporal Order↩︎

The issue raised above, namely the identification of the quantity that plays the role of a Lyapunov function in an open system, belongs to a broader problem in nonequilibrium thermodynamics. It is the problem of how to describe, within a thermodynamic framework, the time evolution of systems far from equilibrium, especially dynamics involving oscillations and periodic motion.

Nonequilibrium thermodynamics began with individual empirical laws established in the nineteenth century, and acquired a unified theoretical framework in the twentieth century through Onsager’s theory of linear response [1]. By assuming detailed balance and linearity near equilibrium, Onsager theory derived reciprocal relations among transport coefficients and successfully described nonequilibrium phenomena in terms of entropy production.

Subsequently, Glansdorff and Prigogine formulated a general evolution criterion for stationary states in the linear regime, thereby advancing the understanding of dissipative structures and self-organization in open systems [2].

However, these frameworks alone do not make it straightforward to describe concrete time evolution in nonlinear far-from-equilibrium states, especially forms of temporal order such as periodic motion and rotational flows, within a thermodynamic framework. Variational principles for the nonlinear regime have also been proposed, including the maximum entropy production principle of the Ziegler type[3] and GENERIC, which treats reversible and irreversible parts in a unified manner [4]. However, the maximum entropy production principle does not include the reversible dynamical part, while GENERIC imposes degeneracy conditions between the reversible and irreversible sectors in order to ensure consistency with the first and second laws. For this reason, these approaches do not always provide the most direct representation of far-from-equilibrium dynamics involving rotational motion structures in open systems.

Oscillations and periodic motion observed in nonequilibrium systems are naturally viewed as the coexistence of conservative rotational flows and dissipative relaxation flows. Recent developments in stochastic thermodynamics and active matter physics have drawn attention to probability currents generated by nonconservative forces and to the geometric structure of entropy production. In stochastic thermodynamics, frameworks have been developed for decomposing entropy production geometrically, and for structurally decomposing probability currents into gradient and nongradient components. In particular, Dechant, Sasa, and Ito [5] and Yoshimura et al. [6] showed that probability currents in nonequilibrium systems can be decomposed into a gradient-flow component associated with relaxation and a nongradient component associated with steady cycle currents. However, at the level of macroscopic phenomenological thermodynamics, frameworks that directly treat the geometric structure of the reversible part, namely rotational flows, remain limited. This observation motivates a macroscopic formulation in which reversible rotational dynamics and dissipative gradient relaxation are separated explicitly.

1.3 Nambu Nonequilibrium Thermodynamics and the Aim of This Paper↩︎

The present work is intended as a continuation of the macroscopic tradition of nonequilibrium thermodynamics initiated by Onsager and further developed by Glansdorff and Prigogine. Its purpose is not to reduce irreversible thermodynamics to microscopic statistical mechanics, but to extend the macroscopic description so that reversible temporal order and irreversible relaxation can be represented within a common geometric framework.

The central question of this paper is therefore not whether entropy production is nonnegative for the total system, but what quantity plays the role of a Lyapunov function for the reduced open-system dynamics.

Motivated by this perspective, we have proposed Nambu nonequilibrium thermodynamics (NNET) as a macroscopic framework for treating reversible and irreversible parts in a covariant and unified manner [7], [8]. In NNET, the reversible part is described by the Nambu bracket, namely a generalized Hamiltonian dynamics, whereas the irreversible part is represented as a gradient flow generated by a scalar function \(S_{NB}\). The mathematical background of this theory is provided by the Helmholtz decomposition and Darboux’s theorem [9]. This structure provides a possible route to describing dissipative systems with periodic dynamics, such as chemical reaction systems and neuronal models [10].

Previous studies introduced NNET as a geometric framework for decomposing nonlinear nonequilibrium dynamics into Nambu-type reversible flow and dissipative gradient flow. The present paper focuses on a more elementary but fundamental issue: the thermodynamic meaning of the dissipation potential itself in an open macroscopic system.

The main contributions are as follows.

First, we formulate the reversible piston dynamics as a Nambu rotational flow generated by the extended energy and the subsystem entropy. This shows that the mechanical oscillation is not part of the dissipative sector, but belongs to the reversible geometric structure.

Second, we construct a dissipative gradient-flow sector generated by \(S_{NB}=S-H_{1}/T_{b}\), where \(H_{1}\) is the extended energy including the pressure-reservoir work term. This identifies \(S_{NB}\) as a free-energy-like dissipation potential rather than the thermodynamic entropy itself.

Third, we prove a conditional Lyapunov criterion: \(S_{NB}\) is monotonic when it is preserved by the reversible sector and generates a positive-semidefinite irreversible gradient flow. Numerical simulations then show the coexistence of damped piston oscillations, nonmonotonic subsystem entropy, and monotonic \(S_{NB}\).

The present formulation should be viewed as complementary to Onsager theory and GENERIC. While Onsager theory emphasizes near-equilibrium linear response and GENERIC ensures thermodynamic consistency through degeneracy conditions, the present model emphasizes a directly visible separation between reversible rotational motion and dissipative gradient relaxation[9].

The remainder of this paper is organized as follows. In Sec. 2, we summarize the basic structure of Nambu nonequilibrium thermodynamics. In Sec. 3, we show that an adiabatic reversible piston system coupled to a pressure bath can be naturally formulated in the Nambu form. In Sec. 4, we introduce a heat bath and friction, construct the irreversible part, and discuss the relation between the dissipation potential and entropy production. In Sec. 5, we present numerical results. Finally, in Sec. 6, we discuss possible extensions to chemical reaction systems and the broader significance of the present framework.

²

2 Basic Structure of Nambu Nonequilibrium Thermodynamics↩︎

Let the state variables of the system be denoted by \[x=(x^{1},\dots,x^{N}).\] In Nambu nonequilibrium thermodynamics (NNET), the time evolution is decomposed into reversible and irreversible parts[8]: \[\dot{x}^{i}=\partial_{t}^{(\mathrm{H})}x^{i}+\partial_{t}^{(\mathrm{S})}x^{i}.\]

Here, \(\partial_{t}^{(H)}x^{i}\) and \(\partial_{t}^{(S)}x^{i}\) denote the reversible and irreversible parts of the time evolution, respectively.

2.1 Nambu Bracket and the Reversible Part↩︎

The reversible part is formulated in terms of \(N-1\) functions \[H_{1},\dots,H_{N-1}\] which are invariants of the reversible sector, and the Nambu bracket[8]. The Nambu bracket, introduced by Yoichiro Nambu in 1973[12] as a generalization of Hamiltonian mechanics, is the completely antisymmetric Jacobian defined by \[\{A_{1},\dots,A_{N}\}\equiv\epsilon^{i_{1}\cdots i_{N}}\frac{\partial A_{1}}{\partial x^{i_{1}}}\cdots\frac{\partial A_{N}}{\partial x^{i_{N}}}.\] Using this bracket, the reversible part of the dynamics can be written as \[\partial_{t}^{(\mathrm{H})}x^{i}=\{x^{i},H_{1},\dots,H_{N-1}\}\] Because of the complete antisymmetry of the Nambu bracket, one immediately obtains \[\partial_{t}^{(H)}H_{m}=0.\] Thus, \(H_{m}\) are conserved by the reversible dynamics. In the full dynamics, however, their conservation depends on the structure of the irreversible sector introduced below; see Ref.[8].

Furthermore, in a regular region where \[\nabla H_{1},\dots,\nabla H_{N-1}\] are linearly independent, the common intersection of the \(N-1\) level surfaces is locally a one-dimensional curve, as schematically illustrated in Fig. 1. The reversible vector field generated by the Nambu bracket is tangent to this curve. In this sense, it represents a circulating, or rotational, flow along the intersection of the conserved level surfaces. In particular, when this intersection forms a closed curve, the motion can be interpreted as periodic motion, in the same manner as in a harmonic oscillator.

2.2 Dissipation Potential and the Irreversible Part↩︎

The irreversible part of NNET is generally written in the following gradient-flow form using a transport coefficient matrix \(L^{ij}\)[8]: \[\partial_{t}^{(\mathrm{S})}x^{i}=L^{ij}\frac{\partial S_{NB}}{\partial x^{j}}.\] Here, \(L^{ij}\) can be decomposed into symmetric and antisymmetric parts. The structure in which irreversible dynamics is described as a gradient flow generated by a thermodynamic potential has recently been reinterpreted from the viewpoint of information geometry. In particular, attempts have been made to understand Onsager’s nonequilibrium thermodynamics as a natural gradient flow on the space of probability distributions[13]. As another geometric direction, formulations have also been proposed in which thermodynamic forces are regarded as gauge fields, and Onsager-type constitutive relations are derived as gauge-fixing conditions. In such gauge-fixing theories, nonlinear constitutive relations and transverse components appear by extending thermodynamic forces from pure gauge fields to physical gauge fields [14], [15]. In the present paper, however, we do not introduce such transverse components as thermodynamic gauge fields. Instead, we explicitly separate the reversible nongradient component as a dynamical sector generated by the Nambu bracket.

In this paper, in order to describe the irreversible part as a gradient flow of \(S_{NB}\), we use the symmetric positive-semidefinite part of the transport coefficient matrix \(L^{ij}\). The reversible rotational flow is not treated as an antisymmetric component of the irreversible part, but is described separately by the Nambu bracket. This separation clarifies the distinct roles of the reversible and irreversible sectors³.

\(S_{NB}\) is a scalar function that generates the irreversible part of the dynamics, and is referred to here as the dissipation potential. In general, it does not coincide with the thermodynamic entropy \(S\). A pure gradient flow alone cannot generate periodic motion. In NNET, however, temporal order in far-from-equilibrium systems can be naturally represented through the coupling of the irreversible gradient flow to the reversible part, namely the rotational flow generated by the Nambu bracket.

This point is one of the central themes of the present paper. The quantity \(S_{NB}\) is not the thermodynamic entropy \(S\) itself that appears in the second law, but the generating potential used to represent the irreversible vector field as a gradient flow. Therefore, the monotonicity of \(S_{NB}\) is not an automatic or universal consequence of its definition. For \(S_{NB}\) to function as a Lyapunov quantity, the reversible part must preserve \(S_{NB}\), and the irreversible part must close as a gradient flow of \(S_{NB}\) with a positive-semidefinite transport coefficient. The piston model studied in this paper provides a minimal example in which these conditions are realized explicitly.

3 Nambu Dynamics of an Adiabatic Reversible Piston System↩︎

In this section, we consider the adiabatic reversible process of an ideal piston system without friction. We show that this process is naturally described by the Nambu bracket and that it contains an intrinsic oscillatory mode.

3.1 State Variables and Thermodynamic Quantities↩︎

Let \(A\) be the cross-sectional area of the piston and \(q\) its position, so that the gas volume is given by

\[V=Aq.\]

We define the physical momentum of the piston by \[p=m\dot{q},\] which gives \[\dot{V}=\frac{A}{m}p.\]

Let \(P_{b}\) denote the external pressure, \(S\) the thermodynamic entropy of the gas, and \(U(S,V)\) its internal energy. The thermodynamic relations are then \[T=\frac{\partial U}{\partial S},\qquad P=-\frac{\partial U}{\partial V}.\] In an adiabatic reversible process, one has \(\dot{S}=0\) . The equation of motion for the piston follows from the pressure difference acting on it:

\[\dot{p}=A(P-P_{b}).\] It is therefore natural to choose the three state variables of the system⁴ as

\[x=(V,p,S).\] Here, because we use the volume \(V=Aq\) and the physical momentum \(p=m\dot{q}\) as state variables, the pair \((V,p)\) is not a canonical pair. Indeed, for the canonical pair\((q,p)\), the Poisson bracket satisfies

\[\{V,p\}=A.\] Therefore, in the coordinates \((V,p,S)\) , we use the normalized Nambu bracket \[\{F,G,K\}_{A}\equiv A\{F,G,K\}\equiv A\frac{\partial(F,G,K)}{\partial(V,p,S)}.\]

3.2 Conserved Quantities and Nambu Formulation↩︎

In the adiabatic reversible dynamics, this system has two conserved quantities. The first is the extended energy \(H_{1}\), which includes the work exchange with the pressure reservoir. The second is the thermodynamic entropy \(H_{2}=S\) , which is conserved in the adiabatic process. They are given by \[H_{1}=\frac{p^{2}}{2m}+U(S,V)+P_{b}V,\qquad H_{2}=S.\] Indeed, using the reversible equations of motion, one obtains \[\dot{H}_{1}^{(\mathrm{H})}=\frac{p}{m}\dot{p}+\partial_{V}U\,\dot{V}+P_{b}\dot{V}=0.\]

Taking these quantities as the Hamiltonians of the Nambu dynamics, the time evolution generated by the normalized Nambu bracket becomes \[\partial_{t}^{(\mathrm{H})}V=\{V,H_{1},H_{2}\}_{A}=A\frac{\partial(V,H_{1},S)}{\partial(V,p,S)}=\frac{Ap}{m},\] \[\partial_{t}^{(\mathrm{H})}p=\{p,H_{1},H_{2}\}_{A}=A\frac{\partial(p,H_{1},S)}{\partial(V,p,S)}=-A(\partial_{V}U+P_{b})=A(P-P_{b}),\] and

\[\partial_{t}^{(\mathrm{H})}S=\{S,\;H_{1},H_{2}\}_{A}=0.\] Thus, the Nambu formulation exactly reproduces the expected reversible dynamics of the adiabatic piston system.

3.3 Linearization Around Equilibrium and the Oscillatory Mode↩︎

To show that the oscillatory motion is generated by the reversible part, we linearize the dynamics around an equilibrium point \[(V_{*},p_{*}=0,S_{*}).\] The equilibrium condition is \[P(S_{*},V_{*})=P_{b}.\] Writing \[V=V_{*}+\delta V,p=\delta p\] we obtain \[\frac{d\delta V}{dt}=\frac{A\delta p}{m},\qquad\frac{d\delta p}{dt}=-A\left.\frac{\partial^{2}U}{\partial V^{2}}\right|_{*}\delta V.\] These equations give the harmonic-oscillator equation \[\dfrac{d^{2}\delta V}{dt^{2}}+\omega_{*}^{2}\,\delta V=0,\qquad\omega_{*}^{2}=\frac{A^{2}}{m}\left.\frac{\partial^{2}U}{\partial V^{2}}\right|_{*}.\] The eigenvalues are therefore \[\mu_{\pm}=\pm i\omega_{*}.\]

Thus, the oscillatory mode near equilibrium, namely the temporal order of the piston motion, originates purely from the reversible part generated by the Nambu bracket.

4 Extension to an Open System: Heat Bath and Irreversible Dynamics↩︎

In real systems, dissipation and heat exchange with the environment are present. In this section, we introduce a heat bath at temperature \(T_{b}\) and piston friction, and clarify the role of the irreversible part.

4.1 Phenomenological Model and Entropy Production↩︎

We consider an explicit model in which the kinetic energy dissipated by friction is converted into heat within the system. The irreversible part is introduced as \[\partial_{t}^{(\mathrm{S})}V=0,\qquad\partial_{t}^{(\mathrm{S})}p=-\lambda p,\qquad T\,\partial_{t}^{(\mathrm{S})}S=\kappa(T_{b}-T)+\frac{\lambda}{m}p^{2}.\] Here, \(\lambda\) is the coefficient characterizing piston friction, and determines the decay rate of the kinetic energy. The second term in \(T\partial_{t}^{(S)}S\) represents the conversion of the kinetic energy lost by friction into internal heat⁵. Indeed, the kinetic energy lost by friction is

\[\frac{\partial}{\partial t}\left(\frac{p^{2}}{2m}\right)=\frac{p\dot{p}}{m}=-\frac{\lambda}{m}p^{2}.\] The coefficient \(\kappa>0\) is a phenomenological parameter characterizing heat exchange with the heat bath, corresponding to Newton’s law of cooling.

With the irreversible terms introduced above, the energy change of the system follows from the total time evolution \[\dot{O}=\partial_{t}^{(\mathrm{H})}O+\partial_{t}^{(\mathrm{S})}O.\] Since \(\partial_{t}^{(H)}H_{1}=0,\) we obtain \[\dot{H_{1}}=\partial_{t}^{(H)}H_{1}+\partial_{t}^{(S)}H_{1}=\partial_{t}^{(S)}H_{1}=-\lambda\frac{p^{2}}{m}+T\partial_{t}^{(S)}S=\kappa(T_{b}-T).\] If the energy change rate of the heat bath is defined as \[\dot{E}_{b}=-\kappa(T_{b}-T),\] then \[\dot{H}_{1}+\dot{E}_{b}=0.\] Thus, the first law, namely energy conservation for the system plus bath, is satisfied.

We next discuss the second law. In this paper, the pressure reservoir is treated as an ideal work reservoir: it exchanges work with the system but carries no thermodynamic entropy, so that \(\dot{S}_{P}=0\). Under this idealization, the total entropy production, given by the entropy change of the system plus that of the heat bath, becomes \[\dot{S}_{\mathrm{tot}}=\dot{S}+\frac{\dot{E}_{b}}{T_{b}}=\kappa\frac{(T_{b}-T)^{2}}{TT_{b}}+\frac{\lambda}{mT}p^{2}\ge0.\] This confirms consistency with the second law of thermodynamics. It should be noted, however, that the thermodynamic entropy \(S\) refers only to the open subsystem, and therefore its time derivative is allowed to become negative.

4.2 Gradient-Flow Representation by the Dissipation Potential \(S_{NB}\)↩︎

To organize the phenomenological model introduced above within the framework of NNET, we define the dissipation potential \(S_{NB}\) , which generates the irreversible part, as \[S_{NB}=S-\frac{H_{1}}{T_{b}}=S-\frac{1}{T_{b}}\left(U(S,V)+\frac{p^{2}}{2m}+P_{b}V\right).\] The gradient of \(S_{\mathrm{NB}}\) is then \[\frac{\partial S_{\mathrm{NB}}}{\partial V}=\frac{P-P_{b}}{T_{b}},\qquad\frac{\partial S_{\mathrm{NB}}}{\partial p}=-\frac{p}{mT_{b}},\qquad\frac{\partial S_{\mathrm{NB}}}{\partial S}=1-\frac{T}{T_{b}}=\frac{T_{b}-T}{T_{b}}.\] The form of \(S_{NB}\) adopted here is not an arbitrary assumption. Since the system is in contact with a heat bath at temperature \(T_{b}\) and a work reservoir at pressure \(P_{b}\), it is natural, under the fixed external conditions \((T_{b},P_{b})\), to construct a free-energy-like potential from the extended energy \[H_{1}=U(S,V)+\frac{p^{2}}{2m}+P_{b}V,\] which includes the internal energy, the kinetic energy of the piston, and the work term associated with the pressure reservoir. Thus, \[S_{NB}=S-\frac{H_{1}}{T_{b}}\] can be interpreted as the negative of the generalized free energy \(H_{1}-T_{b}S\), divided by \(T_{b}\). It is therefore an effective potential with the dimension of entropy. This is precisely the point: \(S_{NB}\) is not the thermodynamic entropy itself, but a Massieu/free-energy-like potential adapted to the imposed bath variables⁶.

To represent both the frictional decay of the momentum and the recovery of the lost kinetic energy as internal heat within the gradient flow of \(S_{NB}\), we adopt the following positive-semidefinite transport coefficient matrix.

The transport coefficient matrix can be decomposed into two physically distinct dissipative channels:

\[L^ {}=L_{\mathrm{fric}}^ {}+L_{heat}^ {}.\]

friction channel↩︎

\[L_{\mathrm{fric}}^ {}=\lambda|v\rangle\langle v|\]

\[|v\rangle=\sqrt{mT}|p\rangle-\frac{p}{\sqrt{mT}}|S\rangle\]

This rank-one contribution describes the conversion of kinetic energy into internal heat.

heat-bath channel \[L_{\mathrm{heat}}=\kappa\frac{T_{b}}{T}|S\rangle\langle S|.\]↩︎

This contribution represents thermal relaxation toward the bath temperature.

positivity↩︎

Since both contributions are positive semidefinite for

\[T>0,\;\lambda\geq0,\;\kappa\geq0,\] their sum is positive semidefinite:

\[\langle x|\left(L_{\mathrm{fric}}+L_{\mathrm{heat}}\right)|x\rangle=\lambda\langle x|v\rangle\langle v|x\rangle+\kappa\frac{T_{b}}{T}\langle x|S\rangle\langle S|x\rangle\geq0.\]

Then one obtains \[\partial_{t}^{(S)}V=0,\qquad\partial_{t}^{(S)}p=-\lambda p,\qquad\partial_{t}^{(S)}S=\kappa\frac{T_{b}-T}{T}+\frac{\lambda p^{2}}{mT}.\label{eq:V44p44SHeatTimeVari}\tag{1}\] Thus, the phenomenological irreversible dynamics introduced in the previous subsection is reproduced.

It is important to note that the reversible part in the present formulation has already been specified by the Nambu bracket, \[\partial_{t}^{(H)}x^{i}=\{x^{i},H_{1},H_{2}\}_{A}.\]

Therefore, it is not necessary to introduce Onsager--Casimir-type antisymmetric cross terms into the irreversible part. In the irreversible sector of the present model, we use only the symmetric positive-semidefinite part of the transport coefficient matrix in order to ensure the monotonicity of \(S_{\mathrm{NB}}\).

The time evolution of \(S_{NB}\) is determined solely by the irreversible part, since the reversible contribution satisfies

\[\partial_{t}^{(H)}S_{NB}=0.\] Therefore, \[\dot{S}_{\mathrm{NB}}=\partial_{t}^{(S)}S_{\mathrm{NB}}=\frac{\lambda p^{2}}{mT}+\kappa\frac{(T-T_{b})^{2}}{TT_{b}}\ge0.\label{eq:SNBTimeVari}\tag{2}\]

The equality \(\dot{S}_{NB}=0\) is attained only when both dissipative channels are inactive. For \(\lambda>0\), \(\kappa>0\), and \(T>0\), Eq. 2 implies \(p=0\) and \(T=T_{b}\), namely the piston has no kinetic motion and the gas is in thermal equilibrium with the heat bath. At a fixed point of the full dynamics, the mechanical equation further requires \(P=P_{b}\). Thus the equilibrium state is characterized by \[p=0,\qquad T=T_{b},\qquad P=P_{b}.\]

In this state, the system entropy also becomes stationary, since Eq. 1 gives \(\dot{S}=0\).

Thus, whereas the thermodynamic entropy \(S\) of the system may oscillate, the dissipation potential \(S_{NB}\) functions as a monotonically increasing Lyapunov function in the explicit dissipative model adopted in this section.

This monotonicity is analogous to a corresponding geometric structure in stochastic thermodynamics. In Fokker--Planck dynamics, the Kullback--Leibler divergence between probability distributions is known to decrease monotonically along a natural gradient flow, and this property has been rigorously established within geometric thermodynamics based on information geometry and optimal transport theory[17]. The monotonic increase of \(S_{NB}\) in the present model may be regarded as a macroscopic counterpart of this structure.

However, this monotonic increase does not follow unconditionally from the definition of \(S_{NB}\).

In the present model, the monotonicity above holds because \(S_{NB}\) is constructed from the conserved quantity \(H_{1}\) and the thermodynamic entropy \(S\), the reversible part leaves \(S_{NB}\) unchanged, and the irreversible part closes as a gradient flow of \(S_{NB}\) with a positive-semidefinite transport coefficient. In more general strongly nonlinear far-from-equilibrium systems, for example when the reversible part does not preserve \(S_{NB}\), or when the irreversible part does not close as a positive-semidefinite gradient flow generated by a single \(S_{NB}\), \(S_{NB}\) itself may exhibit nonmonotonic oscillations. Therefore, the result obtained here is not a claim of universal monotonicity of \(S_{NB}\), but rather demonstrates, in the explicit piston model, sufficient conditions under which \(S_{NB}\) functions as a Lyapunov quantity.

The following proposition does not state that \(S_{\mathrm{NB}}\) is always a Lyapunov function. Rather, it states that the monotonicity of \(S_{\mathrm{NB}}\) is guaranteed under specific geometric conditions, such as those realized in the present piston model.

A sufficient condition for the preservation condition ?? is that \(S_{\mathrm{NB}}\) can be written as a function of the conserved quantities of the reversible sector,

\[H_{1},H_{2},\dots,H_{N-1}.\] Indeed, if

\[S_{NB}=f(H_{1},\dots,H_{N-1})\] then \[\partial_{t}^{(H)}S_{NB}=\sum_{m}\frac{\partial f}{\partial H_{m}}\partial_{t}^{(H)}H_{m}=0,\] and the preservation condition follows. In the present piston model, \[S_{NB}=H_{2}-H_{1}/T_{b}\] so this sufficient condition is satisfied.

Near equilibrium, this condition reduces to the usual extremum condition for a free-energy-type thermodynamic potential under the imposed bath conditions.

4.3 Linearization of the Irreversible System and Crossover of Relaxation Modes↩︎

In Sec. 3.3, we showed that, in the adiabatic reversible limit, the motion near equilibrium appears as a purely oscillatory mode.

In this subsection, we examine by linearization how this oscillatory mode is damped in the irreversible system with a heat bath and friction, and how it competes with the thermal relaxation mode.

We denote the equilibrium point by \[(V_{*},p_{*},S_{*})=(V_{*},0,S_{*}),\] with the equilibrium conditions \[P(S_{*},V_{*})=P_{b},\qquad T(S_{*},V_{*})=T_{b}.\] We write small deviations from equilibrium as

\[V=V_{*}+\delta V,\qquad p=\delta p,\qquad S=S_{*}+\delta S.\] The full time evolution is

\[\dot{V}=\frac{A}{m}p,\] \[\dot{p}=A(P-P_{b})-\lambda p,\] and \[\dot{S}=\kappa\frac{T_{b}-T}{T}+\frac{\lambda p^{2}}{mT}.\]

Here, the \(p^{2}\) term is second order in the deviation from equilibrium and therefore does not contribute to the linearized dynamics.

Thus, the linearized fluctuation equation is \[\frac{d}{dt}\begin{pmatrix}\delta V\\ \delta p\\ \delta S \end{pmatrix}=M(\lambda)\begin{pmatrix}\delta V\\ \delta p\\ \delta S \end{pmatrix},\] where \[M(\lambda)=\begin{pmatrix}0 & A/m & 0\\ AP_{V}^{*} & -\lambda & AP_{S}^{*}\\ -\kappa T_{V}^{*}/T_{b} & 0 & -\kappa T_{S}^{*}/T_{b} \end{pmatrix},\] with \[P_{V}^{*}=\left.\frac{\partial P}{\partial V}\right|_{*},\quad P_{S}^{*}=\left.\frac{\partial P}{\partial S}\right|_{*},\quad T_{V}^{*}=\left.\frac{\partial T}{\partial V}\right|_{*},\quad T_{S}^{*}=\left.\frac{\partial T}{\partial S}\right|_{*}.\]

Let \(\mu_{j}(\lambda)\) denote the eigenvalues of this matrix. We define the corresponding decay rates by \[\Gamma_{j}(\lambda)=-\mathrm{Re}\,\mu_{j}(\lambda).\]

In the reversible limit without the heat bath,\(\lambda=\kappa=0\), the eigenvalues are purely imaginary, as shown in Sec. 3.3: \[\mu_{\pm}=\pm i\omega_{*}.\]

Thus, the time evolution near equilibrium is an undamped oscillation. For \(\lambda>0\), by contrast, the eigenvalues associated with this oscillatory mode acquire negative real parts and the motion becomes a damped oscillation.

Under weak thermal coupling, the mechanical oscillatory mode can be approximated by temporarily neglecting its coupling to the heat bath and fixing \(\delta S\). Then \[\frac{d}{dt}\begin{pmatrix}\delta V\\ \delta p \end{pmatrix}=M_{osc}(\lambda)\begin{pmatrix}\delta V\\ \delta p \end{pmatrix},\] where \[M_{osc}(\lambda)=\begin{pmatrix}0 & A/m\\ AP_{V}^{*} & -\lambda \end{pmatrix}.\] This gives \[\frac{d^{2}}{dt^{2}}\delta V+\lambda\frac{d}{dt}\delta V+\omega_{*}^{2}\delta V=0,\;\omega_{*}^{2}=-\frac{A^{2}}{m}P_{V}^{*}.\] The corresponding eigenvalues are

\[\mu_{\pm}=-\frac{\lambda}{2}\pm i\sqrt{\omega_{*}^{2}-\frac{\lambda^{2}}{4}}.\] Therefore, the decay rate of the oscillatory mode increases approximately as

\[\Gamma_{{\rm osc}}=-\mathrm{Re}\mu_{\pm}\simeq\frac{\lambda}{2}.\]

The estimate \(\Gamma_{\mathrm{osc}}\simeq\lambda/2\) is a weak-coupling and underdamped approximation obtained by neglecting the coupling to the thermal variable. The actual decay rates and the crossover point are determined by the eigenvalues of the full linearized matrix \(M(\lambda)\).

In the irreversible system with a heat bath, however, there exists a nonoscillatory thermal relaxation mode in addition to the oscillatory mechanical mode. This thermal mode describes the relaxation of the temperature toward the bath temperature. Its relaxation rate is mainly determined by the heat-exchange coefficient \(\kappa\), and depends only weakly on \(\lambda\).

For an ideal gas, we have \[S=C_{V}\ln T+R\ln V.\] At the equilibrium point, this gives \[T_{S}^{*}=\frac{T_{b}}{C_{V}},\qquad T_{V}^{*}=-\frac{RT_{b}}{C_{V}V_{*}},\label{eq:Tstar}\tag{3}\] and \[P_{S}^{*}=\frac{P_{b}}{C_{V}},\qquad P_{V}^{*}=-\frac{P_{b}}{V_{*}}\left(1+\frac{R}{C_{V}}\right).\label{eq:Pstar}\tag{4}\]

In the regime where the mechanical pressure equilibration is faster than the heat exchange with the bath, the slow thermal mode may be estimated by imposing the quasi-mechanical equilibrium condition

\[\delta P\simeq0.\]

This approximation should not be interpreted as the asymptotic limit \(\lambda\to\infty\). For extremely large friction, the piston displacement itself may become slow because the motion is overdamped. The approximation used here is instead intended for the intermediate-friction regime in which the pressure imbalance relaxes on a shorter time scale than the thermal exchange with the bath.

Since \[\delta P=P_{V}^{*}\delta V+P_{S}^{*}\delta S=0,\] we obtain \[\delta V=-\frac{P_{S}^{*}}{P_{V}^{*}}\delta S.\] On the other hand, the temperature fluctuation is \[\delta T=T_{V}^{*}\delta V+T_{S}^{*}\delta S.\] Thus, \[\delta T=\left(T_{S}^{*}-T_{V}^{*}\frac{P_{S}^{*}}{P_{V}^{*}}\right)\delta S.\] Using the ideal-gas expressions 3 4 , we find \[\frac{P_{S}^{*}}{P_{V}^{*}}=-\frac{V_{*}}{C_{V}+R}.\] Therefore \[\delta T=\left[\frac{T_{b}}{C_{V}}-\left(-\frac{RT_{b}}{C_{V}V_{*}}\right)\left(-\frac{V_{*}}{C_{V}+R}\right)\right]\delta S=\frac{T_{b}}{C_{V}+R}\delta S.\]

The linearized entropy equation is \[\frac{d}{dt}\delta S=-\frac{\kappa}{T_{b}}\delta T.\] Substituting the relation above gives \[\frac{d}{dt}\delta S=-\frac{\kappa}{C_{V}+R}\delta S.\]

Hence, the thermal relaxation rate is approximately \[\Gamma_{{\rm th}}\simeq\frac{\kappa}{C_{V}+R}.\] Equivalently, this is the relaxation rate determined by the constant-pressure heat capacity \[C_{P}=C_{V}+R.\]

Thus, even when \(\lambda\) is increased, the relaxation rate of this slow thermal mode changes only weakly as long as the heat-exchange coefficient \(\kappa\) is fixed.

The irreversible piston system therefore has two relevant relaxation modes. For small \(\lambda\), friction is weak, and the damping of the mechanical oscillatory mode limits the overall relaxation. As \(\lambda\) increases within the underdamped-to-intermediate regime considered here, the mechanical damping rate increases and can become faster than the thermal relaxation rate. In that case, the remaining slow relaxation is limited mainly by heat exchange with the bath. Thus, in the parameter range examined below, the relaxation process exhibits a crossover, \[\text{friction-limited}\quad\longrightarrow\quad\text{heat-exchange-limited}.\]

This crossover refers to a change of the rate-limiting relaxation mechanism, not to the large-\(\lambda\) asymptotic behavior of an overdamped mechanical oscillator.

For the numerical parameters used below, namely \(m=1.0,C_{V}=1.5,R=1.0,P_{b}=1.0,T_{b}=1.0,A=1\), and \(\kappa=0.20\), the thermal relaxation rate is estimated as \(\Gamma_{\mathrm{th}}\simeq\kappa/(CV+R)=0.08\). The crossover therefore occurs when the decay rate of the mechanical oscillatory mode becomes comparable to this thermal relaxation scale. This crossover should be distinguished from the critical damping of a purely mechanical oscillator; it refers instead to the exchange of the slowest relaxation mode between mechanical damping and thermal relaxation.

4.4 Connection to Near-Equilibrium Linear Response↩︎

Within the linearized full dynamics, the irreversible sector has an Onsager-type linear-response structure.

The linearized matrix \(M(\lambda)\) around the equilibrium point contains both the reversible oscillatory component originating from the Nambu bracket and the irreversible relaxation component generated by the dissipation potential \(S_{NB}\). Therefore, the full matrix \(M(\lambda)\) should not be identified with an Onsager matrix. The correspondence with Onsager-type linear nonequilibrium thermodynamics appears only in the irreversible sector of the full linearized dynamics: \[\frac{d}{dt}\left.\delta x\right|_{irr}=L_{*}\left.\nabla^{2}S_{NB}\right|_{*}\delta x.\] Here, \(L_{*}\) is the positive-semidefinite transport coefficient matrix evaluated at the equilibrium point, and the linearization of \(\nabla S_{NB}\) corresponds to the thermodynamic force. By contrast, the reversible oscillatory mode does not arise from the symmetric dissipative part of an Onsager matrix, but from the reversible sector generated by the Nambu bracket.

Although \(L^{ij}\) generally depends on the state variables, its derivative terms do not contribute to the linearization at equilibrium because

\[\left.\nabla S_{NB}\right|_{*}=0\]

Thus, the irreversible linearized dynamics takes the form given above.

5 Numerical Results↩︎

We numerically examine the dynamics of the piston system introduced above.

Fig. 3 shows the phase portrait in the \((V,p)\) plane, with the reversible and irreversible systems superposed. In the reversible system, the motion traces closed orbits, whereas in the irreversible system the trajectory spirals inward because of damping.

Fig. 4 shows the time evolution of the volume around the equilibrium point. With only the reversible part, the Nambu bracket generates persistent oscillations. When the irreversible part is added, these become naturally damped oscillations.

Fig. 5 provides the central numerical demonstration of the distinction addressed in this paper by comparing the time evolution of the thermodynamic entropy \(S\) and the dissipation potential \(S_{NB}\). Since the system entropy \(S\) belongs to an open subsystem, it is allowed to oscillate, including temporary decreases. By contrast, \(S_{NB}\), constructed so as to incorporate the mechanical energy of the system, increases monotonically. This supports its validity as an indicator of nonequilibrium evolution within the present framework.

Fig. 6 shows the \(\lambda\) dependence of the damping rate \(\gamma\) and the relaxation time. The damping rate is estimated from the exponential envelope of the peak amplitudes of \[\delta V=V-V_{*},\] where \(V_{*}\) is the equilibrium volume, namely \[\gamma=-\frac{d}{dt}\ln|\delta V_{\mathrm{peak}}|,\;|\delta V_{\mathrm{peak}}|\sim e^{-\gamma t}.\] The damping rate \(\gamma\) increases as \(\lambda\) increases.

By contrast, the relaxation time \(t\) is defined as the first time such that \[|\delta V|<\epsilon\] holds thereafter, with

\[\epsilon=10^{-3}V_{*}.\]

The numerical results confirm that the relaxation time decreases as \(\lambda\) increases. It is noteworthy, however, that there exists a region in which the damping rate becomes nearly flat and shows only weak dependence on \(\lambda\).

The relatively large uncertainty observed in the intermediate range of \(\lambda\) should not be interpreted as a transition to an overdamped regime. In the present model the friction force is linear in the momentum, and no static-friction or stick-slip mechanism is included. Rather, this uncertainty reflects the limited robustness of trajectory-based estimators such as the peak-envelope fit and the settling-time criterion. As \(\lambda\) increases, the number of clearly identifiable oscillation peaks decreases, and the long-time behavior becomes increasingly affected by the slow thermal relaxation mode. Therefore, the damping rate estimated from the peak envelope of \(\delta V\) should be regarded as a trajectory-based indicator, not as a direct eigenmode decay rate in the large-\(\lambda\) regime.

The crossover of relaxation mechanisms is more directly characterized by the eigenvalues of the linearized matrix. The decay rate of the mechanical oscillatory mode increases with \(\lambda\), whereas the thermal relaxation rate is mainly controlled by the heat-exchange coefficient \(\kappa\) and remains nearly constant. Consequently, the slowest relaxation mode changes from the friction-limited mechanical mode at small \(\lambda\) to the heat-exchange-limited thermal mode at larger \(\lambda\). This eigenvalue-based analysis provides the clearest interpretation of the saturation-like tendency observed in the settling-time data.

This behavior reflects the crossover of relaxation modes discussed in Sec. 4.3. As shown in Fig. 7, the \(\lambda\) dependence of the decay rates obtained from the linearized irreversible dynamics around equilibrium is nontrivial, and can be understood as the result of competition between two relaxation modes: mechanical damping and thermal relaxation. The decay rate of the oscillatory mechanical mode increases with \(\lambda\), whereas the decay rate of the thermal relaxation mode is determined mainly by the heat-exchange coefficient \(\kappa\) and depends only weakly on \(\lambda\). Consequently, the slowest relaxation mode crosses over from friction-limited mechanical relaxation to heat-exchange-limited thermal relaxation.

6 Discussion and Conclusion↩︎

6.1 Conclusions of This Study↩︎

In this paper, we used a piston system coupled to a pressure reservoir and a heat bath to clarify how NNET separates and describes the dynamics of a macroscopic nonequilibrium system. The reversible temporal order, namely oscillatory motion, is supported by the geometric structure of the Nambu bracket, whereas irreversible dissipation and relaxation are described as a gradient flow generated by the dissipation potential. This structural decomposition into rotational flow and gradient flow provides a natural way to describe far-from-equilibrium dynamics that is not easily captured by frameworks based primarily on gradient-flow descriptions, such as the maximum entropy production principle.

Another conclusion of this study is the explicit clarification of the relation between \(S_{NB}\) and the thermodynamic entropy \(S\). In the open piston model, the thermodynamic entropy \(S\) of the system alone can exhibit nonmonotonic time evolution through exchange with the heat bath. By contrast, in the explicit dissipative model constructed here, \(S_{NB}\) is not the thermodynamic entropy itself, but a dissipation potential that generates the irreversible part. Under the conditions stated in Proposition 1, it therefore becomes a monotonic quantity. Thus, \(S_{NB}\) should not in general be identified with the entropy \(S\), nor should it be regarded as a universal Lyapunov function. Rather, the monotonicity of \(S_{NB}\) is a conditional property that appears when its preservation by the reversible part and the positive-semidefinite gradient-flow structure of the irreversible part are simultaneously satisfied.

From the viewpoint of the broader NNET series, the piston model studied here serves as a benchmark model for considering local reductions and global obstructions that may appear in general far-from-equilibrium systems. In a general nonlinear system, Nambu Hamiltonians and dissipation potentials need not exist globally, and the dynamics may have to be described by patching together local charts. In contrast, in the present model, \(H_{1}\), \(H_{2}\), and \(S_{NB}\) are defined globally, and the conditions of Proposition 1 are explicitly satisfied. The model therefore provides a solvable reference point in which \(S_{NB}\) functions as a global Lyapunov function.

6.2 Discussion: Idealization of the Pressure Reservoir↩︎

In this paper, we idealized the pressure reservoir as a work reservoir that supplies a constant external pressure, and assumed that the pressure reservoir itself carries no entropy exchange, \[\dot{S}_{P}=0.\] Under this assumption, the conserved quantity of the reversible part closes as \(H_{1}\), and the second law can be traced unambiguously as the increase of the total entropy including the heat bath.

A more detailed modeling would be possible in which the pressure reservoir has a finite temperature and the work received by the reservoir is dissipated internally. In that case, however, the definition of the reservoir entropy may depend on how far one includes the thermalization of work inside the reservoir. For the purpose of the present paper, namely to visualize the distinct roles of the Nambu bracket and the dissipation potential, the idealization of the pressure reservoir as a work reservoir is minimal and transparent.

6.3 Application to General Open Systems Such as Chemical Reaction Networks↩︎

The framework presented in this paper can also be applied to chemical reaction networks that generate dissipative structures. For a constant-volume reaction system with state variables \[x=(c^{1},\dots,c^{M},S),\] conserved quantities such as elemental conservation laws and the total internal energy can be assigned to the Nambu Hamiltonians \(H_{i}\) . In this way, rotational cycles in the adiabatic reversible limit can be extracted as Nambu dynamics.

For the irreversible part, let \(\nu_{r}^{\alpha}\) be the stoichiometric coefficient of reaction \(r\), and let \(\mu_{\alpha}\) be the chemical potential of species \(\alpha\). The reaction affinity is then defined by

\[A_{r}=-\sum_{\alpha}\nu_{r}^{\alpha}\mu_{\alpha}.\] Near equilibrium, the reaction flux can be written in the Onsager form \[J_{r}=L_{rs}A_{s}.\] When the system is coupled to chemical reservoirs, or chemostats, one may introduce a Massieu-type generating potential that is analogous to the piston-system expression \(S_{NB}=S-H_{1}/T_{b}\) . Including the chemical potentials of the reservoirs, this potential takes the form \[S_{NB}=S-\frac{1}{T_{b}}(U-\sum\mu_{i}^{(b)}N_{i}),\] where \(\mu_{i}^{(b)}\) denotes the chemical potential of the reservoir. In NNET, the irreversible part is then represented as a gradient flow generated by this \(S_{NB}\).

Complex temporal order observed in nonlinear reaction systems, such as limit cycles in the Belousov--Zhabotinsky reaction, cannot be explained by simple gradient descent alone. From the viewpoint of NNET, the main component responsible for temporal order is the reversible part described by the Nambu bracket, while the irreversible part determines its amplitude, stability, and transitions. This geometric division of roles is the same as the structure demonstrated in the piston system studied in this paper.

However, in such strongly nonlinear reaction systems, the global conditions of Proposition 1 need not hold. In particular, a single \(S_{NB}\) need not function as a global Lyapunov function for the entire time evolution. Rather, through the coupling between reversible circulation and nonlinear dissipation, \(S_{NB}\) itself may exhibit nonmonotonic time evolution. The open piston is the simplest macroscopic thermodynamic system in which this separation can be tested explicitly.

6.4 Perspective on Connections to Modern Nonequilibrium Statistical Mechanics↩︎

The following discussion is not required for the macroscopic thermodynamic formulation developed above, but indicates possible connections to modern stochastic thermodynamics.

In recent years, stochastic thermodynamics has actively investigated the relation between entropy production and nonconservative probability currents in Langevin systems and Markov jump processes. The monotonic increase of the macroscopic dissipation potential \(S_{\mathrm{NB}}\) demonstrated in the present model can be understood as a structure corresponding to the monotonic decrease of relative entropy expressed by the Kullback--Leibler divergence, or to the positivity of nonadiabatic entropy production, in stochastic thermodynamics. Extensions toward fluctuating Nambu nonequilibrium thermodynamics, which incorporates microscopic fluctuations, and the role of the Nambu bracket in nonequilibrium steady states (NESSs) between multiple heat baths have been discussed in Ref.[7]. The macroscopic model studied in the present paper can be regarded as a deterministic limit of such a formulation. A systematic clarification of the relation between these descriptions remains an important subject for future work.

When the theory is extended to a stochastic description, \(S_{\mathrm{NB}}\)may also be related to Shannon entropy and self-information. For example, if a Nambu-type distribution of the form \[\phi\propto\exp\left[\frac{S_{\mathrm{NB}}}{\alpha k_{B}}\right]\] appears as a stationary distribution of Fokker--Planck type, then the self-information \(\log\phi\) corresponds to \(S_{\mathrm{NB}}/(\alpha k_{B})\), up to an additive normalization constant. This relation is an important issue in extending the deterministic macroscopic model treated in this paper to stochastic NNET.

6.5 Future work↩︎

Another natural extension is to replace the ideal-gas equation of state by an interacting-fluid equation of state, such as a van der Waals fluid. The Nambu formulation of the reversible sector and the bath-relative potential \(S_{NB}=S-H_{1}/T_{b}\) can be written formally for a general thermodynamic potential \(U(S,V)\). Therefore, the main geometric structure is not restricted to an ideal gas. However, the explicit linearized relaxation rates and the numerical examples presented in this paper use the ideal-gas equation of state. For interacting fluids, the derivatives of \(P(S,V)\) and \(T(S,V)\) would be modified, and additional issues such as metastability, phase coexistence, and possible spatial inhomogeneities near coexistence regions may have to be considered. A systematic analysis of such interacting-fluid extensions is left for future work.

Acknowledgments↩︎

The author would like to thank Akio Sugamoto and Yoshiki Matsuoka for their collaboration in the development of Nambu nonequilibrium thermodynamics and for valuable comments on the present work. The author is deeply grateful to Tatsuaki Wada for carefully reading the manuscript and for many helpful discussions. The author also thanks Shiro Komata for reading the manuscript and for constructive questions, in particular on the relation to Onsager theory and the meaning of the dissipation potential \(S_{\mathrm{NB}}\). The author is also grateful to the members of the Field Theory Seminar at the Open University of Japan for stimulating discussions.

References↩︎

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1. So.Katagiri@gmail.com↩︎

2. It should be noted that, in port-thermodynamic systems based on contact geometry and symplectization, a gas-piston-damper system has been described as homogeneous Hamiltonian/contact dynamics on an extended space \((q,S,E,p,p_{S},p_{E})\)[11]. In contrast, the present work describes the reversible part by the Nambu bracket and the irreversible part by the gradient flow of \(S_{NB}\) directly on the original macroscopic state variables\((V,p,S)\), thereby allowing a direct comparison between \(S_{NB}\) and the thermodynamic entropy \(S\).↩︎

3. In Heimburg’s formulation[16], isentropic oscillations are generated by the antisymmetric part of an Onsager-type linear phenomenological coefficient. In the terminology of the present paper, this corresponds to the near-equilibrium linearization of the reversible sector. Here, the reversible part is separated from the outset as a geometric rotational flow generated by the Nambu bracket, rather than being included as an antisymmetric component of the irreversible transport coefficient. Thus, adding Onsager--Casimir-type antisymmetric cross terms to the irreversible part would amount to double counting the reversible motion.↩︎

4. The present model should be understood as a lumped macroscopic description in which the gas is assumed to remain sufficiently close to local equilibrium and is represented only by the global variables \(V\), \(p\), and \(S\). Spatially inhomogeneous hydrodynamic modes, such as vortical flows, viscous boundary layers, acoustic transients, and temperature gradients inside the gas, are not included explicitly. This approximation is appropriate when these internal relaxation modes decay on time scales shorter than, or are otherwise coarse-grained over, the piston relaxation time considered here. If such modes become comparable to the piston time scale, the state space must be enlarged to include hydrodynamic fields, and the present minimal model should be regarded as the reduced slow-variable description.↩︎

5. Strictly speaking, the frictional heat may be distributed between the gas, the piston, and the external reservoir, depending on the microscopic realization of the contact. In the present minimal model, the thermal degrees of freedom of the piston are not included explicitly, and we assume that the kinetic energy dissipated by friction is effectively recovered as heat in the gas subsystem. If part of this heat is instead transferred directly to the heat bath, the bath temperature is still kept fixed in the ideal-reservoir limit; such a redistribution would correspond to a different coarse-grained heat partition and does not affect the geometric point addressed here.↩︎

6. The temperature appearing in this potential is the reservoir temperature \(T_{b}\), not the instantaneous system temperature \(T\). Thus \(S_{NB}\) should be understood as a bath-relative Massieu potential under the imposed external condition \(T_{b}=\mathrm{const}\)., rather than as an approximation obtained by replacing \(T\) with \(T_{b}\). The deviation of the system temperature from the bath temperature appears instead as the thermodynamic force \(\partial S_{NB}/\partial S=1-T/T_{b}\).↩︎