Exergetic Port-Hamiltonian Framework

• Exergetic Port-Hamiltonian Systems (EPHS) are a modeling paradigm that defines system dynamics using exergy, unifying energy conservation with entropy production.
• EPHS employs Dirac (power-preserving) and Onsager (dissipative) structures, along with bond-graph–inspired wiring diagrams, to enable modular, hierarchical modeling of multiphysical systems.
• EPHS guarantees thermodynamic consistency and is effectively applied to complex systems like fluid dynamics and multibody mechanics, supporting robust simulation and optimal control.

The Exergetic Port-Hamiltonian Systems (EPHS) framework provides a compositional, energy-based modeling paradigm for thermodynamic and multiphysical systems that encapsulates both the first and second laws of thermodynamics as structural invariants. EPHS extends classical port-Hamiltonian systems by choosing exergy—not total energy—as the Hamiltonian, together with Dirac (power-preserving) and Onsager (irreversible dissipative) structures. The framework features a categorical, bond-graph–inspired graphical syntax based on undirected wiring diagrams, enabling modular and hierarchical composition of lumped and distributed systems across diverse domains, including fluid dynamics, multibody mechanics, and multiphysics environments. By construction, EPHS guarantees thermodynamic consistency, compositionality, and the enforceability of conservation/dissipation principles, making it suitable for integration, optimization, and control of complex open systems (Lohmayer et al., 2022, Lohmayer et al., 2024, Lohmayer et al., 2024, Lohmayer et al., 2020, Lohmayer et al., 2022, Lohmayer et al., 2024).

1. Exergy and the EPHS Hamiltonian

In EPHS, the Hamiltonian is the system's exergy: the maximum useful work extractable as the system is brought reversibly to a reference environment (fixed temperature $\theta_0$, chemical potentials, etc.). For a generic system with internal energy $E(x)$, entropy $S(x)$, and other extensive variables (e.g., mass $M(x)$), the exergy functional takes the form

$H(x) = E(x) - \theta_0 S(x) - \mu_0 M(x)$

where $\theta_0$, $\mu_0$ are environmental references (Lohmayer et al., 2022, Lohmayer et al., 2020, Lohmayer et al., 2024). This choice ensures that both conservation of energy and non-negativity of exergy destruction (second law) are embedded into the system's mathematical structure, as dissipation manifests as exergy destruction. In the case of purely mechanical systems, exergy coincides with energy, but for thermodynamic systems, this construction is essential for encoding entropy production and irreversibility (Lohmayer et al., 2024).

2. Mathematical Structure and Laws of Thermodynamics

Formally, an EPHS is specified by a tuple $(X, H, J, R, G)$, where $X$ is the state space (often a Banach manifold of fields), $H$ the exergy functional, $J(x)$ a skew-symmetric (Dirac) structure, $R(x)$ a symmetric positive semi-definite (Onsager) structure, and $G(x)$ an external input map (Lohmayer et al., 2024, Lohmayer et al., 2024, Lohmayer et al., 2022). The evolution is governed by

$$\dot{x} = [J(x) - R(x)] \nabla H(x) + G(x) u,\quad y = G(x)^\top \nabla H(x)$$

• The reversible part $J(x) \nabla H(x)$ (Dirac structure) conserves exergy.
• The irreversible part $R(x) \nabla H(x)$ ensures non-negative exergy destruction.
• Ports $(u, y)$ connect the system to its environment or other subsystems.

The exergy balance law,

$$\frac{d}{dt}H(x) = y^\top u - (\nabla H)^\top R \nabla H,$$

guarantees:

• First Law: Reversible interconnections (skew $J$) preserve exergy.
• Second Law: Dissipations ($R \ge 0$) always decrease exergy unless $u$ injects it.

These structures embed Onsager symmetry, and the framework directly aligns with the GENERIC (General Equation for the Non-Equilibrium Reversible-Irreversible Coupling) formalism, providing a Poisson (Hamiltonian)–dissipation (gradient) splitting of the dynamics (Lohmayer et al., 2020, Lohmayer et al., 2024).

3. Graphical Syntax: Bond-Graph–Inspired and Wiring Diagrams

EPHS employs a graphical syntax rooted in the operad of undirected wiring diagrams. Primitive subsystems are modeled as boxes with typed ports (e.g., velocity–force, entropy–temperature), interconnected via junctions enforcing either equality of effort (potential) or conservation of flow (current) (Lohmayer et al., 2022, Lohmayer et al., 2024, Lohmayer et al., 2022). Bond-graph elements correspond to:

• C-elements: Storage (exergy: capacitive)
• I-elements: Inertial storage (momentum/kinetic)
• 0-junctions/1-junctions: Enforcement of effort/flow continuity
• R-elements: Dissipative Onsager structures, encoding irreversible processes

Each subsystem contributes a local state manifold, exergy function, Dirac/Onsager relations, and ports, and diagrams specify composition patterns. The semantics is functorial: hierarchical and modular composition of diagrams yields composition of physical models with preservation of exergy balancing and non-negative entropy production (Lohmayer et al., 2024, Lohmayer et al., 2022).

4. Compositional Modeling of Multiphysical Systems

Hierarchical (and recursive) composition is central: EPHS allows complex multiphysical systems to be constructed from nested, reusable components. For example, in fluid dynamics, models for kinetic energy, internal energy, thermal conduction, and viscosity are assembled via port interconnections, yielding the full Navier–Stokes–Fourier (NSF) evolution (Lohmayer et al., 2022, Lohmayer et al., 2024). Similarly, multibody dynamics models are built from primitive blocks—storage, reversible couplings (Dirac), irreversible couplings (Onsager), and environments—plainly reflecting which interfaces are conservative, which are dissipative, and ensuring correct energy/exergy accounting (Lohmayer et al., 2024).

The compositional paradigm supports encapsulation, parameterization, and polymorphic reuse: subsystems (e.g., ideal fluids) are embedded in more complex ones (e.g., NSF), which are then in turn subsystems of even more involved constructs (e.g., magneto-hydrodynamics, EMHD) (Lohmayer et al., 2024). This supports not only physical insight and modularity but also tractable model synthesis and extension.

5. Detailed Examples: Fluid and Multibody Dynamics

A canonical EPHS instance for the Navier–Stokes–Fourier fluid on a domain $\Omega$ comprises five primary interconnected subsystems:

1. Kinetic energy storage ($H_k$): $\upsilon$, $m$, Dirac structure $D_k$
2. Internal energy storage ($H_i$): $s$, $m$, Dirac structure $D_i$
3. Thermal conduction ($R_t$): Fourier's law, entropy production
4. Bulk viscosity ($R_b$) and Shear viscosity ($R_s$): Newtonian dissipation, symmetric Onsager structures

Global balances of mass, momentum, and entropy are encoded as algebraic consequences of the port interconnection rules (0- and 1-junctions and functorial semantics) (Lohmayer et al., 2022, Lohmayer et al., 2024). All irreversible mechanisms are represented as explicit Onsager blocks, and local entropy production rates or exergy destruction rates are non-negative by construction.

For multibody dynamics, assemblies of rigid bodies and joints exploit the same four EPHS primitives: storage, reversible (Dirac) coupling, Onsager dissipation, and environment. Joint constraints and frictional dissipation translate into Onsager-type blocks; conservation is automatic, and exergy lost to friction is captured in the entropy ports (Lohmayer et al., 2024).

6. Thermodynamic Guarantees, Numerical and Control Implications

In EPHS, thermodynamic laws are structural:

• Energy/exergy conservation is assured by Dirac interconnection (skew $J$).
• Nonnegative entropy production is guaranteed by the symmetric, positive-semidefinite Onasger structure $R$ (Lohmayer et al., 2024, Lohmayer et al., 2024).
• Onsager reciprocity for irreversible couplings: $R$ symmetric.
• Power balance at the port level: incoming/outgoing exergy flows match change of the internal exergy.

Numerical discretization of EPHS inherits these guarantees if primitive models are discretized with structure-preserving methods (e.g., discrete exterior calculus, discrete gradients). This ensures thermodynamically consistent time integration and avoids spurious energy or entropy artifacts (Lohmayer et al., 2022). Additionally, exergy-based optimal control formulations arise naturally: cost functionals combine work/exergy input, entropy production, or state tracking, with long-horizon turnpike properties leading to trajectories close to thermodynamic equilibria (Philipp et al., 2023).

7. Significance, Comparison with Other Modeling Paradigms, and Extensibility

EPHS extends classical port-Hamiltonian and bond-graph methodologies by constructing exergy as the central network “currency,” thereby unifying passivity, conservation, and the second law of thermodynamics. Unlike equation-based languages (e.g., Modelica), which admit arbitrary user-specified components and may inadvertently violate physical laws, EPHS restricts primitives to four geometrically defined block types. Modularity and type-checked composition are formalized via operads of undirected wiring diagrams, enhancing safety, scalability, and extensibility—particularly for cross-domain and control-oriented modeling (Lohmayer et al., 2024, Lohmayer et al., 2022).

A plausible implication is that EPHS provides an operational foundation for large-scale, robust, and thermodynamically consistent modeling of cyber-physical energy systems, climate control, electromechanical devices, and beyond, where structural preservation of conservation and dissipation is critical. Extensibility to irreversible, stochastic, and distributed-parameter systems is natural and has been demonstrated across multiple application domains (Lohmayer et al., 2020, Lohmayer et al., 2024, Lohmayer et al., 2022).