Energetic Variational Approaches

• Energetic variational approaches are a unified mathematical framework that derive evolution equations from free energy functionals, dissipation potentials, and kinematic constraints.
• They couple conservative forces via the least action principle with dissipative forces from Onsager’s principle to ensure full thermodynamic consistency.
• These methods are applied in multiphase flow, electrokinetics, reaction–diffusion, and active matter, and support structure-preserving numerical discretizations.

Energetic variational approaches (EVA, or EnVarA) comprise a unified mathematical and physical framework for the derivation, analysis, and numerical approximation of evolution equations in complex, multi-physics systems. The critical principle is that the structure and evolution of such systems are fully determined by a postulated energetic (free energy) functional, a dissipation potential, and thermodynamically consistent kinematic constraints. Systems spanning continuum mechanics, kinetic theory, electrokinetics, reaction–diffusion, multiphase flow, and active matter have been formulated and analyzed within the energetic variational paradigm.

1. Mathematical Foundation: Energy–Dissipation Law and Variational Principles

Energetic variational approaches are built upon two core variational principles:

• Least Action Principle (LAP): Determines conservative (reversible) forces by insisting that system trajectories are stationary points of an action functional, typically of the form

$$\mathcal{A} = \int_{t_0}^{t_1} [ \text{Kinetic Energy} - \text{Potential (Free) Energy} ]\, dt$$

Hamilton’s principle (δA = 0) yields Euler-Lagrange (evolution) equations for the flow or system configuration.

• Maximum Dissipation Principle (MDP, or Onsager Principle): Determines dissipative (irreversible) forces as the steepest-descent direction of a prescribed dissipation functional. Typically, the dissipation is quadratic in rates (e.g., velocities, reaction fluxes), yielding equations such as

$$\mathrm{Dissipation}[\mathsf{u}] = \frac{1}{2} \int \eta(\rho) |\mathsf{u}|^2\, dx$$

The system’s evolution is governed by the balance of conservative and dissipative forces.

The energy–dissipation law combines the first and second laws of thermodynamics in the general form

$$\frac{d}{dt}\, \mathcal{F}[\rho] = -2\, \mathcal{D}[\mathsf{u},\rho],$$

where $\mathcal{F}$ is the free energy and $\mathcal{D}$ is the dissipation functional, subject to kinematic constraints such as mass conservation $\rho_t + \operatorname{div}(\rho\,\mathsf{u})=0$ (Liu et al., 2019).

2. Generic EVA Workflow: From Energies to Partial Differential Equations

The standard EVA modeling procedure consists of the following steps (Brannick et al., 2014, Liu et al., 2017, Wang et al., 2020):

1. Selection of the Free Energy: Identify all relevant sources of internal, mechanical, chemical, and interface energy in the system. Examples include mixing (Ginzburg–Landau) energies for multiphase systems, Helmholtz free energy for fluids or reacting mixtures, elastic stored energy for solids, etc.
2. Prescription of Dissipation Functional: Assign a dissipation potential quadratic in rates (velocities, fluxes, reaction rates, etc.), consistent with the physical character of dissipative processes (e.g., viscosity, diffusion, drag, reaction friction).
3. Specification of Kinematic Constraints: Fix conservation laws (mass, charge, polymer length, etc.) or geometry (e.g., incompressibility, constraint on surface divergence).
4. Application of Variational Principles:
   • LAP: Vary the action to obtain the reversible (conservative/Hamiltonian) parts of the PDEs.
   • MDP: Vary the dissipation to obtain the dissipative (relaxational/gradient flow) parts.
5. Force–Balance Law: The total system is closed by enforcing

$$\text{(Conservative Forces)} + \text{(Dissipative Forces)} = 0$$

yielding a generally nonlinear, coupled PDE system.

3. Canonical Examples: Applications Across Fields

The breadth and flexibility of energetic variational approaches is illustrated by several canonical examples:

3.1. Multiphase Flow and Capillarity

Systems of phase fields (labeling variables) and hydrodynamics are governed by free energies of Ginzburg–Landau type (describing interfaces), with mixing energies incorporating coefficients reflecting surface tension and interfacial thicknesses. Viscous dissipation is classical, possibly phase-field dependent, and Onsager’s principle prescribes relaxational (Allen–Cahn or Cahn–Hilliard–type) dissipative terms (Brannick et al., 2014). Table 1 summarizes the main energetic components in diffuse-interface models for three-phase flow:

Component	Expression	Role
Free Energy	$$\int [\tfrac{1}{2}\rho |\mathsf{u}|^2 + \lambda W(\cdot)]dx$$	Kinetic + mixing
Mixing Energy $W$	$$W = \gamma_1[(\psi-1)/2]^2(\ldots) + \gamma_2(\ldots)$$	Interfacial tension
Dissipation	$$\int\mu |\nabla \mathsf{u}|^2 + \lambda M_i| \delta \mathcal{E}/\delta \phi|^2 dx$$	Viscosity + interface relax.

3.2. Ion Transport and Electrokinetics

The energetic variational framework naturally yields the coupled Poisson–Nernst–Planck–Navier–Stokes (PNP–NS) system:

• Energy: entropic (mixing), electrostatic, kinetic
• Dissipation: frictional drag (mathematically, mobility-weighted velocities)
• Kinematic constraint: mass conservation for each ion species, incompressibility for the fluid (Xu et al., 2014, Liu et al., 2017).

3.3. Reaction–Diffusion with Mass Action Kinetics

The mass-action reaction rates and their dynamics are not prescribed but emerge as the unique force–flux relations from the variational structure. For isothermal reactions, the equilibrium arises from the minimization of free energy under stoichiometric constraints, while the dynamical approach to equilibrium is determined by the chosen dissipation functional (Wang et al., 2020).

3.4. Complex Fluid Rheology and Micro–Macro Models

For non-Newtonian fluids or polymeric solutions, both the mechanical energy (kinetic/internal/macromolecular) and the configurational entropy of microstructure molecules (dumbbells or micelles) are included. The dissipation includes both macroscopic viscoelastic (generalized viscosity) and microscopic drag. Microscale closure via maximum-entropy arguments enables deriving nonlinear stress–strain relationships and capturing phenomena such as shear banding (Koba et al., 2017, Wang et al., 2021).

3.5. Porous Medium and Degenerate Diffusion Equations

EVA enables the derivation of Lagrangian trajectory equations for nonlinear degenerate parabolic PDEs, such as the porous medium equation. These models can be discretized into numerical schemes that guarantee conservation, positivity, and energy dissipation at the discrete level (Duan et al., 2018, Duan et al., 2020, Duan et al., 2019, Liu et al., 2019).

3.6. Active Matter and Chemo-Mechanical Systems

In active nematics, chemo-mechanical coupling between chemical reactions (e.g., ATP hydrolysis) and mechanical transport/order is incorporated into the energy and dissipation functionals, resulting in full thermodynamic consistency and capturing energy transduction in biological contexts (Wang, 29 Jun 2025).

4. Numerical and Discrete Variational Formulations

Energetic variational principles are not only theoretical but provide the foundation for structure-preserving numerical discretizations:

• Discrete energetic variational schemes: Discretize the flow map or order parameter, assemble discrete analogs of the energy and dissipation, and enforce force-balance at the discrete level (Liu et al., 2020, Liu et al., 2019).
• Convex splitting and optimization: Implicitly or semi-implicitly split convex and concave parts of the discrete energy, ensuring unconditional energy stability and unique solvability per time-step (often via damped Newton or variational minimization techniques).
• Preservation of invariants: Proper discretization preserves conservation laws, monotonic decay of discrete energy, and often allows the natural resolution of free boundaries, Dirac singularities, or waiting-time phenomena (Duan et al., 2018, Duan et al., 2018, Duan et al., 2019).

Energetic variational inference (EVI) illustrates the extension of EVA to Bayesian computation, where the Kullback–Leibler divergence plays the role of free energy and the variational optimization proceeds via particle-based dynamics consistent with an energy–dissipation law (Wang et al., 2020).

5. Thermodynamic and Structural Implications

Energetic variational approaches guarantee:

• Thermodynamic consistency: Derived evolution equations satisfy exact energy–dissipation (entropy) balances reflecting the first and second laws of thermodynamics at both continuous and discrete levels (Liu et al., 2017, Wang, 29 Jun 2025).
• Emergence of invariants: The variational structure can produce additional invariants (e.g., cross-helicity in kinetic–MHD models) not present in ad hoc schemes (Close et al., 2018).
• Reduction to classical models: Upon appropriate limiting procedures (e.g., no dissipation, sharp interface limits, vanishing viscosity), EVA-derived systems recover classical models such as inviscid Euler, parabolic reaction–diffusion, Navier–Stokes, and Maxwell equations (Koba, 2022, Koba, 2022).

6. Extensions, Limitations, and Ongoing Developments

Energetic variational approaches are under active development with several recognized directions:

• Multiscale and coarse-grained modeling: Incorporation of molecular-scale free energies and dissipation into continuum PDEs, often requiring phase-field or level-set methods for interface evolution, or maximum-entropy closures for complex fluids (Wei et al., 2016, Wang et al., 2021).
• Surface and interfacial phenomena: Extensions to account for moving interfaces with mass, surface flows, and curvature-driven capillarity or phase transition (Koba, 2022, Koba, 2022).
• Thermo-chemo-mechanical coupling: Non-isothermal, chemically active, and active-matter extensions—for example, ATP-driven nematics and non-isothermal electrokinetics—are naturally accommodated provided one augments the energetic and dissipative functionals accordingly (Liu et al., 2017, Wang, 29 Jun 2025).
• Rigorous convergence and analysis: Convexity properties of the discrete energy often ensure unique solvability and facilitate optimal error analysis, while in higher dimensions or strongly nonlinear settings, regularity and well-posedness remain nontrivial (Duan et al., 2019, Liu et al., 2019).

7. Significance and Research Impact

Energetic variational approaches have profoundly influenced modern modeling and simulation of continuum and discrete systems. Key contributions include:

• Unifying disparate physical models under a single variational-dissipative framework
• Clarifying connections and distinctions among equilibrium minimization, gradient flows, and generalized Onsager systems
• Enabling robust structure-preserving numerical discretizations applicable to degenerate, nonlinear, and interface-dominated dynamics
• Revealing hidden invariants, guaranteeing compliance with fundamental thermodynamic laws, and providing extensibility to new domains (active matter, non-isothermal flows, coarse-grained closures).

These methods underpin a substantial body of contemporary research in mathematical physics, applied analysis, computational mathematics, and engineering, with ongoing efforts to generalize the approach to ever more complex, multiscale, and strongly coupled systems (Brannick et al., 2014, Liu et al., 2017, Koba et al., 2017, Koba, 2022, Koba, 2022, Wang et al., 2020, Wang et al., 2021, Wang, 29 Jun 2025).