Energetic Variational Approach (EnVarA)

• EnVarA is a thermodynamically structured approach that derives governing PDEs by balancing energy variations and dissipation under kinematic constraints.
• It combines the Least Action Principle for conservative dynamics and Maximum Dissipation Principle for irreversible processes to state force balances.
• EnVarA has broad applications including ion transport, electrokinetics, multiphase flows, and biomolecular modeling, and supports structure-preserving numerical schemes.

Searching arXiv for recent EnVarA-related papers to support the article. {"query":"all: \"Energetic Variational Approach\" OR all: EnVarA", "max_results": 10, "sort_by": "submittedDate", "sort_order": "descending"} Energetic Variational Approach (EnVarA) is a thermodynamically structured methodology for deriving governing equations of dissipative systems from an energy-dissipation law. In the formulations represented here, the conservative part is obtained from the Least Action Principle (LAP), the dissipative part from the Maximum Dissipation Principle (MDP), or Onsager principle, and the final PDE system from a force balance constrained by the system kinematics, such as continuity equations, incompressibility, flow maps, or interfacial transport laws (Xu et al., 2014). Across the recent literature, this architecture is used for ion transport, non-isothermal electrokinetics, multiphase flow, phase-field dynamics, porous-medium diffusion, reaction-diffusion, electromechanics, non-Newtonian fluids, biomolecular modeling, and structure-preserving numerics (Liu et al., 2017).

1. Variational architecture

A standard EnVarA formulation begins from the thermodynamic statement

$\frac{d}{dt}E^{total}=-\Delta,$

with $E^{total}=\mathcal K+\mathcal F$, where $\mathcal K$ is kinetic energy, $\mathcal F$ is free energy, and $\Delta\ge 0$ is dissipation. LAP is then applied to an action such as

$$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$

while MDP is applied to the dissipation functional, producing conservative and dissipative forces whose balance determines the PDEs (Xu et al., 2014). In reduced or overdamped settings, kinetic energy may be omitted and the action becomes the negative time integral of free energy, but the underlying logic remains the same: energy specifies reversible structure, dissipation specifies irreversible structure, and the kinematic constraints determine how state variables are transported (Cheng et al., 2019).

This architecture is not restricted to isothermal gradient flows. In non-isothermal electrokinetics, the same program is extended so that the First Law and Second Law are enforced simultaneously: conservative forces come from variation of the action, dissipative forces from variation of entropy production, and a temperature equation is derived together with the mechanical equations, with Fourier’s law $j=-k\nabla T$ appearing as part of the closure (Liu et al., 2017). A common simplification is to say that EnVarA is “just free-energy minimization.” The surveyed literature does not support that simplification. In its standard usage, EnVarA is a force-balance framework coupling conservative variation, dissipative variation, and explicit kinematics.

The kinematic part is decisive. In transport problems, densities satisfy continuity laws. In Lagrangian phase-field reformulations, the order parameter can be made constant along trajectories, so that the dynamics are transferred to the flow map rather than the field itself. In chemical reaction systems, the role of configuration is played by the reaction trajectory or extent of reaction $R(t)$, with concentrations expressed through stoichiometric constraints. This suggests that EnVarA is best regarded not as a single constitutive model but as a derivational schema: choose energy, choose dissipation, choose kinematics, then impose force balance (Wang et al., 2020).

2. Energies, dissipation functionals, and constitutive meaning

The central constitutive objects in EnVarA are the free energy and the dissipation. Their concrete form varies widely by application. In ion transport, one canonical free energy is

$$E^{total} =\int_{\Omega}\frac{\rho}{2}|\mathbf{u}|^2\,dx +\int_{\Omega}\left[ K_B T\left(n\ln \frac{n}{n_\infty}+p\ln\frac{p}{p_\infty}\right) +\frac{\varepsilon}{2}|\nabla \phi|^2 \right]dx,$$

with dissipation

$$\Delta=\int_{\Omega}\left[ \frac{K_B T}{D_n}n|\mathbf u_n-\mathbf u|^2 +\frac{K_B T}{D_p}p|\mathbf u_p-\mathbf u|^2 +\eta |\nabla \mathbf u|^2 \right]dx,$$

so that entropy, electrostatics, ionic drag, and solvent viscosity enter the derivation in explicitly separated roles (Xu et al., 2014). In generalized diffusion, the free energy may take the entropy-plus-potential form

$E^{total}=\mathcal K+\mathcal F$0

with dissipation

$E^{total}=\mathcal K+\mathcal F$1

which leads to a fluctuation-dissipation-compatible Fokker–Planck equation (Lu et al., 2024).

In reaction systems, the literature makes a particularly explicit distinction between equilibrium and kinetics. For the reaction $E^{total}=\mathcal K+\mathcal F$2, the free energy

$E^{total}=\mathcal K+\mathcal F$3

determines the equilibrium constant, while the dissipation functional determines the reaction law. One dissipation recovers the classical law of mass action, while a quadratic Onsager dissipation yields a logarithmic rate law. The paper’s explicit formulation is that free energy determines equilibrium and dissipation determines dynamics (Wang et al., 2020). This is a characteristic EnVarA statement: identical equilibrium states can coexist with distinct nonequilibrium kinetics if the dissipation changes.

Electromechanical formulations make the same point in constitutive form. In polarization theory, the electric free energy is written as

$E^{total}=\mathcal K+\mathcal F$4

and the electric displacement is defined variationally by

$E^{total}=\mathcal K+\mathcal F$5

Classical dielectrics arise only for the quadratic choice $E^{total}=\mathcal K+\mathcal F$6, so dielectric constants appear as constitutive outputs rather than primitive assumptions (Wang et al., 2021).

In generalized non-Newtonian fluid systems, the constitutive content is encoded by scalar functions $E^{total}=\mathcal K+\mathcal F$7 of the invariants of strain, divergence, vorticity, thermal gradient, and concentration gradient: $E^{total}=\mathcal K+\mathcal F$8 together with

$E^{total}=\mathcal K+\mathcal F$9

EnVarA then converts these scalar energetic choices into bulk stresses and fluxes (Koba et al., 2017).

3. Electrokinetics as a canonical EnVarA application

Electrokinetics is one of the most developed EnVarA domains. A representative model is the coupled Poisson–Nernst–Planck–Navier–Stokes system for two ionic species,

$\mathcal K$0

derived by combining the free energy, the ionic and viscous dissipation, and the incompressibility and species-transport kinematics (Xu et al., 2014). The resulting energy law states that the chosen total energy is dissipated exactly by ionic drag and solvent viscosity. In this literature, the electric body force in the fluid momentum equation is not inserted phenomenologically; it appears as the reaction counterpart of the electrochemical transport law.

A second major development is the non-isothermal extension. In that setting, the state variables include species densities, velocities, electrostatic potential, and temperature $\mathcal K$1, with free energy built from local thermodynamic densities $\mathcal K$2, Coulomb interaction, and external electric potential. The heat equation contains conduction, drag heating, bulk viscous heating, shear viscous heating, and thermo-pressure work, while the mechanical equations are derived from the same variational framework (Liu et al., 2017). An explicit caution from that literature is that, once $\mathcal K$3 is dynamic, the system is generally not equivalent to a simple gradient flow in chemical potentials. The correct balances are total energy conservation and entropy production, not merely free-energy decay.

That non-isothermal program has been specialized to electric double-layer capacitors. In a one-dimensional EDLC model with steric effects, the free energy combines Coulomb interaction with lattice-gas entropy, and EnVarA yields modified non-isothermal Nernst–Planck equations together with a temperature equation whose source terms are thermal-pressure work and electrostatic interaction work. The model predicts temperature oscillation during charging and discharging, captures both reversible and irreversible heat generation, and produces delayed thermal-electrokinetic response and hysteresis at fast scan rates (Ji et al., 2022).

Electrokinetics has also driven major advances in discrete EnVarA. For the Poisson–Nernst–Planck system, a second-order accurate finite-difference scheme has been constructed from the EnVarA reformulation as a non-constant mobility $\mathcal K$4 gradient flow. Its continuum free energy is

$\mathcal K$5

with chemical potentials and dissipation law derived variationally. The fully discrete method preserves positivity, unique solvability, energy stability, and optimal $\mathcal K$6 convergence, and each time step can be written as minimization of a strictly convex discrete functional (Liu et al., 2022).

4. Interfaces, geometry, and multiphase systems

Interfacial geometry is a recurrent EnVarA theme. In biomolecular modeling, the energetic variational principle is used to organize structure, function, and dynamics around essential energetic components and interface-associated free energies. The supplied abstract identifies solute–solvent interfaces, molecular binding interfaces, lipid domain interfaces, and membrane surfaces, and emphasizes the inclusion of interface geometry, particularly curvatures, in the parametrization of free energies. The same abstract cites applications to biomolecular electrostatics and solvation with curvature energy of the molecular surface, lipid membrane microdomain formation, and mean-curvature-driven protein localization, and it notes that phase-field representations can reduce degrees of freedom and computational complexity (Wei et al., 2016). This supports a broad view of EnVarA as a framework for geometry-sensitive biomolecular continuum models.

Diffuse-interface multiphase systems provide a more explicit PDE realization. For three materials described by two phase fields $\mathcal K$7 and $\mathcal K$8, the mixing energy

$\mathcal K$9

with

$\mathcal F$0

produces, via the principle of virtual work, the reversible stress

$\mathcal F$1

while Allen–Cahn relaxations provide the dissipative part. The full system satisfies a total energy dissipation law and reproduces three-phase contact-line balance, bubble penetration through a fluid–fluid interface, and slip-induced effects in contact-line motion and pinch-off (Brannick et al., 2014).

EnVarA also supports sharp-interface models with phase transition. In a compressible–incompressible two-phase system, the interface $\mathcal F$2 is treated as a mass-carrying object with constant surface density $\mathcal F$3 and constant surface tension $\mathcal F$4. Variation of the interface work term yields the capillary force $\mathcal F$5, and total mass conservation, now including interface mass, determines bulk source terms that represent evaporation and condensation. This produces interface conditions of Laplace–Young type together with nonstandard density equations for the two bulks (Koba, 2022).

A complementary line replaces Eulerian phase evolution by flow-map dynamics. For Allen–Cahn,

$\mathcal F$6

the transport relation

$\mathcal F$7

freezes $\mathcal F$8 in Lagrangian coordinates and transfers the dynamics to the trajectory equation. In one dimension this leads to an interface-capturing method in which thin interfaces are resolved by mesh concentration rather than Eulerian mesh refinement, and both first- and second-order time schemes are shown to be energy dissipative and uniquely solvable (Cheng et al., 2019).

5. Lagrangian diffusion, reaction coordinates, and generalized transport

A large EnVarA subliterature reformulates degenerate transport equations as trajectory problems. For the porous medium equation

$\mathcal F$9

one introduces a particle map $\Delta\ge 0$0 with

$\Delta\ge 0$1

Using the energy

$\Delta\ge 0$2

and the corresponding dissipation, EnVarA yields the trajectory equation

$\Delta\ge 0$3

which underlies first-order and second-order structure-preserving schemes. The second-order method is proved to be uniquely solvable and unconditionally energy stable, with optimal error

$\Delta\ge 0$4

and the analysis relies on higher-order asymptotic expansion, rough and refined error estimates, and discrete $\Delta\ge 0$5 control of the nonlinear terms (Duan et al., 2020). A closely related paper emphasizes that distinct dissipative energy laws can generate distinct trajectory formulations for the same PME, including one based on $\Delta\ge 0$6 and another based on $\Delta\ge 0$7; both preserve discrete dissipation and handle free boundaries and waiting time naturally, while the second is more efficient because it is linear (Duan et al., 2018).

This trajectory viewpoint extends beyond PME. A discrete energetic variational approach has been formulated for porous-medium-type generalized diffusion equations at the semidiscrete level, with the flow map as the primary unknown. The resulting minimization-based backward-Euler update preserves mass, positivity through the admissible deformation constraint $\Delta\ge 0$8, and a discrete energy law, and it has been used to capture free boundaries and waiting times in one and two dimensions (Liu et al., 2019).

In population genetics, the Wright–Fisher random genetic drift equation is rewritten as a trajectory system derived from the free energy

$\Delta\ge 0$9

and dissipation

$$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$0

The associated convex-splitting scheme is uniquely solvable on an admissible convex set, dissipates the discrete energy, and captures fixation-induced Dirac delta singularities at the boundaries close to machine precision on an equidistant reference grid (Duan et al., 2018).

Reaction systems require a different kinematic variable. For reversible mass-action kinetics, the reaction trajectory $$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$1 is introduced through

$$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$2

and the free energy determines equilibrium while the dissipation determines the kinetic law. This same framework extends to reaction-diffusion, where the transport law

$$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$3

and the reaction force

$$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$4

are derived from one common energy-dissipation structure (Wang et al., 2020).

More recently, EnVarA has been used as an inverse-modeling principle. For generalized diffusion with

$$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$5

law discovery is posed not by direct PDE residual fitting but by minimizing the mismatch in the energy-dissipation law. This yields learning procedures for $$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$6 or $$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$7 from either density data or particle data, with the EnVarA formulation used as a thermodynamic inductive bias (Lu et al., 2024).

6. Discrete EnVarA, scope, and interpretive issues

One of the most persistent themes in the literature is that EnVarA is as much a numerical design principle as a continuum derivation principle. Discrete chemical potentials, weighted inverse-mobility operators, convex incremental energies, and minimizing-movement updates are all discrete analogues of the continuous force-balance structure. In the PNP setting this leads to discrete positivity, unique solvability, exact nonlinear entropy balance, and optimal convergence (Liu et al., 2022). In PME and generalized diffusion it yields minimization problems on admissible deformation sets, discrete free-energy dissipation, and natural handling of support motion (Liu et al., 2019). In genetic drift it produces convex-splitting schemes that preserve mass and capture singular fixation states (Duan et al., 2018).

A second interpretive issue concerns representational nonuniqueness. The porous-medium literature explicitly states that different dissipative energy laws can lead to different trajectory equations for the same Eulerian PDE (Duan et al., 2018). This suggests that EnVarA is not merely a formal restatement of a given PDE; the chosen energy-dissipation split affects constitutive interpretation, numerical stability, and sometimes the most natural state variables.

A third issue is the status of gradient-flow language. Many EnVarA systems can be written as gradient flows, but the literature summarized here also emphasizes situations where that description is incomplete. Non-isothermal electrokinetics is the clearest example: when temperature evolves, the equations are not generally equivalent to

$$A=\int_0^{t^*}\int_\Omega (\mathcal K-\mathcal F)\,dx\,dt,$$8

and total energy conservation plus entropy production replaces a single isothermal free-energy dissipation law as the governing thermodynamic principle (Liu et al., 2017).

Finally, the phrase “energetic variational principle” is used at different levels of specificity. The biomolecular modeling paper clearly aligns with the energy-based variational tradition and explicitly discusses energetic variational principles, interfaces, curvature, and phase-field reduction, but the supplied record gives only abstract-level detail for that work (Wei et al., 2016). A careful reading therefore distinguishes broad energy-based variational modeling from the stricter LAP-plus-MDP force-balance formulation documented explicitly in the electrokinetic, porous-medium, phase-field, reaction, and electromechanical papers cited above.

Within that range, EnVarA functions as a unifying methodology for continuum model construction, geometric reformulation, thermodynamic closure, and structure-preserving computation. Its characteristic claim, repeated across these arXiv contributions, is that physically meaningful choices of energy, dissipation, and kinematics can determine coupled PDEs, interface laws, and numerical schemes in a systematic and thermodynamically consistent way (Koba et al., 2017).