Onsager's Variational Principle

• Onsager's Variational Principle is a framework that unifies conservative and dissipative processes by minimizing the sum of free energy change and dissipation.
• It is applied in phase-field, diffuse-interface, and hydrodynamic models to ensure thermodynamic consistency and accurate energy decay in multiphase systems.
• The principle guides the development of numerical methods that preserve energy stability and robust simulation of complex fluid dynamics.

Onsager's Variational Principle is a universal framework for generating thermodynamically consistent equations governing the coupled evolution of conservative and dissipative processes in complex fluids, soft condensed matter, and multiphase systems. Crucially, Onsager's principle prescribes both the reversible (energy-conserving) and irreversible (dissipative) contributions to the dynamics by minimizing the sum of the rate of change of the free energy and the instantaneous dissipation functional. It thus recovers, unifies, and generalizes many existing models for hydrodynamics, phase transitions, and interfacial phenomena.

1. Fundamental Formulation

In Onsager's framework, the dynamics of a system with state variables $\mathbf{q} = (q_1, \dots, q_n)$ are determined by specifying:

• A free energy functional $F[\mathbf{q}]$, typically incorporating bulk, interfacial, or elastic contributions.
• A quadratic dissipation functional $\Phi[\dot{\mathbf{q}}]$, encoding the irreversible response (e.g., viscosity, mobility, friction).

The principle states that the time evolution $\dot{\mathbf{q}}$ is the minimizer of the functional

$$\mathcal{R}[\dot{\mathbf{q}}] = \frac{d}{dt}F[\mathbf{q}] + \Phi[\dot{\mathbf{q}}]$$

with respect to $\dot{\mathbf{q}}$, subject to constraints (e.g., incompressibility, mass conservation). This guarantees that the resulting PDE system is consistent with the Second Law—the total energy decays monotonically, and entropy production is nonnegative.

The application of Onsager's principle thus produces, for each variable,

• Conservative (Euler-Lagrange) terms from the variational derivative $\delta F/\delta q_j$,
• Dissipative (relaxation/transport) terms from $\delta\Phi/\delta \dot{q}_j$.

This approach underpins a wide range of phase-field, Cahn–Hilliard, and hydrodynamic models in contemporary multiphase-flow research, including those derived via energetic variational approaches (Brannick et al., 2014).

2. Onsager Principle in Diffuse-Interface and Hydrodynamic Models

Energetic variational principles rooted in Onsager's formulation are explicit in the construction of diffuse-interface models (DIMs) for multiphase flow, where complex interface and bulk physics must be captured consistently.

For binary or multicomponent fluid mixtures (e.g., NSCH, Cahn–Hilliard–Navier–Stokes systems (Abels et al., 2012, Demont et al., 2022, Garcke et al., 2016, Brannick et al., 2014)), the standard procedure is:

• Free energy: $F[\phi, \mathbf{u}]$ is typically the sum of kinetic energy, interfacial energy (e.g., Ginzburg–Landau double-well plus gradient terms), possibly generalized to several phase fields for multicomponent systems.
• Dissipation functional: $\Phi$ is quadratic in the rates (shear, diffusion, interface mobility), with, e.g.,

$$\Phi = \int_\Omega [2\eta(\phi) D(\mathbf{u}):D(\mathbf{u}) + m(\phi) |\nabla\mu|^2]\,dx$$

where $D(\mathbf{u})$ is the rate-of-strain tensor, $\mu$ is the chemical potential, and $m(\phi)$ the (possibly degenerate) mobility.

The ensuing evolution equations are:

• Momentum equation (from virtual work/least action): Navier–Stokes with capillary (Korteweg) force from $\delta F/\delta \mathbf{u}$, as in

$$\rho(\phi)\left(\partial_t\mathbf{u} + \mathbf{u}\cdot\nabla\mathbf{u}\right) = -\nabla p + \nabla\cdot[\eta(\phi) (\nabla\mathbf{u} + \nabla\mathbf{u}^T)] + \mu\,\nabla\phi$$

• Order parameter evolution: Cahn–Hilliard (or Allen–Cahn) equation,

$$\partial_t\phi + \mathbf{u}\cdot\nabla\phi = \nabla\cdot(m(\phi)\nabla\mu)$$

with $\mu = \delta F/\delta\phi$.

• Energy law: The total energy decays precisely as dictated by the structure of $\Phi$ (see also [(Abels et al., 2012), Eq. (energy law)]).

3. Energetic Variational and Onsager-Principle-Based Approaches in Complex Multiphase Systems

For multiphase and multi-material systems (solid-solid, solid-fluid, reactive-fluid) the Onsager principle extends naturally:

• The free energy may include contributions from multiple fields: e.g., several phase-fields for different phases, strain energy for solids, mixing or mixing–entropy terms, wetting energies (Brannick et al., 2014, Zhan et al., 2023).
• The dissipation functional captures viscous, plastic, chemical-relaxation, and interface-mobility effects.

For example, in the energetic variational derivation for multiphase flow (Brannick et al., 2014), conservative dynamics are encoded via the least-action principle (virtual work), while dissipation is imposed via Onsager's minimum dissipation principle. The resulting coupled system in the three-phase case,

$$\begin{aligned} \rho(\mathbf{u}_t + \mathbf{u}\cdot\nabla\mathbf{u}) + \nabla p &= \lambda \nabla\cdot(\sigma^e + \sigma^v) \ \phi_t + \mathbf{u}\cdot\nabla\phi &= -M_1 \frac{\delta E}{\delta\phi} \ \psi_t + \mathbf{u}\cdot\nabla\psi &= -M_2 \frac{\delta E}{\delta\psi} \end{aligned}$$

arises by minimizing the Onsagerian Rayleighian functional, integrating both variational and dissipative contributions (Brannick et al., 2014).

In more general settings—multiphysics, solid–fluid coupling, or reaction–diffusion–mechanics—the methodology is systematic. Each physical mechanism (elasticity, viscosity, diffusion, phase change, etc.) is represented either in $F$ or $\Phi$. For instance, in phase-field models of solidification, tumor growth, or reactive-fluid blends, Onsager’s principle governs the construction of all constitutive and evolution equations (Dai et al., 2015, Wallis et al., 2020).

4. Sharp-Interface Limits and the Role of Mobility Scaling

Onsager's principle constrains not only the evolution equations but the allowed limiting behavior under vanishing interfacial thickness $\varepsilon\to0$. For diffuse-interface models, the detailed scaling of the mobility (within $\Phi$) strongly influences the sharp-interface limit (Abels et al., 2012, Abels et al., 2010):

• Moderate (nonvanishing or weakly vanishing) mobility: The limiting equations yield classical two-phase hydrodynamics (Navier–Stokes or Cahn–Hilliard–Navier–Stokes), with sharp interfaces obeying the Young–Laplace law and, when appropriate, a Mullins–Sekerka or similar kinetic relation (e.g., (Abels et al., 2012), Theorem 1).
• Strongly vanishing mobility: The irreversible transport disappears in the limit; sharp-interface dynamics degenerate to pure advection, but pressure jumps across the interface may no longer satisfy the classical Young–Laplace law, and a “discrepancy measure” quantifies the departure from equilibrium capillarity (Abels et al., 2012).

This dichotomy reflects Onsager's requirement that the dissipative structure (encoded in $\Phi$) persists at the limiting scale for thermodynamic consistency of the sharp-interface model.

5. Onsager Principle in Variational and Numerical Methods

Numerical and variational methods for phase-field/DIM systems incorporate Onsager’s principle to preserve structure discretely:

• Time discretization schemes (e.g., convex–concave splitting, stabilized semi-implicit methods) enforce a discrete version of the energy law, with dissipation matching that of the continuous system. For example, in (Garcke et al., 2016), the two-step BDF scheme for the NSCH system rigorously inherits energy-stability from the Onsager formulation.
• Galerkin and finite-element discretizations are constructed to respect Onsager-variational structure at the discrete level, permitting convergence analyses and guaranteeing thermodynamic consistency (Demont et al., 2022, Nochetto et al., 2011).

Partitioned solvers and adaptive mesh refinement algorithms often build on this structure, maintaining conservation and dissipation properties despite algebraic and spatial approximation (Demont et al., 2022). In control and optimization settings, the adjoint systems are derived variationally within the Onsager framework (Garcke et al., 2016).

6. Practical Implications and Extensions

The universality of Onsager's principle allows existing models to be systematically extended:

• To multi-phase, multi-component, or multi-field systems, simply add corresponding variables to $F$ and $\Phi$.
• To incorporate additional physics (elasticity, electrochemistry, wetting, anisotropic effects), add energy or dissipation terms as prescribed by the physical mechanism, preserving the variational–dissipative balance.

Notably, Onsager's principle remains valid for degenerate mobilities and singular potentials (see e.g., singular potentials in (Dai et al., 2015, Frigeri et al., 2013)), provided dissipation remains nonnegative and the regularity sufficient for variational derivatives to be defined.

In summary, Onsager's Variational Principle is a foundational theoretical tool for constructing, analyzing, and discretizing thermodynamically consistent continuum models of complex multiphase dynamical systems, as rigorously elaborated and applied in up-to-date arXiv literature (Abels et al., 2012, Brannick et al., 2014, Demont et al., 2022, Garcke et al., 2016).