Fourth-Order DLSS Equation

• Fourth-Order DLSS equation is a nonlinear, fourth-order parabolic PDE derived from chemical reaction networks, capturing both fast diffusion and porous medium dynamics.
• The equation connects discrete microscopic jump processes on periodic lattices to macroscopic gradient flows driven by entropy and nonlinear mobility.
• Its analysis unifies entropy principles with nonlinearity in mobility, revealing distinct regimes where solution behavior shifts between algebraic tails and compact support.

The fourth-order DLSS equation is a nonlinear, fourth-order parabolic evolution equation with origins in statistical physics (interface fluctuations), kinetic theory, and quantum drift-diffusion. Recent work rigorously derives this equation and a broad class of its nonlinear-mobility generalizations from reversible microscopic chemical reaction networks on discrete periodic lattices, connecting microscopic jump processes and macroscopic gradient-flow structures driven by entropy with generalized diffusive transport and nonlinear mobility (Mielke et al., 8 Oct 2025). The equation admits a rich spectrum of behaviors, with variants exhibiting features of both fast diffusion and porous medium equations depending on the parameterization of the mobility.

1. Microscopic Chemical Reaction Network Derivation

The approach models the density ρ as emerging from concentrations $c_k$ indexed on a discretized circle (periodic lattice, the discrete torus). The core reaction process involves simultaneous hopping of particle pairs: pairs occupying the same site transition to the two nearest neighbors; conversely, pairs at adjacent sites can reconvene at the original site. The rates are modulated by a symmetric, positively homogeneous function $\sigma_\alpha$, characterized by the parameter α (homogeneity degree α – 2, yielding overall flux homogeneity α).

The discrete net flux at site k is

$$J_{\alpha, k} = \sigma_\alpha(c_{k-1}, c_k, c_{k+1})\, (c_k^2 - c_{k-1}c_{k+1})$$

with time evolution given by the discrete continuity equation:

$$\dot{c}_k = N^2 [ J_{\alpha, k-1} - 2J_{\alpha, k} + J_{\alpha, k+1} ]$$

where $N$ is the number of lattice points (mesh size δ = 1/N).

2. Discrete Rate Equation, Entropy, and Gradient Flow

A discrete free-energy (entropy) functional is introduced:

$$E_N(c) = \frac{1}{N} \sum_k (c_k \log c_k - c_k + 1)$$

This entropy acts as the Lyapunov functional for the reaction network, driving evolution via an energy-dissipation balance (EDB): $\frac{d}{dt} E_N(c) = - D_{\alpha, N}(c, J)$ where $D_{\alpha, N}$ is a discrete dissipation functional capturing both "primal" and "slope" terms related to chemical kinetics and mobility structure. The dissipation is formulated through a Legendre-Fenchel dual of a convex function, e.g., $C^*(r) = 4 (\cosh(r/2) - 1)$.

3. Vanishing-Mesh-Size Limit: Continuum Fourth-Order DLSS Equation

The continuum limit is approached by embedding discrete densities into $ρ^N(x)$ via piecewise-constant reconstructions: $ρ^N(x) = c_k^N$ for $x \in [k/N, (k+1)/N)$. Expanding $N^2(c_k^2 - c_{k-1}c_{k+1})$ in powers of δ yields: $$N^2(c_k^2 - c_{k-1}c_{k+1}) \approx -\rho^2 \partial_{xx} \log \rho + O(N^{-2})$$ Under scaling (diffusive rescaling: $N^4$ in time), and appropriate choice of $\sigma_\alpha$, the limiting PDE is

$$\partial_t \rho = -\partial_{xx}\left(\rho^\alpha\, \partial_{xx} \log \rho\right)$$

where α = 1 recovers the classical DLSS equation; power-type mobility generalizations arise for other choices.

4. Energy-Dissipation Principle (EDP) and Gradient Structure

Both discrete and continuum models are formulated as gradient flows of entropy in the continuity equation format. On the continuum, the entropy becomes

$$\mathcal{E}(\rho) = \int_{\mathbb{T}} (\rho \log \rho - \rho + 1)\,dx$$

and the quadratic dual dissipation potential is

$$\mathcal{R}_\alpha^*(\rho,\eta) = \frac{1}{2} \int_{\mathbb{T}} \rho^\alpha \eta^2\,dx$$

yielding the constitutive relation for the flux $j = -\rho^\alpha \partial_{xx} \log \rho$. EDP convergence is established: the discrete energy-dissipation functional converges variationally to its continuum analog. The chain rule and integration by parts in modified variables ($V = \rho^{-\alpha/2} j$, $$\Sigma = -\frac{2}{\alpha} (\Delta \rho^{\alpha/2} - 4|\nabla \rho^{\alpha/4}|^2)$$) ensure the evolution satisfies

$$E(\rho(s)) - E(\rho(r)) = -\int_r^s \int_{\mathbb{T}} \Sigma V\,dx\,dt$$

validating the limiting weak solution to the nonlinear fourth-order DLSS equation.

5. Nonlinear Mobility: Fast Diffusion and Porous Medium Analogy

For $\alpha \neq 1$, the nonlinear mobility term $\rho^\alpha$ fundamentally alters solution behavior:

• $\alpha < 1$ (“fast diffusion” regime): traveling wave solutions have algebraic tails, indicating long-range effects and noncompact support.
• $\alpha > 1$ (“porous medium” regime): traveling waves exhibit compact support and polynomial profiles. This demonstrates an interpolation between fast-diffusive and porous-medium dynamics in the fourth-order context, with solution regularity, support, and propagation properties strongly dependent on α.

6. Key Formulas and Structural Summary

Discrete/Continuum	Formula	Description
Discrete flux	$$J_{\alpha, k} = \sigma_\alpha(c_{k-1}, c_k, c_{k+1})(c_k^2 - c_{k-1}c_{k+1})$$	Chemical reaction network flux
Discrete rate equation	$$\dot{c}_k = N^2 [J_{\alpha,k-1} - 2J_{\alpha,k} + J_{\alpha,k+1}]$$	Continuity-form master equation
Embedding	$$\rho^N(x) = \sum_k c_k^N \mathbf{1}_{[k/N, (k+1)/N)}(x)$$	Lattice-to-continuum interpolation
Continuum PDE	$$\partial_t \rho = -\partial_{xx}(\rho^\alpha\, \partial_{xx} \log \rho)$$	Nonlinear-mobility DLSS equation
Dissipation potential	$$\mathcal{R}_\alpha^*(\rho, \eta) = \frac{1}{2} \int_{\mathbb{T}} \rho^\alpha \eta^2 dx$$	Gradient-flow structure

7. Implications and Generalization

The methodology rigorously connects reversible microscopic lattice models governed by local jump processes and entropy-driven kinetics to macroscopic fourth-order, nonlinear-mobility gradient flows. The framework is general, encompassing both classical DLSS ($\alpha=1$) and nonlinear-mobility equations, and demonstrates how fast-diffusion and porous-medium analogs are unified at the fourth order. The EDP convergence framework ensures not only the limiting equation structure but also the preservation of entropy principles and the gradient-flow character in the continuum.

This systematic derivation clarifies both the microscopic origins and the macroscopic structure of the fourth-order DLSS equation and its nonlinear extensions (Mielke et al., 8 Oct 2025). The qualitative similarities with second-order fast diffusion and porous medium equations suggest a broader universality of nonlinear-mobility behaviors in higher-order, entropy-driven evolution.