Reaction–Diffusion with Potentials

• Reaction–diffusion systems with potentials are mathematical models combining localized reaction kinetics, diffusion processes, and spatial energy landscapes to drive species concentration dynamics.
• They employ reaction, external, and singular potentials to influence pattern formation, blow‐up control, and phase coexistence in both stochastic and deterministic settings.
• Advanced computational methods, including hybrid particle simulations and Fredholm integral approaches, enable precise analysis of these complex models in biophysical and chemical applications.

Reaction–diffusion systems with potentials encompass a broad class of mathematical and physical models where reaction kinetics, stochastic or deterministic diffusion, and one or more potential energy landscapes jointly govern the spatiotemporal evolution of species' concentrations. “Potentials” appear in several capacities: as reaction potentials (encoding finite reaction rates within prescribed zones), as external or pairwise interaction potentials (affecting drift and spatial encounter probabilities), as effective “energy” landscapes in deterministic and stochastic settings, and as singular or spatially structured contributions modulating global existence or pattern selection. This article surveys the rigorous frameworks, main results, and key applications of reaction–diffusion with potentials, emphasizing exact mathematical relationships, computational strategies, and implications for biophysical modeling.

1. Reaction Potentials in Stochastic and Deterministic Models

Reaction potentials represent local reaction kinetics by introducing a spatially localized reaction term. In the Doi model (Agbanusi et al., 2013), pairs of diffusively moving particles are allowed to react only when within a reaction radius $r_b$, with propensity (rate per unit time) %%%%1%%%%. The probability density $p_{\rm D}(r,t)$ of an unreacted pair at separation $r$ evolves as

$$\frac{\partial p_{\rm D}}{\partial t}(r,t) = D\,\Delta_r\,p_{\rm D}(r,t) - \lambda\, \mathbf{1}_{\{0 < r < r_b\}} p_{\rm D}(r,t),$$

where $\Delta_r$ is the radial Laplacian and $D$ the sum of individual diffusivities. Here $\lambda \mathbf{1}_{\{r < r_b\}}$ acts as the reaction potential, removing probability mass only within the reactive domain.

The Smoluchowski model represents the limiting case of instantaneous reaction, replacing the reaction potential with a perfect-absorption boundary at $r = r_b$: $$\frac{\partial p_{\rm S}}{\partial t}(r,t) = D\,\Delta_r\,p_{\rm S}(r,t), \quad \text{with } p_{\rm S}(r_b, t) = 0.$$ The Doi model converges in the $\lambda \to \infty$ limit to the Smoluchowski model, with rigorous rate $O(\lambda^{-1/2+\epsilon})$ for all $\epsilon > 0$ (Agbanusi et al., 2013).

2. External and Pair Interaction Potentials

The inclusion of potential energy landscapes $V(x)$ or pairwise potentials $U(r)$ leads to drift-diffusion (reaction-drift-diffusion) models. The corresponding Fokker–Planck equations take the gradient-flow form: $$\frac{\partial P}{\partial t} = D\,\partial_x^2 P + \frac{D}{k_B T}\,\partial_x\bigl(P\,V'(x)\bigr) - k_0 S(x) P - k_r P,$$ where $D$ is diffusivity, $S(x)$ a (possibly spatially extended) reactive sink, and $k_0$ the local sink strength (Samanta et al., 2020, Spendier et al., 2013). When $V(x)$ is harmonic, the propagator is Ornstein–Uhlenbeck, yielding explicit survival probabilities and mean first-passage times (Spendier et al., 2013). For arbitrary $V(x)$ and $S(x)$, the Fredholm integral method in the Laplace domain provides a systematic semi-analytic framework (Samanta et al., 2020).

For pair potentials, the reaction–diffusion equation for spherically symmetric overdamped motion is: $$\frac{\partial p}{\partial t}(r, t) = \frac{1}{r^2} \frac{\partial}{\partial r}\left[ r^2 D e^{-\beta U(r)} \frac{\partial}{\partial r}\left( e^{\beta U(r)}p(r, t)\right) \right] - a(r) p(r, t),$$ with $a(r) = \lambda \theta(R - r)$ for a Doi-type reaction volume, and $U(r)$ any isotropic pairwise potential. The macroscopic rate constant $k$ decomposes as

$$k = \left(k_e^{-1} + k_f^{-1}\right)^{-1}, \qquad k_e = 4\pi D \left[\int_R^\infty g(s)\,ds\right]^{-1}$$

where $g(r) = e^{\beta U(r)}/r^2$, with $k_e$ (encounter rate) set by diffusion against $U(r)$, and $k_f$ (formation rate) determined by the local reaction propensity inside $r < R$ (Dibak et al., 2019).

3. Effective Potential Frameworks and Energy Landscapes

Reaction–diffusion systems with scalar or vector potential landscapes are naturally connected to “energy functional” frameworks. The classical case is the scalar reaction–diffusion equation admitting an energy (Lyapunov) functional: $$E[u] = \int_\Omega \frac{1}{2} |\nabla u|^2 + V(u)\, dx, \quad V(u) = -\int^u f(s)\, ds,$$ with $$u_t = -\frac{\delta E}{\delta u} = D \Delta u - V'(u)$$ (Marquez-Lago et al., 2013). The pattern selection problem (spots vs. stripes) is controlled by the double-well structure of $V(u)$:

• Unequal minima $V(a) \neq V(b)$: spots are favored.
• Equal minima $V(a) = V(b)$: stripes or interfaces ("Maxwell construction" or equal-area rule).

Multi-component systems lacking strict gradient-flow structure admit an auxiliary “effective potential” from the stationary solution of the associated Fokker–Planck equation in state space. If the steady FP density $p_\infty(u,v)$ has a unique peak, spots are favored; if two peaks of equal height, stripes or labyrinths emerge (Marquez-Lago et al., 2013).

4. Singular and Nonlinear Potentials: Blow-Up and Global Existence

Singular spatially dependent potentials, such as $|x|^\sigma$ with $\sigma < 0$, can radically alter reaction–diffusion blow-up behavior. For the equation

$u_t = \Delta (u^m) + |x|^\sigma u^p,$

finite-time blow-up present in standard homogeneous models (with $\sigma = 0$, $p > 1$) may be prevented if $1 < p < 1 - \sigma(m-1)/2$ and $\sigma$ is sufficiently negative. The existence and uniqueness of global-in-time self-similar solutions that grow up but do not blow up are established via dynamical systems techniques and barrier constructions (Iagar et al., 2021). This constitutes the first rigorous instance where a spatially singular potential enforces global existence in a quasilinear reaction–diffusion problem that otherwise blows up.

5. Statistical Mechanics and Stochastic Phase Coexistence

In weakly stochastic, spatially discrete networks described by coupled bistable reactors (e.g., the Schlögl model), the phase coexistence criterion between stable concentrations depends crucially on the diffusion (hopping) rate between nodes. In the limit of fast hopping ($r \gg$ kinetic rates), the deterministic mean-field potential $V_{\mathrm{high}}(\xi) = \int^{\xi} -F(\eta) d\eta$ governs coexistence via $$V_{\mathrm{high}}(\xi_-) = V_{\mathrm{high}}(\xi_+)$$, with $F$ the rate function (Yanagisawa et al., 2024). In the slow-hopping regime ($r \ll$ kinetic rates), the relevant criterion shifts to the stochastic quasi-potential (large deviation rate function) $$V_{\mathrm{low}}(\xi) = \int^{\xi}\ln[w_-(\eta)/w_+(\eta)] d\eta$$, with coexistence at $V_{\mathrm{low}}(\xi_-) = V_{\mathrm{low}}(\xi_+)$.

This distinction generalizes the Maxwell equal-area rule to nonequilibrium and stochastic settings; $V_{\mathrm{high}}$ is a Lyapunov functional of the deterministic PDE, whereas $V_{\mathrm{low}}$ encodes the large deviation principle for stochastic fluctuations in a well-mixed reactor.

6. Computational Methods for Reaction–Drift–Diffusion with Potentials

Exact simulation of reaction–drift–diffusion with arbitrary potentials typically requires hybrid computational schemes. In 1D or pairwise settings, analytical or semi-analytical solutions exist (via propagator or Fredholm integral approaches) for certain classes of $V(x)$ or interaction $U(r)$ (Samanta et al., 2020, Spendier et al., 2013).

Stochastic particle-based simulation methods such as the Dynamic Lattice First-Passage Kinetic Monte Carlo (DL-FPKMC) algorithm (Mauro et al., 2013) generalize first-passage strategies to include drift from fixed potentials. Diffusion and reactions are approximated by continuous-time random walks on dynamically adapted lattices, with jump rates derived from the discretized Fokker–Planck equation: $$a_{ij} = \frac{D}{h^2} \frac{V_j - V_i}{e^{V_j - V_i} - 1}$$ for uniform meshes, preserving detailed balance and achieving high accuracy for both smooth and discontinuous $V(x)$. These methods demonstrate that drift/local potential energy landscapes strongly modulate both reaction locations and times, with reaction events clustering near potential minima and potential barriers leading to long-tail survival distributions.

7. Applications and Implications

Reaction–diffusion systems with potentials underpin diverse biophysical, chemical, and engineering phenomena:

• Biomolecular association: Models with reaction potentials and pairwise interactions quantify how molecular structure, steric exclusion, and soft or Lennard-Jones-like forces modulate association rates in crowded environments (Dibak et al., 2019).
• Pattern formation: Effective potentials arising from kinetics govern the emergence of spatial motifs (spots vs. stripes) in developmental systems, synthetic biology constructs, and tissue engineering matrices (Marquez-Lago et al., 2013).
• Cellular electrophysiology: Coupled reaction–diffusion–potential systems are deployed in minimal models of cardiac tissue, with gating variables and external stimuli encoded as potentials that affect action potential propagation and arrhythmogenesis (Richter et al., 2017).
• Energy transduction: Models combining diffusion, reactions, and electrostatic potentials via energetic variational principles capture self-regulated switching and feedback, as in the mitochondrial electron transport chain (Xu et al., 2023).
• Blow-up control and nonlinear PDE theory: Singular or spatially inhomogeneous potentials may suppress catastrophic concentration increases, offering rigorous prevention of finite-time blow-up (Iagar et al., 2021).
• Phase separation and coexistence: Stochastic and deterministic potentials mediate spatial phase segregation and the coexistence of distinct concentration domains in large stochastic reactor networks (Yanagisawa et al., 2024).
• Diffusion-influenced exciton/energy transport: Survival probabilities and rates in the presence of Gaussian sinks and external potentials unify solution-phase electronic relaxation and photosynthetic yield predictions (Samanta et al., 2020).

In all these areas, the explicit or implicit structure of the potential is as central as the reaction or diffusion terms themselves, dictating not only quantitative rates but also qualitative behaviors such as pattern selection, stability, and global existence.