Pattern stability in reaction-diffusion systems depends on path entropy

Abstract

Reaction-diffusion systems driven far from thermodynamic equilibrium through the injection of energy can support multiple distinct spatial patterns that persist as long-lived dynamical phases. The stability of these metastable phases is not determined by thermodynamics, but by the transition paths connecting them. At finite particle numbers, intrinsic stochasticity induces rare transitions between competing patterns, rendering continuum mean-field descriptions insufficient, while exact stochastic simulations become computationally prohibitive in spatially extended systems. Here, we develop a nonequilibrium instanton framework that enables efficient computation of transition rates between metastable patterns from a single optimal transition path and its fluctuations. Using this theoretical framework, we show that an effective entropy in path space can qualitatively alter stability at finite particle numbers by increasing the exit rates of metastable patterns. By studying models of varying complexity, this work establishes path entropy as a key organizing principle for nonequilibrium pattern formation.

Figures (4)

• Figure 1: Overview of how transition path entropy can alter stability of metastable states in nonequilibrium systems. (a) Effective-action or quasipotential landscape along a one-dimensional order parameter, indicating that the high-density state ($\rho_\text{H}$) is favored in the absence of noise, $\Omega \rightarrow \infty$. (b) A representative stochastic trajectory and (c) the corresponding steady-state probability density show that, at sufficiently strong noise, the low-concentration state ($\rho_\text{L}$) is more stable due to entropic enhancement of the $\rho_\text{H} \to \rho_\text{L}$ transition. (d) Two-dimensional representation of the system in concentration space, showing an underlying double-well potential (color map) together with a nonconservative force field (stream plot). The forward (blue) and backward (red) optimal transition paths differ because detailed balance is broken. Broader fluctuations (transparent tubes) around the backward path indicate a larger statistical weight in path space, favoring it entropically at finite noise. (e) Schematic nonequilibrium phase diagram as a function of a system parameter $\kappa$ and the noise strength $\Omega^{-1}$. In the zero-noise limit ($\Omega^{-1} \to 0$), stability follows from the action, whereas at finite noise, path entropy can qualitatively alter stability and eliminate the phase transition altogether.
• Figure 2: (a) Instanton actions for the forward (blue) and backward (red) transitions between the high- and low-concentration states of the 2$d$ Schlögl model defined in the text. (b) Stability diagram, indicating the more stable phase as a function of $\kappa=D\tau/\xi^2$ and $\Omega$ based on the NEQI rate constants for the forward and backward transitions. Snapshots along the instantons of the forward (c) and backward (d) transitions at $\kappa=1.5$ with the color map indicating molecule concentration, where $\rho_{-}\xi^2 \approx 0.37$ and $\rho_{+}\xi^2 \approx 1.77$. The coordinates $x_0$ and $x_1$ span the two-dimensional domain in units of $\xi$.
• Figure 3: Schematic representation of the competing enzyme model studied in this work. Lipid membrane containing PIP$_1$ and PIP$_2$ in contact with kinase and phosphatase enzymes in solution that can bind and catalyze lipid interconversion under ATP consumption.
• Figure 4: (a) Instanton actions for the forward (blue) and backward (red) transitions between the uniform PIP$_1$- and PIP$_2$-dominated states of the 2D competitive enzyme model defined in the text. (b) Stability diagram, indicating the more stable phase as a function of $\kappa=D\tau/\xi^2$ and $\Omega$ based on the NEQI rate constants for the forward and backward transitions. Snapshots along the instantons of the forward (c) and backward (d) transitions at $\kappa=0.15$ with the color map indicating the PIP$_2$ concentration, where the PIP$_1$- and PIP$_2$-dominated states exhibit a PIP$_2$ concentration of $\rho^{\text{PIP}_2} \xi^2\approx 0.05$ and $\rho^{\text{PIP}_2} \xi^2\approx 0.87$. The coordinates $x_0$ and $x_1$ span the two-dimensional domain in units of $\xi$.