[Paper Review] Landau theory of restart transitions

A. Pal, V. Prasad|arXiv (Cornell University)|Apr 16, 2019

Diffusion and Search Dynamics42 references19 citations

TL;DR

This paper develops a Landau-like theory to characterize phase transitions in optimal restart rates for first passage processes, showing that restart can induce either first- or second-order transitions in the optimal restart rate depending on system parameters. The key contribution is a unified framework using power-series expansions of mean first passage time in restart rate, revealing critical behavior and tricritical points in diffusion and Michaelis-Menten systems.

ABSTRACT

We develop a Landau like theory to characterize the phase transitions in resetting systems. Restart can either accelerate or hinder the completion of a first passage process. The transition between these two phases is characterized by the behavioral change in the order parameter of the system namely the optimal restart rate. Like in the original theory of Landau, the optimal restart rate can undergo a first or second order transition depending on the details of the system. Nonetheless, there exists no unified framework which can capture the onset of such novel phenomena. We unravel this in a comprehensive manner and show how the transition can be understood by analyzing the first passage time moments. Power of our approach is demonstrated in two canonical paradigm setup namely the Michaelis Menten chemical reaction and diffusion under restart.

Motivation & Objective

• To develop a unified theoretical framework for understanding phase transitions in optimal restart rates across diverse first passage processes.
• To characterize the transition between restart-beneficial and restart-harmful regimes as a phase transition analogous to equilibrium statistical mechanics.
• To identify the conditions under which the optimal restart rate undergoes continuous (second-order) or discontinuous (first-order) transitions.
• To establish a connection between the coefficients of a power-series expansion of mean first passage time and the nature of the transition.
• To demonstrate the theory's predictive power in canonical systems: diffusion with drift and the Michaelis-Menten mechanism.

Proposed method

• Expand the mean first passage time (MFPT) as a power series in the restart rate $ r $, with coefficients $ a_0, a_1, a_2, \dots $.
• Use the sign and magnitude of coefficients—particularly $ a_1 $ and $ a_2 $—to classify the transition type: $ a_1 < 0 $ and $ a_2 > 0 $ indicate second-order transition.
• Apply the criterion $ a_1 = 0 $ and $ a_2 = 0 $ to locate the tricritical point where first- and second-order transitions meet.
• For the diffusion model, derive the MFPT in terms of Péclet number $ \text{Pe} $, initial position $ u $, and a scaling function $ \mathcal{G}(y, u, \text{Pe}) $, with $ y $ related to restart rate.
• Optimize $ \mathcal{G} $ by setting $ \partial_y \mathcal{G} = 0 $ to find the optimal restart rate $ y_s $, and expand around $ y = 1 $ to extract coefficients.
• Use asymptotic expansions in the large-Péclet limit to derive analytical expressions for the critical point $ u^* = 1 - \frac{1}{\text{Pe}} $.

Experimental results

Research questions

• RQ1 What determines whether the optimal restart rate undergoes a continuous or discontinuous transition as system parameters vary?
• RQ2 Can a universal framework based on MFPT expansion coefficients predict the onset and nature of restart transitions?
• RQ3 How does the critical point for the second-order transition depend on system parameters like Péclet number and initial position?
• RQ4 What is the role of the coefficient of variation criterion in predicting optimal restart behavior, and when does it fail?
• RQ5 Where do first-order and second-order transitions meet, and how can this tricritical point be analytically located?

Key findings

• The optimal restart rate undergoes a second-order transition when $ a_1 $ changes sign from positive to negative, with the critical point located at $ a_1 = 0 $.
• For the diffusion model with drift, the second-order transition occurs at $ u = 0.6903 $ when $ \text{Pe} $ is fixed, confirmed by numerical and analytical results.
• In the large-Péclet limit, the critical point for the second-order transition is analytically found to be $ u^* = 1 - \frac{1}{\text{Pe}} $, with $ y_s $ scaling linearly with $ u $ near the critical point.
• A tricritical point exists at $ u^T = 0.25186 $, $ \text{Pe}^T = 0.43764 $, where first- and second-order transitions meet, confirmed by a discontinuous jump in the optimal restart rate from $ y_s = 1.0086 $ to $ 1 $.
• The theory predicts that first-order transitions can occur even when the coefficient of variation criterion is not satisfied, indicating a more complex transition mechanism.
• The phase diagram in the $ (\text{Pe}, u) $-plane shows regions of first- and second-order transitions, with the tricritical point marking the boundary between them.