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Optimal strategies for the growth of dual-seeded lattice structures

Maike C. de Jongh, Cristian Spitoni, Emilio N. M. Cirillo

TL;DR

The paper addresses how to optimally steer growth of dual-seeded lattice structures governed by a zero-temperature 2D Ising dynamics by formulating the process as a Markov decision process and optimizing external plus-spin insertions to reach the all-plus absorbing state. It introduces an Ising MDP restricted to robust configurations and analyzes three geometric regimes (stripe-stripe, stripe-droplet, and droplet-droplet) using auxiliary MDPs to compare a small set of policies, yielding both numerical and rigorous results. A key finding is a critical discount factor $λ_c = \tfrac{15}{17}$ that separates distance-1 versus distance-2 stripe-growth optimality in the two-stripe case, while stripe-droplet and droplet-droplet regimes favor fast-front and diagonal growth, respectively. The framework provides a principled, decision-theoretic lens for spatiotemporal control of stochastic lattice growth with potential applications to material synthesis, biofilm patterning, and microfluidic networks, and points to extensions to higher dimensions and partial observability as promising directions.

Abstract

Optimal growth of structures governed by spatially stochastic dynamics arises in many scientific settings, for example in processes such as solution-based crystallization and the formation of microbial biofilms on patterned substrates or microfluidic networks. In this work, we investigate lattice growth using a two-dimensional, zero-temperature stochastic model of short-range spin interactions. Our goal is to determine how external perturbations can be optimized to steer the system efficiently toward the uniformly positive state, starting from two initial clusters of positive sites. To achieve this, we cast the problem as a Markov decision process adapted for a two-dimensional Ising model with zero-temperature dynamics. Within this framework, we compare alternative growth geometries and identify the structure of optimal strategies across three representative regimes.

Optimal strategies for the growth of dual-seeded lattice structures

TL;DR

The paper addresses how to optimally steer growth of dual-seeded lattice structures governed by a zero-temperature 2D Ising dynamics by formulating the process as a Markov decision process and optimizing external plus-spin insertions to reach the all-plus absorbing state. It introduces an Ising MDP restricted to robust configurations and analyzes three geometric regimes (stripe-stripe, stripe-droplet, and droplet-droplet) using auxiliary MDPs to compare a small set of policies, yielding both numerical and rigorous results. A key finding is a critical discount factor that separates distance-1 versus distance-2 stripe-growth optimality in the two-stripe case, while stripe-droplet and droplet-droplet regimes favor fast-front and diagonal growth, respectively. The framework provides a principled, decision-theoretic lens for spatiotemporal control of stochastic lattice growth with potential applications to material synthesis, biofilm patterning, and microfluidic networks, and points to extensions to higher dimensions and partial observability as promising directions.

Abstract

Optimal growth of structures governed by spatially stochastic dynamics arises in many scientific settings, for example in processes such as solution-based crystallization and the formation of microbial biofilms on patterned substrates or microfluidic networks. In this work, we investigate lattice growth using a two-dimensional, zero-temperature stochastic model of short-range spin interactions. Our goal is to determine how external perturbations can be optimized to steer the system efficiently toward the uniformly positive state, starting from two initial clusters of positive sites. To achieve this, we cast the problem as a Markov decision process adapted for a two-dimensional Ising model with zero-temperature dynamics. Within this framework, we compare alternative growth geometries and identify the structure of optimal strategies across three representative regimes.

Paper Structure

This paper contains 31 sections, 4 theorems, 71 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The model
  3. The stochastic two-dimensional Ising model at zero temperature
  4. The Markov decision process
  5. Remarks on the value function
  6. Definition of the Ising MDP
  7. Numerical study of the double seeded Ising MDP
  8. The two-stripe case
  9. The auxiliary MDP
  10. Candidates for optimality
  11. Discussion of results
  12. The stripe-droplet case
  13. The auxiliary MDP
  14. Candidates for optimality
  15. Discussion of results
  16. ...and 16 more sections

Key Result

Theorem 1

Given a policy $\pi$ and $s,\bar{s}\in S$ such that $s\neq\bar{s}$ and $\bar{s}$ is an absorbing state. Then,

Figures (14)

  • Figure 1: Illustration of fragile versus robust configurations.
  • Figure 2: Illustration of the state space $S^x$ (left) and of the action space $A^x$ (right).
  • Figure 3: Visualization of the action sets defined in \ref{['eq:action-ssc']}.Top: $(i,j)=(2,2),(3,2),(4,2)$. Bottom: $(i,j)=(3,3),(3,4),(4,4)$
  • Figure 4: Visualization of the action sets defined in \ref{['eq:action-ssnc']}. Top: sets $A_1(i,j)$ for $i\ge5$ and $j=2,3,4$ and $j\ge5$ (from the left to the right). Bottom: as above for $A_2(i,j)$.
  • Figure 5: Illustration of policies $\pi_1$ (upper row) and $\pi_2$ (bottom row) for $N=100$ with initial seeds two stripe of width $3$ at distance $47$. Left group: $\kappa = 5000$ at times $t = 200, 400,600$ (from left to right). Right group: $\kappa = 20,000$ at times $t = 50,100,150$ (from left to right).
  • ...and 9 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Theorem 3
  • proof