Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal strategies for controlled growth in metastable Kawasaki dynamics

Simone Baldassarri, Maike C. de Jongh

Abstract

In this paper, we develop a Markov decision process (MDP) formulation for the low--temperature metastable Ising model evolving according to Kawasaki dynamics in a finite box of the two--dimensional square lattice. We analyze how an external controller can guide the system to the all--occupied state by appropriately adding and moving particles at specified moments in time. To this end, we construct a reduced MDP on a constrained family of configurations having a single cluster, a regime where particle attachment is more likely than detachment. We investigate two reward structures: one that depends solely on the time to reach the target configuration, and another that incorporates action--dependent energy costs. Within this MDP framework, we characterize the exact optimal policies under both reward structures, which turn out to have a different behavior: while a purely efficiency--based criterion promotes the growth from the boundary centers of the cluster, an energy--based reward function favours the growth at the corners of the cluster.

Optimal strategies for controlled growth in metastable Kawasaki dynamics

Abstract

In this paper, we develop a Markov decision process (MDP) formulation for the low--temperature metastable Ising model evolving according to Kawasaki dynamics in a finite box of the two--dimensional square lattice. We analyze how an external controller can guide the system to the all--occupied state by appropriately adding and moving particles at specified moments in time. To this end, we construct a reduced MDP on a constrained family of configurations having a single cluster, a regime where particle attachment is more likely than detachment. We investigate two reward structures: one that depends solely on the time to reach the target configuration, and another that incorporates action--dependent energy costs. Within this MDP framework, we characterize the exact optimal policies under both reward structures, which turn out to have a different behavior: while a purely efficiency--based criterion promotes the growth from the boundary centers of the cluster, an energy--based reward function favours the growth at the corners of the cluster.
Paper Structure (27 sections, 4 theorems, 63 equations, 3 figures)

This paper contains 27 sections, 4 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main contributions
  4. Outline
  5. Model and results
  6. The Markov decision process
  7. Definition of the Ising MDP for Kawasaki dynamics
  8. The auxiliary MDP
  9. Main results
  10. Proof of the main results
  11. Proof of Theorem \ref{['thm1']}
  12. Proof of expression (\ref{['ineqq1']}):
  13. Proof of expression (\ref{['ineqq2']}):
  14. Proof of expression (\ref{['ineqq3']}):
  15. Proof of expression (\ref{['ineqq4']}):
  16. ...and 12 more sections

Key Result

Theorem 2.1

Puterman A policy $\pi^* \in \Pi$ is optimal if and only if $v^{\pi^*}_{\lambda}$ is a solution to the optimality equations.

Figures (3)

  • Figure 2.1: Cost of adding or removing a row of length $\ell$. This figure is taken from denHollander2000.
  • Figure 2.2: Post--decision configurations corresponding to the actions $b_1$, $b_2$, $b_1'$ and $b_2'$.
  • Figure 2.3: Post-decision configurations after taking actions $b_1$ and $b_1'$ from a state $(i,j)$, $j < L-2$ (first row) and from state $(i, L-2)$ (second row).

Theorems & Definitions (6)

  • Theorem 2.1
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3