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Stochastic Analysis of Retention Time of Coupled Memory Topology

Anirudh Bangalore Shankar, Avhishek Chatterjee, Bhaswar Chakrabarti, Anjan Chakravorty

TL;DR

This work addresses the costly evaluation of retention times in coupled memory topologies by introducing a Glauber-dynamics-inspired, physically grounded Markov framework that models memory units as Ising-like spins in a thermal bath. The energy of a configuration includes a local field term and pairwise couplings, yielding a Markov process on the spin hypercube; retention time is treated as a first-passage time toward the majority flip, with analytical expressions derived for elementary topologies. Key contributions include closed-form retention-time expressions for single and three-dipole topologies, and analytical insight into how coupling strength $s_f$ and external field $H$ affect retention across topologies, validated against simulations. The framework enables rapid design-space exploration to guide material choice and topology for desired retention targets and offers a foundation for investigating topological error-correction concepts in classical memory systems.

Abstract

Recently, it has been experimentally demonstrated that individual memory units coupled in certain topology can provide the intended performance. However, experimental or simulation based evaluation of different coupled memory topologies and materials are costly and time consuming. In this paper, inspired by Glauber dynamics models in non-equilibrium statistical mechanics, we propose a physically accurate generic mathematical framework for analyzing retention times of various coupled memory topologies and materials. We demonstrate efficacy of the proposed framework by deriving closed form expressions for a few popular coupled and uncoupled memory topologies, which match simulations. Our analysis also offers analytical insights helping us estimate the impact of materials and topologies on retention time.

Stochastic Analysis of Retention Time of Coupled Memory Topology

TL;DR

This work addresses the costly evaluation of retention times in coupled memory topologies by introducing a Glauber-dynamics-inspired, physically grounded Markov framework that models memory units as Ising-like spins in a thermal bath. The energy of a configuration includes a local field term and pairwise couplings, yielding a Markov process on the spin hypercube; retention time is treated as a first-passage time toward the majority flip, with analytical expressions derived for elementary topologies. Key contributions include closed-form retention-time expressions for single and three-dipole topologies, and analytical insight into how coupling strength and external field affect retention across topologies, validated against simulations. The framework enables rapid design-space exploration to guide material choice and topology for desired retention targets and offers a foundation for investigating topological error-correction concepts in classical memory systems.

Abstract

Recently, it has been experimentally demonstrated that individual memory units coupled in certain topology can provide the intended performance. However, experimental or simulation based evaluation of different coupled memory topologies and materials are costly and time consuming. In this paper, inspired by Glauber dynamics models in non-equilibrium statistical mechanics, we propose a physically accurate generic mathematical framework for analyzing retention times of various coupled memory topologies and materials. We demonstrate efficacy of the proposed framework by deriving closed form expressions for a few popular coupled and uncoupled memory topologies, which match simulations. Our analysis also offers analytical insights helping us estimate the impact of materials and topologies on retention time.

Paper Structure

This paper contains 13 sections, 44 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Memory Topology and Temporal Evolution
  4. Single dipole
  5. Coupled $N$-dipoles
  6. Analysis of Different Memory Topologies
  7. Single dipole
  8. Three dipoles
  9. No connection between any two nodes
  10. Triangular graph
  11. Linear graph
  12. Comparisons across Memory Topologies
  13. Conclusion

Figures (5)

  • Figure 1: Markov transition diagram for single dipole.
  • Figure 2: Markov transition diagram for three dipoles.
  • Figure 3: Markov chain evolution for linear graph.
  • Figure 4: Normalized coupling co-efficient dependent normalized retention time: comparison of results obtained from simulation and analytical expression.
  • Figure 5: Normalized coupling coefficient dependent normalized retention time at different external field: comparison among the results obtained for linear array of 3 dipoles (solid), $3$ dipoles in a triangular arrangement (dashed), and single dipole (dotted).