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  • Open Access

Embedding memory in coarse-grained Hamiltonian systems

Pep Español*

  • Department of Fundamental Physics, UNED, Madrid, Spain

  • *Contact author: pep@fisfun.uned.es
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Phys. Rev. Research 8, 033133 – Published 3 August, 2026

DOI: https://doi.org/10.1103/dlzj-gtx3

Abstract

Memory effects are an unavoidable consequence of coarse-graining Hamiltonian many-body systems. Although Markovian embeddings with auxiliary variables are widely used to approximate such dynamics, their consistency with the exact projected microscopic evolution is rarely examined. Here, we analyze this consistency problem at the structural level. Starting from a general Markovian Fokker-Planck equation defined on an extended state space comprising coarse-grained variables and auxiliary degrees of freedom, we derive the exact non-Markovian evolution governing the marginal probability of the coarse-grained variables. We then identify the conditions under which this marginal equation reproduces the structural form of the generalized Fokker-Planck equation obtained from Zwanzig's projection formalism. These conditions follow from microscopic Hamiltonian dynamics and constrain the admissible form of Markovian embeddings, ensuring consistency with equilibrium structure, conservation laws, positivity, and microscopic reversibility. The resulting framework provides a systematic route for constructing microscopically consistent Markovian representations of non-Markovian coarse-grained dynamics. We illustrate the approach in coarse-grained particle models, ranging from isothermal dynamics to fully energy-conserving nonisothermal formulations.

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Phys. Rev. Research 8, 033133– Published 3 August, 2026
  • Received 23 February 2026
  • Accepted 1 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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