A standard form of master equations for general non-Markovian jump processes: the Laplace-space embedding framework and asymptotic solution

ArXiv ID: 2312.05475 “View on arXiv”

Authors: Unknown

Abstract

We present a standard form of master equations (ME) for general one-dimensional non-Markovian (history-dependent) jump processes, complemented by an asymptotic solution derived from an expanded system-size approach. The ME is obtained by developing a general Markovian embedding using a suitable set of auxiliary field variables. This Markovian embedding uses a Laplace-convolution operation applied to the velocity trajectory. We introduce an asymptotic method tailored for this ME standard, generalising the system-size expansion for these jump processes. Under specific stability conditions tied to a single noise source, upon coarse-graining, the Generalized Langevin Equation (GLE) emerges as a universal approximate model for point processes in the weak-coupling limit. This methodology offers a unified analytical toolset for general non-Markovian processes, reinforcing the universal applicability of the GLE founded in microdynamics and the principles of statistical physics.

Keywords: Master Equations, Non-Markovian Processes, Generalized Langevin Equation, Statistical Physics

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: The paper is dense with advanced mathematical formalisms including master equations, Laplace-space embeddings, system-size expansions, and functional analysis, with no mention of backtesting or implementation details.
  flowchart TD
    Goal["Research Goal<br>Create standard form of Master Equations<br>for non-Markovian jump processes"] --> Method["Methodology<br>Laplace-Space Embedding Framework<br>Markovianization via Auxiliary Fields"] --> Data["Input<br>General 1D Non-Markovian<br>Jump Process History"]
    Data --> Comp["Computational Process<br>System-Size Expansion<br>Asymptotic Solution"]
    Comp --> F1["Outcome 1<br>Standard Master Equation<br>General Form"]
    Comp --> F2["Outcome 2<br>Emergence of Generalized<br>Langevin Equation (GLE)<br>Weak-coupling/Coarse-graining limit"]
    F1 & F2 --> Unified["Unified Analytical Toolset<br>Statistical Physics & Microdynamics"]