Regensburg 2025 – scientific programme
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DY: Fachverband Dynamik und Statistische Physik
DY 5: Statistical Physics: General
DY 5.3: Talk
Monday, March 17, 2025, 10:00–10:15, H47
Long-term behavior of master equations on a countable system — •Bernd Michael Fernengel1, Thilo Gross1, and Wolfram Just2 — 1HIFMB, Oldenburg, Germany — 2University of Rostock, Rostock, Germany
Master equations play a crucial role in natural science, as they describe the time evolution of probability distributions of all systems that can be modeled as directed, weighted graphs. Despite their essential role, computing a solution is often avoided and authors refer to numerical methods or approximation techniques instead.
We present both a mathematically sound framework for master equations on a discrete, countable configuration space as well as sufficient conditions the generator of the master equation must have for the time limit t -> infinity to converge, which is not guaranteed on an infinite dimensional space.
We discuss the assumptions for the possibility of interchanging the thermodynamic limit and the time limit. This makes it possible to obtain the long-term behavior of an infinite system from a thermodynamic limit of stationary solutions of corresponding finite subnetworks.
Our method is demonstrated by a few examples of master equations on linear, infinitely long chains, with one- and two open ends.
Keywords: master equation; long-term behavior; stationary solution; thermodynamic limit