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In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
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Adiabatic theorem for Markov jump processes.

¹Chengdu Academy of Education Sciences, Chengdu 610036, China.

Physical Review. E

|April 18, 2026

Summary

This summary is machine-generated.

We proved the adiabatic theorem for Markov jump processes. A system stays in its steady state if the transition rates change slowly, simplifying complex dynamics.

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Area of Science:

Statistical Physics
Quantum Mechanics
Non-Hermitian Dynamics

Background:

The adiabatic theorem is crucial in quantum mechanics, describing system evolution under slow changes.
Markov jump processes model systems with discrete states and probabilistic transitions.
Extending adiabatic principles to non-Hermitian systems like Markov jump processes presents unique challenges.

Purpose of the Study:

To present an elementary proof of the adiabatic theorem specifically for Markov jump processes.
To demonstrate the analogy between quantum adiabatic theorems and behavior in Markov jump processes.
To provide a foundational understanding of adiabatic dynamics in systems governed by non-Hermitian matrices.

Main Methods:

Adapting techniques from established quantum adiabatic theorems.
Developing a proof tailored to the properties of transition rate matrices in Markov jump processes.
Addressing the complexities introduced by non-Hermitian dynamics inherent in these processes.

Main Results:

An elementary proof of the adiabatic theorem for Markov jump processes is established.
The theorem confirms that systems remain in instantaneous steady states under slow parameter changes.
The proof leverages unique properties of transition matrices, simplifying the analysis.

Conclusions:

The adiabatic theorem for Markov jump processes is rigorously proven using accessible methods.
This work bridges quantum and classical stochastic dynamics, offering new insights.
The findings facilitate a deeper understanding of adiabatic behavior in diverse scientific models.