General Theory of Static Response for Markov Jump Processes

Area of Science:

• Statistical Physics
• Network Theory
• Stochastic Processes

Background:

• Markov jump processes are fundamental to modeling dynamic systems.
• Understanding system responses to control parameters is crucial.
• Graph topology significantly influences system dynamics.

Purpose of the Study:

• Derive explicit expressions for static responses of edge currents and steady-state probabilities.
• Investigate constraints imposed by graph topology on these responses.
• Apply findings to stochastic thermodynamics for analyzing dissipation and forces.

Main Methods:

• Analysis of Markov jump processes on graphs.
• Derivation of explicit expressions for static responses.
• Development of response relations and topology-dependent bounds.
• Application of graph theory, specifically incidence matrices.

Main Results:

• Explicit formulas for static responses of currents and probabilities derived.
• Response relations and bounds established, linking responses to graph topology.
• For unicyclic networks, scaled current responses are bounded [0, 1] and sum to 1.
• Static responses of fundamental currents to thermodynamic forces derived.

Conclusions:

• Graph topology fundamentally constrains Markov jump process dynamics.
• The derived response relations and bounds offer predictive power.
• This work provides a framework for analyzing dissipation in complex systems.