Statistics of trajectories in two-state master equations

Area of Science:

• Statistical Physics
• Chemical Kinetics
• Computational Chemistry

Background:

• Master equations are fundamental for describing the time evolution of systems with discrete states.
• Calculating trajectory probabilities and related observables can be computationally intensive, especially for complex systems.
• Existing methods may not be optimal for systems with a small number of states or for analyzing entire trajectory functionals.

Purpose of the Study:

• To derive a simplified analytical expression for the probability of trajectories governed by a master equation.
• To enable the calculation of observables defined as functionals of entire system trajectories.
• To demonstrate the utility of the derived expression using a practical two-state system example.

Main Methods:

• Derivation of a general expression for trajectory probabilities in the context of master equations.
• Application of the derived expression to a two-state master equation model.
• Analytical calculation of specific trajectory-dependent observables, such as time-in-state and transition counts.

Main Results:

• A simple, generalizable expression for master equation trajectory probabilities was successfully derived.
• The method was illustrated with a two-state system, yielding analytical results.
• The distributions of time spent in a state and the number of transitions were calculated using modified Bessel functions.

Conclusions:

• The derived expression offers an efficient way to compute trajectory probabilities and related observables for small systems.
• The analytical results for the two-state model demonstrate the practical applicability of the method.
• This approach provides valuable insights into the dynamics of systems described by master equations.