Path-integral formulation of stochastic processes for exclusive particle systems

Area of Science:

• Statistical Mechanics
• Quantum Field Theory
• Many-Body Physics

Background:

• Understanding systems far from equilibrium is crucial in statistical mechanics.
• Hard-core particle systems exhibit complex dynamics not easily described by equilibrium theories.
• Stochastic processes are fundamental to modeling many-body systems.

Purpose of the Study:

• To develop a systematic formalism for path-integral formulation of hard-core particle systems.
• To analyze the long-time behavior of such systems, particularly reaction-diffusion models.
• To identify universal properties in systems far from equilibrium.

Main Methods:

• Derivation of the master equation using annihilation and creation operators.
• Calculation of Kramers-Moyal coefficients to obtain the Fokker-Planck equation (FPE).
• Mapping the stochastic differential equation (SDE) to a path-integral formulation for field-theoretic analysis.
• Application of renormalization group methods.

Main Results:

• A systematic path-integral formalism for hard-core particle systems far from equilibrium.
• Identification of Kramers-Moyal coefficients and derivation of the corresponding FPE and SDE.
• The SDE is successfully mapped to a field theory with a path-integral action.
• Application to a two-species reaction-diffusion system revealed a universal decay exponent for average particle concentration in arbitrary dimensions.

Conclusions:

• The developed formalism provides a powerful tool for studying non-equilibrium statistical mechanics.
• The findings highlight universal long-time behaviors in complex reaction-diffusion systems.
• This work bridges stochastic processes, field theory, and renormalization group methods for non-equilibrium systems.