[Paper Review] Path Integral Formulation of Anomalous Diffusion Processes

Rudolf Friedrich, Stephan Eule|arXiv (Cornell University)|Oct 26, 2011

Diffusion and Search Dynamics1 references3 citations

TL;DR

This paper presents a path integral formulation for generalized anomalous diffusion processes, including Continuous Time Random Walks (CTRWs), by introducing a stochastic time-rescaling process. It derives exact path probability densities and a generalized Feynman-Kac formula, with a closed-form expression for CTRW path distributions in terms of waiting time distributions via a Dyson equation solution.

ABSTRACT

We present the path integral formulation of a broad class of generalized diffusion processes. Employing the path integral we derive exact expressions for the path probability densities and joint probability distributions for the class of processes under consideration. We show that Continuous Time Random Walks (CTRWs) are included in our framework. A closed expression for the path probability distribution of CTRWs is found in terms of their waiting time distribution as the solution of a Dyson equation. As the formalism naturally includes the treatment of functionals a generalized Feynman-Kac formula is derived.

Motivation & Objective

• To develop a path integral framework for generalized diffusion processes driven by an additional stochastic process, extending beyond standard Brownian motion.
• To address the lack of a path integral formulation for non-Markovian CTRW processes, which are widely used but poorly described by single-point statistics.
• To derive exact expressions for path probability densities and joint distributions in this generalized class of processes.
• To establish a generalized Feynman-Kac formula for functionals of anomalous diffusion paths.
• To provide a closed-form solution for the path probability of CTRWs in terms of their waiting time distribution.

Proposed method

• Formulates a discrete Langevin equation with an additional stochastic term αₖrₖ to model anomalous diffusion, where αₖ governs the waiting time distribution.
• Derives the transition amplitude using a Fourier representation of the Gaussian noise, leading to a Martin-Siggia-Rose (MSR) action in the path integral formalism.
• Averages over the Gaussian-distributed rₖ variables to obtain a modified effective diffusion coefficient depending on αₖ².
• Introduces a characteristic function Z(ηₖ) for the αₖ² process to encode the statistical properties of the waiting time distribution.
• Solves the path probability via a Dyson equation for the CTRW case, expressing the path distribution as a series expansion in terms of events.
• Derives a generalized Fokker-Planck equation and a generalized Feynman-Kac formula by analyzing functionals of the path through generating functionals and time-discrete evolution equations.

Experimental results

Research questions

• RQ1 Can a path integral formulation be consistently developed for non-Markovian anomalous diffusion processes such as CTRWs?
• RQ2 What is the exact form of the path probability density for CTRWs in terms of their waiting time distribution?
• RQ3 How can functionals of anomalous diffusion paths be evaluated systematically using path integral techniques?
• RQ4 What is the structure of the joint probability distribution for multi-point paths in CTRW processes?
• RQ5 Can a generalized Fokker-Planck equation be derived from the path integral formalism for CTRWs with internal dynamics?

Key findings

• A closed-form expression for the path probability density of CTRWs is derived in terms of the waiting time distribution K₀ and the waiting time kernel V, solved via a Dyson equation.
• The path probability distribution is expanded in powers of the number of events, with the zeroth-order term corresponding to paths with no events after the first step.
• The first-order correction term explicitly includes contributions from paths with one event, showing a formal analogy to self-energy corrections in quantum field theory.
• The generalized Feynman-Kac formula is derived, enabling the evaluation of path functionals such as exponentials of time-integrated functions.
• A generalized Fokker-Planck equation is obtained in the continuous-time limit, with a memory kernel Q(t−t′) and a modified generator H = L₀ + ipU(q), consistent with known results for CTRWs.
• The formalism naturally includes non-equilibrium conditions, such as aging CTRWs, by considering paths that start with an event.