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Efficient Computation of Large Deviation Rate Functions for Entropy Production in High-Dimensional Diffusion Processes


Core Concepts
The core message of this paper is to present an efficient interacting particle method (IPM) for numerically computing the large deviation rate function of entropy production for high-dimensional diffusion processes, with a focus on the vanishing-noise limit.
Abstract
The paper presents an interacting particle method (IPM) for numerically computing the large deviation rate function of entropy production for diffusion processes. The key aspects are: The method is based on a discretization of the Feynman-Kac semigroup associated with the principal eigenvalue problem, which is then accessed through the spectral radius of the discretized semigroup. The IPM naturally handles unbounded domains, high dimensions, and singular behaviors in the vanishing-noise limit, which are challenges for traditional mesh-based numerical methods. Numerical examples in dimensions up to 16 demonstrate the scalability and robustness of the method, with the numerical results converging to the analytical vanishing-noise limit. Techniques for setting the initial measure of the particles are introduced to obtain faster convergence of the method. The empirical density of particles at the final time accurately captures the singular behavior of the vanishing-noise limit, as predicted by the theory.
Stats
The paper does not contain any explicit numerical data or statistics to support the key claims. The focus is on the development and analysis of the numerical method.
Quotes
None.

Deeper Inquiries

How can the proposed IPM be extended to handle more general drift and diffusion coefficients beyond the specific forms considered in the paper

To extend the proposed Interacting Particle Method (IPM) to handle more general drift and diffusion coefficients beyond the specific forms considered in the paper, we can introduce flexibility in the algorithm. One approach is to allow for user-defined functions for the drift and diffusion coefficients, enabling the IPM to adapt to a wider range of stochastic processes. By parameterizing the drift and diffusion functions, the IPM can be applied to various systems with different dynamics. Additionally, incorporating adaptive techniques to adjust the particle distribution based on the characteristics of the specific coefficients can enhance the method's versatility. This adaptability would enable the IPM to handle a broader class of diffusion processes with diverse drift and diffusion behaviors.

What are the theoretical convergence rates of the IPM approximation to the true principal eigenvalue and large deviation rate function

The theoretical convergence rates of the IPM approximation to the true principal eigenvalue and large deviation rate function can be analyzed based on the stability properties of the discrete-time semigroup used in the method. By studying the convergence properties of the semigroup iterations and the resampling steps in the IPM, one can derive bounds on the rate of convergence of the numerical approximation to the exact values. The convergence rates may depend on factors such as the time step size, the number of particles, and the complexity of the system being studied. Analyzing the convergence rates theoretically would provide insights into the accuracy and efficiency of the IPM in approximating the principal eigenvalue and large deviation rate function.

Can the IPM framework be adapted to study large deviations in other stochastic thermodynamic observables beyond entropy production

The IPM framework can be adapted to study large deviations in other stochastic thermodynamic observables beyond entropy production by modifying the observables of interest in the algorithm. By adjusting the quantities being computed in the IPM to reflect different thermodynamic observables, such as heat flux, work distribution, or entropy flow, the method can be tailored to analyze the large deviation behavior of these variables. The key lies in formulating the appropriate observables and their associated rate functions within the IPM framework. This adaptation would allow researchers to investigate a wide range of stochastic thermodynamic processes and explore the large deviation principles governing various observables in non-equilibrium systems.
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