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Computing large deviation rate functions of entropy production for diffusion processes by an interacting particle method

Zhizhang Wu, Renaud Raquépas, Jack Xin, Zhiwen Zhang

TL;DR

This work develops an interacting particle method (IPM) to compute the large-deviation rate function $I^bepsilon$ for entropy production in diffusion processes by approximating the principal eigenvalue $bb^{bepsilon,balpha}$ of a non-self-adjoint elliptic generator via a discretized Feynman--Kac semigroup. The authors establish stability and convergence results for the time-discretized semigroup and its spectral radius, and implement an IPM that handles unbounded domains and high dimensions, using burn-in and cross-$bepsilon$ initialization to accelerate convergence. Numerical experiments demonstrate convergence to the vanishing-noise limits $ bb^{0,balpha}$ and $I^0(s)$ in dimensions up to $d=16$, and show the method's ability to capture singular behaviors and multimodal invariant measures even when explicit formulas are unavailable. The results provide a scalable tool for evaluating large-deviation rate functions in stochastic thermodynamics and related areas, with potential extensions to higher-order discretizations and error analysis.

Abstract

We develop an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient to obtain the rate function is the computation of the principal eigenvalue $λ$ of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator, which is obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size. We also show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, which is suitable for computation using the IPM. The IPM applies naturally to problems in unbounded domains and scales easily to high dimensions. We show numerical examples of dimensions up to 16, and the results show that our numerical approximation of $λ$ converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. It is numerically shown that the IPM can adapt to singular behaviors in the vanishing-noise limit. We also apply the IPM to explore situations with no explicit formulas of the vanishing-noise limit. Our paper appears to be the first one to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.

Computing large deviation rate functions of entropy production for diffusion processes by an interacting particle method

TL;DR

This work develops an interacting particle method (IPM) to compute the large-deviation rate function for entropy production in diffusion processes by approximating the principal eigenvalue of a non-self-adjoint elliptic generator via a discretized Feynman--Kac semigroup. The authors establish stability and convergence results for the time-discretized semigroup and its spectral radius, and implement an IPM that handles unbounded domains and high dimensions, using burn-in and cross- initialization to accelerate convergence. Numerical experiments demonstrate convergence to the vanishing-noise limits and in dimensions up to , and show the method's ability to capture singular behaviors and multimodal invariant measures even when explicit formulas are unavailable. The results provide a scalable tool for evaluating large-deviation rate functions in stochastic thermodynamics and related areas, with potential extensions to higher-order discretizations and error analysis.

Abstract

We develop an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient to obtain the rate function is the computation of the principal eigenvalue of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator, which is obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size. We also show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, which is suitable for computation using the IPM. The IPM applies naturally to problems in unbounded domains and scales easily to high dimensions. We show numerical examples of dimensions up to 16, and the results show that our numerical approximation of converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. It is numerically shown that the IPM can adapt to singular behaviors in the vanishing-noise limit. We also apply the IPM to explore situations with no explicit formulas of the vanishing-noise limit. Our paper appears to be the first one to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.
Paper Structure (12 sections, 3 theorems, 53 equations, 12 figures, 1 algorithm)

This paper contains 12 sections, 3 theorems, 53 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Continuous-time Feynman--Kac semigroups and large deviations
  4. Time discretization using discrete-time semigroups
  5. Interacting particle methods
  6. The empirical measure of particles at the final time
  7. Choice of the initial measure
  8. Numerical examples
  9. The principal eigenvalue and the rate function in the vanishing-noise limit
  10. Convergence tests
  11. Conclusions
  12. Proof sketches

Key Result

Proposition 2.1

Let $W(x) = \mathrm{e}^{\theta|x|^2}$ and suppose that Assumptions assumption: quadratic growth of potential--assumption: bounded velocity hold. For $\theta>0$ small enough, there exists a unique measure $\mu^{\star}_U \in \mathcal{P}(\mathbb R^d)$ with $(\mu^{\star}_U,W) < +\infty$ and a constant $ for all $\varphi \in L^{\infty}_W(\mathbb R^d)$ and $t > 0$. Moreover, the leading eigenvalue $\lam

Figures (12)

  • Figure 1: In Example \ref{['example: 2D_single_well']}, we plot our numerical approximation $\widehat{\lambda}^{\varepsilon,\alpha}_{\Delta t}$ of the principal eigenvalue $\lambda^{\varepsilon,\alpha}$ and the resulting approximation $\widehat{I}^{\varepsilon}_{\Delta t}(s)$ of the rate function $I^\varepsilon(s)$, compared respectively to the limit $\lambda^{0,\alpha}$ in \ref{['eq:lim-E1']} and its Legendre transform $I^0(s)$. Note the consistency of the symmetries mentioned in Section \ref{['sec: intro']}. Also note that the restriction of $\widehat{I}^{\varepsilon}_{\Delta t}(s)$ to certain values of $s$ is due to our restriction of $\widehat{\lambda}^{\varepsilon,\alpha}_{\Delta t}$ and how it interacts with the derivatives.
  • Figure 2: In Example \ref{['example: 2D_single_well']}, we plot the empirical density of particles at $T$ with $\alpha \approx 0.6742$. Note the concentration of the mass of the measure around $(0, 0)$ as $\varepsilon$ decreases.
  • Figure 3: In Example \ref{['example: 2D_double_well']}, we plot our numerical approximation $\widehat{\lambda}^{\varepsilon,\alpha}_{\Delta t}$ of the principal eigenvalue $\lambda^{\varepsilon,\alpha}$ and the resulting approximation $\widehat{I}^{\varepsilon}_{\Delta t}(s)$ of the rate function $I^\varepsilon(s)$, compared respectively to the limit $\lambda^{0,\alpha}$ in \ref{['eq: eigenvalue_2D_double_well']} and its Legendre transform $I^0(s)$. Note that the maximum in \ref{['eq: eigenvalue_2D_double_well']} causes a discontinuity of the derivative of the limit of the eigenvalue in $\alpha = 0$ and $\alpha = 1$, in turn causing flat regions in the limit of the rate function.
  • Figure 4: In Example \ref{['example: 2D_double_well']}, we plot the empirical density of particles at $T$ with $\alpha \approx 0.5968$. Note the concentration of the mass of the measure around $(-1, 0)$ as $\varepsilon$ decreases; no mass could be observed near $(1,0)$ at $\varepsilon = 0.001$.
  • Figure 5: In Example \ref{['example: 2D_double_well']}, we plot the empirical density of particles at $T$ with $\alpha \approx 1.0613$. Note the concentration of the mass of the measure around $(1, 0)$ as $\varepsilon$ decreases; no mass could be observed near $(-1,0)$ at $\varepsilon = 0.001$.
  • ...and 7 more figures

Theorems & Definitions (15)

  • Proposition 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Remark 3.1
  • Remark 4.1
  • Example 5.1
  • Example 5.2
  • Remark 5.1
  • Example 5.3
  • Example 5.4
  • ...and 5 more