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Linear approximations of large deviations: Cubic diffusion test

Pelerine Tsobgni Nyawo, Hugo Touchette

TL;DR

This work addresses the challenge of computing large-deviation rate functions for time-integrated observables of diffusion processes. It introduces a linear, Gaussian approximation of the effective drift, $\bar{F}(x) = -\alpha (x-\beta)$, to bound the rate function $I(a)$ and the SCGF $\lambda(k)$, yielding tractable expressions and an optimization framework for the two parameters $\alpha$ and $\beta$ that depend on the fluctuation level. In a one-dimensional cubic-drift test, the method accurately captures the tails of both $\lambda(k)$ and $I(a)$, with accuracy linked to how concentrated the effective process is around its fixed point; optimal parameters align well with the local linearization around that point. The approach provides a fast, interpretable first-approximation for large deviations, offering bounds and practical guidance for more sophisticated methods, and it lays the groundwork for extensions to higher-dimensional diffusions and more complex observables.

Abstract

We propose a method for approximating the large deviation rate function of time-integrated observables of diffusion processes, used in statistical physics to characterize the fluctuations of nonequilibrium systems. The method is based on linearizing the effective process associated with the large deviations of the process and observable considered, and is tested for a simple one-dimensional nonlinear diffusion model involving a cubic drift. The results show that the linear approximation compares well with the exact rate function, especially in the large fluctuation regime, and that its accuracy is related to the way the linearized process localizes in space. Possible extensions and applications to more complex diffusion models are proposed for future work.

Linear approximations of large deviations: Cubic diffusion test

TL;DR

This work addresses the challenge of computing large-deviation rate functions for time-integrated observables of diffusion processes. It introduces a linear, Gaussian approximation of the effective drift, , to bound the rate function and the SCGF , yielding tractable expressions and an optimization framework for the two parameters and that depend on the fluctuation level. In a one-dimensional cubic-drift test, the method accurately captures the tails of both and , with accuracy linked to how concentrated the effective process is around its fixed point; optimal parameters align well with the local linearization around that point. The approach provides a fast, interpretable first-approximation for large deviations, offering bounds and practical guidance for more sophisticated methods, and it lays the groundwork for extensions to higher-dimensional diffusions and more complex observables.

Abstract

We propose a method for approximating the large deviation rate function of time-integrated observables of diffusion processes, used in statistical physics to characterize the fluctuations of nonequilibrium systems. The method is based on linearizing the effective process associated with the large deviations of the process and observable considered, and is tested for a simple one-dimensional nonlinear diffusion model involving a cubic drift. The results show that the linear approximation compares well with the exact rate function, especially in the large fluctuation regime, and that its accuracy is related to the way the linearized process localizes in space. Possible extensions and applications to more complex diffusion models are proposed for future work.

Paper Structure

This paper contains 5 sections, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Large deviation problem
  3. Linear approximation
  4. Accuracy assessment
  5. Conclusion

Figures (3)

  • Figure 1: (a) SCGF $\lambda(k)$ (black) compared with the linear approximation $\bar{\lambda}(k)$ (red). Inset: Error $\lambda(k)-\bar{\lambda}(k)$. (b) Rate function $I(a)$ (black) compared with the linear approximation $\bar{I}(a)$ (red). Inset: Error $\bar{I}(a)-I(a)$. (c) Effective drift $\tilde{F}_k(x)$ for different values of $k$ compared with the original drift $F(x)$ in black.
  • Figure 2: Optimal parameters of the linear approximation $\bar{F}(x)$ compared with the linearization of the effective drift $\tilde{F}_k(x)$. (a) Fixed point $x_k^*$ of $\tilde{F}_k(x)$ compared with the center $\beta_k$. The blue curve shows the minimum of $V_k(x)$. (b) Local friction $\gamma_k=-F'_k(x_k^*)$ compared with the friction $\alpha_k$.
  • Figure 3: (a) Variance of $p_k^*(x)$ as a function of $k$. (b) Relation between the error $\lambda(k)-\bar{\lambda}(k)$ and the variance, plotted parametrically in $k$.