Nonequilibrium ensemble averages using nonlinear response relations

Abstract

The transient time correlation function (TTCF) method is widely used in molecular fluids to compute non-equilibrium transport quantities, providing improved signal-to-noise ratios in ensemble averages without requiring prohibitively large sample sizes. In spite of its success in molecular and turbulent fluid systems, the method has not been systematically explored for more general non-equilibrium dynamical systems, including geophysical applications where the invariant measure is typically unknown. In this work, we present an analytical and numerical investigation of the TTCF method for computing nonlinear response functions in systems far from equilibrium. We discuss its relation to the spectral theory of stochastic systems, highlighting regimes where linear theory is insufficient and the advantages of TTCF. The aim of this work is to provide a framework for studying transient and steady-state responses using the TTCF approach in a broad class of nonequilibrium systems.

Figures (4)

• Figure 1: In blue we show the direct averages of the response of $\Psi_i$ relative to perturbations in $\varepsilon$ in Eq. \ref{['eq:ou1d']}. In red, we show the TTCF calculations. The observable analysed and number of ensemble members is indicated in the titles of each panel. The black lines are the ground truth computed using the Fokker-Planck equation associated with Eq. \ref{['eq:ou1d']}.
• Figure 2: Each panel shows the same comparison of direct averages and TTCF as Fig. \ref{['fig:4']}, but this time for Eq. \ref{['eq:ou_2d']} and different values of both $N$ and $b$. Here, we show the response of the observable $\Psi(\mathbf{x})=x_1$, in the system Eq. \ref{['eq:ou_2d']} with respect to the forcing $\varepsilon\mathbf{f}$. Each row corresponds, respectively, to $b=0.5,1.5$ and $5$ and each column corresponds to $N=50,500$ and $5000$. The parameters are selected to be: $a=-1$, $\sigma=0.4$, $\mathbf{f}=(1,1)^{\ast}$ and $\varepsilon = 0.1$
• Figure 3: Direct averages vs TTCF: $\varepsilon=0.1$. We show the response of the observable $\Psi_j$ defined Eq. \ref{['eq:observable_l96']} in response to $F\mapsto F +\varepsilon$. In blue, we show the direct averages, in yellow, the TTCF estimation using a Gaussian approximation and in green, the TTCF estimation employing the KDMD method. The number of ensemble members is indicated in the titles together with the forcing strength $\varepsilon$. The black lines correspond to the ground-truth computed from direct averages using $10^7$ independent ensemble members.
• Figure 4: Direct averages vs TTCF: $\varepsilon=0.1$. We show the response of the observable $\Psi_j$ defined Eq. \ref{['eq:observable_l96']} that results form $F\mapsto F +\varepsilon$. In blue, we show the direct averages, in yellow, the TTCF estimation using a Gaussian approximation and in green, the TTCF estimation employing the Kernel method together with the forcing strength $\varepsilon$. The number of ensemble members is indicated in the titles. The black lines correspond to the ground-truth computed from direct averages using $10^7$ independent ensemble members.