Transport coefficients, such as the mobility, thermal conductivity and shear viscosity, are quantities of prime interest in statistical physics. At the macroscopic level, transport coefficients relate an external forcing of magnitude $\eta$, with $\eta \ll 1$, acting on the system to an average response expressed through some steady-state flux. In practice, steady-state averages involved in the linear response are computed as time averages over a realization of some stochastic differential equation. Variance reduction techniques are of paramount interest in this context, as the linear response is scaled by a factor of $1/\eta$, leading to large statistical error. One way to limit the increase in the variance is to allow for larger values of $\eta$ by increasing the range of values of the forcing for which the nonlinear part of the response is sufficiently small. In theory, one can add an extra forcing to the physical perturbation of the system, called synthetic forcing, as long as this extra forcing preserves the invariant measure of the reference system. The aim is to find synthetic perturbations allowing to reduce the nonlinear part of the response as much as possible. We present a mathematical framework for quantifying the quality of synthetic forcings, in the context of linear response theory, and discuss various possible choices for them. Our findings are illustrated with numerical results in low-dimensional systems.

翻译：输运系数，如迁移率、热导率和剪切粘度，是统计物理学中备受关注的核心量。在宏观层面，输运系数将大小为$\eta$（$\eta \ll 1$）且作用于系统的外部驱动与通过稳态通量表达的平均响应联系起来。在实际计算中，线性响应涉及的稳态平均值通过随机微分方程实现的时间平均来求解。在此背景下，方差缩减技术具有极其重要的意义，因为线性响应被缩放因子$1/\eta$所影响，导致统计误差显著增大。限制方差增长的一种途径是允许更大的$\eta$值，即通过增大驱动值的范围使得响应的非线性部分足够小。理论上，可以在系统的物理扰动上添加一个额外的驱动，称为合成驱动，只要该额外驱动能保持参考系统的不变测度即可。目标是找到能够最大限度减小响应非线性部分的合成扰动。我们在线性响应理论框架下，提出了量化合成驱动质量的数学方法，并讨论了多种可能的候选方案。通过在低维系统中的数值模拟结果验证了我们的发现。