Jung Gerhard, Schmid Friederike
Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria.
Institut für Physik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany.
Soft Matter. 2021 Jul 7;17(26):6413-6425. doi: 10.1039/d1sm00521a.
Fluctuation-dissipation relations or "theorems" (FDTs) are fundamental for statistical physics and can be rigorously derived for equilibrium systems. Their applicability to non-equilibrium systems is, however, debated. Here, we simulate an active microrheology experiment, in which a spherical colloid is pulled with a constant external force through a fluid, creating near-equilibrium and far-from-equilibrium systems. We characterize the structural and dynamical properties of these systems, and reconstruct an effective generalized Langevin equation (GLE) for the colloid dynamics. Specifically, we test the validity of two FDTs: The first FDT relates the non-equilibrium response of a system to equilibrium correlation functions, and the second FDT relates the memory friction kernel in the GLE to the stochastic force. We find that the validity of the first FDT depends strongly on the strength of the external driving: it is fulfilled close to equilibrium and breaks down far from it. In contrast, we observe that the second FDT is always fulfilled. We provide a mathematical argument why this generally holds for memory kernels reconstructed from a deterministic Volterra equation for correlation functions, even for non-stationary non-equilibrium systems. Motivated by the Mori-Zwanzig formalism, we therefore suggest to impose an orthogonality constraint on the stochastic force, which is in fact equivalent to the validity of this Volterra equation. Such GLEs automatically satisfy the second FDT and are unique, which is desirable when using GLEs for coarse-grained modeling.
涨落耗散关系或“定理”（FDTs）是统计物理学的基础，并且可以针对平衡系统进行严格推导。然而，它们在非平衡系统中的适用性存在争议。在此，我们模拟了一个主动微观流变学实验，其中一个球形胶体在恒定外力作用下被拉过一种流体，从而创建了近平衡和远离平衡的系统。我们表征了这些系统的结构和动力学性质，并为胶体动力学重建了一个有效的广义朗之万方程（GLE）。具体而言，我们测试了两个FDTs的有效性：第一个FDT将系统的非平衡响应与平衡关联函数联系起来，第二个FDT将GLE中的记忆摩擦核与随机力联系起来。我们发现第一个FDT的有效性强烈依赖于外部驱动的强度：它在接近平衡时成立，而在远离平衡时失效。相比之下，我们观察到第二个FDT始终成立。我们给出了一个数学论证，说明为什么这对于从关联函数的确定性沃尔泰拉方程重建的记忆核一般成立，即使对于非平稳非平衡系统也是如此。受森 - 万兹格形式主义的启发，我们因此建议对随机力施加一个正交性约束，这实际上等同于这个沃尔泰拉方程的有效性。这样的GLEs自动满足第二个FDT并且是唯一的，这在使用GLEs进行粗粒化建模时是很理想的。
