arXiv 1202.10732012-02-06cond-mat.stat-mechnlin.CDphysics.chem-ph

超越线性涨落耗散定理：因果关系的作用

Beyond the linear Fluctuation-Dissipation Theorem: the Role of Causality

Valerio Lucarini, Matteo Colangeli

DOI：: 10.1088/1742-5468/2012/05/P05013

摘要：

在本文中，我们重新审视传统的问题，连接系统的内部波动，其对外部强迫的响应，并扩展了经典理论，以便能够包括非线性过程。有了这个目标，我们试图加入的结果久保统计力学系统接近平衡，即其未扰动状态可以描述的正则系综，理论的色散关系，和响应理论最近开发的Ruelle为非平衡系统配备了一个不变的SRB措施。我们的推导突出了因果关系之间的紧密联系，以及在线性和非线性水平上明确连接波动和响应的可能性。我们首先展示了在一个相当一般的设置如何的形式主义的Ruelle响应理论可以用来推导出一种新的方式Kramers-Kronig关系连接的实部和虚部的线性和非线性响应外部扰动。然后，我们提供了一个正式的扩展，在每个顺序的非线性波动耗散定理（FDT）的一般系统具有光滑不变的措施。最后，我们专注于物理相关的情况下，系统接近平衡，我们提出了明确的波动耗散关系的磁化率描述的$n^{th}$阶响应的系统与适当定义的相关性的期望值的平衡合奏中采取的$n+1$观测。虽然FDT在线性情况下具有特别紧凑的结构，但在非线性情况下，将系统波动的统计特性与其对外部扰动的响应相结合需要变量的线性变化、简单的代数和和乘法以及多重卷积积分。这些操作虽然繁琐，但可以很容易地在数字上实现。

英文摘要：

In this paper we re-examine the traditional problem of connecting the internal fluctuations of a system to its response to external forcings and extend the classical theory in order to be able to encompass also nonlinear processes. With this goal, we try to join on the results by Kubo on statistical mechanical systems close to equilibrium, i.e. whose unperturbed state can be described by a canonical ensemble, the theory of dispersion relations, and the response theory recently developed by Ruelle for non-equilibrium systems equipped with an invariant SRB measure. Our derivations highlight the strong link between causality and the possibility of connecting unambiguously fluctuation and response, both at linear and nonlinear level. We first show in a rather general setting how the formalism of the Ruelle response theory can be used to derive in a novel way Kramers-Kronig relations connecting the real and imaginary part of the linear and nonlinear response to external perturbations. We then provide a formal extension at each order of nonlinearity of the fluctuation-dissipation theorem (FDT) for general systems possessing a smooth invariant measure. Finally, we focus on the physically relevant case of systems close to equilibrium, for which we present explicit fluctuation-dissipation relations linking the susceptibility describing the $n^{th}$ order response of the system with the expectation value of suitably defined correlations of $n+1$ observables taken in the equilibrium ensemble. While the FDT has an especially compact structure in the linear case, in the nonlinear case joining the statistical properties of the fluctuations of the system to its response to external perturbations requires linear changes of variables, simple algebraic sums and multiplications, and a multiple convolution integral. These operations, albeit cumbersome, can be easily implemented numerically.

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摘要：

英文摘要：

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