A generalized fluctuation-response relation is found for thermal systems driven out of equilibrium. Its derivation is independent of many details of the dynamics, which is only required to be firstorder. The result gives a correction to the equilibrium fluctuation-dissipation theorem, in terms of the correlation between observable and excess in dynamical activity caused by the perturbation. Previous approaches to this problem are recovered and extended in a unifying scheme.PACS numbers: 05.70. Ln, The fluctuation-dissipation theorem is a standard chapter in statistical mechanics [1,2,3]. A system in thermal equilibrium has statistical fluctuations proportional to its response to external perturbations: a small impulse changing the potential U → U − h s V at time s, will produce a response R eq QV (t, s) = δ Q(t) /δh s in a quantity Q at time t ≥ s of the formwhere V (s) Q(t) eq is a correlation function quantifying the equilibrium fluctuations in absence of any perturbation, and the proportionality constant β = 1/k B T is the inverse temperature. An early example of this theorem is present in Einstein's treatment of Brownian motion, where the diffusion constant, expressed as a velocity autocorrelation function, is found proportional to the mobility. Other famous examples include the JohnsonNyquist formula for electronic white noise and the Onsager reciprocity for linear response coefficients. So far, approaches deriving a fluctuation-dissipation relation (FDR) for nonequilibrium [4,5,6,7,8,9,10,11,12,13] have not found a physical unification and do not appear as textbook material. One reason may be that previous work has not been seen to identify a sufficiently general structure with a clear corresponding statistical thermodynamic interpretation. Today, such an interpretation has become available from advances in dynamical fluctuation theory for nonequilibrium systems.Aiming to provide a simple and general approach to FDRs, in this Letter we put forward a FDR for nonequilibrium regimes in a framework that may represent a unifying scheme for previous formulations. Our main result can be found in a general formula, Eq. (6) below, which can also sometimes be rewritten as (7) or (11).In order to go beyond equilibrium and beyond formal perturbation theory, it is important to recognize in the right-hand side of (1) the role of entropy production, as usual governing close-to-equilibrium considerations. Eq. (1) expresses the correlation between the dissipation represented by entropy production (energy change divided by temperature) and the observable Q(t). Here, for perturbations of a nonequilibrium system (that has already a non-vanishing set of flows and hence a non-zero entropy production) we will rather speak of the excess of entropy produced by the perturbation h s V .The second ingredient that is essential in nonequilibrium is still a less known quantity, called traffic [14] or dynamical activity [15], first introduced in [16]. Perhaps activity has been somewhat overlooked in the past because it plays a truly sign...

We propose a metric to quantify correlations between earthquakes. The metric consists of a product involving the time interval and spatial distance between two events, as well as the magnitude of the first one. According to this metric, events typically are strongly correlated to only one or a few preceding ones. Thus a classification of events as foreshocks, main shocks, or aftershocks emerges automatically without imposing predetermined space-time windows. In the simplest network construction, each earthquake receives an incoming link from its most correlated predecessor. The number of aftershocks for any event, identified by its outgoing links, is found to be scale free with exponent gamma=2.0(1). The original Omori law with p=1 emerges as a robust feature of seismicity, holding up to years even for aftershock sequences initiated by intermediate magnitude events. The broad distribution of distances between earthquakes and their linked aftershocks suggests that aftershock collection with fixed space windows is not appropriate.

Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the correlation between observable and excess entropy flux yielding a relation with energy dissipation like in equilibrium. The second term comes with a new meaning: it is the correlation between the observable and the excess in dynamical activity or reactivity, playing an important role in dynamical fluctuation theory out-of-equilibrium. It appears as a generalized escape rate in the occupation statistics. The resulting response formula holds for all observables and allows direct numerical or experimental evaluation, for example in the discussion of effective temperatures, as it only involves the statistical averaging of explicit quantities, e.g. without needing an expression for the nonequilibrium distribution. The physical interpretation and the mathematical derivation are independent of many details of the dynamics, but in this first part they are restricted to Markov jump processes and overdamped diffusions

The unique fluctuation-dissipation theorem for equilibrium stands in contrast with the wide variety of nonequilibrium linear response formulae. Their most traditional approach is 'analytic', which, in the absence of detailed balance, introduces the logarithm of the stationary probability density as observable. The theory of dynamical systems offers an alternative with a formula that continues to work even when the stationary distribution is not smooth. We show that this method works equally well for stochastic dynamics, and we illustrate it with a numerical example for the perturbation of circadian cycles. A second 'probabilistic' approach starts from dynamical ensembles and expands the probability weights on path space. This line suggests new physical questions, as we meet the frenetic contribution to linear response, and the relevance of the change in dynamical activity in the relaxation to a (new) nonequilibrium condition.