Motivated by stochastic models of climate phenomena, the steady-state of a linear stochastic model with additive Gaussian white noise is studied. Fluctuation theorems for nonequilibrium steady-states provide a constraint on the character of these fluctuations. The properties of the fluctuations which are unconstrained by the fluctuation theorem are investigated and related to the model parameters. The irreversibility of trajectory segments, which satisfies a fluctuation theorem, is used as a measure of nonequilibrium fluctuations. The moments of the irreversibility probability density function (pdf) are found and the pdf is seen to be non-Gaussian. The average irreversibility goes to zero for short and long trajectory segments and has a maximum for some finite segment length, which defines a characteristic timescale of the fluctuations. The initial average irreversibility growth rate is equal to the average entropy production and is related to noise-amplification. For systems with a separation of deterministic timescales, modes with timescales much shorter than the trajectory timespan and whose noise amplitudes are not asymptotically large, do not, to first order, contribute to the irreversibility statistics, providing a potential basis for dimensional reduction.
Deep Dive into Fluctuation Properties of Steady-State Langevin Systems.
Motivated by stochastic models of climate phenomena, the steady-state of a linear stochastic model with additive Gaussian white noise is studied. Fluctuation theorems for nonequilibrium steady-states provide a constraint on the character of these fluctuations. The properties of the fluctuations which are unconstrained by the fluctuation theorem are investigated and related to the model parameters. The irreversibility of trajectory segments, which satisfies a fluctuation theorem, is used as a measure of nonequilibrium fluctuations. The moments of the irreversibility probability density function (pdf) are found and the pdf is seen to be non-Gaussian. The average irreversibility goes to zero for short and long trajectory segments and has a maximum for some finite segment length, which defines a characteristic timescale of the fluctuations. The initial average irreversibility growth rate is equal to the average entropy production and is related to noise-amplification. For systems with a se
Recent advances in nonequilibrium statistical mechanics have investigated fluctuation theorems in a variety of contexts [1]. The fluctuation theorem quantifies the probability of finding fluctuations in nonequilibrium systems that violate the Second Law of Thermodynamics. Fluctuation theorems take many forms. The formulation we will focus on is in terms of the probability of observing finite time trajectory segments of a system [2,3]. In this context, the fluctuation theorem provides a constraint that such trajectory segments must satisfy. Here we investigate the fluctuations in a nonequilibrium steadystate governed by Langevin dynamics. We go beyond the fluctuation theorem and study those properties of the fluctuations that are not constrained by the fluctuation theorem. These properties are not generic. They depend on the details of the specific dynamical system, and we investigate the relationship between the nonequilibrium fluctuations and the parameters defining the Langevin dynamics.
Our motivation for studying specific details of nonequilibrium fluctuations comes from work in theoretical climate dynamics. In recent years, linear stochastic dynamical systems have been successfully used to model many phenomena in the climate system such as El-Niño [4,5,6], the North Atlantic Gulf Stream [7], and a variety of atmospheric phenomena [8,9,10,11,12]. We shall refer to these phenomena as climate subsystems in that they are often considered to be dynamical systems that are separable from the larger climate system, at least on some set of spatial and temporal scales. Since these fluctuations have macroscopic timescales, it is important to * Electronic address: jeffrey.weiss@colorado.edu investigate the character of individual fluctuations and the statistics of their properties.
In the work on climate subsystems, the focus has been on two considerations: the utility of linear stochastic systems in forecasting [4,6,13], and the potential for the deterministic part of the dynamics to amplify the random noise [6,14,15,16,17]. It is often assumed that the large amplitudes of these phenomena requires them to be the result of dynamical instabilities. The recognition that deterministic dynamics can amplify small noise forcing, which in meteorology goes back to Lorenz [18], provides an alternative view of such phenomena. This amplification occurs when the deterministic matrix is non-normal [14,15]. One common critique of the noise-amplification view is that non-normality and the resulting amplification depends on the subjective choice of coordinate system, and can be removed by an appropriate coordinate transformation. Recently this objection has been answered by noting that underlying the property of nonnormality is the more fundamental, coordinate invariant property of detailed balance. Linear stochastic climate subsystem models share the property that they violate detailed balance, and this is what is responsible for the noise amplification [17]. Thus, a wide range of phenomena in the climate system can be interpreted as fluctuations in a nonequilibrium steady-state. For climate fluctuations such as El-Niño, understanding the character of the fluctuations is extremely important. Further, due to global warming, the steady-state is changing. Understanding how phenomena such as El-Niño will change as climate changes is a major uncertainty in climate change predictions [19]. Thus, improved understanding of how nonequilibrium fluctuations depend on the properties of the steady-state could lead to improvements in climate change forecasts.
As a concrete example, we will focus on El-Niño. El-Niño is a coupled atmosphere-ocean phenomenon that is centered in the tropical Pacific Ocean and has global impacts. One key aspect of El-Niño is that the atmosphere evolves on a faster timescale than the ocean. The turbulent dynamics of the atmosphere has a predictability limit of about two weeks [20]. The ocean, on the other hand, has timescales of months. Thus, on the monthly timescale of El-Niño, the atmosphere is unpredictable and can be considered a random forcing [21]. While this parameterization of fast chaos as random noise is typically done empirically, there are some theoretical results [22,23,24,25].
The phenomenon of El-Niño is described in terms of the sea surface temperature (SST) of the tropical Pacific. Although the SST is a continuous field, both observations and models use a finite number N of SST values. Thus, the state vector x of the system is an N -dimensional vector of real numbers representing the discretized SST field. Often, the dimensionality is reduced by truncating to some number of leading modes. Typically, the mean SST is removed, so x represents the SST anomaly and can be positive or negative. A linear stochastic Langevin model for El-Niño is then
where A is an N × N real matrix representing the linear deterministic dynamics, F is an N × N real matrix representing the noise forcing, ξ is N dimension
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