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Register a new userApplications of large deviation theory in geophysical fluid dynamics and climate science
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The climate system is a complex, chaotic system with many degrees of freedom and variability on a vast range of temporal and spatial scales. Attaining a deeper level of understanding of its dynamical processes is a scientific challenge of great urgency, especially given the ongoing climate change and the evolving climate crisis. In statistical physics, complex, many-particle systems are studied successfully using the mathematical framework of Large Deviation Theory (LDT). A great potential exists for applying LDT to problems relevant for geophysical fluid dynamics and climate science. In particular, LDT allows for understanding the fundamental properties of persistent deviations of climatic fields from the long-term averages and for associating them to low-frequency, large scale patterns of climatic variability. Additionally, LDT can be used in conjunction with so-called rare events algorithms to explore rarely visited regions of the phase space and thus to study special dynamical configurations of the climate. These applications are of key importance to improve our understanding of high-impact weather and climate events. Furthermore, LDT provides powerful tools for evaluating the probability of noise-induced transitions between competing metastable states of the climate system or of its components. This in turn essential for improving our understanding of the global stability properties of the climate system and of its predictability of the second kind in the sense of Lorenz. The goal of this review is manifold. First, we want to provide an introduction to the derivation of large deviation laws in the context of stochastic processes. We then relate such results to the existing literature showing the current status of applications of LDT in climate science and geophysical fluid dynamics. Finally, we propose some possible lines of future investigations.
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In this article, we address both recent advances and open questions in some mathematical and computational issues in geophysical fluid dynamics (GFD) and climate dynamics. The main focus is on 1) the primitive equations (PEs) models and their related mathematical and computational issues, 2) climate variability, predictability and successive bifurcation, and 3) a new dynamical systems theory and its applications to GFD and climate dynamics.
In these notes we present a pedagogical account of the population dynamics methods recently introduced to simulate large deviation functions of dynamical observables in and out of equilibrium. After a brief introduction on large deviation functions and their simulations, we review the method of Giardin`a emph{et al.} for discrete time processes and that of Lecomte emph{et al.} for the continuous time counterpart. Last we explain how these methods can be modified to handle static observables and extract information about intermediate times.
The theory of large deviations constitutes a mathematical cornerstone in the foundations of Boltzmann-Gibbs statistical mechanics, based on the additive entropy $S_{BG}=- k_Bsum_{i=1}^W p_i ln p_i$. Its optimization under appropriate constraints yields the celebrated BG weight $e^{-beta E_i}$. An elementary large-deviation connection is provided by $N$ independent binary variables, which, in the $Ntoinfty$ limit yields a Gaussian distribution. The probability of having $n e N/2$ out of $N$ throws is governed by the exponential decay $e^{-N r}$, where the rate function $r$ is directly related to the relative BG entropy. To deal with a wide class of complex systems, nonextensive statistical mechanics has been proposed, based on the nonadditive entropy $S_q=k_Bfrac{1- sum_{i=1}^W p_i^q}{q-1}$ ($q in {cal R}; ,S_1=S_{BG}$). Its optimization yields the generalized weight $e_q^{-beta_q E_i}$ ($e_q^z equiv [1+(1-q)z]^{1/(1-q)};,e_1^z=e^z)$. We numerically study large deviations for a strongly correlated model which depends on the indices $Q in [1,2)$ and $gamma in (0,1)$. This model provides, in the $Ntoinfty$ limit ($forall gamma$), $Q$-Gaussian distributions, ubiquitously observed in nature ($Qto 1$ recovers the independent binary model). We show that its corresponding large deviations are governed by $e_q^{-N r_q}$ ($propto 1/N^{1/(q-1)}$ if $q>1$) where $q= frac{Q-1}{gamma (3-Q)}+1 ge 1$. This $q$-generalized illustration opens wide the door towards a desirable large-deviation foundation of nonextensive statistical mechanics.
The standard Large Deviation Theory (LDT) represents the mathematical counterpart of the Boltzmann-Gibbs factor which describes the thermal equilibrium of short-range Hamiltonian systems, the velocity distribution of which is Maxwellian. It is generically applicable to systems satisfying the Central Limit Theorem (CLT). When we focus instead on stationary states of typical complex systems (e.g., classical long-range Hamiltonian systems), both the CLT and LDT need to be generalized. Specifically, when the N->infinity attractor in the space of distributions is a Q-Gaussian related to a Q-generalized CLT (Q=1 recovers Gaussian attractors), we expect the LDT probability distribution to approach a q-exponential (where q=f(Q) with f(1)=1, thus recovering the standard LDT exponential distribution) with an argument proportional to N, consistently with thermodynamics. We numerically verify this conjectural scenario for the standard map, the coherent noise model for biological extinctions and earthquakes, the Ehrenfest dog-flea model, and the random-walk avalanches.
The fluctuation-dissipation theorem is a central result in statistical mechanics and is usually formulated for systems described by diffusion processes. In this paper, we propose a generalization for a wider class of stochastic processes, namely the class of Markov processes that satisfy detailed balance and a large-deviation principle. The generalized fluctuation-dissipation theorem characterizes the deterministic limit of such a Markov process as a generalized gradient flow, a mathematical tool to model a purely irreversible dynamics via a dissipation potential and an entropy function: these are expressed in terms of the large-deviation dynamic rate function of the Markov process and its stationary distribution. We exploit the generalized fluctuation-dissipation theorem to develop a new method of coarse-graining and test it in the context of the passage from the diffusion in a double-well potential to the jump process that describes the simple reaction $A rightleftarrows B$ (Kramers escape problem).
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