Multivariate truncated moments problems and maximum entropy

We characterize the existence of the Lebesgue integrable solutions of the truncated problem of moments in several variables on unbounded supports by the existence of some maximum entropy -- type representing densities and discuss a few topics on their approximation in a particular case, of two variables and 4th order moments.Comment: Revised version, to appear in Analysis and Mathematical Physic

Response operators for Markov processes in a finite state space: radius of convergence and link to the response theory for Axiom A systems

Using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state Markov processes. The results can be used for studying empirically constructed—e.g. from observations or through coarse graining of model simulations—finite state approximation of statistical mechanical systems. Recent results concerning the convergence of the statistical properties of finite state Markov approximation of the full asymptotic dynamics on the SRB measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of Axiom A system. Our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language, plus providing bounds on the radius of convergence of the perturbative theory. In particular, we relate the convergence of the response theory to the rate of mixing of the unperturbed system. One can use the formulas derived for finite state Markov processes to recover previous findings obtained on the response of continuous time Axiom A dynamical systems to perturbations, by considering the generator of time evolution for the measure and for the observables. A very basic, low-tech, and computationally cheap analysis of the response of the Lorenz ’63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state Markov processes to compute the system’s response

Mechanics and thermodynamics of a new minimal model of the atmosphere

The understanding of the fundamental properties of the climate system has long benefitted from the use of simple numerical models able to parsimoniously represent the essential ingredients of its processes. Here, we introduce a new model for the atmosphere that is constructed by supplementing the now-classic Lorenz ’96 one-dimensional lattice model with temperature-like variables. The model features an energy cycle that allows for energy to be converted between the kinetic form and the potential form and for introducing a notion of efficiency. The model’s evolution is controlled by two contributions—a quasi-symplectic and a gradient one, which resemble (yet not conforming to) a metriplectic structure. After investigating the linear stability of the symmetric fixed point, we perform a systematic parametric investigation that allows us to define regions in the parameters space where at steady-state stationary, quasi-periodic, and chaotic motions are realised, and study how the terms responsible for defining the energy budget of the system depend on the external forcing injecting energy in the kinetic and in the potential energy reservoirs. Finally, we find preliminary evidence that the model features extensive chaos. We also introduce a more complex version of the model that is able to accommodate for multiscale dynamics and that features an energy cycle that more closely mimics the one of the Earth’s atmosphere

The Effects of Irreversible Electroporation (IRE) on Nerves

Background: If a critical nerve is circumferentially involved with tumor, radical surgery intended to cure the cancer must sacrifice the nerve. Loss of critical nerves may lead to serious consequences. In spite of the impressive technical advancements in nerve reconstruction, complete recovery and normalization of nerve function is difficult to achieve. Though irreversible electroporation (IRE) might be a promising choice to treat tumors near or involved critical nerve, the pathophysiology of the nerve after IRE treatment has not be clearly defined. Methods: We applied IRE directly to a rat sciatic nerve to study the long term effects of IRE on the nerve. A sequence of 10 square pulses of 3800 V/cm, each 100 ms long was applied directly to rat sciatic nerves. In each animal of group I (IRE) the procedure was applied to produce a treated length of about 10 mm. In each animal of group II (Control) the electrodes were only applied directly on the sciatic nerve for the same time. Electrophysiological, histological, and functional studies were performed on immediately after and 3 days, 1 week, 3, 5, 7 and 10 weeks following surgery. Findings: Electrophysiological, histological, and functional results show the nerve treated with IRE can attain full recovery after 7 weeks. Conclusion: This finding is indicative of the preservation of nerve involving malignant tumors with respect to the application of IRE pulses to ablate tumors completely. In summary, IRE may be a promising treatment tool for any tumor involving nerves

Predicting climate change using response theory: global averages and spatial patterns

The provision of accurate methods for predicting the climate response to anthropogenic and natural forcings is a key contemporary scientific challenge. Using a simplified and efficient open-source general circulation model of the atmosphere featuring O(105105) degrees of freedom, we show how it is possible to approach such a problem using nonequilibrium statistical mechanics. Response theory allows one to practically compute the time-dependent measure supported on the pullback attractor of the climate system, whose dynamics is non-autonomous as a result of time-dependent forcings. We propose a simple yet efficient method for predicting—at any lead time and in an ensemble sense—the change in climate properties resulting from increase in the concentration of CO22 using test perturbation model runs. We assess strengths and limitations of the response theory in predicting the changes in the globally averaged values of surface temperature and of the yearly total precipitation, as well as in their spatial patterns. The quality of the predictions obtained for the surface temperature fields is rather good, while in the case of precipitation a good skill is observed only for the global average. We also show how it is possible to define accurately concepts like the inertia of the climate system or to predict when climate change is detectable given a scenario of forcing. Our analysis can be extended for dealing with more complex portfolios of forcings and can be adapted to treat, in principle, any climate observable. Our conclusion is that climate change is indeed a problem that can be effectively seen through a statistical mechanical lens, and that there is great potential for optimizing the current coordinated modelling exercises run for the preparation of the subsequent reports of the Intergovernmental Panel for Climate Change

Beyond forcing scenarios: predicting climate change through response operators in a coupled general circulation model

Global Climate Models are key tools for predicting the future response of the climate system to a variety of natural and anthropogenic forcings. Here we show how to use statistical mechanics to construct operators able to flexibly predict climate change for a variety of climatic variables of interest. We perform our study on a fully coupled model - MPI-ESM v.1.2 - and for the first time we prove the effectiveness of response theory in predicting future climate response to CO2 increase on a vast range of temporal scales, from inter-annual to centennial, and for very diverse climatic quantities. We investigate within a unified perspective the transient climate response and the equilibrium climate sensitivity and assess the role of fast and slow processes. The prediction of the ocean heat uptake highlights the very slow relaxation to a newly established steady state. The change in the Atlantic Meridional Overturning Circulation (AMOC) and of the Antarctic Circumpolar Current (ACC) is accurately predicted. The AMOC strength is initially reduced and then undergoes a slow and only partial recovery. The ACC strength initially increases as a result of changes in the wind stress, then undergoes a slowdown, followed by a recovery leading to a overshoot with respect to the initial value. Finally, we are able to predict accurately the temperature change in the Northern Atlantic

Analysis of a bistable climate toy model with physics-based machine learning methods

We propose a comprehensive framework able to address both the predictability of the first and of the second kind for high-dimensional chaotic models. For this purpose, we analyse the properties of a newly introduced multistable climate toy model constructed by coupling the Lorenz ’96 model with a zero-dimensional energy balance model. First, the attractors of the system are identified with Monte Carlo Basin Bifurcation Analysis. Additionally, we are able to detect the Melancholia state separating the two attractors. Then, Neural Ordinary Differential Equations are applied to predict the future state of the system in both of the identified attractors

Introduction to the special issue on the statistical mechanics of climate

We introduce the special issue on the Statistical Mechanics of Climate by presenting an informal discussion of some theoretical aspects of climate dynamics that make it a topic of great interest for mathematicians and theoretical physicists. In particular, we briefly discuss its nonequilibrium and multiscale properties, the relationship between natural climate variability and climate change, the different regimes of climate response to perturbations, and critical transitions

Beyond equilibrium climate sensitivity

ISSN:1752-0908ISSN:1752-089