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Anion Trapping and Ionic Conductivity Enhancement in PEO-Based Composite Polymer-Li<inf>7</inf>La<inf>3</inf>Zr<inf>2</inf>O<inf>12</inf> Electrolytes: The Role of the Garnet Li Molar Content
No full textThe successful development of all-solid-state batteries will provide solutions for many problems facing current Li-ion batteries, such as high flammability, limited energy density, poor cyclability and low cation transference number. In this quest, the development of high-performance solid-state electrolytes is critical. Composite polymer electrolytes (CPE), comprising ion-conducting (active) inorganic fillers and polymer matrices, have emerged as a promising strategy to yield better conductivity, interfacial stability, and mechanical strength than their single-phase counterparts. Recent experiments indicate that active garnet fillers may enhance the ionic conductivity of CPEs by inducing anion trapping onto their surface. Moreover, substitutions that modify the lithium molar content within the filler were shown to impact this enhancement. However, the molecular underpinning behind this phenomenon is poorly understood, hindering the development of strategies to exploit it optimally. In this study, we use an enhanced hybrid Monte Carlo technique in combination with extensive molecular dynamics simulations to bridge this gap. By focusing on the archetypal CPE formed by Ga-doped Li7-3xGaxLa3Zr2O12 (Gax-LLZO) embedded within a poly(ethylene oxide) (PEO) and lithium bis(trifluoromethane sulfonyl) imide (LiTFSI) polymer matrix, we describe how the dynamic electrostatic trapping of anions leads to overall conductivity enhancement by increasing the lithium transference number and tracer diffusivity in the polymer phase. The extent of this enhancement can be fine-tuned by modulating the Li molar content of LLZO through the doping of Ga. We predict an optimal Li molar content of 5.95, which is lower than the optimal 6.50 reported in the literature for single LLZO
WEIGHTED LORENTZ SPACES: SHARP MIXED Ap − A∞ ESTIMATE FOR MAXIMAL FUNCTIONS
Get PDFWe prove the sharp mixed Ap − A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely
11 ∥M∥ p,q ≲p,q,n [w]p [σ]min(p,q) ,
L (w) Ap A∞
1
where σ = w 1−p . Our method is rearrangement free and can also be
used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting
C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics
Get PDFThe present book grew from lecture notes we have written for participants of lectures on applications of C^{∗}-algebra theory to the foundations of quantum statistical mechanics, as well as a mini-course on thermodynamic equilibrium of quantum lattice systems with mean-field interactions, we held at the Institute of Physics of the University of São Paulo and at the Basque Center for Applied Mathematics (BCAM), in the last few years. In both cases the audience was rather heterogeneous, composed by students at graduate and undergraduate level, from the Institute of Physics, the Institute of Mathematics and Statistics of University of São Paulo and the BCAM. Most participants from the Institute of Physics had only very modest previous knowledge on fundamental mathematical disciplines like analysis, topology and functional analysis. Thus, it was necessary to provide friendly and self-contained material, in order to allow them to follow the main ideas presented. In this sense, one important feature of our book is a good compromise between conceptual depth and technical simplicity. In fact, our book is mainly addressed to students stemming from physics departments, who are interested in mathematical foundations of physics and aim at studying physical theories in a mathematically rigorous way. From the point of view of a graduate student in mathematics, a considerable part of the material presented here is rather elementary, but, in contrast to many textbooks in mathematics, we systematically discuss the physical significance of each single (abstract) mathematical structure we use. Thus, mathematicians interested in quantum statistical physics can use our book as a quick introduction to the subject, written in a language that they easily understand, in particular specialists from the domains of C^{∗}-algebras and convex analysis. Based on our experience as supervisors of master and PhD theses of students from physics departments, there is a lack of a textbook on the subject addressed to students that we refer to above. There are indeed excellent books available (for instance, by Bratteli and Robinson, Israel, Simon, and others), but they require close acquaintance with different mathematical disciplines, what is very frequently not the case for those students. We thus aim, among other things, at bringing physics students to a sufficient level to fruitfully read those (nowadays) classical books. In other words, the present work has a propaedeutic character. Notice, additionally, that these classical books are aged of about forty years and do not cover (at least not in a systematic way) cases that are nowadays of great relevance in research, like interacting fermions and mean-field models. In fact, the "working case" of previous books are the quantum spin lattices. Our book will close this gap by presenting interacting fermions and mean-field models on the same theoretical ground as the quantum spin lattices of previous works. However, we do not discuss bosonic systems, for they are, from a technical point of view, quite peculiar and do not fit naturally in our setting. Another technical novelty of our exposition is that we present the C^{∗}-algebras for fermions and quantum spins in the context of universal C^{∗}-algebras of polynomial relations. This allows us, in particular, to construct all important algebra (∗-)automorphisms, related to physical symmetries of the systems under consideration (like space translations, gauge and parity transformations, Bogoliubov automorphisms, etc.), in a technically simple and conceptually transparent way.CNPq (309723/2020-5). Project of Basque Government through the grant IT1615-22. COST Action CA18232 financed by the European Cooperation in Science and Technology (COST). Project PID2020-112948GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe"
From short-range to mean-field models in quantum lattices
Get PDFRealistic effective interparticle interactions of quantum many-body systems are widely seen as being short-range. However, the rigorous mathematical analysis of this type of model turns out to be extremely difficult, in general, with many important fundamental questions remaining open still nowadays. By contrast, mean-field models come from different approximations or Ansätze, and are thus less realistic, in a sense, but are technically advantageous, by allowing explicit computations while capturing surprisingly well many real physical phenomena. Here, we establish a precise mathematical relation between mean-field and short-range models, by using the long-range limit that is known in the literature as the Kac, or van der Waals, limit. If both attractive and repulsive longrange forces are present then it turns out that the limit mean-field model is not necessarily what one traditionally guesses. One important innovation of our study, in contrast with previous works on the subject, is the fact that we are able to show the convergence of equilibrium states, i.e., of all correlation functions. This paves the way for studying phase transitions, or at least important fingerprints of them like strong correlations at long distances, for models having interactions whose ranges are finite, but very large. It also sheds a new light on mean-field models. Even on the level of pressures, our results go considerably further than previous ones, by allowing, for instance, a continuum of long-range interaction components, as well as very general short-range Hamiltonians for the “free” part of the model. The present results were made possible by the variational approach of [1] for equilibrium states of mean-field models, as well as the game theoretical characterization of these states. Our results are obtained in an abstract, model-independent, way.CNPq (309723/2020-5). Project of Basque Government through the grant IT1615-22. COST Action CA18232 financed by the European Cooperation in Science and Technology (COST). Project PID2020-112948GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe"
Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
Get PDFWe establish an asymptotic formula for the number of lattice points in the sets Sh1,h2,h3(λ):={x∈Z+3:⌊h1(x1)⌋+⌊h2(x2)⌋+⌊h3(x3)⌋=λ} with λ∈Z+; where functions h1, h2, h3 are constant multiples of regularly varying functions of the form h(x) : = xcℓh(x) , where the exponent c> 1 (but close to 1) and a function ℓh(x) is taken from a certain wide class of slowly varying functions. Taking h1(x) = h2(x) = h3(x) = xc we will also derive an asymptotic formula for the number of lattice points in the sets Sc3(λ):={x∈Z3:⌊|x1|c⌋+⌊|x2|c⌋+⌊|x3|c⌋=λ}withλ∈Z+;which can be thought of as a perturbation of the classical Waring problem in three variables. We will use the latter asymptotic formula to study, the main results of this paper, norm and pointwise convergence of the ergodic averages 1#Sc3(λ)∑n∈Sc3(λ)f(T1n1T2n2T3n3x)asλ→∞;where T1, T2, T3: X→ X are commuting invertible and measure-preserving transformations of a σ-finite measure space (X, ν) for any function f∈ Lp(X) with p>11-4c11-7c. Finally, we will study the equidistribution problem corresponding to the spheres Sc3(λ).Foundation for Polish Science via the START Scholarship,
the Juan de la Cierva Incorporaci´on 2019, grant number IJC2019-039661-I, the
Agencia Estatal de Investigaci´on, grant PID2020-113156GB-I00/AEI/10.13039/501100011033,
the Basque Government through the BERC 2022-2025 program,
and by the Spanish Ministry of Sciences, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718
Effects of optogenetic and visual stimulation on gamma activity in the visual cortex
Get PDFStudying brain functions and activity during gamma oscillations can be a challenge because it requires careful planning to create the necessary conditions for a controlled experiment. Such an experiment consists of placing the brain into a gamma state and investigating cognitive processing with a careful design. Cortical oscillations in the gamma frequency range (30–80 Hz) play an essential role in a variety of cognitive processes, including visual processing and cognition. The present study aims to investigate the effects of a visual stimulus on the primary visual cortex under gamma oscillations. Specifically, we sought to explore the behavior of gamma oscillations triggered by optogenetic stimulation in the II and IV layers of the visual cortex, both with and without concurrent visual stimulation. Our results show that optogenetic stimulation increases the power of gamma oscillation in both layers of the visual cortex. However, the combined stimuli resulted in a reduction of gamma power in layer II and an increase and reinforcement in gamma power in layer IV. Modelling the results with the Wilson-Cowan model suggests changes in the input of the excitatory population due to the combined stimuli. In addition, our analysis of the data using the Lempel-Ziv complexity method supports our interpretations from the modeling. Thus, our results suggest that optogenetic stimulation enhances low gamma power in both layers of the visual cortex, while simultaneous visual stimulation has differing effects on the two layers, reducing gamma power in layer II and increasing it in layer IV.Elkartek project SILICON BURMUIN no. KK-2023/00090
An ETD method for multi-asset American option pricing under jump-diffusion model
Get PDFIn this paper, we propose a numerical method for American multi-asset options under jump-diffusion model based on the combination of the exponential time differencing (ETD) technique for the differential operator and Gauss–Hermite quadrature for the integral term. In order to simplify the computational sten- cil and improve characteristics of the ETD-scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods
Numerical simulations of Brownian suspensions using Smoothed Dissipative Particle Dynamics: Diffusion, rheology and microstructure
No full textIn this work, a Smoothed Dissipative Particle Dynamics (SDPD) model is presented to simulate dilute-to-concentrated colloidal suspensions with a Newtonian matrix. The Brownian solvent medium is simulated explicitly with the SDPD model and it mediates the long-range fluctuating hydrodynamic interaction between suspended particles. To account for the short-range lubrication interactions, interparticle correction terms are included. In particular, in the Brownian regime the GENERIC framework is used to introduce the stochastic contribution to the lubrication, complying the Fluctuation-Dissipation Theorem. The resulting stochastic equations are solved implicitly by using a splitting technique. The SDPD scheme allows to accurately and efficiently simulate dilute to highly concentrated Brownian suspensions of spheres. Diffusivity, rheology and microstructure of the Brownian particulate system are discussed and compared with previous simulations and experimental results. The effect of the stochastic lubrication model is also analyzed and discussed
