arXiv subjects

Stefan Adams

Publications and source records attributed to Stefan Adams.

At least 19 recordsLinked to original sources

arXiv2024

Large systems of symmetrized trapped Brownian Bridges and Schrodinger processes

Consider a large system of $N$ Brownian motions in $\R ^d$ fixed on a time interval $[0,\beta]$ with symmetrized initial and terminal conditions, under the influence of a trap potential. Such systems describe systems of bosons at positive temperatures confined in a spatial domain. We describe the large $N$ behavior of the averaged path (that is, their empirical path measure) and its connection with a well known optimal transport problem formulated by Erwin Schr\"odinger. We also explore the asymptotic behavior of the Brownian motions in terms of Large Deviations. In particular, the rate function that governs the mean of occupation measures turns out to be the well-known Donsker-Varadhan rate function. We therefore prove a simple formula for the large $N$ asymptotic of the symmetrized trace of $e^{-\beta \Hcal_N}$, where $\Hcal_N$ is an $N$ particle Hamilton operator in a trap

math.PR↗

arXiv2023

Characterising the infinite volume scaling limit of gradient models with non-convex energy

We study the scaling limit of statistical mechanics models with non-convex Hamiltonians that are gradient perturbations of Gaussian measures. Characterising features of our gradient models are the imposed boundary tilt and the surface tension (free energy) as a function of tilt. In the regime of low temperatures and bounded tilt, we prove the scaling limit with respect to infinite volume Gibbs states for macroscopic functions on the continuum, and we show that the limit is a continuum Gaussian Free Field with covariance (diffusion) matrix given as the Hessian of surface tension. Our proof of this longstanding conjecture for non-convex energy complements recent studies in [Hil16, ABKM], as well as the proof for strictly convex Hamiltonians in [AW22].

math.PR↗

arXiv2021

Lithium-ion Battery State of Health Estimation based on Cycle Synchronization using Dynamic Time Warping

The state of health (SOH) estimation plays an essential role in battery-powered applications to avoid unexpected breakdowns due to battery capacity fading. However, few studies have paid attention to the problem of uneven length of degrading cycles, simply employing manual operation or leaving to the automatic processing mechanism of advanced machine learning models, like long short-term memory (LSTM). As a result, this causes information loss and caps the full capability of the data-driven SOH estimation models. To address this challenge, this paper proposes an innovative cycle synchronization way to change the existing coordinate system using dynamic time warping, not only enabling the equal length inputs of the estimation model but also preserving all information. By exploiting the time information of the time series, the proposed method embeds the time index and the original measurements into a novel indicator to reflect the battery degradation status, which could have the same length over cycles. Adopting the LSTM as the basic estimation model, the cycle synchronization-based SOH model could significantly improve the prediction accuracy by more than 30% compared to the traditional LSTM.

cs.LG↗

arXiv2021

Large deviations analysis for random combinatorial partitions with counter terms

In this paper, we study various models for random combinatorial partitions using large deviation analysis for diverging scale of the reference process. Scaling limits of similar models have been studied recently \cite{FSa,FSb} going back to \cite{Ver96}. After studying the reference model, we provide a complete analysis of two mean field models, one of which is well-know \cite{BCMP05} and the other one is the cycle mean field model. Both models show critical behaviour despite their rate functions having unique minimiser. The main focus is then a model with negative counter term, the probabilistic version of the so-called \emph{Huang-Yang-Luttinger} (HYL) model \cite{BLP}. Criticality in this model is the existence of a critical parameter for which two simultaneous minimiser exists. At criticality an order parameter is introduced as the double limits for the density of cycles with diverging length, and as such it extends recent work \cite{AD21}.

math.PR↗

arXiv2021

Phase transition for Gibbs Delaunay Tessellations with geometric hardcore conditions

In this paper, we prove the existence of infinite Gibbs Delaunay Potts tessellations for marked particle configurations. The particle systems has two types of interaction, a so-called \emph{background potential} ensures that small and large triangles are excluded in the Delaunay tessellation, and is similar to the so-called hardcore potential introduced in \cite{Der08}. Particles carry one of $q$ separate marks. Our main result is that for large activities and high \emph{type interaction} strength the model has at least $ q$ distinct translation-invariant Gibbs Delaunay Potts tessellations. The main technique is a coarse-graining procedure using the scales in the system followed by comparison with site percolation on $ \Z^2 $.

math.PR↗

arXiv2021

A Transfer Learning-based State of Charge Estimation for Lithium-Ion Battery at Varying Ambient Temperatures

Accurate and reliable state of charge (SoC) estimation becomes increasingly important to provide a stable and efficient environment for Lithium-ion batteries (LiBs) powered devices. Most data-driven SoC models are built for a fixed ambient temperature, which neglect the high sensitivity of LiBs to temperature and may cause severe prediction errors. Nevertheless, a systematic evaluation of the impact of temperature on SoC estimation and ways for a prompt adjustment of the estimation model to new temperatures using limited data have been hardly discussed. To solve these challenges, a novel SoC estimation method is proposed by exploiting temporal dynamics of measurements and transferring consistent estimation ability among different temperatures. First, temporal dynamics, which is presented by correlations between the past fluctuation and the future motion, is extracted using canonical variate analysis. Next, two models, including a reference SoC estimation model and an estimation ability monitoring model, are developed with temporal dynamics. The monitoring model provides a path to quantitatively evaluate the influences of temperature on SoC estimation ability. After that, once the inability of the reference SoC estimation model is detected, consistent temporal dynamics between temperatures are selected for transfer learning. Finally, the efficacy of the proposed method is verified through a benchmark. Our proposed method not only reduces prediction errors at fixed temperatures (e.g., reduced by 24.35% at -20°C, 49.82% at 25°C) but also improves prediction accuracies at new temperatures.

cs.LG↗

arXiv2020

Invariant learning based multi-stage identification for Lithium-ion battery performance degradation

By informing accurate performance (e.g., capacity), health state management plays a significant role in safeguarding battery and its powered system. While most current approaches are primarily based on data-driven methods, lacking in-depth analysis of battery performance degradation mechanism may discount their performances. To fill in the research gap about data-driven battery performance degradation analysis, an invariant learning based method is proposed to investigate whether the battery performance degradation follows a fixed behavior. First, to unfold the hidden dynamics of cycling battery data, measurements are reconstructed in phase subspace. Next, a novel multi-stage division strategy is put forward to judge the existent of multiple degradation behaviors. Then the whole aging procedure is sequentially divided into several segments, among which cycling data with consistent degradation speed are assigned in the same stage. Simulations on a well-know benchmark verify the efficacy of the proposed multi-stages identification strategy. The proposed method not only enables insights into degradation mechanism from data perspective, but also will be helpful to related topics, such as stage of health.

eess.SP↗

arXiv2019

An explicit large deviation analysis of the spatial cycle Huang-Yang-Luttinger model

Here we introduce a family of spatial cycle HYL-type models in which the counter-term only affects cycles longer than some cut-off that diverges in the thermodynamic limit. We derive large deviation principles and explicit pressure expressions for these models, and use the zeroes of the rate functions to study Bose-Einstein condensation.

math-ph↗

arXiv2019

Commutative diagram of the Gross-Pitaevskii approximation

It is well-known that the {\it Gross-Pitaevskii} variational formula describes the the ground state energy of of $N$-indistinguishable trapped particles (bosons) in a dilute state in the large system size $N\to\infty$. The goal of the present article is to prove that the Gross-Pitaevskii formula also appears in the {\it iterative limit} of zero temperature and large system size of the {\it product ground state energy} of the $N$-particle Hamiltonian operator.

math.PR↗

arXiv2019

Cauchy-Born Rule from Microscopic Models with Non-convex Potentials

We study gradient field models on an integer lattice with non-convex interactions. These models emerge in distinct branches of physics and mathematics under various names. In particular, as zero-mass lattice (Euclidean) quantum field theory, models of random interfaces, and as mass-string models of nonlinear elasticity.Our attention is mostly devoted to the latter with random vector valued fields as displacements for atoms of crystal structures,where our aim is to prove the strict convexity of the free energy as a function of affine deformations for low enough temperatures and small enough deformations. This claim can be interpreted as a form of verification of the Cauchy-Born rule at small non-vanishing temperatures for a class of these models. We also show that the scaling limit of the Laplace transform of the corresponding Gibbs measure (under a proper rescaling) corresponds to the Gaussian gradient field with a particular covariance. The proofs are based on a multi-scale (renormalisation group analysis) techniques needed in view of strong correlations of studied gradient fields. To cover sufficiently wide class of models, we extend these techniques from the standard case with rotationally symmetric nearest neighbour interaction to a more general situation with finite range interactions without any symmetry. Our presentation is entirely self-contained covering the details of the needed renormalisation group methods.

math-ph↗

arXiv2019

Space-time random walk loop measures

In this work, we investigate a novel setting of Markovian loop measures and introduce a new class of loop measures called Bosonic loop measures. Namely, we consider loop soups with varying intensity $ μ\le 0 $ (chemical potential in physics terms), and secondly, we study Markovian loop measures on graphs with an additional "time" dimension leading to so-called space-time random walks and their loop measures and Poisson point loop processes. Interesting phenomena appear when the additional coordinate of the space-time process is on a discrete torus with non-symmetric jump rates. The projection of these space-time random walk loop measures onto the space dimensions is loop measures on the spatial graph, and in the scaling limit of the discrete torus, these loop measures converge to the so-called [Bosonic loop measures]. This provides a natural probabilistic definition of [Bosonic loop measures]. These novel loop measures have similarities with the standard Markovian loop measures only that they give weights to loops of certain lengths, namely any length which is multiple of a given length $ β> 0 $ which serves as an additional parameter. We complement our study with generalised versions of Dynkin's isomorphism theorem (including a version for the whole complex field) as well as Symanzik's moment formulae for complex Gaussian measures. Due to the lacking symmetry of our space-time random walks, the distributions of the occupation time fields are given in terms of complex Gaussian measures over complex-valued random fields ([B92,BIS09]. Our space-time setting allows obtaining quantum correlation functions as torus limits of space-time correlation functions.

math.PR↗

arXiv2018

Large deviation analysis for classes of interacting Bosonic cycle counts

This paper studies probabilistic mean-field models for interacting bosons at a positive temperature in the thermodynamic limit with random particle density. In particular, we prove large deviation principles for empirical cycle counts in all our models, and as such generalise recent work in \cite{ACK} where upper and lower large deviation bounds do not match. Namely, on the one hand we generalise to the so-called grand canonical ensemble, and on the other hand, we consider classes of interaction potentials depending on the cycle counts which are not restricted to be positive. Our large deviation results provide representation formulae for the thermodynamic limit of the pressure which in turn leads to proving Bose-Einstein-condensation (BEC) in all our models. A primary focus and novelty is the pressure representation via extended large deviation analysis for the so-called hard-sphere, or HYL-model (Huang-Yang-Luttinger) studied in \cite{Lew86}. This model has negative counter terms in the Hamiltonian and shows BEC depending on the coupling constants.

math.PR↗

arXiv2018

The Widom-Rowlinson Model on the Delaunay Graph

We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the length of the Delaunay edges. We show that a phase transition occurs for sufficiently large activities and for sufficiently large potential parameter proving an old conjecture of Lebowitz and Lieb extended to the Delaunay structure. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transition manifests itself in the mixed site-bond percolation of the corresponding random-cluster model. Our proofs rely mainly on geometric properties of Delaunay tessellations in $\mathbb{R}^2 $ and on recent studies [DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is a uniform bound on the number of connected components in the Delaunay graph which provides a novel approach to Delaunay Widom Rowlinson models based on purely geometric arguments. The interaction potential ensures that shorter Delaunay edges are more likely to be open and thus offsets the possibility of having an unbounded number of connected components.

math.PR↗

arXiv2016

Sample path large deviations for Laplacian models in $(1+1)$-dimensions

For Laplacian models in dimension $(1+1)$ we derive sample path large deviations for the profile height function, that is, we study scaling limits of Gaussian integrated random walks and Gaussian integrated random walk bridges perturbed by an attractive force towards the zero-level, called pinning. We study in particular the regime when the rate functions of the corresponding large deviation principles admit more than one minimiser, in our models either two, three, or five minimiser depending on the pinning strength and the boundary conditions. This study complements corresponding large deviation results for gradient systems with pinning for Gaussian random walk bridges in $ (1+1) $-dimension (\cite{FS04}) and in $(1+d) $-dimension (\cite{BFO}), and recently in higher dimensions in \cite{BCF}. In particular it turns out that the Laplacian cases, i.e., integrated random walks, show richer and more complex structures of the minimiser of the rate functions which are linked to different phases.

math.PR↗

arXiv2016

Strict Convexity of the Surface Tension for Non-convex Potentials

We study gradient models on the lattice $\mathbb{Z}^d$ with non-convex interactions. These Gibbs fields (lattice models with continuous spin) emerge in various branches of physics and mathematics. In quantum field theory they appear as massless field theories. Even though our motivation stems from considering vector valued fields as displacements for atoms of crystal structures and the study of the Cauchy-Born rule for these models, our attention here is mostly devoted to interfaces, with the gradient field as an \emph{effective} interface interaction. In this case we prove the strict convexity of the surface tension (interface free energy) for low temperatures and sufficiently small interface tilts using muli-scale (renormalisation group analysis) techniques following the approach of Brydges and coworkers \cite{B07}. This is a complement to the study of the high temperature regime in \cite{CDM09} and it is an extension of Funaki and Spohn's result \cite{FS97} valid for strictly convex interactions.

math-ph↗

arXiv2015

Phase transitions in Delaunay Potts models

We establish phase transitions for classes of continuum Delaunay multi-type particle systems (continuum Potts models) with infinite range repulsive interaction between particles of different type. In one class of the Delaunay Potts models studied the repulsive interaction is a triangle (multi-body) interaction whereas in the second class the interaction is between pairs (edges) of the Delaunay graph. The result for the edge model is an extension of finite range results in \cite{BBD04} for the Delaunay graph and in \cite{GH96} for continuum Potts models to an infinite range repulsion decaying with the edge length. This is a proof of an old conjecture of Lebowitz and Lieb. The repulsive triangle interactions have infinite range as well and depend on the underlying geometry and thus are a first step towards studying phase transitions for geometry-dependent multi-body systems. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transitions manifest themselves in the percolation of the corresponding random-cluster model. Our proofs rely on recent studies \cite{DDG12} of Gibbs measures for geometry-dependent interactions.

math.PR↗

arXiv2015

Representation and poly-time approximation for pressure of $\mathbb{Z}^2$ lattice models in the non-uniqueness region

We develop a new pressure representation theorem for nearest-neighbour Gibbs interactions and apply this to obtain the existence of efficient algorithms for approximating the pressure in the $2$-dimensional ferromagnetic Potts, multi-type Widom-Rowlinson and hard-core models. For Potts, our results apply to every inverse temperature but the critical. For Widom-Rowlinson and hard-core, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of our work is in the latter.

math.DS↗

arXiv2012

Finite range decomposition for families of gradient Gaussian measures

Let a family of gradient Gaussian vector fields on $ \mathbb{Z}^d $ be given. We show the existence of a uniform finite range decomposition of the corresponding covariance operators, that is, the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters $ \sim L^k $. In addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient Gaussian measures.

math-ph↗