Marco Lenci, Luc Rey-Bellet
We give large deviation upper bounds, and discuss lower bounds, for the Gibbs-KMS state of a system of quantum spins or an interacting Fermi gas on the lattice. We cover general interactions and general observables, both in the high temperature regime and in dimension one.
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We discuss the Donsker-Varadhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the Donsker-Varadhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary state as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic cou…
We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.
We study run and tumble particles on the one-dimensional lattice $\mathbb{Z}$. We explicitly compute the Fourier-Laplace transform of the position of the particle and as a consequence obtain explicit expressions for the diffusion constant and the large deviation free energy function. We also do the same computations in a corresponding continuum model. In the latter, when adding an external field, we can explicitly compute the large deviation free energy, and the deviation from the Einstein relation due to activity. Finally, we generalize the model to the $d$-dimensional lattice $\mathbb{Z}^d$,…
Large Deviations for Ablowitz-Ladik lattice, and the Schur flow
本文研究 Ablowitz-Ladik 格点与 Schur 流的广义吉布斯系综，推导经验测度的大偏差原理，并证明其几乎必然收敛，极限由圆与 Jacobi beta 系综的平衡测度刻画。
We consider a lattice gas on the discrete d-dimensional torus $(\mathbb{Z}/N\mathbb{Z})^d$ with a generic translation invariant, finite range interaction satisfying a uniform strong mixing condition. The lattice gas performs a Kawasaki dynamics in the presence of a weak external field E/N. We show that, under diffusive rescaling, the hydrodynamic behavior of the lattice gas is described by a nonlinear driven diffusion equation. We then prove the associated dynamical large deviation principle. Under suitable assumptions on the external field (e.g., E constant), we finally analyze the variationa…
We prove the dynamical large deviations for a particle system in which particles may have different velocities. We assume that we have two infinite reservoirs of particles at the boundary: this is the so-called boundary driven process. The dynamics we considered consists of a weakly asymmetric simple exclusion process with collision among particles having different velocities.
We introduce the multiplicative Ising model and prove basic properties of its thermodynamic formalism such as existence of pressure and entropies. We generalize to one-dimensional "layer-unique" Gibbs measures for which the same results can be obtained. For more general models associated to a $d$-dimensional multiplicative invariant potential, we prove a large deviation theorem in the uniqueness regime for averages of multiplicative shifts of general local functions. This thermodynamic formalism is motivated by the statistical properties of multiple ergodic averages.
Dynamical phase transitions are crucial features of the fluctuations of statistical systems, corresponding to boundaries between qualitatively different mechanisms of maintaining unlikely values of dynamical observables over long periods of time. They manifest themselves in the form of non-analyticities in the large deviation function of those observables. In this paper, we look at bulk-driven exclusion processes with open boundaries. It is known that the standard asymmetric simple exclusion process exhibits a dynamical phase transition in the large deviations of the current of particles flowi…
The properties of quantum probabilities are linked to the geometry of quantum mechanics, described by the Birkhoff-von Neumann lattice. Quantum probabilities violate the additivity property of Kolmogorov probabilities, and they are interpreted as Dempster-Shafer probabilities. Deviations from the additivity property are quantified with the Mobius (or non-additivity) operators which are defined through Mobius transforms, and which are shown to be intimately related to commutators. The lack of distributivity in the Birkhoff-von Neumann lattice Lambda , causes deviations from the law of the total…
A "lifting" method for exponential large deviation estimates and an application to certain non-stationary 1D lattice Anderson models
本文提出一种“提升”方法，将平稳模型的指数大偏差估计推广到非平稳模型，并据此推导一维晶格上特定非平稳势的Anderson局域化。
We consider high temperature KMS states for quantum spin systems on a lattice. We prove a large deviation principle for the distribution of empirical averages $\frac{1}{|Λ|} \sum_{i\inΛ} X_i$, where the $X_i$'s are copies of a self-adjoint element $X$ (level one large deviations). From the analyticity of the generating function, we obtain the central limit theorem. We generalize to a level two large deviation principle for the distribution of $\frac{1}{|Λ|}\sum_{i\inΛ} δ_{X_i}$.
Effect of slow bonds on current fluctuations in the symmetric simple exclusion process
本文通过精确大偏差函数和稀有事件模拟，研究了对称简单排斥过程中局部慢键对电流波动的影响，并给出了半无限情况下的初等推导。
We contribute an extension of large-deviation results obtained in [N.J.B. Aza, J.-B. Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math. Pures Appl. 125 (2019) 209] on conductivity theory at atomic scale of free lattice fermions in disordered media. Disorder is modeled by (i) a random external potential, like in the celebrated Anderson model, and (ii) a nearest-neighbor hopping term with random complex-valued amplitudes. In accordance with experimental observations, via the large deviation formalism, our previous paper showed in this case that quantum uncertainty of microscopic electric…
Quantum Large Deviations
本文重新审视了Varadhan定理的量子类比，利用量子版大偏差分析和Trotter乘积公式给出了新证明，并将其推广至非交换变量情形，得出一般平均场量子自旋系统的变分公式。
The optimal state-independent lower bounds for the sum of variances or deviations of observables are of significance for the growing number of experiments that reach the uncertainty limited regime. We present a framework for computing the tight uncertainty relations of variance or deviation via determining the uncertainty regions, which are formed by the tuples of two or more of quantum observables in random quantum states induced from the uniform Haar measure on the purified states. From the analytical formulae of these uncertainty regions, we present state-independent uncertainty inequalitie…
Mechanism of production and deviation from the standard Gutenberg-Richter law of the big earthquakes (An analysis of big earthquakes)
本文描述了两种自复制能量积累机制，它们修改了大地震区域的古登堡-里克特定律，引入了临界震级区域作为额外拟合参数，并可能解释“自停止”和“龙王”地震现象。
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice for decaying initial data in the soliton region. In addition, we point out how to reduce the problem in the remaining region to the known case without solitons.
A two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged regular closed region K whose charge density is determined by its equilibrium potential at an inverse temperature βis investigated. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a point process in the complex plane. We describe the weak* limits of the joint intensities of this point process and show that it is exponentially likely to find the process in a neighborhood of the equ…
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the periodic (and slightly more generally of the quasi-periodic finite-gap) Toda lattice for decaying initial data in the soliton region. In addition, we show how to reduce the problem in the remaining region to the known case without solitons.
Probing the large deviations for the Beta random walk in random medium
本文研究了随机介质中Beta随机游走样本概率的大偏差速率函数，论证其与连续介质模型结果一致，并通过数值模拟验证了理论预测及一阶相变现象。
Schrödinger Operators with Potentials Generated by Hyperbolic Transformations: II. Large Deviations and Anderson Localization
本文研究由一致双曲变换轨道的赫尔德连续采样生成势的离散一维薛定谔算子，建立了大能量区域的一致大偏差估计，并证明了全谱安德森局域化。
We consider homogeneous open quantum random walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated…
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