scholarly journals Multiparticle localization for disordered systems on continuous space via the fractional moment method

2015 ◽  
Vol 27 (04) ◽  
pp. 1550010 ◽  
Author(s):  
Michael Fauser ◽  
Simone Warzel
Keyword(s):  
Disordered Systems ◽  
Weak Interactions ◽  
Particle System ◽  
Pair Potential ◽  
Dynamical Localization ◽  
Continuous Space ◽  
Fractional Moment Method ◽  
Particle Localization ◽  
Infinite Range ◽  
Number Of Particles

We investigate the spectral and dynamical localization of a quantum system of n particles on ℝd which are subject to a random potential and interact through a pair potential which may have infinite range. We establish two conditions which ensure spectral and dynamical localization near the bottom of the spectrum of the n-particle system: (i) localization is established in the regime of weak interactions supposing one-particle localization, and (ii) localization is also established under a Lifshitz-tail type condition on the sparsity of the spectrum. In case of polynomially decaying interactions, we provide an upper bound on the number of particles up to which these conditions apply.

scholarly journals Absence of dynamical localization in interacting driven systems

2017 ◽  
Vol 3 (4) ◽  
Author(s):  
David J. Luitz ◽  
Yevgeny Bar Lev ◽  
Achilleas Lazarides
Keyword(s):  
Diffusion Coefficient ◽  
Weak Interactions ◽  
Interaction Strength ◽  
Dynamical Localization ◽  
Driven Systems ◽  
Periodically Driven ◽  
Diffusive Regime ◽  
The Stability ◽  
Small Interactions ◽  
Number Of Particles

Using a numerically exact method we study the stability of dynamical localization to the addition of interactions in a periodically driven isolated quantum system which conserves only the total number of particles. We find that while even infinitesimally small interactions destroy dynamical localization, for weak interactions density transport is significantly suppressed and is asymptotically diffusive, with a diffusion coefficient proportional to the interaction strength. For systems tuned away from the dynamical localization point, even slightly, transport is dramatically enhanced and within the largest accessible systems sizes a diffusive regime is only pronounced for sufficiently small detunings.

scholarly journals Direct Scaling Analysis of Fermionic Multiparticle Correlated Anderson Models with Infinite-Range Interaction

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
Victor Chulaevsky
Keyword(s):  
Dynamical Localization ◽  
Scaling Analysis ◽  
Range Interaction ◽  
Direct Scaling ◽  
Strongly Mixing ◽  
Optimal Decay ◽  
Decay Bounds ◽  
Multiparticle Systems ◽  
Infinite Range ◽  
Anderson Models

We adapt the method of direct scaling analysis developed earlier for single-particle Anderson models, to the fermionic multiparticle models with finite or infinite interaction on graphs. Combined with a recent eigenvalue concentration bound for multiparticle systems, the new method leads to a simpler proof of the multiparticle dynamical localization with optimal decay bounds in a natural distance in the multiparticle configuration space, for a large class of strongly mixing random external potentials. Earlier results required the random potential to be IID.

scholarly journals An infinite particle system with additive interactions

1979 ◽  
Vol 11 (02) ◽  
pp. 355-383 ◽  
Author(s):  
Richard Durrett
Keyword(s):  
Necessary Conditions ◽  
Particle System ◽  
Particle Systems ◽  
Compact Set ◽  
Translation Invariant ◽  
Limiting Behavior ◽  
Branching Random Walks ◽  
Sufficient And Necessary Conditions ◽  
Infinite Particle ◽  
Number Of Particles

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set ast→ ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.

Explicit sufficient invariants for an interacting particle system

1998 ◽  
Vol 35 (3) ◽  
pp. 633-641 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Colin Mallows ◽  
Larry Shepp
Keyword(s):  
Markov Chain ◽  
Finite Time ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
The Other ◽  
New Class ◽  
Interacting Particle ◽  
Death State ◽  
Number Of Particles

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are Ni(0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, ηi = Ni(∞), as a function of Ni(0).We are able to obtain, for some special graphs, the limiting distribution of Ni if the total number of particles N → ∞ in such a way that the fraction, Ni(0)/S = ξi, at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S2, the two-leaf star which has three vertices and two edges.

scholarly journals Multiparticle Localization at Low Energy for Multidimensional Continuous Anderson Models

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Trésor Ekanga
Keyword(s):  
Probability Distribution ◽  
Anderson Model ◽  
Multiscale Analysis ◽  
Random Potential ◽  
Particle Interaction ◽  
Low Energy ◽  
Dynamical Localization ◽  
Hölder Continuous ◽  
The Continuum ◽  
Number Of Particles

We study the multiparticle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multiparticle lower spectral edges are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multiparticle Anderson model at low energy. The proof based on the multiparticle multiscale analysis bounds needs the values of the external random potential to be independent and identically distributed, whose common probability distribution is at least Log-Hölder continuous.

An infinite particle system with continual input and random death

1978 ◽  
Vol 10 (04) ◽  
pp. 764-787
Author(s):  
J. N. McDonald ◽  
N. A. Weiss
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Limit Theorems ◽  
Particle System ◽  
Countable State Space ◽  
Infinite Particle System ◽  
Large Numbers ◽  
Countable State ◽  
Infinite Particle ◽  
Number Of Particles

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

Explicit sufficient invariants for an interacting particle system

1998 ◽  
Vol 35 (03) ◽  
pp. 633-641 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Colin Mallows ◽  
Larry Shepp
Keyword(s):  
Markov Chain ◽  
Finite Time ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
The Other ◽  
New Class ◽  
Interacting Particle ◽  
Death State ◽  
Number Of Particles

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N i (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, η i = N i (∞), as a function of N i (0). We are able to obtain, for some special graphs, the limiting distribution of N i if the total number of particles N → ∞ in such a way that the fraction, N i (0)/S = ξ i , at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S 2, the two-leaf star which has three vertices and two edges.

scholarly journals Approximation of a free Poisson process by systems of freely independent particles

2018 ◽  
Vol 21 (03) ◽  
pp. 1850020 ◽  
Author(s):  
Marek Bożejko ◽  
José Luís da Silva ◽  
Tobias Kuna ◽  
Eugene Lytvynov
Keyword(s):  
Poisson Process ◽  
Radon Measure ◽  
Poisson Point Process ◽  
Particle System ◽  
Intensity Measure ◽  
Bounded Set ◽  
Lévy White Noise ◽  
Free Independence ◽  
Bounded Sets ◽  
Number Of Particles

Let [Formula: see text] be a non-atomic, infinite Radon measure on [Formula: see text], for example, [Formula: see text] where [Formula: see text]. We consider a system of freely independent particles [Formula: see text] in a bounded set [Formula: see text], where each particle [Formula: see text] has distribution [Formula: see text] on [Formula: see text] and the number of particles, [Formula: see text], is random and has Poisson distribution with parameter [Formula: see text]. If the particles were classically independent rather than freely independent, this particle system would be the restriction to [Formula: see text] of the Poisson point process on [Formula: see text] with intensity measure [Formula: see text]. In the case of free independence, this particle system is not the restriction of the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. Nevertheless, we prove that this is true in an approximative sense: if bounded sets [Formula: see text] ([Formula: see text]) are such that [Formula: see text] and [Formula: see text], then the corresponding particle system in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. We also prove the following [Formula: see text]-limit: Let [Formula: see text] be a deterministic sequence of natural numbers such that [Formula: see text]. Then the system of [Formula: see text] freely independent particles in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.

THERMAL VIBRATION ANALYSIS OF RuO2 BY EXAFS

2006 ◽  
Vol 20 (25n27) ◽  
pp. 4111-4116 ◽  
Author(s):  
KEI-ICHIRO MURAI ◽  
YASUHIRO AKUNE ◽  
YOHEI SUZUKI ◽  
TOSHIHIRO MORIGA ◽  
ICHIRO NAKABAYASHI
Keyword(s):  
Thermal Expansion ◽  
Nearest Neighbor ◽  
Weak Interactions ◽  
Negative Thermal Expansion ◽  
Pair Potential ◽  
Waller Factor ◽  
Thermal Vibration ◽  
Debye Waller Factor ◽  
Effective Pair Potential ◽  
Effective Pair

Ruthenium dioxide which has a rutile-type structure is an important material in a viewpoint of electronic and magnetic properties. RuO 2 has a negative thermal expansion along c-axis in the high temperature region above room temperature. In this study, we could obtain detailed information about the thermal vibration of atoms by the analysis of EXAFS Debye-Waller factors. EXAFS analysis provides an effective pair potential with temperature dependent shape with Debye-Waller factor. The distance between second-nearest neighbor atoms ( Ru - Ru ) are equal to the length of c-axis in unit cell. It has become apparent that Ru - O bonds in RuO 6 octahedron are much stronger than the interaction between the second-nearest neighbor atoms ( Ru - Ru ) as same as FeF 2 in fluorides. Those results suggest that the negative thermal expansion along c-axis is caused by those weak interactions between second-nearest neighbor atoms ( Ru - Ru ).

scholarly journals Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

2021 ◽  
Author(s):  
David P. Bourne ◽  
Riccardo Cristoferi
Keyword(s):  
Probability Measure ◽  
Triangular Lattice ◽  
Best Approximation ◽  
Particle System ◽  
Asymptotic Optimality ◽  
Optimal Location ◽  
Two Dimensions ◽  
Particle Systems ◽  
Discrete Measure ◽  
Number Of Particles

AbstractWe prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure $$f \mathrm {d}x$$ f d x by a discrete probability measure $$\sum _i m_i \delta _{z_i}$$ ∑ i m i δ z i , subject to a constraint on the particle sizes $$m_i$$ m i . The locations $$z_i$$ z i of the particles, their sizes $$m_i$$ m i , and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

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