System Of Independent Particles Research Articles

Enhanced metrology at the critical point of a many-body Rydberg atomic system

The spectral properties of an interacting many-body system may display critical character and have potential applications in precision metrology. Here, we demonstrate such many-body enhanced metrology for microwave (MW) electric fields in a non-equilibrium Rydberg atomic gas. Near criticality the high sensitivity of Rydberg atoms to external MW electric fields, combined with many-body enhancement induces significant changes in the optical transmission. We quantify this behavior using the Fisher information. For continuous optical transmission at the critical point, the Fisher information is three orders of magnitude larger than in independent particle systems, the measured data provides an equivalent sensitivity of 49 nV/cm/$\textrm{Hz}^{1/2}$. The reported results constitute a milestone towards the application of many-body effects in precision metrology.

Quantitative mean-field limit for interacting branching diffusions

We establish an explicit rate of convergence for some systems of mean-field interacting diffusions with logistic binary branching, towards solutions of nonlinear evolution equations with non-local self-diffusion and logistic mass growth, which were shown to describe their large population limits in [12]. The proof relies on a novel coupling argument for binary branching diffusions based on optimal transport, allowing us to sharply mimic the trajectory of the interacting binary branching population by means of a system of independent particles with suitably distributed random space-time births. We are thus able to derive an optimal convergence rate, in the dual bounded-Lipschitz distance on finite measures, for the empirical measure of the population, from the convergence rate in 2-Wasserstein distance of empirical distributions of i.i.d. samples. Our approach and results extend propagation of chaos techniques and ideas, from kinetic models to stochastic systems of interacting branching populations, and appear to be new in this setting, even in the simple case of pure binary branching diffusions.

Flux Large Deviations of Independent and Reacting Particle Systems, with Implications for Macroscopic Fluctuation Theory

We consider a system of independent particles and a system of reacting particles on a discrete state space. For the independent case, we rigorously prove a dynamic large-deviation principle for the empirical measure-integrated flux pair, taking the one-way fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We formally calculate similar one-way flux large deviations for the more general reacting particles case. Next, we use these results to derive a number of principles from Macroscopic Fluctuation Theory like time-reversal principles and generalised gradient structures.

Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment

Nous considérons un système de particules indépendantes évoluant dans un milieu aléatoire commun. Auparavant, la limite hydrodynamique de ce système de particules en milieu aléatoire a été obtenue quand les particules sont transientes avec une vitesse positive (Electron. J. Probab. 15 (2010) 1024–1040). Dans cet article nous considérons le cas où les particules sont transientes mais ont une vitesse sous-linéaire d’ordre $n^{\kappa}$ pour $\kappa\in(0,1)$ et nous montrons l’existence d’une limite hydrodynamique du système de particules avec une échelle du temps $n^{1/\kappa}$ et une échelle spatiale $n$. La propriété la plus intéressante de cette limite hydrodynamique est que le milieu n’est pas moyenné par la limite ; c’est-à-dire, la densité asymptotique des particules dépend de la réalisation du milieu choisi. La limite hydrodynamique du système de particules est déduite à partir de la limite hydrodynamique d’un système de particules indépendantes dans un milieu aléatoire dirigé.

Response calculations based on an independent particle system with the exact one-particle density matrix: polarizabilities.

Recently, we have demonstrated that the problems finding a suitable adiabatic approximation in time-dependent one-body reduced density matrix functional theory can be remedied by introducing an additional degree of freedom to describe the system: the phase of the natural orbitals [K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, Phys. Rev. Lett. 105, 013002 (2010); K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, J. Chem. Phys. 133, 174119 (2010)]. In this article we will show in detail how the frequency-dependent response equations give the proper static limit (ω → 0), including the perturbation in the chemical potential, which is required in static response theory to ensure the correct number of particles. Additionally we show results for the polarizability for H2 and compare the performance of two different two-electron functionals: the phase-including Löwdin-Shull functional and the density matrix form of the Löwdin-Shull functional.

Response calculations based on an independent particle system with the exact one-particle density matrix: Excitation energies

Adiabatic response time-dependent density functional theory (TDDFT) suffers from the restriction to basically an occupied → virtual single excitation formulation. Adiabatic time-dependent density matrix functional theory allows to break away from this restriction. Problematic excitations for TDDFT, viz. bonding-antibonding, double, charge transfer, and higher excitations, are calculated along the bond-dissociation coordinate of the prototype molecules H(2) and HeH(+) using the recently developed adiabatic linear response phase-including (PI) natural orbital theory (PINO). The possibility to systematically increase the scope of the calculation from excitations out of (strongly) occupied into weakly occupied ("virtual") natural orbitals to larger ranges of excitations is explored. The quality of the PINO response calculations is already much improved over TDDFT even when the severest restriction is made, to virtually the size of the TDDFT diagonalization problem (only single excitation out of occupied orbitals plus all diagonal doubles). Further marked improvement is obtained with moderate extension to allow for excitation out of the lumo and lumo+1, which become fractionally occupied in particular at longer distances due to left-right correlation effects. In the second place the interpretation of density matrix response calculations is elucidated. The one-particle reduced density matrix response for an excitation is related to the transition density matrix to the corresponding excited state. The interpretation of the transition density matrix in terms of the familiar excitation character (single excitations, double excitations of various types, etc.) is detailed. The adiabatic PINO theory is shown to successfully resolve the problematic cases of adiabatic TDDFT when it uses a proper PI orbital functional such as the PILS functional.

Quenched scaling limits of trap models

In this paper, we study Bouchaud’s trap model on the discrete d-dimensional torus ${\mathbb{T}}^{d}_{n}=({\mathbb{Z}}/n{\mathbb{Z}})^{d}$. In this process, a particle performs a symmetric simple random walk, which waits at the site $x\in {\mathbb{T}}^{d}_{n}$ an exponential time with mean ξx, where $\{\xi_{x},x\in {\mathbb{T}}^{d}_{n}\}$ is a realization of an i.i.d. sequence of positive random variables with an α-stable law. Intuitively speaking, the value of ξx gives the depth of the trap at x. In dimension d=1, we prove that a system of independent particles with the dynamics described above has a hydrodynamic limit, which is given by the degenerate diffusion equation introduced in [Ann. Probab. 30 (2002) 579–604]. In dimensions d>1, we prove that the evolution of a single particle is metastable in the sense of Beltrán and Landim [Tunneling and Metastability of continuous time Markov chains (2009) Preprint]. Moreover, we prove that in the ergodic scaling, the limiting process is given by the K-process, introduced by Fontes and Mathieu in [Ann. Probab. 36 (2008) 1322–1358].

Response Calculations with an Independent Particle System with an Exact One-Particle Density Matrix

We use the natural orbitals to define an independent particle system, from which the exact one-particle density matrix can be obtained with an ensemble of degenerate determinantal ground states. Also defining explicit phases for the orbitals, and admitting functionals that are dependent on those phases, time-dependent equations for the orbitals and occupation numbers are obtained from an action principle. The wrong polarizability and lack of double excitations of straightforward adiabatic time-dependent density matrix functional theory are then corrected, and the important symmetry χ(ω)=χ{*}(-ω), lost in previous ad hoc improvements, is restored. The extension of the response calculations beyond the occupied-virtual pairs, which are the only ones admitted in time-dependent density functional theory, leads to greatly improved response properties.

Small-angle neutron scattering analysis of a water-based magnetic fluid with charge stabilization: contrast variation and scattering of polarized neutrons

Structure analysis of a magnetic fluid (nanoparticles of maghemite dispersed in water with charge stabilization and without surfactant) by means of small-angle neutron scattering is presented. A combination of the contrast variation technique and scattering of polarized neutrons was applied. In the first case, the scattering curves obtained for the unmagnetized fluid with variation of the heavy water content in the carrier are treated in terms of the basic functions approach. The almost homogeneous character of the nanoparticles with respect to the nuclear scattering length density makes it possible to separate information about their characteristic nuclear and magnetic radii. Polarized neutrons are then used to separate and analyze independently the nuclear and magnetic scattering contributions for the fully magnetized fluid. Both methods reveal a significant excess of the apparent nuclear size over the magnetic one, which is explained by a difference in the nonmagnetic and magnetic interactions in the system. The results indicate that from the viewpoint of magnetic interaction the studied fluid behaves under a magnetic field as a purely superparamagnetic system of independent particles. The magnetic scattering length density of the maghemite nanoparticles is found to be ∼25% less than the bulk value, which is in agreement with the data of the magnetization analysis.

PARTICLE PICTURE APPROACH TO THE SELF-INTERSECTION LOCAL TIME OF BRANCHING DENSITY PROCESSES IN ${\mathcal S}' ({\mathbb R}^d)$

The problems studied in this paper are associated with a critical branching particle system in [Formula: see text], where the particle motion is described by a Lévy process. We define the intersection local time (ILT) of two independent trees, i.e. two independent particle systems, each starting from a single particle and we give sufficient conditions for its existence. The [Formula: see text]-valued density process arises as the high density limit of a "charged" particle system, where the initial positions of particles are given by a Poisson random measure. We express the self-intersection local time of this density process by means of ILTs of pairs of trees.

Occupation time limits of inhomogeneous Poisson systems of independent particles

We prove functional limits theorems for the occupation time process of a system of particles moving independently in R d according to a symmetric α -stable Lévy process, and starting from an inhomogeneous Poisson point measure with intensity measure μ ( d x ) = ( 1 + | x | γ ) − 1 d x , γ > 0 , and other related measures. In contrast to the homogeneous case ( γ = 0 ) , the system is not in equilibrium and ultimately it becomes locally extinct in probability, and there are more different types of occupation time limit processes depending on arrangements of the parameters γ , d and α . The case γ < d < α leads to an extension of fractional Brownian motion.

Phase-space overlap measures. I. Fail-safe bias detection in free energies calculated by molecular simulation

We consider ways to quantify the overlap of the parts of phase space important to two systems, labeled A and B. Of interest is how much of the A-important phase space lies in that important to B, and how much of B lies in A. Two measures are proposed. The first considers four total-energy distributions, formed from all combinations made by tabulating either the A-system or the B-system energy when sampling either the A or B system. Measures for A in B and B in A are given by two overlap integrals defined on pairs of these distributions. The second measure is based on information theory, and defines two relative entropies which are conveniently expressed in terms of the dissipated work for free-energy perturbation (FEP) calculations in the A-->B and B-->A directions, respectively. Phase-space overlap is an important consideration in the performance of free-energy calculations. To demonstrate this connection, we examine bias in FEP calculations applied to a system of independent particles in a harmonic potential. Systems are selected to represent a range of overlap situations, including extreme subset, subset, partial overlap, and nonoverlap. The magnitude and symmetry of the bias (A-->B vs B-->A) are shown to correlate well with the overlap, and consequently with the overlap measures. The relative entropies are used to scale the amount of sampling to obtain a universal bias curve. This result leads to develop a simple heuristic that can be applied to determine whether a work-based free-energy measurement is free of bias. The heuristic is based in part on the measured free energy, but we argue that it is fail-safe inasmuch as any bias in the measurement will not promote a false indication of accuracy.

Dynamic response of a strongly perturbed electron gas

The theoretical description of many-body systems in their ground state has nowadays become an affordable task because of recent advances in the numerical implementation of ab initio methods, as well as huge improvements in computing capabilities. However, an accurate evaluation of excitedstates properties in such systems is still a major theoretical challenge. The problem is of obvious importance as most experimental techniques are based on testing of excited state. Apart from its intrinsic fundamental interest, the knowledge of the system excitation spectrum is useful to understand a large variety of physical phenomena, such as ~i! photoabsorption of atoms 1,2 and plasmas, 3,4 ~ii! optical response of clusters, 5‐ 8 ~iii! core polarization and dielectric response in solids, 9,10 ~iv! resonant photoemission, 11 or ~v! neutralization of ions at surfaces. 12,13 Common solid-state approaches to study excited states are many-body perturbation and time-dependent densityfunctional theories. The application of these methods to real systems often relies on system symmetries, such as twodimensional or three-dimensional periodicity. The theoretical description of electronic excitations turns out to be more intricate when a localized impurity breaks the spatial symmetry. Numerical methods based on system periodicity are not efficient in this case. Furthermore, localized perturbations in solids, such as impurities in bulk, inner-shell holes, or adsorbates at surfaces strongly modify their environment. The appearance of new states bound to the impurity 14 and/or resonances in the continuum 15 can change dramatically the local properties of the system. The influence of impurity atoms in the excitation spectrum of a 2D electron gas ~where any attractive potential has a bound state! has been considered recently. 16 Kondo-type effects arise for single magnetic impurities as well. 17,18 In metallic systems, valence-band electrons are highly mobile and the presence of an atomic impurity introduces a strong distortion in the electronic density. Our purpose in this work is to show that this rearrangement of electronic charge determines the local dynamic response of the valence electrons to external perturbations. We study the case of an atomic impurity embedded in a free electron gas ~FEG!, which represents the valence band of a simple metal ~such as Na or Al!. Therefore, band-structure effects are not considered as we are interested in determining the local dynamic response of the system. We use density-functional theory to calculate the ground state of the system and obtain the valence-electron response function in linear theory afterwards. We show that the dynamic screening of the Coulomb interaction is highly modified in the vicinity of the impurity and has a complex behavior determined by two competing effects. As an example, we calculate transition probabilities for hole-filling processes in the embedded impurity. We compare some of our results with those obtained in calculations which do not include the presence of the impurity to remark the specific effects introduced by the latter. For small external dynamic perturbations of the system, the many-body response is well described by the linear density response function x(r,r8,v). The self-consistent response function x(r,r8,v) can be obtained at the randomphase approximation ~RPA! level in terms of the response function of a system of independent particles x 0 (r,r8,v) ~atomic units are used throughout unless otherwise stated!: x~ r,r8,v!5x 0 ~ r,r8,v!

Ward identities for interacting electronic systems

A Ward-Takahashi identity, as a consequence of gauge invariance and in a form that relates self-energy to the two-particle Bethe-Salpeter scattering kernel, was first derived by Vollhardt and Wölfle for a system of independent particles moving in a random medium. This is generalized to a class of interacting electronic systems in materials with or without random impurities, following a procedure previously used for classical wave transport in disordered media. This class of systems also possesses other symmetry properties such as invariance under time translations and local spin rotations, which imply local conservation laws for energy and spin current. They imply additional Vollhardt-Wölfle type identities. We present non-perturbative derivations of these identities, and consider the constraints they impose on the relationship between the self-energy and the two-particle scattering kernel.

Large deviations for the occupation times of independent particle systems

prove a large deviation principle for the density field of independent particle systems in an infinite volume. We then deduce from the one-dimensional case of this result the large deviations for the occupation times of various sets (from microscopic to macroscopic scales) and we recover the theorem established by Cox and Griffeath. An expression of the rate function is given using the Brownian motion local time as in Deuschel and Wang.

On angular momentum and parity dependence of nuclear level densities in a simple random sampling approach

Abstract Based on simple random sampling (SRS), we propose a Monte Carlo method for faster computation of the smoothed part of the density of nuclear states. To test the applicability of the SRS approach we study in this framework the excitation energy ( E ), angular momentum ( J ) and parity dependence of nuclear level densities for an independent particle system. As an illustrative example, we consider a pf -shell nucleus, 48 Cr. It is found that the values of a few lower order moments for the state density I ( E ) calculated using SRS and combinatorial (or direct counting) methods are almost the same and a locally smoothed part of the state density can be constructed using these moments in a univariate Edgeworth expansion. We calculate the energy dependent spin-cutoff factor and parity asymmetry and find that for both cases the SRS approach works quite well. We use the SRS moments to construct different forms of the bivariate distribution for I ( E , M ) ( M is the z -component of J ) namely (a) a bivariate Edgeworth expansion, (b) a product of the univariate Edgeworth expansion ( I ( E )) and a Gaussian form for conditional M distribution I( M E ) and (c) a product of the univariate Edgeworth expansions for both I ( E ) and I( M E ) and compare the resulting fixed- J level density I l ( E , J ) with the corresponding combinatorial results.

Convergence of independent particle systems

We consider a system of particles moving independently on a countable state space, according to a general (non-space-homogeneous) Markov process. Under mild conditions, the number of particles at each site will converge to a product of independent Poisson distributions; this corresponds to settling to an ideal gas. We derive bounds on the rate of this convergence. In particular, we prove that the variation distance to stationarity decreases proportionally to the sum of squares of the probabilities of each particle to be at a given site. We then apply these bounds to some examples. Our methods include a simple use of the Chen-Stein lemma about Poisson convergence. Our results require certain strong hypotheses, which further work might be able to eliminate.

Lagrangian description and entropy of magnetic Vlasov systems

In this paper we study the relation between the collective magnetic entropy of the Vlasov systems, introduced previously on a phenomenological basis, and the Lagrangian description of the motion of a system of independent particles in a magnetic field, subject to a constraint that introduces collective behaviour. The main result is the equivalence of the first variations of the action integral and of the collective entropy with respect to a certain family of variations of the Lagrangian co-ordinates corresponding to reversible variations of the collective system considered as isolated. In the case of irreversible transformations the collective entropy is not determined dynamically, but acquires its meaning through the procedure of maximum probability assignment according to the standard formalism of information theory, under the requirement of macroscopic reproducibility.

Long-range spatial correlations for anisotropic zero-range processes

The spatial correlations are investigated for a homogeneous system of indistinguishable particles undergoing stochastic anisotropic hopping dynamics on the d-dimensional lattice, d>or=2. The interaction is zero range, i.e. the rate at which particles leave a given site only depends on the occupation number at that site. A series expansion around the independent particle system is given for the equal time correlations and is shown to converge for small times t. The formal t to infinity limiting expansion is analysed termwise from which a quadrupole type decay ( approximately r-d) is derived for the stationary two-points function ( eta (0) eta (r))-( eta (0))2. This phenomenon of self-organized criticality is a direct consequence of the anisotropy causing the system to violate the condition of detailed balance, combined with the conservation law forcing a diffusive decay ( approximately t-d2/) of the temporal correlations.

Simulation of electron motion in a single valley semiconductor: GaAs

Simulation of electron motion in single valley semiconductors such as GaAs using the Monte Carlo method is analyzed, with confirmation of earlier theoretical work and the Gunn effect in the semiconductor. The actual system can be considered as a quasi system of independent particles. Threshold field is rather insensitive to temperature change. Mean energy increases with field. Time of flight between two virtual scattering processes is large. The total change in k-space and E-space position is linear.