Diffusive Behavior for Randomly Kicked Newtonian Particles in a Spatially Periodic Medium

We prove a central limit theorem for the momentum distribution of a particle undergoing an unbiased spatially periodic random forcing at exponentially distributed times without friction. The start is a linear Boltzmann equation for the phase space density, where the average energy of the particle grows linearly in time. Rescaling time, the momentum converges to a Brownian motion, and the position is its time-integral showing superdiffusive scaling with time t ^3/2. The analysis has two parts: (1) to show that the particle spends most of its time at high energy, where the spatial environment is practically invisible; (2) to treat the low energy incursions where the motion is dominated by the deterministic force, with potential drift but where symmetry arguments cancel the ballistic behavior.     

Velocity Multistability vs. Ergodicity Breaking in a Biased Periodic Potential

Multistability, i.e., the coexistence of several attractors for a given set of system parameters, is one of the most important phenomena occurring in dynamical systems. We consider it in the velocity dynamics of a Brownian particle, driven by thermal fluctuations and moving in a biased periodic potential. In doing so, we focus on the impact of ergodicity—A concept which lies at the core of statistical mechanics. The latter implies that a single trajectory of the system is representative for the whole ensemble and, as a consequence, the initial conditions of the dynamics are fully forgotten. The ergodicity of the deterministic counterpart is strongly broken, and we discuss how the velocity multistability depends on the starting position and velocity of the particle. While for non-zero temperatures the ergodicity is, in principle, restored, in the low temperature regime the velocity dynamics is still affected by initial conditions due to weak ergodicity breaking. For moderate and high temperatures, the multistability is robust with respect to the choice of the starting position and velocity of the particle.     

Distribution of time-averaged observables for weak ergodicity breaking

We find a general formula for the distribution of time-averaged observables for systems modeled according to the subdiffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly nonergodic statistical mechanical framework is constructed, which is based on Lévy's generalized central limit theorem. As an example we calculate the distribution of X, the time average of the position of the particle, for unbiased and uniformly biased particles, and show that X exhibits large fluctuations compared with the ensemble average .     

Unstable trajectories and the quantum mechanical uncertainty

There is still an ongoing discussion about various seemingly contradictory aspects of classical particle motion and its quantum mechanical counterpart. One of the best accepted viewpoints that intend to bridge the gap is the so-called Copenhagen Interpretation. A major issue there is to regard wave functions as probability amplitudes (usually for the position of a particle). However, the literature also reports on approaches that claim a trajectory for any quantum mechanical particle, Bohmian mechanics probably being the most prominent one among these ideas. We introduce a way to calculate trajectories as well, but our crucial ingredient is their well controlled local (thus also momentaneous) degree of instability. By construction, at every moment their unpredictability, i.e., their local separation rates of neighboring trajectories, is governed by the local value of the given modulus square of a wave function. We present extensive numerical simulations of the H and He atom, and for some velocity-related quantities, namely angular momentum and total energy, we inspect their agreement with the values appearing in wave mechanics. Further, we interpret the archetypal double slit interference experiment in the spirit of our findings. We also discuss many-particle problems far beyond He, which guides us to a variety of possible applications.     

变参数自由电子激光放大器的设计方法

据自由电子激光原理,利用变参数摇摆器设计可以大幅度提高自由电子激光放大器的效率,在分析文献中所提出的“单电子共振”和“磁场指数下降法”基础上,提出了变参数摇摆器的“多电子自洽设计和最佳选择”的设计方法。经数值计算证明,所提出的方法明显优于其它方法。     

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Some dynamical properties for a time dependent Lorentz gas considering both the dissipative and non dissipative dynamics are studied. The model is described by using a four-dimensional nonlinear mapping. For the conservative dynamics, scaling laws are obtained for the behavior of the average velocity for an ensemble of non interacting particles and the unlimited energy growth is confirmed. For the dissipative case, four different kinds of damping forces are considered namely: (i) restitution coefficient which makes the particle experiences a loss of energy upon collisions; and in-flight dissipation given by (ii) F=-ηV(2); (iii) F=-ηV(μ) with μ≠1 and μ≠2 and; (iv) F=-ηV, where η is the dissipation parameter. Extensive numerical simulations were made and our results confirm that the unlimited energy growth, observed for the conservative dynamics, is suppressed for the dissipative case. The behaviour of the average velocity is described using scaling arguments and classes of universalities are defined.     

Quantum theory with an energy operator defined as a quartic form of the momentum

Quantum theory of the non-harmonic oscillator defined by the energy operator proposed by Yurke and Buks (2006) is presented. Although these authors considered a specific problem related to a model of transmission lines in a Kerr medium, our ambition is not to discuss the physical substantiation of their model. Instead, we consider the problem from an abstract, logically deductive, viewpoint. Using the Yurke–Buks energy operator, we focus attention on the imaginary-time propagator. We derive it as a functional of the Mehler kernel and, alternatively, as an exact series involving Hermite polynomials. For a statistical ensemble of identical oscillators defined by the Yurke–Buks energy operator, we calculate the partition function, average energy, free energy and entropy. Using the diagonal element of the canonical density matrix of this ensemble in the coordinate representation, we define a probability density, which appears to be a deformed Gaussian distribution. A peculiarity of this probability density is that it may reveal, when plotted as a function of the position variable, a shape with two peaks located symmetrically with respect to the central point.     

On Measuring and Modeling Local Average Particle Velocities for particle ensembles in classifying systems

In most multi-phase flow problems, the particulate phase is exposed to an external field which causes dispersion. Therefore, local velocity measurements of the disperse phase are no longer equivalent with respect to averaging in time and averaging in volume. While the local time-averaged velocity still characterizes the transport of the ensemble in the Eulerian sense, one has to be be careful in modeling this velocity average by considering the ensemble's composition. It is shown for different particle ensembles that the conventional particle velocity average M_1,0 calculated with respect to the dispersion relationship and a particle size number density distribution is far below the measured ensemble average; the deviation depends on the width of the particle size distribution. It is deduced that Eulerian particle velocity values referring to a certain time interval can be modeled by a ratio of velocity moments M_2,0/M_1,0 calculated with particle size number distributions referring to a certain probe volume. This relationship was confirmed by measurements.     

受碾区域内颗粒轴向流动特性的离散元模拟

为探讨受碾状态颗粒的稳定流动, 在碾辊轴与筛筒组成的受碾区域内, 建立了轴向运动的颗粒流离散元物理模型. 研究结果表明: 受碾区域内各颗粒沿轴向运动能力的差异造成了颗粒流密度不均匀; 颗粒与筛筒间的静摩擦系数影响颗粒轴向流动的形态、速率及集散程度, 受碾区域内单层颗粒的轴向均方偏差与流动时间的平方正相关, 属于“super”扩散; 整体分析受碾区域发现, 颗粒的轴向平均速度沿轴向坐标逐渐增大, 而颗粒的三轴合成平均速度沿轴向坐标逐渐降低; 受碾区域内各轴向位置处颗粒运动的剧烈程度不同, 沿轴向坐标颗粒的波动速度平方呈现先增大后降低而后又增大的趋势; 单颗粒的碰撞总能量损失能谱也表明了颗粒运动程度不同, 即轴向流动时在受碾区域的前半段碰撞剧烈, 能量损失多, 在后半段碰撞程度弱, 能量损失较少. 通过对受碾区域内颗粒流动的数值模拟分析, 明晰了颗粒在受碾条件下稳定流动特性, 有益于碾磨工业对产品品质控制及设备参数优化的研究.     

Absorption and scattering properties of arbitrarily shaped particles in the Rayleigh domain: A rapid computational method and a theoretical foundation for the statistical approach

We provide a theoretical foundation for the statistical approach for computing the absorption properties of particles in the Rayleigh domain. We present a general method based on the discrete dipole approximation to compute the absorption and scattering properties of particles in the Rayleigh domain. The method allows to separate the geometrical aspects of a particle from its material properties. Doing the computation of the optical properties of a particle once, provides them for any set of refractive indices, wavelengths and orientations. This allows for fast computations of e.g. absorption spectra of arbitrarily shaped particles. Other practical applications of the method are in the interpretation of atmospheric and radar measurements as well as computations of the scattering matrix of small particles as a function of the scattering angle. In the statistical approach, the optical properties of irregularly shaped particles are represented by the average properties of an ensemble of particles with simple shapes. We show that the absorption cross section of an ensemble of arbitrarily shaped particles with arbitrary orientations can always be uniquely represented by the average absorption cross section of an ensemble of spheroidal particles with the same composition and fixed orientation. This proves for the first time that the statistical approach is generally viable in the Rayleigh domain.     

基于POD方法开缝圆柱绕流流场的研究

利用粒子成像测速技术（particle image velocimetry,PIV）,在水槽中探究缝隙对圆柱流场结构的影响,应用频谱分析和本征正交分解（proper orthogonal decomposition,POD）方法,研究了开缝圆柱流场相干结构.实验Reynolds数范围内,缝隙的"吹吸"作用从根本上改变了圆柱绕流近区尾流结构,前6阶模态形态是流场中最主要的相干结构.第1,2阶模态形态控制着圆柱绕流流场涡街相继脱落过程,1或2阶模态系数为尾迹涡的固有频率;第3,4阶模态形态控制着脱落旋涡沿流向方向能量运输;第5,6阶模态形态中的同向涡旋结构作用于旋涡缓慢脱离柱体这一过程,并对旋涡能量起着衰减作用.      

Competing ordered structures formed by particles with a regular tetrahedral patch decoration

We study the ordered equilibrium structures of patchy particles where the patches are located on the surface of the colloid such that they form a regular tetrahedron. Using optimization techniques based on ideas of evolutionary algorithms we identify possible candidate structures. We retain not only the energetically most favourable lattices but also include a few energetically less favourable particle arrangements (i.e., local minima on the enthalpy landscape). Using suitably developed Monte Carlo based simulation techniques in an NPT ensemble we evaluate the thermodynamic properties of these candidate structures along selected isobars and isotherms and identify thereby the respective ranges of stability. We demonstrate on a quantitative level that the equilibrium structures at a given state point result from a delicate compromise between entropy, energy (i.e., the lattice sum) and packing.     

Effect of Ergodic and Non-Ergodic Fluctuations on a Charge Diffusing in a Stochastic Magnetic Field

In this paper, we study the basic problem of a charged particle in a stochastic magnetic field. We consider dichotomous fluctuations of the magnetic field where the sojourn time in one of the two states are distributed according to a given waiting-time distribution either with Poisson or non-Poisson statistics, including as well the case of distributions with diverging mean time between changes of the field, corresponding to an ergodicity breaking condition. We provide analytical and numerical results for all cases evaluating the average and the second moment of the position and velocity of the particle. We show that the field fluctuations induce diffusion of the charge with either normal or anomalous properties, depending on the statistics of the fluctuations, with distinct regimes from those observed, e.g., in standard Continuous-Time Random Walk models.     

Feedback control in a collective flashing ratchet

An ensemble of Brownian particles in a feedback controlled flashing ratchet is studied. The ratchet potential is switched on and off depending on the position of the particles, with the aim of maximizing the current. We study in detail a protocol which maximizes the instant velocity of the center of mass of the ensemble at any time. This protocol is optimal for one particle and performs better than any periodic flashing for ensembles of moderate size, but is defeated by a random or periodic switching for large ensembles.     

Collective modes in a slab of interacting nuclear matter

We study the properties of a slab of nuclear matter. The behaviour with the slab thickness of the particle density, kinetic energy density and surface tension are given in the non-interacting case, together with the slab free response to an external field. Next we introduce a zero-range isovector interaction among the nucleons and analyze the slab collective excitations. For moderate momenta hard and soft modes are found, which exhaust most of the excitation strength. Their position and splitting in energy favourably compares with the splitted giant dipole resonance experimentally seen in deformed nuclei.     

Particle Dynamics of a Phase Space Distribution Function

No Heading A hydrodynamic analogy for quantum mechanics is used to develop a phase-space representation in terms of a quasi-probability distribution function. Averages over phase space using this approach agree with the usual expectation values of quantum mechanics for a certain class of observables. We also derive the equations of motion that particles in an ensemble would have in phase space in order to mimic the time development of this probability distribution, thus giving the position and momentum of particles in the ensemble as a function of time. The equations of motion separate into position and momentum components. The position component reproduces the de Broglie-Bohm equation of motion. As a simple example, we calculate the phase space trajectories and entropy of a free particle wave packet.     

对于“包含在连续谱量子体系中的决定论性”一文的讨论

对于具有连续能谱的单粒子量子体系,“包含在连续谱量子体系中的决定论性”一文用所谓“双波函数”来描述处于能量本征态的粒子系综中各粒子的量子行为,并且在所谓的“等价定理”中称:双波函数描述在经典极限下将化为经典力学描述.然而,此描述所给出的系综力学量观测值统计分布的预言与通常量子力学不相容;并且,该文对其“等价定理”的证明是不正确的,这个“定理”实际上不成立 关键词： 连续能谱量子体系 双波函数 经典极限     

Magnetic moment of a two-dimensional degenerate electron gas in mesoscopic samples

The orbital magnetism of two-dimensional electrons in mesoscopic samples is studied in models where the interaction between electrons is neglected. Various geometries are considered as there are disc, plaquette, bracelet with hard wall confinement and also a confinement with a parabolic potential. We calculate the average magnetic moment which means an average with respect to size fluctuations and de Haas-van Alphen oscillations which arise in the case of a sharp Fermi cutoff. We see three distinct ranges in the magnetic field: (i) small field region where perturbation theory applies; (ii) moderate fields where edge currents play a prominent role; and (iii) the high field range with a Landau type susceptibility. In a quasiclassical picture, the electronic orbits are not qualitatively changed by a magnetic field in (i); skipping orbits are important in (ii); and in (iii), the cyclotron radius is smaller than the sample size. As a rule, we find an enhancement of the magnetic response which increases with k_FL, that is, with sample size divided by the Fermi wave length. Also, we have found out that the quasiclassical approximation fails in the calculation of the magnetic properties; on the other hand, we have seen no essential differences between the canonical ensemble (fixed particle number) and the grand canonical ensemble (chemical potential given). In the case of plaquettes, in particular for samples in the form of squares, we have found agreement with experimental results by Lévy, Reich, Pfeiffer and West.     

Classical dissipation and asymptotic equilibrium via interaction with chaotic systems

We study the energy flow between a one-dimensional oscillator and a chaotic system with two degrees of freedom in the weak coupling limit. The oscillator's observables are averaged over an initially microcanonical ensemble of trajectories of the chaotic system, which plays the role of an environment for the oscillator. We show numerically that the oscillator's average energy exhibits irreversible dynamics and ‘thermal’ equilibrium at long times. We use linear response theory to describe the dynamics at short times and we derive a condition for the absorption or dissipation of energy by the oscillator from the chaotic system. The equilibrium properties at long times, including the average equilibrium energies and the energy distributions, are explained with the help of statistical arguments. We also check that the concept of temperature defined in terms of the ‘volume entropy’ agrees very well with these energy distributions.     

Magnetization switching in single-domain particles and a response to a field pulse

An equation in the Gilbert form that describes the motion of the magnetization vector in intense high-frequency magnetic fields is solved numerically. The solution obtained is used to study switching of the magnetization of a single-domain ferromagnetic particle that has the shape of an ellipsoid of revolution and possesses cubic anisotropy from the position parallel to an easy axis to the position normal to this axis. The ranges of amplitudes and frequencies of the magnetic field where magnetization switching is observed are determined. An expression for the response of an ensemble of variously oriented particles is derived. It is shown that a particle ensemble generated by an rf field may serve as a data carrier on which information is written and read out by means of nonlinear and linear ferromagnetic resonances.