Ensemble Of Classical Particles Research Articles

A variational framework for the inverse Henderson problem of statistical mechanics

The inverse Henderson problem refers to the determination of the pair potential which specifies the interactions in an ensemble of classical particles in continuous space, given the density and the equilibrium pair correlation function of these particles as data. For a canonical ensemble in a bounded domain, it has been observed that this pair potential minimizes a corresponding convex relative entropy functional, and that the Newton iteration for minimizing this functional coincides with the so-called inverse Monte Carlo (IMC) iterative scheme. In this paper, we show that in the thermodynamic limit analogous connections exist between the specific relative entropy introduced by Georgii and Zessin and a proper formulation of the IMC iteration in the full space. This provides a rigorous variational framework for the inverse Henderson problem, valid within a large class of pair potentials, including, for example, Lennard-Jones-type potentials. It is further shown that the pressure is strictly convex as a function of the pair potential and the chemical potential, and that the specific relative entropy at fixed density is a strictly convex function of the pair potential. At a given reference potential and a corresponding density in the gas phase, we determine the gradient and the Hessian of the specific relative entropy, and we prove that the Hessian extends to a symmetric positive semidefinite quadratic functional in the space of square integrable perturbations of this potential.

Diffusion treatment of quantum mechanics and its consequences

Localized ensemble of free microparticles spreads out as in a frictionless diffusion satisfying the principle of relativity. An ensemble of classical particles in a fluctuating classical scalar field diffuses in a similar way, and this analogy is used to formulate diffusion quantum mechanics (DQM). DQM reproduces quantum mechanics for homogeneous and gravity for inhomogeneous scalar field. Diffusion flux and probability density are related by Fick’s law, diffusion coefficient is constant and invariant. Hamiltonian includes a “thermal” energy, kinetic energies of drift and diffusion flux. The probability density and the action function of drift form a canonical pair and canonical equations for them lead to the Hamilton-Jacobi-Madelung and continuity equations. At canonical transformation to a complex probability amplitude they form a linear Schrödinger equation. DQM explains appearance of quantum statistics, rest energy (“thermal” energy) and gravity (“thermal” diffusion) and leads to a low mass mechanism for composite particles.

Kernel density estimation-based solution of the nuclear Schrödinger equation

Solving the time-dependent Schrödinger equation for nuclear motion remains a challenge. Despite novel approaches based on Bohmian mechanics, the long-time stability and generalization to multiple dimensions remains an open question. In the present work a method based on an ensemble of classical particles instead of a wave function is employed to evolve the system. Quantum effects are introduced through forces derived from the quantum potential Q and the necessary derivatives are obtained from a density estimate using kernel density estimation. Application of the procedure to typical 1- and 2-dimensional problems yields good agreement with numerically exact solutions and favourable scaling with the number of particles is found.

Transport and dynamical properties for a bouncing ball model with regular and stochastic perturbations

Some statistical properties related to the diffusion in energy for an ensemble of classical particles in a bouncing ball model are studied. The particles are confined to bounce between two rigid walls. One of them is fixed while the other oscillates. The dynamics is described by a two dimensional nonlinear map for the velocity of the particle and time at the instant of the collision. Two different types of change of momentum are considered: (i) periodic due to a sine function and; (ii) stochastic. For elastic collisions case (i) leads to finite diffusion in energy while (ii) produces unlimited diffusion. On the other hand, inelastic collisions yield either (i) and (ii) to have limited diffusion. Scaling arguments are used to investigate some properties of the transport coefficient in the chaotic low energy region. Scaling exponents are also obtained for both conservative and dissipative case for cases (i) and (ii). We show that the parameter space has complicated structures either in Lyapunov as well as period coordinates. When stochasticity is introduced in the dynamics, we observed the destruction of the parameter space structures.

Generation of the ballistic particle transport in a periodic Hamiltonian system subjected to small time-dependent perturbation

The problem of the motion of an ensemble of classical particles in a periodic potential field has been considered. A method is proposed for generating directed ballistic transport by means of a perturbation oscillating in time and space. This method makes it possible to significantly reduce the perturbation intensity required to generate a particle flux. In particular, it has been shown that, even if the ensemble of particles is initially near the stable-equilibrium states, a directed flux appears at a perturbation amplitude of about 10−2 of the potential barrier height. The flux generation mechanism is associated with the creation of global chaotic diffusion due to resonances between spatial and time oscillations of perturbation. A nonlinear pendulum is considered as an example.

Exact uncertainty approach in quantum mechanics and quantum gravity

The assumption that an ensemble of classical particles is subject to nonclassical momentum fluctuations, with the fluctuation uncertainty fully determined by the position uncertainty, has been shown to lead from the classical equations of motion to the Schrodinger equation. This ‘exact uncertainty’ approach may be generalised to ensembles of gravitational fields, where nonclassical fluctuations are added to the field momentum densities, of a magnitude determined by the uncertainty in the metric tensor components. In this way one obtains the Wheeler-DeWitt equation of quantum gravity, with the added bonus of a uniquely specified operator ordering. No a priori assumptions are required concerning the existence of wave functions, Hilbert spaces, Planck's constant, linear operators, etc. Thus this approach has greater transparency than the usual canonical approach, particularly in regard to the connections between quantum and classical ensembles. Conceptual foundations and advantages are emphasised.

Wave-packet dynamics in quantum wells.

It has been recently recognized that in bulk semiconductors the displacement current caused by ultrafast optical generation of ``polarized pairs'' in the applied dc field is an important mechanism of charge transport in addition to the usual transport current. In quantum-well systems, this polarized pair creation is thought to be the only source of photocurrent at the early stages of photoexcitation since the bulklike transport current is inhibited by the barriers. In this work we perform a full quantum-mechanical analysis of ultrafast optical excitation in a dc-biased quantum well. We take into account the multiple transitions that become allowed in the dc field which breaks the \ensuremath{\Delta}n=0 selection rule. As a result, the carriers are created as wave packets formed by coherent superposition of several eigenstates. When the characteristic size of these wave packets (coherence length) is much larger than the well width (for long pulses and/or narrow wells), we recover the polarized pairs behavior of the photocurrent. For shorter pulses, when the coherence length becomes comparable to the well width, the photocurrent exhibits quantum beats. Finally, for very short pulses (around 10 fs) we find that the carriers in a quantum well can behave as an ensemble of classical particles and produce a transportlike photocurrent.