One-particle Dynamics Research Articles

Dynamical Thermalization of Interacting Fermionic Atoms in a Sinai Oscillator Trap

We study numerically the problem of dynamical thermalization of interacting cold fermionic atoms placed in an isolated Sinai oscillator trap. This system is characterized by a quantum chaos regime for one-particle dynamics. We show that, for a many-body system of cold atoms, the interactions, with a strength above a certain quantum chaos border given by the Åberg criterion, lead to the Fermi–Dirac distribution and relaxation of many-body initial states to the thermalized state in the absence of any contact with a thermostate. We discuss the properties of this dynamical thermalization and its links with the Loschmidt–Boltzmann dispute.

More quantum centrifugal effect in rotating frame

The behaviour of quantum systems in non-inertial frames is revisited from the point of view of affine coherent state (ACS) quantization. We restrict our approach to the one-particle dynamics confined in a rotating plane about a fixed axis. This plane is considered as punctured due to the existence of the rotation center, which is viewed as a singularity. The corresponding phase space is the affine group of the plane and the ACS quantization enables us to quantize the system by respecting the affine symmetry of the true phase space. Our formulation predicts the appearance of an additional quantum centrifugal term, besides the usual angular-momentum one, which prevents the particle to reach the singular rotation center. Moreover it helps us to understand why two different non-inertial Schrödinger equations are obtained in previous works. The validity of our equation can be confirmed experimentally by observing the harmonic oscillator bound states and the critical angular velocity for their existence.

Dynamics of observables and exactly solvable quantum problems: Using time-dependent density-functional theory to control quantum systems

We use analytic (current) density-potential maps of time-dependent (current) density functional theory (TD(C)DFT) to inverse engineer analytically solvable time-dependent quantum problems. In this approach the driving potential (the control signal) and the corresponding solution of the Schr\"odinger equation are parametrized analytically in terms of the basic TD(C)DFT observables. We describe the general reconstruction strategy and illustrate it with a number of explicit examples. First we consider the real space one-particle dynamics driven by a time-dependent electromagnetic field and recover, from the general TDDFT reconstruction formulas, the known exact solution for a driven oscillator with a time-dependent frequency. Then we use analytic maps of the lattice TD(C)DFT to control quantum dynamics in a discrete space. As a first example we construct a time-dependent potential which generates prescribed dynamics on a tight-binding chain. Then our method is applied to the dynamics of spin-1/2 driven by a time dependent magnetic field. We design an analytic control pulse that transfers the system from the ground to excited state and vice versa. This pulse generates the spin flip thus operating as a quantum NOT gate.

Short-range order and dynamics of atoms in liquid gallium

The features of the microscopic structure, as well as one-particle and collective dynamics of liquid gallium in the temperature range from T = 313 to 1273 K, are studied on the p = 1.0 atm isobar. Detailed analysis of the data on diffraction of neutrons and X-rays, as well as the results of atomic dynamics simulation, lead to some conclusions about the structure. In particular, for preset conditions, gallium is in the equilibrium liquid phase showing no features of any stable local crystalline clusters. The pronounced asymmetry of the principle peak of the static structure factor and the characteristic “shoulder” in its right-hand part appearing at temperatures close to the melting point, which are clearly observed in the diffraction data, are due to the fact that the arrangement of the nearest neighbors of an arbitrary atom in the system is estimated statistically from the range of correlation length values and not by a single value as in the case of simple liquids. Compactly located dimers with a very short bond make a significant contribution to the statistics of nearest neighbors. The temperature dependence of the self-diffusion coefficient calculated from atomic dynamics simulation agrees well with the results obtained from experimental spectra of the incoherent scattering function. Interpolation of the temperature dependence of the self-diffusion coefficient on a logarithmic scale reveals two linear regions with a transition temperature of about 600 K. The spectra of the dynamic structure factor and spectral densities of the local current calculated by simulating the atomic dynamics indicate the existence of acoustic vibrations with longitudinal and transverse polarizations in liquid gallium, which is confirmed by experimental data on inelastic scattering of neutrons and X-rays. It is found that the vibrational density of states is completely reproduced by the generalized Debye model, which makes it possible to decompose the total vibrational motion into individual contributions associated with the formation of acoustic waves with longitudinal and transverse polarizations. Comparison of the heights of the low-frequency component and of the high-frequency peak in the spectral density of vibrational states also indicates a temperature of T ≈ 600 K, at which the diffusion type of one-particle dynamics changes to the vibrational type upon a decrease in temperature. It is demonstrated that the modified Einstein–Stokes relation can be derived using the generalized Debye model.

Complex singularities of the fluid velocity autocorrelation function

There are intensive debates regarding the nature of supercritical fluids: if their evolution from liquid-like to gas-like behavior is a continuous multistage process or there is a sharp well-defined crossover. Velocity auto-correlation function Z is the established detector of evolution of fluid particles dynamics. Usually, complex singularities of correlation functions give more information. For this reason, we investigate Z in complex plane of frequencies using numerical analytic continuation. We have found that naive picture with few isolated poles fails describing Z(ω) of one-component Lennard-Jones (LJ) fluid. Instead, we see the singularity manifold forming branch cuts extending approximately parallel to the real frequency axis. That suggests LJ velocity autocorrelation function is a multivalued function of complex frequency. The branch cuts are separated from the real axis by the well-defined “gap” whose width corresponds to an important time scale of a fluid characterizing crossover of system dynamics from kinetic to hydrodynamic regime. Our working hypothesis is that the branch cut origin is related to competition between one-particle dynamics and hydrodynamics. The observed analytic structure of Z is very stable under changes in the temperature; it survives at temperatures two orders of magnitude higher than the critical one.

Mean Field Limits in Strongly Confined Systems

We consider the dynamics of $N$ interacting bosons in three dimensions which are strongly confined in one or two directions. We analyze the two cases where the interaction potential $w$ is rescaled by either $N^{-1}w(\cdot)$ or $a^{3\theta-1}w(a^\theta \cdot)$ and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit $N\rightarrow \infty $ the dynamics of the $N$-particle system are described by a nonlinear equation in one or two dimensions. In the case of the scaling $N^{-1}w(\cdot)$ this equation is the Hartree equation and for the scaling $a^{3\theta-1}w(a^\theta \cdot) $ the nonlinear Schrodinger equation. In both cases we obtain explicit bounds for the rate of convergence of the $N$-particle dynamics to the one-particle dynamics.

A geometric approach to the Landauer-Büttiker formula

We consider an ideal Fermi gas confined to a geometric structure consisting of a central region – the sample – connected to several infinitely extended ends—the reservoirs. Under physically reasonable assumptions on the propagation properties of the one-particle dynamics within these reservoirs, we show that the state of the Fermi gas relaxes to a steady state. We compute the expected value of various current observables in this steady state and express the result in terms of scattering data, thus obtaining a geometric version of the celebrated Landauer-Büttiker formula.

Mode-coupling approximation in a fractional-power generalization: Particle dynamics in supercooled liquids and glasses

We describe the particle dynamics in supercooled liquids in the mode-coupling approximation in a fractional-power generalization, where the kinetic integro-differential equations are exactly derived from the equations of motion of the dynamical variables by projection operator methods. We show that in the case of separated time scales in the particle dynamics and with the nonlinear interaction between the stochastic and translation motion modes taken into account, the solution of the equations gives a well-defined picture of singularities of the one-particle dynamics of supercooled liquids and glasses. Comparison with the data for the metal alloy Fe 50 Cr 50 atomic dynamics simulation demonstrates a good agreement in the entire temperature range corresponding to the supercooled liquid and glass phases.

Power-law tailed statistical distributions and Lorentz transformations

The present Letter, deals with the statistical theory [G. Kaniadakis, Phys. Rev. E 66 (2002) 056125; G. Kaniadakis, Phys. Rev. E 72 (2005) 036108], which predicts the probability distribution p(E)∝expκ(−I), where, I∝βE−βμ, is the collision invariant, and expκ(x)=(1+κ2x2+κx)1/κ, with κ2<1. This, experimentally observed distribution, at low energies behaves as the Maxwell–Boltzmann exponential distribution, while at high energies presents power law tails. Here we show that the function expκ(x) and its inverse lnκ(x), can be obtained within the one-particle relativistic dynamics, in a very simple and transparent way, without invoking any extra principle or assumption, starting directly from the Lorentz transformations. The achievements support the idea that the power law tailed distributions are enforced by the Lorentz relativistic microscopic dynamics, like in the case of the exponential distribution which follows from the Newton classical microscopic dynamics.

Out-of-Equilibrium Phase Transitions and Time-Asymptotic One-Particle Dynamics in the Vlasov Limit of The Hamiltonian Mean Field Model

When starting from specific initial conditions, the ferromagnetic-like XY Hamiltonian mean field (HMF) model evolves toward quasistationary states, with lifetimes diverging with the number N of degrees of freedom that violate equilibrium statistical mechanics predictions. Phase transitions have been reported between low-energy magnetized quasistationary states and large energy unexpected, antiferromagnetic-like ones with low, but nonvanishing, magnetization. This issue is addressed here in the Vlasov N→∞ limit. It is argued that the time asymptotic states emerging in the Vlasov limit can be related to simple generic time asymptotic forms for the force field. The proposed picture unveils the nature of the out-of-equilibrium phase transitions reported for the ferromagnetic HMF in the second order regime: This is a bifurcation point connecting an effective integrable Vlasov one-particle time-asymptotic dynamic to a partly ergodic one, which means an abrupt open-up of the Vlasov one-particle phase space. This is proposed as a mechanism for second-order phase transitions compatible with nonvanishing time-asymptotic values of the order parameter in mean-field long-range systems.

Cubic nonlinear Schrödinger equation with vorticity

In this paper, we introduce a new class of nonlinear Schrödinger equations (NLSEs), with an electromagnetic potential , both depending on the wavefunction Ψ. The scalar potential Φ depends on |Ψ|2, whereas the vector potential satisfies the equation of magnetohydrodynamics with coefficient depending on Ψ.In Madelung variables, the velocity field comes to be not irrotational in general and we prove that the vorticity induces dissipation, until the dynamical equilibrium is reached. The expression of the rate of dissipation is common to all NLSEs in the class.We show that they are a particular case of the one-particle dynamics out of dynamical equilibrium for a system of N identical interacting Bose particles, as recently described within stochastic quantization by Lagrangian variational principle.The cubic case is discussed in particular. Results of numerical experiments for rotational excitations of the ground state in a finite two-dimensional trap with harmonic potential are reported.

Finite-size particles, advection, and chaos: A collective phenomenon of intermittent bursting

We consider finite-size particles colliding elastically, advected by a chaotic flow. The collisionless dynamics has a quasiperiodic attractor and particles are advected towards this attractor. We show in this work that the collisions have dramatic effects in the system's dynamics, giving rise to collective phenomena not found in the one-particle dynamics. In particular, the collisions induce a kind of instability, in which particles abruptly spread out from the vicinity of the attractor, reaching the neighborhood of a coexisting chaotic saddle, in an autoexcitable regime. This saddle, not present in the dynamics of a single particle, emerges due to the collective particle interaction. We argue that this phenomenon is general for advected, interacting particles in chaotic flows.

SEMICLASSICAL MOTION OF DRESSED ELECTRONS

We consider an electron coupled to the quantized radiation field and subject to a slowly varying electrostatic potential. We establish that over sufficiently long times radiation effects are negligible and the dressed electron is governed by an effective one-particle Hamiltonian. In the proof only a few generic properties of the full Pauli–Fierz Hamiltonian H PF enter. Most importantly, H PF must have an isolated ground state band for |p|&lt;p c ≤∞ with p the total momentum and p c indicating that the ground state band may terminate. This structure demands a local approximation theorem, in the sense that the one-particle approximation holds until the semiclassical dynamics violates |p|&lt;p c . Within this framework we prove an abstract Hilbert space theorem which uses no additional information on the Hamiltonian away from the band of interest. Our result is applicable to other time-dependent semiclassical problems. We discuss semiclassical distributions for the effective one-particle dynamics and show how they can be translated to the full dynamics by our results.

One-particle dynamics in the Hubbard model at half-filling

The one-particle Green function for the 2d Hubbard model at half-filling and zero temperature is considered. By use of projection technique we derive equations of motion for the Green functions related to the four Hubbard operators c ̂ kσ α and c ̂ kσ α . Here, the index α = 1, 2indicates the two sublattices of the antiferromagnetic phase. The operator c ̂ kσ α creates a hole on sublattice α at singly occupied sites. Similarly, c ̂ kσ α annihilates an electron at doubly occupied sites. The self-energy matrix is decoupled and a self-consisten scheme for the Green functions is found. This scheme can be considered a generalization of one derived recently for the t- J model. For the hole band, described by the operators c ̂ kσ α , our result for U⪢ t shows strong coupling of the hole to spin excitations, similar to that found for the t- j model, with the additional feature that propagation of up-spin holes on the down sublattice is taken into account. For decreasing U the gap decreases roughly proportional to the staggered magnetization m. For U → 0, where m vanishes, we find the correct limit of uncorrelated electrons.

How certain is the uncertainty principle?

Abstract For the ground problem of Newtonian mechanics, recent work is reviewed where it is shown that there is a one-parameter family of Lagrangians all producing the same equations of motion, all non-equivalent canonically, and all satisfying Galilean covariance. The canonical momentum becomes parameter dependent and therefore ambiguous, and then so too do the canonical commutation rules and the Heisenberg uncertainty principle (e.g. the coordinate of one particle may become complementary not to its usual mechanical momentum but to the momentum of a different particle). A way of resolving the ambiguity of commutation rules, by reformulating many-particle dynamics as higher-order one-particle dynamics, is pointed out and includes as well a means to generalize relativistically a non-relativistic dynamics.

An essential ambiguity of quantum theory

It is shown that there is a substantial family of Lagrangians for one and the same ground problem of Newtonian mechanics where linear momentum is conserved. These alternative Lagrangians involve an arbitrary function of conserved momentum, which reduces to a one-parameter family when it is required that the Galilean group be canonically represented. Then the canonical momentum becomes parameter-dependent: and this leads quantally to ambiguous commutation rules and Heisenberg uncertainty principles—for example the coordinate of one particle may be complementary not to its own mechanical momentum (as in standard treatments) but to that of a different particle. One possible way of resolving the ambiguity is by reformulating many-particle dynamics as higher-order one-particle (HOOP) dynamics by decoupling the standard lower-order equations of motion. This also admits relativistic generalization in which the Poincare group is canonically represented.

Time scales and diffusion mechanisms in the Kramers equation with periodic potentials (I)

Noise-assisted diffusion on a periodic substrate is studied, solving the Klein-Kramers equation (KKE) by the matrix continued fraction method. Assuming a periodic cosine potential and a homogeneous friction, the one-particle dynamics is completely characterized, both at short and long times, calculating, up to large momentum transfer, the dynamic structure factor, its full width at half maximum, the velocity self-correlation spectrum and the mean square displacement. Four different dynamical regimes are found as the potential strength or the friction vary. At high potential barriers, diffusion proceeds by single or multiple jumps at high and low friction respectively, while two different kinds of continuous unactivated diffusion are found at lower barriers. These dynamical features are discussed in terms of three characteristic time scales whose relations determine the diffusion mechanisms.

The low-density limit of quantum systems

The present paper concludes a research programme developed, in the past three years, by various authors in Several papers. The basic idea of this programme is to expiain the irreversible behaviour of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system interacling with a reservoir and, in the first attempts to deal with this problem, only limits of observables of the system and deduced master equations were considered. In our approach we study the limits of quantities related to the whole compound syrtem. As a corollary we obtain an explanation of the physical origins of the quantum Brownian motion. In the present paper we study state at invene temperature p and fugacity r, through an interaction of the scattering type, i.e. one which preserves the total number of particles of the reservoir. We obtain a macroscopic equation, far the limit of the compound system, which is a quantum stochastic differential equation of the Poisson type, in the Frigerio and Maassen sense, whose coefficients are uniquely determined by the one-particle scattering operator of the original Hamiltonian system and whose driving noises are the Creation annihilation and number (or gauge) processes living in the space of a Fock quantum Brownian motion over the space L'(R, dt, where KO,, is an Hilbert space depending on the inverse temperature p, the one-particle reservoir dynamics, the free-system dynamics and the interaction. :he )ny&-si!g limit of2 system co-p!ed q-asi.f:ee bnsnn reje~~oi: 1 the eq:i!ibri:m

Molecular Dynamics Studies on Molten Alkali Hydroxides. III. One-particle Dynamics of Ions in Molten LiOH

Abstract One-particle dynamics of Li+ and OH− ions has been investigated by analysing ion trajectories calculated by molecular dynamics simulation of molten LiOH. Direct observation of ionic motions has elucidated that reorientational motion of OH− ion possesses a strong correlation with the translational motion of the surrounding Li+ ions. Significantly large forces or torques could not be found just before the large displacement or the reorientation of the ions. The energy fluctuation of the particle was very large and frequent. The potential energy fluctuation, which is a few tens of times larger than the kinetic energy fluctuation, seems to play an important role as the trigger of the large displacement and reorientational motion.

The self-consistent pendulum picture of the free electron laser revised

We reconsider the one-particle dynamics of a Free Electron Laser, adopting the so-called universal scaling. By a fully hamiltonian treatment of the electron and radiation field variables, we show that the electron phase-plane is never that of a pendulum. Actually, besides an elliptic and a hyperbolic pendulum-like fixed point, an extra elliptic point is present at the same phase value as the hyperbolic one, for large values of the detuning parameter δ. On decreasing δ, these two points collapse, which implies a dramatic change in the electron orbit topology, at a value of the detuning parameter which coincides with the instability threshold for exponential gain in the many-electron system.