Particle Density Operator Research Articles

A SOLVABLE MODEL OF INTERACTING MANY BODY SYSTEMS EXHIBITING A BREAKDOWN OF THE BOLTZMANN EQUATION

In a particular exactly solvable model of an interacting system, the Boltzmann equation predicts a constant single particle density operator, whereas the exact solution gives a single particle density operator with a nontrivial time dependence. All of the time dependence of the single particle density operator is generated by the correlations.

Structure Factor and Distribution Function of Degenerate Bose Gases Without Anomalous Averages

The structure factor and single-particle distribution function of degenerate Bose gas are calculated within the canonical ensemble without using anomalous averages. The hypothesis about the C-number behavior of the particle density operator with zero momentum (in the Bose condensate) is accepted. The poles of the “density-density” Green function and the single-particle Green’s function are identical and correspond to the well known spectrum of phonon-roton excitations.

Configuration Path Integral Monte Carlo

AbstractA novel path integral Monte Carlo (PIMC) approach for correlated many‐particle systems with arbitrary pair interaction in continuous space at low temperatures is presented. It is based on a representation of the N ‐particle density operator in a basis of (anti‐)symmetrized N ‐particle states (configurations of occupation numbers). The path integral is transformed into a sum over trajectories with the same topology and, finally, the limit of M → ∞, where M is the number of high‐temperature factors, is analytically performed. This yields exact expressions for the thermodynamic quantities and allows to perform efficient simulations for fermions at low temperature and weak to moderate coupling. Our method is expected to be applicable to dense quantum plasmas in the regime of strong degeneracy where conventional PIMC fails due to the fermion sign problem (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Kohn–Sham system at zero temperature

A one-dimensional Kohn–Sham system for spin particles is considered which effectively describes semiconductor nanostructures, and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn–Sham potential and obtain W1,2-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow us to apply Schauder's fixed point theorem. In the case of vanishing exchange–correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.

Irreversible evolution of many-electron systems coupled to a statistical environment

We study the evolution of a many-electron system that is confined in a finite spatial region and coupled to a statistical environment. The latter may be composed of several independent bath subsystems, which are held at some statistical equilibrium. From the master equations describing the evolution of the coarse-grained $N$-particle density matrix, we obtain the equations, which describe the evolution of the $n$-particle density operators ${\overline{D}}^{(n)}$ for $n<N$. These equations are hierarchically coupled through the electron-electron interaction. We show that the hierarchy can be truncated under the assumption that the residual interaction of the electrons in the considered system with the environment introduces a memory loss, which hinders the electronic system to build up more than two-particle correlations. We first consider a weakly excited electronic system, which looses memory but where energy exchange with the statistical environment can be neglected. This is the quantum analog of the classical Boltzmann gas in a box. We derive the master equations, which describe the irreversible evolution of the coarse-grained one-particle density matrix. Based on this result, we show that, in accord with the second law of thermodynamics, the corresponding von Neumann entropy either increases with time or it remains constant. Finally, allowing also for energy exchange with one or more bath subsystems provided, e.g., by phonons or photons, we obtain the corresponding general master equations that describe the evolution of a spatially confined interacting electron gas of metallic density.

The fermion density operator in the droplet bosonization picture

We derive the phase space particle density operator in the ‘droplet’ picture of bosonization in terms of the boundary operator. We demonstrate that it satisfies the correct algebra and acts on the proper Hilbert space describing the underlying fermion system, and therefore it can be used to bosonize any Hamiltonian or related operator. As a demonstration we show that it reproduces the correct excitation energies for a system of free fermions with arbitrary dispersion relations.

Simple model for the detection of a particle by a point detector

A model for a one-dimensional quantum particle interacting with a point detector is developed and analyzed. The analysis leads to any simple, intuitively pleasing expression for the final density operator of the particle in terms of the freely propagating particle density operator, a projection operator representing the detector volume, and a probability density for the time at which the detector fires. This probability density is found explicitly to be the presence-time density given by ${\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}∕\ensuremath{\int}{\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}dt$ where ${\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)$ is the free particle wave function evaluated at the detector position $(x=0)$. Moreover, it is shown that the resultant density operator transformation is a nonlinear transformation, not describable as a positive operator valued measure.

Friedel oscillations in a gas of interacting one-dimensional fermionic atoms confined in aharmonic trap

Using an asymptotic phase representation of the particle density operator in the one-dimensional harmonic trap, the part which describes the Friedel oscillations is extracted. The expectation value with respect to the interacting ground state requires the calculation of the mean squareaverage of a properly defined phase operator. This calculation is performed analytically forthe Tomonaga–Luttinger model with harmonic confinement. It is found that the envelopeof the Friedel oscillations at zero temperature decays with the boundary exponentν = (K+1)/2 away from the classical boundaries. This value differs from that known for open boundaryconditions or strong pinning impurities. The soft boundary in the present case thusmodifies the decay of Friedel oscillations. The case of two components is also discussed.

FERMION AND BOSON RANDOM POINT PROCESSES AS PARTICLE DISTRIBUTIONS OF INFINITE FREE FERMI AND BOSE GASES OF FINITE DENSITY

The aim of this paper is to show that fermion and boson random point processes naturally appear from representations of CAR and CCR which correspond to gauge invariant generalized free states (also called quasi-free states). We consider particle density operators ρ(x), x ∈ ℝd, in the representation of CAR describing an infinite free Fermi gas of finite density at both zero and finite temperature [6], and in the representation of CCR describing an infinite free Bose gas at finite temperature [5]. We prove that the spectral measure of the smeared operators ρ(f) = ∫ dx f(x) ρ(x) (i.e., the measure μ which allows to realize the ρ(f)'s as multiplication operators by <·, f> in L2(dμ)) is a well-known fermion, respectively boson process on the space of all locally finite configurations in ℝd

Reaction-Diffusion Processes with Multi-Species of Particles

We study reaction-diffusion processes with multi-species of particles on a hypercubic lattice. We define the particle density operators for multi-species case and express the time evolution equation of correlation functions in quantum Hamiltonian formalism. We consider the conditions that the time evolution of correlation functions does not generate higher-order correlation functions. These conditions are called the closed conditions. Moreover we generalize the dual processes which are known in the symmetric exclusion processes and find that the Hamiltonian of the dual processes fixes the time evolution of correlation functions of processes with multi-species of particles.

A density-corrected quantum Boltzmann equation

Binary correlations are a recognized part of the pair density operator, but the influence of binary correlations on the singlet density operator is usually not emphasized. Here free motion and binary correlations are taken as independent building blocks for the structure of the nonequilibrium singlet and pair density operators. Binary correlations are assumed to arise from the collision of twofree particles. Together with the first BBGKY equation and a retention of all terms that are second order in gas density, a generalization of the Boltzmann equation is obtained. This is an equation for thefree particle density operator rather than for the (full) singlet density operator. The form for the pressure tensor calculated from this equation reduces at equilibrium to give the correct (Beth-Uhlenbeck) second virial coefficient, in contrast to a previous quantum Boltzmann equation, which gave only part of the quantum second virial coefficient. Generalizations to include higher-order correlations and collision types are indicated.

Free-Fermionic Description of the Repulsive Nonlinear Schrödinger Model

It is shown that in an infinite volume, the nonlinear Schrodinger model (repulsive case) is equivalent to a nonrelativistic free-fermion field theory. This is accomplished by expressing the Hamiltonian, momentum operator and particle density operator in terms of free-fermion operators constructed out of reflection operators. It is also shown that a thermodynamic trace of the density operator can be consistently regularized so as to yield a Fermi-Dirac distribution.

On the hydrodynamics of a charged superfluid

The microscopic derivation of the nonlinear gauge-invariant equations of a charged superfluid in external fields coupled with the energy-, momentum- and particle-density operators in suggested based on two concepts of N.N. Bogloliubov - quasi averages and a contraction of the description. Galilean invariance is not necessarily assumed.

Single-particle widths induced by axially symmetric vibrations in extended fermion matter

We examine the single-particle widths in extended fermion matter originated in the coupling to a harmonic vibration that presents axial symmetry. This is a primary model for the broadening in the nucleon spectrum in an axially symmetric nucleus where a giant collective mode becomes excited. We resort to the Quantal Brownian Motion model of the coupled dynamics of collective and intrinsic degrees of freedom in fermion systems and propose a modified kinetic equation for the particle density operator. This equation is solved in the relaxation time approximation from where the single-particle widths are extracted as the collision rates in the fermionic environment. The geometry and dynamics of the collisions are analyzed, as well as the relevance of those sources of energy spread other than the interaction between the particles and the oscillation, such as the temperature and the inelasticity parameter.

On the pressure dependence of the polarization of diamagnetic muons and muonium in pure noble gases: Is there a missing fraction?

The amplitudes of the signals in μSR exhibit pressure dependencies which are associated with the stopping dynamics of muonium atoms and diamagnetic muon species observed when muons are thermalized in pure noble gases. To explain this effect, a set of coupled rate equations, with time dependent rates and based upon quantal Boltzmann equations, have been developed to describe the spin dynamics for the thermalization of the two species. These, by definition positive, rates depend upon time through the translational single particle density operators associated with each species. Thus, to exactly solve the spin dynamics, the coupled kinetic Boltzmann equations for the stopping process must also be solved. Furthermore, the rate equations also contain spin dynamics generated by the muonium hyperfine interaction. It is the presence of this hyperfine interaction which leads to the loss of polarization for low pressure gases. The coupled quantal rate equations have been solved for a model of the stopping dynamics in which the rates, taken as square box functions of time, describe the charge exchange regime wherein muonium is both formed and ionized by subsequent collisions. Two post charge exchange extensions of this model are now considered. Following the charge exchange region, in the loss model, it is assumed that the rate of muonium formation is zero while the ionization rate is not. On the other hand, the capture model assumes that the ionization rate is zero while the muonium formation rate is not. Fits are presented to the available data for argon, krypton, xenon and neon. Since argon has both diamagnetic and paramagnetic signals then independent fits to each may be compared. A single set of fitting parameters has been found which describes both signals. This single fit requires a further extension of the models. In particular, a missing fraction must be assumed! The missing fraction model is also required when fits are made to the krypton and xenon data. On the other hand, the fit to the neon data is inconclusive. This is the first suggestion that a missing fraction may exist in the pure noble gases. Such missing fractions have been well established in condensed phases.

Number conserving relaxation-time approximation for second-order quantum response

The effect of damping on the quantum response of an electronic system is studied up to second order in an applied electro-magnetic field. This is accomplished by introducing a relaxation term into the equation of motion for the single particle density operator. This relaxation term contains a local chemical potential which is determined by requirements of gauge invariance and number conservation.

Transport properties of intermediate valence compounds studied within the alloy analog approximation of the Periodic Anderson Model

We use the Periodic Anderson Model (PAM) to describe the transport properties of intermediate valence compounds. The transport quantities of interest are related to the current-current response function. Therefore, a current operator being consistent with the PAM Hamiltonian must be defined. This aim is achieved by defining a proper particle density operator and using the continuity equation. The PAM is treated within the alloy analog approximation, i.e. it is replaced by the sum of two effective single-particle “alloy”-Hamiltonians, which are treated within the coherent potential approximation (CPA). Then the CPA for transport quantities is used to calculate the current-current response function. It is shown that for all reasonable assumptions for the conduction band and the hybridization dispersion the current vertex corrections vanish within the CPA. For different assumptions concerning the hybridization dispersion, we present numerical results for the temperature and parameter dependence of some transport quantities, in particular the static resistivity. The relevance of these results for the understanding of the typical experimental resistivity behaviour obtained for different intermediate valence compounds and possible shortcomings of our approach are discussed.

A density functional representation of quantum chemistry. III. Rigorous realization of the program in lattice space

AbstractA mathematically rigorous reformulation of molecular quantum mechanics in terms of the particle density operator and a canonically conjugated phase field is given. Using a momentum cutoff, it is shown that the usual molecular Hamiltonian can be expressed in terms of the particle density operator and a rigorously defined phase operator. It is shown that this Hamiltonian converges strongly to the cutoff‐free Hamiltonian. In spite of the fact that this Hamiltonian is of second order in the phase operators, all hitherto published expressions are not correct. Unfortunately, the correct formulation destroys the intuitive appeal of using the particle density operator as a coordinate for the many‐body problems of quantum chemistry. Unless somebody provides an essential new and clever idea, we propose to resist the fascination of a local quantum field theory of molecular matter in terms of the particle density operator.

A density functional representation of quantum chemistry. II. Local quantum field theories of molecular matter in terms of the charge density operator do not work

AbstractA comprehensive review of the attempts to rephrase molecular quantum mechanics in terms of the particle density operator and the current density or phase density operator is given. All pertinent investigations which have come to our attention suffer from severe mathematical inconsistencies and are not adequate to the few‐body problem of quantum chemistry. The origin of the failure of these attempts is investigated and it is shown that a realization of a local quantum field theory of molecular matter in terms of observables would presuppose the solution of many highly nontrivial mathematical problems.

Comments on Velocity Operators of Bose Fluids

It was perhaps F. London1> who first pointed· out the difficulty of defining local yelocity operators to transcribe Landau's quantum hydrodynamics2> literally ·into microscopic theory. It is related with the fact that the particle density operator has no inverse. In connection with their collective description 'of He 11,8> however, Yamasaki, Kebukawa, and Sunakawa4> have recently claimed that onecan define velocity operators which are commutative and rotation-free in the case of Bose fluids. The aim of the present letter is to point out that the mathematical consistency of their proof is dubious. They ··define a number of operators by iteration or equivalently by integral equa. tions. To make our argument explicit and definite, let us take the operator