Time-dependent Projection Research Articles

Saturation in the quasiadiabatical limit: a time-dependent projection operator formalism approach

This paper presents a new theoretical approach for the description of quasiadiabatic evolution of thermodynamic observables. The new method extends the projection operator technique by considering time-dependent projectors. A master equation is derived in the limit of a slow or adiabatic evolution and is applied to calculate the rate of saturation of dipolar order in a slowly rotating sample. To cite this article: T. Charpentier et al., C. R. Physique 5 (2004).

Optimal Masks for Solarg‐Mode Detection

The detection of gravity (g) modes of solar oscillations is important for probing the physical conditions in the Sun's energy-generating core. We have developed a new method of spatial masks optimized to reveal solar g-modes of angular degree l = 1-3 and applied it to Michelson Doppler Imager data in the frequency range of 50-500 μHz. These masks take into account the horizontal component of g-mode velocity eigenfunctions and the variations in the level of noise across the solar disk and adjust for the time-dependent mode projection properties caused by the inclination of the Sun's axis of rotation. They allow us to optimize the signal-to-noise ratio in the oscillation power spectra for potential g-modes of various angular order and degree. The peaks in the resulting spectra are analyzed in terms of their instrumental origin, long-term stability, and correspondence to the theoretically predicted g-mode spectrum. As a consequence of failing to detect any g-mode candidates, new upper limits for the surface amplitude of g-modes are obtained. The lowest upper limits in the range of 5-6 mm s-1 are found for sectorial g-modes (l = m). These limits are an order of magnitude higher than the theoretical prediction of Kumar et al. in 1996.

Direct Observation of Two-Dimensional Electron Solvation at Alcohol/Ag(111) Interfaces

Time-resolved two-photon photoemission was used to investigate the two-dimensional electron solvation by methanol, 1-propanol, 1-butanol, and 1-pentanol overlayers on a Ag(111) surface. For each system at coverages higher than one monolayer, several image potential state series with time-dependent energies were observed by using two-photon photoemission indicating that multiple time-dependent local work functions can originate from multiple coverages. The time-dependent energy relaxation seen is attributed to the rotation of the adsorbate molecular dipoles to solvate the electron. This rotation lowers the electron-layer interaction energy, causing a dynamic reduction of the local work function which is a signature of the solvation of the electron by the adsorbate layer. A classical electron-disk dipole model supports the conclusion that the dynamic variation in the image potential state energies results from a local work function effect and indicates that the evolution of the energy results from the time-dependent projection of the molecular dipole onto the surface normal on the femtosecond time scale.

Derivation of Transport Equations Using the Time-Dependent Projection Operator Method

We develop a formalism to carry out coarse-grainings in quantum field theoretical systems by using a time-dependent projection operator in the Heisenberg picture. A systematic perturbative expansion with respect to the interaction part of the Hamiltonian is given, and a Langevin-type equation without a time-convolution integral term is obtained. This method is applied to a quantum field theoretical model, and coupled transport equations are derived. Obtaining descriptions of nonequilibrium processes in quantum field theory is an important problem and has been actively studied. In the treatment of such systems, it is often important to carry out reductions or coarse-grainings of irrelevant degrees of freedom. In classical systems, it is widely believed that a system of macroscopic size composed of many microscopic variables exhibits rather simple macroscopic behavior that can be described in terms of only a few macroscopic variables. It is therefore not unreasonable to conjecture that such an elimination of the irrelevant information can be developed for application to quantum systems and help in the analysis of nonequilibrium quantum processes. The projection operator method is one well-known method for carrying out coarse-grainings systematically. 1) - 5) In this

Quantum Zeno and anti-Zeno paradoxes

Continuous observation of a time independent projection operator is known to prevent change of state (the quantum Zeno paradox). We discuss the recent result that generic continuous measurement of time dependent projection operators will in fact ensure change of state: an anti-Zeno paradox.

Application of the time-dependent projection operator technique: The nonlinear quantum master equation

The time-dependent projection operator technique according to Willis and Picard (Phys. Rev. A 9 (1974) 1343) offers a unique quantum statistical description of two interacting subsystems. The technique is used here to go beyond the standard quantum master equation (QME) for a small system coupled to a reservoir. Applying the time-dependent projection operator approach, one is able to overcome a perturbational treatment of the system–reservoir coupling and can incorporate, (i) how the dynamics of the system may drive the reservoir out of the equilibrium, and (ii) how this nonequilibrium state reacts back on the system dynamics. The case of a reservoir of harmonic oscillators coupled linearly by its coordinates to the small system is studied in detail. The derivation of a closed nonlinear equation of motion for the reduced statistical operator of the small system is demonstrated. The method is used to describe the motion of a quantum particle in a molecular system, e.g. a tunneling electron or an exciton, which interacts strongly with its environment.

Fokker–Planck and non-linear hydrodynamic equations of an inelastic system of several Brownian particles in a non-equilibrium bath

The Fokker–Planck equation for the translational modes of an inelastic system of Brownian particles in a non-equilibrium bath of light particles is derived from first principles of statistical mechanics. The bath and internal modes relax on a time scale that is much shorter than that of the translational modes and they are eliminated using time-dependent projection operators techniques and expansions in several small parameters. These small parameters reflect the difference in masses between the Brownian and bath particles, the weak coupling of the bath to the internal modes, the difference in mean bath and Brownian velocities and the macroscopic gradients of the system. The Fokker–Planck equation is expressed in terms of correlation functions over homogeneous local equilibrium averages and is valid up to second order in the smallness parameters and for times greater than the relaxation time of the fast modes. The non-linear hydrodynamic equations for the translational modes are derived using time-dependent projection operators and the effective Liouvillian obtained from the Fokker–Planck equation. The momentum and energy density hydrodynamic equations are not conserved and present terms which reflect the non-equilibrium nature of the bath and of the internal modes, as well as the irreversible processes occurring in the system.

Fokker-Planck equation and non-linear hydrodynamic equations of a system of several Brownian particles in a non-equilibrium bath

The Fokker-Planck equation of a system of several Brownian particles immersed in a non-equilibrium bath of light particles is derived from first principles of statistical mechanics using time-dependent projection operators. The Fokker-Planck equation contains the usual equilibrium streaming and dissipative terms as well as terms reflecting spatial variations in the bath pressure, temperature and velocity. We make use of the effective Liouvillian obtained from the Fokker-Plank equation and of time-dependent projection operators involving properties of local equilibrium distribution functions to derive the exact non-linear hydrodynamic equations of the Brownian particles. The exact equations are simplified using the fact that the thermodynamic forces vary slowly on a molecular timescale and the resulting local transport equations are expressed in terms of homogeneous local equilibrium averages. The non-equilibrium conditional distribution for the bath is obtained from the Fokker-Planck equation using time-dependent projection operators and is used to derive the non-linear hydrodynamic equations of the bath. The number density hydrodynamic equations for the bath and the Brownian particles remain unchanged from the case of a system of isolated particles, but the momentum and energy density expressions are no longer conserved and contain additional terms accounting for the non-equilibrium nature of the bath and for the irreversible processes occurring in the system. The non-linear hydrodynamic equations for the bath and Brownian densities are combined to yield the conserved non-linear hydrodynamic equations for the total densities of the system.

Mean field and collisional dynamics of interacting fermion-boson systems in a soluble model

A general time-dependent projection technique is applied to the study of the dynamics of quantum correlations in a system consisting of interacting fermionic and bosonic subsystems, described by the Jaynes-Cummings Hamiltonian. The amplitude modulation of the Rabi oscillations which occur for a strong, coherent initial bosonic field is obtained from the spin intrinsic depolarization resulting from collisional corrections to the mean-field approximation.

Initial-condition problem for a chiral Gross-Neveu system.

A time-dependent projection technique is used to treat the initial-value problem for self-interacting fermionic fields. On the basis of the general dynamics of the fields, we derive formal equations of kinetic-type for the set of one-body dynamical variables. A nonperturbative mean-field expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Gaussian mean-field approximation, for a uniform system described by the chiral Gross-Neveu Hamiltonian. Standard stationary features of the model, such as dynamical mass generation due to chiral symmetry breaking and a phenomenon analogous to dimensional transmutation, are reobtained in this context. The mean-field time evolution of nonequilibrium initial states is discussed.

Fokker−Planck Equation and Langevin Equation for One Brownian Particle in a Nonequilibrium Bath

The Brownian motion of a large spherical particle of mass M immersed in a nonequilibrium bath of N light spherical particles of mass m is studied. A Fokker−Planck equation and a generalized Langevin equation for an arbitrary function of the position and momentum of the Brownian particle are derived from first principles of statistical mechanics using time-dependent projection operators. These projection operators reflect the nonequilibrium nature of the bath, which is described by the exact nonequilibrium distribution function of Oppenheim and Levine [Oppenheim, I.; Levine, R. D. Physica A 1979, 99, 383]. The Fokker−Planck equation is obtained by eliminating the fast bath variables of the system [Van Kampen, N. G.; Oppenheim, I. Physica A 1986, 138, 231], while the Langevin equation is obtained using a projection operator which averages over these variables [Mazur, P.; Oppenheim, I. Physica 1970, 50, 241]. The two methods yield equivalent results, valid to second order in the small parameters e = (m/M)1/2...

The concentration equation of polymer solution in nonhomogeneous flow field

We have derived formally exact equations for the concentration of polymer in an externally imposed field using a time-dependent projection operator formalism. Two different bases are chosen as projectors; the initial state basis (equilibrium) at t=0 and the final state basis (steady state) at t→∞. The initial state basis is natural for studying transient behavior at short time and the final state basis is good for long time asymptotic behavior. Several methods have been suggested to obtain overall time dependences of the concentration, such as an interpolation model for the approximate initial and final state formulae. The general framework will be useful in studying the macroscopic characteristics of polymer solutions exposed to arbitrary flow fields in confined geometries using the molecular inputs. A simple illustration of polymer migration caused by the flow geometry and hydrodynamic interactions is provided. Calculation of the steady state probability function of polymer chains is also given.

Evolution Equations for the Nonequilibrium LO-Phonon Dynamics in Laser Excited Polar Semiconductors

Starting from an extensire model Hamiltonian a system of evolution equations is derived by means of a time-dependent projection operator technique which describes the nonequilibriurn dynamics of the LO-phonon system in a laser excited polar semiconductor. The carrier system is described in a hydrodynamic stage, whereas the phonon system is treated in an a thermal stage. Third-order anharmonic interactions between optical and acoustic phonons as well as between the acoustic phonons are contained in the theory

Kinetic approach to the initial-value problem in phi4 field theory.

A time-dependent projection technique is used to develop kinetic equations for simple observables in the context of ${\ensuremath{\varphi}}^{4}$ field theory. A mean-field expansion can be written for these equations which are numerically tractable in the few lower orders. The procedure is applied to the case of the spatially uniform system in 1 + 1 dimensions, including numerical solutions.

The time dependent projection operator method for tunneling in Josephson junctions

The time-dependent projector method of quantum statistics is applied to study the process of Josephson tunneling, being considered in a simplified microscopic model. Owing to the symmetry of the Hamiltonian, the evolution equations for dynamic variables, governing the motion in the junction, are derived exactly within the framework of different observation levels. The results show that the treatment of the Josephson junction by the projector technique illustrates the flexibility and power of this approach for an adequate description of dynamic processes in solids.

Linear response theory for a highly excited exciton-electron-hole-pair system in direct gap semiconductors in the athermal stage of evolution

The evolution of a highly excited exciton-electron-hole-pair system in the athermal stage is examined in a first principle theory by use of a time-dependent projection operator technique. The expectation values of excitons and electrons are cast out under the assumption of a Gaussian pulse shape in a theory, suitable for the description of transient properties. The linear response with respect to a white probe pulse is investigated by a Green's function technique, being an extension of the Kubo formalism for reference systems far off equilibrium. This formalism results in a correct description of the time-dependent behavior of effects like blue-shift and bleaching of exciton levels.

The calculation of product quantum state distributions and partial cross-sections in time-dependent molecular collision and photodissociation theory

A new method of analyzing the results of multidimensional time-dependent quantum dynamical wavepacket calculations in terms of the product quantum state distributions is presented. The method requires knowledge only of the time-dependent projection coefficients of the wavepacket onto individual product quantum states along a cut in the exit valley of the photodissociation or reaction process. The squares of the Fourier transforms of these coefficients then directly yield the cross-sections of interest. The great advantage of time-dependent quantum dynamics, namely that a single wavepacket calculation yields the cross-sections at all energies of interest, is fully exploited.

Climate Change Scenarios for the UK

The purpose of this paper is to present time-dependent scenarios of climate change for the UK, including possible changes in extreme climatic events. Initially, the methods of scenario construction are described. These involve the scaling of the regional patterns of equilibrium climate change produced by general circulation models (GCMs) by the time-dependent projection of global-mean temperature change. A 'Business-as-Usual' forcing scenario is then used to project global warming to the year 2050. This projection is used to develop scenarios of the UK temperature and precipitation change. The implications for sea level are also discussed. Finally, the climate change 'commitment' for the UK, due to past radiative forcing changes and the lags in the climate system, is considered.

The second kind of fluctuation-dissipation theorem for a nonequilibrium open system in interaction with a reservoir

Abstract By using the time-dependent projection operator method the second kind of fluctuation-dissipation theorem is derived for an open system interacting with a reservoir. The theorem is given in the form of correlation function of fluctuations of random force and interacting random force and brought into the Volterra equation formalism.

KINETIC APPROACH TO THE INITIAL VALUE PROBLEM IN QUANTUM FIELD THEORY

Time-dependent projection techniques developed to derive kinetic equations in the context of the quantum many-body problem are applied to Φ4 field theory. The approach is illustrated by working out the (0+1)-dimensional case explicitly, including numerical solutions of the kinetic equations. Extension to higher dimensions is briefly discussed.