Projection Operator Technique Research Articles

First-principle approach to rescale the dynamics of simulated coarse-grained macromolecular liquids

We present a detailed derivation and testing of our approach to rescale the dynamics of mesoscale simulations of coarse-grained polymer melts (I. Y. Lyubimov, J. McCarty, A. Clark, and M. G. Guenza, J. Chem. Phys. 132, 224903 (2010)). Starting from the first-principle Liouville equation and applying the Mori-Zwanzig projection operator technique, we derive the generalized Langevin equations (GLEs) for the coarse-grained representations of the liquid. The chosen slow variables in the projection operators define the length scale of coarse graining. Each polymer is represented at two levels of coarse graining: monomeric as a bead-and-spring model and molecular as a soft colloid. In the long-time regime where the center-of-mass follows Brownian motion and the internal dynamics is completely relaxed, the two descriptions must be equivalent. By enforcing this formal relation we derive from the GLEs the analytical rescaling factors to be applied to dynamical data in the coarse-grained representation to recover the monomeric description. Change in entropy and change in friction are the two corrections to be accounted for to compensate the effects of coarse graining on the polymer dynamics. The solution of the memory functions in the coarse-grained representations provides the dynamical rescaling of the friction coefficient. The calculation of the internal degrees of freedom provides the correction of the change in entropy due to coarse graining. The resulting rescaling formalism is a function of the coarse-grained model and thermodynamic parameters of the system simulated. The rescaled dynamics obtained from mesoscale simulations of polyethylene, represented as soft-colloidal particles, by applying our rescaling approach shows a good agreement with data of translational diffusion measured experimentally and from simulations. The proposed method is used to predict self-diffusion coefficients of new polyethylene samples.

RAMAN PERTURBATION AND SPATIAL SOLITON PROPAGATION IN AN OPTICAL LATTICE

A new approach recently developed by one of the authors has been utilised to study the effect of Raman perturbation on the propagation characteristics of spatial soliton in an optical lattice. The time-dependent nature of perturbation effectively induces a spatio-temporal behaviour of the pulse. The variation of the amplitude, width, chirp and frequency of the pulse are obtained through the projection operator technique. Important consequences are seen to follow due to the mutual influence of the periodicity of the lattice and the strength of Raman perturbations. Optimum configurations are sorted out with the help of an external pumping at fixed spatial interval. It is observed that in the absence of any dispersion management, it is possible to restore the profile of the pulse by pumping at regular series of intervals.

Non-Markovian dynamics of a qubit in a reservoir: different solutions of non-Markovian master equation

We present a non-Markovian master equation for a qubit interacting with a general reservoir, which is derived according to the Nakajima—Zwanzig and the time convolutionless projection operator technique. The non-Markovian solutions and Markovian solution of dynamical decay of a qubit are compared. The results indicate the validity of non-Markovian approach in different coupling regimes and also show that the Markovian master equation may not precisely describe the dynamics of an open quantum system in some situation. The non-Markovian solutions may be effective for many qubits independently interacting with the heated reservoirs.

Projection iterative methods for solving some systems of general nonconvex variational inequalities

In this article, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We use the projection operator technique to establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem. This alternative equivalent formulation is used to suggest and analyse some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of nonconvex variational inequalities, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and an improvement of the previously known results for variational inequalities.

The operations of quantum logic gates with pure and mixed initial states

The implementations of quantum logic gates realized by the rovibrational states of a C(12)O(16) molecule in the X((1)Σ(+)) electronic ground state are investigated. Optimal laser fields are obtained by using the modified multitarget optimal theory (MTOCT) which combines the maxima of the cost functional and the fidelity for state and quantum process. The projection operator technique together with modified MTOCT is used to get optimal laser fields. If initial states of the quantum gate are pure states, states at target time approach well to ideal target states. However, if the initial states are mixed states, the target states do not approach well to ideal ones. The process fidelity is introduced to investigate the reliability of the quantum gate operation driven by the optimal laser field. We found that the quantum gates operate reliably whether the initial states are pure or mixed.

Quantum mechanical generalized phase-shift approach to atom-surface scattering: A Feshbach projection approach to dealing with closed channel effects

We have developed a new method for solving quantum dynamical scattering problems, using the time-independent Schrödinger equation (TISE), based on a novel method to generalize a "one-way" quantum mechanical wave equation, impose correct boundary conditions, and eliminate exponentially growing closed channel solutions. The approach is readily parallelized to achieve approximate N(2) scaling, where N is the number of coupled equations. The full two-way nature of the TISE is included while propagating the wave function in the scattering variable and the full S-matrix is obtained. The new algorithm is based on a "Modified Cayley" operator splitting approach, generalizing earlier work where the method was applied to the time-dependent Schrödinger equation. All scattering variable propagation approaches to solving the TISE involve solving a Helmholtz-type equation, and for more than one degree of freedom, these are notoriously ill-behaved, due to the unavoidable presence of exponentially growing contributions to the numerical solution. Traditionally, the method used to eliminate exponential growth has posed a major obstacle to the full parallelization of such propagation algorithms. We stabilize by using the Feshbach projection operator technique to remove all the nonphysical exponentially growing closed channels, while retaining all of the propagating open channel components, as well as exponentially decaying closed channel components.

Reduced Density Matrix Theory for Vibrational Absorption Line Shape in Energy Transfer Systems: Non-Condon Effects in Water

Using the projection operator technique, a reduced density matrix theory for linear absorption spectrum of energy transfer systems is developed for the theoretical absorption line shape of the systems with non-Condon transitions. As an application, we considered a model system of OH vibrations of water. In the present model calculation, the OH vibration modes are coupled to each other via intra-molecular coupling mechanism while their intermolecular couplings are turned off. The time-correlation functions appearing in the formulation are calculated from a mixed quantum/classical mechanics method. The present theory is successful in reproducing the exact absorption line shape. Also the present theory was improved from an existing approximate theory, time-averaged approximation approach.

Transient Kinetics of Nucleotide Binding to the P-glycoprotein Multidrug Transporter

The P-glycoprotein multidrug transporter (Pgp) is a member of the ABC (ATP-binding cassette) superfamily of proteins, and contains two highly conserved cytosolic nucleotide-binding domains (NBDs). These domains couple ATP hydrolysis to the export of a variety of hydrophobic natural products, chemotherapeutic drugs and peptides. Pgp ATPase activity has been extensively characterized, but many details of the catalytic cycle of ATP hydrolysis remain unexplored. The fluorescent nucleotide analogue, TNP-ADP, interacts with the NBDs of Pgp, and the stoichiometry and affinity of binding were previously determined in our laboratory. Transient kinetic methods are commonly used to investigate the mechanism of molecular processes over a time-scale from milliseconds to hundreds of seconds. In this work, we studied the binding of TNP-ADP to Pgp using rapid stopped-flow kinetics. In these experiments, the binding reaction was monitored by following different fluorescent signals, under pseudo-first-order conditions (excess ligand). The time course of TNP-ADP binding displayed five relaxation times (τi's) over a time-span of 300 sec, which correspond to at least five different transitions. All the relaxation times presented a strong temperature-dependence. The time course of the reaction was analyzed by the computational tool Global Kinetic Explorer (KinTek, USA), using several models of sequential isomerizations after/before the fast bimolecular binding reaction. Also, the parameters were analyzed using the matrix projection operator technique for kinetic data (Bujalowski et al., 2000, J. Mol. Biol. 295:831). Both approaches provide, for the first time, information about the rate constants and fluorescent properties of the diverse intermediates formed during binding of TNP-ADP to Pgp. Elucidation of the details of the interaction of nucleotides with Pgp is of prime importance for formulation of a detailed mechanism of action of the transporter. This research is supported by a grant from the Canadian Cancer Society.

Projection algorithms for the system of mixed variational inequalities in Banach spaces

In this paper, the system of mixed variational inequalities is introduced and considered in Banach spaces, which includes some known systems of variational inequalities and the classical variational inequalities as special cases. Using the projection operator technique, we suggest some iterative algorithms for solving the system of mixed variational inequalities and prove the convergence of the proposed iterative methods under suitable conditions. Our theorems generalize some known results shown recently.

Time-dependent closure relations for relativistic collisionless fluid equations

Linear fluid equations for relativistic and collisionless plasmas are derived. Closure relations for the fluid equations are analytically computed from the relativistic Vlasov equation in the Fourier space (ω,k), where ω and k are the conjugate variables of time t and space x variables, respectively. The mathematical method used is based on the projection operator techniques and the continued fraction mathematical tools. The generalized heat flux and stress tensor are calculated for arbitrary parameter ω/kc where c is the speed of light, and for arbitrary relativistic parameter z=mc²/T , where m is the particle rest mass and T, the plasma temperature in energy units.

Transport in the three-dimensional Anderson model: an analysis of the dynamics at scales below the localization length

Single-particle transport in disordered potentials is investigated at scales below the localization length. The dynamics at those scales is concretely analyzed for the three-dimensional Anderson model with Gaussian on-site disorder. This analysis particularly includes the dependence of characteristic transport quantities on the amount of disorder and the energy interval, e.g. the mean free path that separates ballistic and diffusive transport regimes. For these regimes mean velocities and diffusion constants are quantitatively given. Using the Boltzmann equation in the limit of weak disorder, we reveal the known energy dependences of transport quantities. By the application of the time-convolutionless projection operator technique in the limit of strong disorder, we obtain evidence for much less pronounced energy dependences. All our results are partially confirmed by the numerically exact solution of the time-dependent Schrödinger equation or by approximative numerical integrators. A comparison with other findings in the literature is also provided.

Quantum dot Rabi rotations beyond the weak exciton–phonon coupling regime

We study the excitonic dynamics of a driven quantum dot under the influence of a phonon environment, going beyond the weak exciton–phonon coupling approximation. By combining the polaron transform and time-local projection operator techniques, we develop a master equation that can be valid over a much larger range of exciton–phonon coupling strengths and temperatures than in the case of the standard weak-coupling approach. For the experimentally relevant parameters considered here, we find that the weak-coupling and polaron theories give very similar predictions for low temperatures (below 30 K), while at higher temperatures we begin to see discrepancies between the two. This is because, unlike the polaron approach, the weak-coupling theory is incapable of capturing multiphonon effects, while it also does not properly account for phonon-induced renormalization of the driving frequency. In particular, we find that the weak-coupling theory often overestimates the damping rate when compared to the polaron theory. Finally, we extend our theory to include non-Markovian effects and find that, for the parameters considered here, they have little bearing on the excitonic Rabi rotations when plotted as a function of pulse area.

Projection operator approach to spin diffusion in the anisotropic Heisenberg chain at high temperatures

Spin transport in the anisotropic Heisenberg chain is typically investigated theoretically with respect to the finiteness of transport coefficients only. Assuming their finiteness at high temperatures, we develop a concrete quantitative picture of the diffusion constant/(dc-)conductivity as a function of both the anisotropy parameter Δ and the spin quantum number s, going beyond the most commonly considered case s=1/2. Using this picture, we enable the comparison of finite transport coefficients from complementary theoretical methods on a quantitative level, having more significance than the finiteness alone. Our method is essentially based on an application of the time-convolutionless projection operator technique to current autocorrelations. This technique, although being a perturbation theory in Δ, is found to be applicable, even if Δ is not small. This finding supports the applicability to a wider class of strongly interacting many-particle quantum systems.

Non-Markovian master equation for a damped driven two-state system

We present a detailed microscopic derivation for a non-Markovian master equation for a driven two-state system interacting with a general structured reservoir. The master equation is derived using the time-convolutionless projection operator technique in the limit of weak coupling between the two-state quantum system and its environment. We briefly discuss the Markov approximation, the secular approximation and their validity.

Generalized projective representations for [formula omitted

It is well known that n-dimensional projective group gives rise to a non-homogeneous representation of the Lie algebra sl ( n + 1 ) on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known as Larsson functor), we generalize the above representation of sl ( n + 1 ) to a non-homogeneous representation on the tensor space of any finite-dimensional irreducible g l ( n ) -module with the polynomial space. Moreover, the structure of such a representation is completely determined by employing projection operator techniques and well-known Kostant's characteristic identities for certain matrices with entries in the universal enveloping algebra. In particular, we obtain a new one parameter family of infinite-dimensional irreducible sl ( n + 1 ) -modules, which are in general not highest-weight type, for any given finite-dimensional irreducible sl ( n ) -module. The results could also be used to study the quantum field theory with the projective group as the symmetry.

Distance and angular holonomic constraints in molecular simulations

Finding the energy minima of systems with constraints is a challenging problem. We develop a minimization method based on the projection operator technique to enforce distance and angle constraints in minimization and reaction-path dynamics. The application of the projection operator alone does not maintain the constraints, i.e., they are slightly violated. Therefore, we use the SHAKE-methodology to enforce the constraints after each minimization step. We have extended theta -SHAKE for bend angles and introduce phi-SHAKE and chi-SHAKE to constrain dihedral and out-of-plane angles, respectively. Two case studies are presented: (1) A mode analysis of united-atom n-butane with various internal degrees of freedom kept frozen and (2) the minimization of chromene at a fixed approach toward the catalytic site of a (salen)Mn. The obtained information on energetics can be used to explain why specific enantioselectivity is observed. Previous minimization methods work for the free molecular case, but fail when molecules are tightly confined.

On general quasi-variational inequalities

A new class of general quasi-variational inequalities involving two operators is introduced and studied. Using essentially the projection operator technique, we establish the equivalence between the general quasi-variational inequalities and the fixed-point problem and the Wiener–Hopf equations. These alternative equivalent formulations have been used to suggest and analyze several iterative methods for solving the general quasi-variational inequalities. We also discuss the convergence criteria of these iterative methods under some suitable conditions. Several special cases are also discussed.

The spin-1 two-dimensional [formula omitted]– [formula omitted] Heisenberg antiferromagnet on a triangular lattice

The spin-1 Heisenberg antiferromagnet on a triangular lattice with the nearest- and next-nearest-neighbor couplings, J 1 = ( 1 − p ) J and J 2 = p J , J > 0 , is studied in the entire range of the parameter p. Mori's projection operator technique is used as a method which retains the rotation symmetry of spin components and does not anticipate any magnetic ordering. For zero temperature four second-order phase transitions are observed. At p ≈ 0.038 the ground state is transformed from the long-range ordered 120° spin structure into a state with short-range ordering, which in its turn is changed to a long-range ordered state with the ordering vector Q ′ = ( 0 , − 2 π 3 ) at p ≈ 0.2 . For p ≈ 0.5 a new transition to a state with a short-range order occurs. This state has a large correlation length which continuously grows with p until the establishment of a long-range order happens at p ≈ 0.65 . In the range 0.5 < p < 0.96 , the ordering vector is incommensurate. With growing p it moves along the line Q ′ – Q 1 to the point Q 1 = ( 0 , − 4 π 3 3 ) which is reached at p ≈ 0.96 . The obtained state with a long-range order can be conceived as three interpenetrating sublattices with the 120° spin structure on each of them.

Projection-operator approach to master equations for coarse-grained occupation numbers in nonideal quantum gases

We aim at deriving an equation of motion for specific sums of momentum mode occupation numbers from models for electrons in periodic lattices experiencing elastic scattering, electron-phonon scattering, or electron-electron scattering. These sums correspond to "grains" in momentum space. This equation of motion is supposed to involve only a moderate number of dynamical variables and/or exhibit a sufficiently simple structure such that neither its construction nor its analyzation or solution requires substantial numerical effort. To this end we compute, by means of a projection operator technique, a linear(ized) collision term which determines the dynamics of the above grain sums. This collision term results as nonsingular finite-dimensional rate matrix and may thus be inverted regardless of any symmetry of the underlying model. This facilitates calculations of, e.g., transport coefficients, as we demonstrate for a three-dimensional Anderson model featuring weak disorder.

Stability criterion for Gaussian pulse propagation through negative index materials

We analyze the dynamics of propagation of a Gaussian light pulse through a medium having a negative index of refraction employing the recently reported projection operator technique. The governing modified nonlinear Schr\"odinger equation, obtained by taking into account the Drude dispersive model, is expressed in terms of the parameters of Gaussian pulse, called collective variables, such as width, amplitude, chirp, and phase. This approach yields a system of ordinary differential equations for the evolution of all the pulse parameters. We demonstrate the dependence of stability of the fixed-point solutions of these ordinary differential equations on the linear and nonlinear dispersion parameters. In addition, we validate the analytical approach numerically utilizing the method of split-step Fourier transform.