Adiabatic eigenfunction-based approach for coherent excitation transfer: An almost analytical treatment of the Fenna-Matthews-Olson complex

We suggest a method of studying coherence in finite-level systems coupled to the environment and use it for the Hamiltonian that has been used to describe the light-harvesting pigment-protein complex. The method works with the adiabatic states and transforms the Hamiltonian to a form in which the terms responsible for decoherence and population relaxation are separated out. Decoherence is then accounted for nonperturbatively and population relaxation using a Markovian master equation. Almost analytical results can be obtained for the seven-level system, and the calculations are very simple for systems with more levels. We apply the treatment to the seven-level system, and the results are in excellent agreement with the exact numerical results of Nalbach et al. Nalbach, Braun, and Thorwart, Phys. Rev. E 84, 041926 (2011)]. Our approach is able to account for decoherence and population relaxation separately. It is found that decoherence causes only damping of oscillations and does not lead to transfer to the reaction center. Population relaxation is necessary for efficient transfer to the reaction center, in agreement with earlier findings. Our results show that the transformation to the adiabatic basis followed by a Redfield type of approach leads to results in good agreement with exact simulation.

Adiabatic Eigenfunction Based Approach to Coherent Transfer: Application to the Fenna-Matthews-Olson (FMO) Complex and the Role of Correlations in the Efficiency of Energy Transfer

We have recently suggested a method (Pallavi Bhattacharyya and K. L. Sebastian, Physical Review E 2013, 87, 062712) for the analysis of coherence in finite-level systems that are coupled to the surroundings and used it to study the process of energy transfer in the Fenna-Matthews-Olson (FMO) complex. The method makes use of adiabatic eigenstates of the Hamiltonian, with a subsequent transformation of the Hamiltonian into a form where the terms responsible for decoherence and population relaxation could be separated out at the lowest order. Thus one can account for decoherence nonperturbatively, and a Markovian type of master equation could be used for evaluating the population relaxation. In this paper, we apply this method to a two-level system as well as to a seven-level system. Comparisons with exact numerical results show that the method works quite well and is in good agreement with numerical calculations. The technique can be applied with ease to systems with larger numbers of levels as well. We also investigate how the presence of correlations among the bath degrees of freedom of the different bacteriochlorophyll a molecules of the FMO Complex affect the rate of energy transfer. Surprisingly, in the cases that we studied, our calculations suggest that the presence of anticorrelations, in contrast to correlations, make the excitation transfer more facile.

The effect of surface tension in porous wave maker problems

Using a mixed-type Fourier transform of a general form in the case of water of infinite depth and the method of eigenfunction expansion in the case of water of finite depth, several boundary-value problems involving the propagation and scattering of time harmonic surface water waves by vertical porous walls have been fully investigated, taking into account the effect of surface tension also. Known results are recovered either directly or as particular cases of the general problems under consideration.

A study on boundary-layer transition induced by free-stream turbulence

Boundary-layer transition at different free-stream turbulence levels has been investigated using the particle-image velocimetry technique. The measurements show organized positive and negative fluctuations of the streamwise fluctuating velocity component, which resemble the forward and backward jet-like structures reported in the direct numerical simulation of bypass transition. These fluctuations are associated with unsteady streaky structures. Large inclined high shear-layer regions are also observed and the organized negative fluctuations are found to appear consistently with these inclined shear layers, along with highly inflectional instantaneous streamwise velocity profiles. These inflectional velocity profiles are similar to those in the ribbon-induced boundary-layer transition. An oscillating-inclined shear layer appears to be the turbulent spot-precursor. The measurements also enabled to compare the actual turbulent spot in bypass transition with the simulated one. A proper orthogonal decomposition analysis of the fluctuating velocity field is carried out. The dominant flow structures of the organized positive and negative fluctuations are captured by the first few eigenfunction modes carrying most of the fluctuating energy. The similarity in the dominant eigenfunctions at different Reynolds numbers suggests that the flow prevails its structural identity even in intermittent flows. This analysis also indicates the possibility of the existence of a spatio-temporal symmetry associated with a travelling wave in the flow.

A note on the slow rotation of a disc in a bounded fluid with a surfactant surface layer

A method involving eigenfunction expansion and collocation is employed to solve the axisymmetric problem of a slowly and steadily rotating circular disc in a fluid of finite extent whose surface is covered with a surfactant film. The present method (originally described by Wang (Acta Mech. 94, 97, 1992)) is observed to produce results of practical importance associated with the problem more quickly and more easily than the one used earlier by Shail and Gooden (Int. J. Multiphase Flow 7, 245, 1992). (C) 1994 Academic Press, Inc.

Stability of spatially developing boundary layers in pressure gradients

A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow and utilizing a special coordinate transformation. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms nominally of order R(-1) in the boundary-layer Reynolds number R. In Blasius flow, the present approach is consistent with that of Bertolotti et al. (1992) to O(R(-1)) but simpler (i.e. has fewer terms), and may best be seen as providing a parametric differential equation which can be solved without having to march in space. The computed neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld) boundary. For this quantity, solutions for the Falkner-Skan flows show the effects of spatial growth to be striking only in the presence of strong adverse pressure gradients. As a rational analysis to O(R(-1)) demands inclusion of higher-order corrections on the mean flow, an illustrative calculation of one such correction, due to the displacement effect of the boundary layer, is made, and shown to have a significant destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of relatively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approach, a method of dealing with non-similar flows is also presented and illustrated. However, inherent in the transformation underlying the present approach is a lower-order non-parallel theory, which is obtained by dropping all terms of nominal order R(-1) except those required for obtaining the lowest-order solution in the critical and wall layers. It is shown that a reduced Orr-Sommerfeld equation (in transformed coordinates) already contains the major effects of non-parallelism.

Direct numerical simulation of turbulent spots

A group of high-order finite-difference schemes for incompressible flow was implemented to simulate the evolution of turbulent spots in channel flows. The long-time accuracy of these schemes was tested by comparing the evolution of small disturbances to a plane channel flow against the growth rate predicted by linear theory. When the perturbation is the unstable eigenfunction at a Reynolds number of 7500, the solution grows only if there are a comparatively large number of (equispaced) grid points across the channel. Fifth-order upwind biasing of convection terms is found to be worse than second-order central differencing. But, for a decaying mode at a Reynolds number of 1000, about a fourth of the points suffice to obtain the correct decay rate. We show that this is due to the comparatively high gradients in the unstable eigenfunction near the walls. So, high-wave-number dissipation of the high-order upwind biasing degrades the solution especially. But for a well-resolved calculation, the weak dissipation does not degrade solutions even over the very long times (O(100)) computed in these tests. Some new solutions of spot evolution in Couette flows with pressure gradients are presented. The approach to self-similarity at long times can be seen readily in contour plots.

A simple numerical method for three-dimensional analysis of simple expansion chamber mufflers of rectangular as well as circular cross-section with a stationary medium

Three-dimensional effects are a primary source of discrepancy between the measured values of automotive muffler performance and those predicted by the plane wave theory at higher frequencies. The basically exact method of (truncated) eigenfunction expansions for simple expansion chambers involves very complicated algebra, and the numerical finite element method requires large computation time and core storage. A simple numerical method is presented in this paper. It makes use of compatibility conditions for acoustic pressure and particle velocity at a number of equally spaced points in the planes of the junctions (or area discontinuities) to generate the required number of algebraic equations for evaluation of the relative amplitudes of the various modes (eigenfunctions), the total number of which is proportional to the area ratio. The method is demonstrated for evaluation of the four-pole parameters of rigid-walled, simple expansion chambers of rectangular as well as circular cross-section for the case of a stationary medium. Computed values of transmission loss are compared with those computed by means of the plane wave theory, in order to highlight the onset (cutting-on) of various higher order modes and the effect thereof on transmission loss of the muffler. These are also compared with predictions of the finite element methods (FEM) and the exact methods involving eigenfunction expansions, in order to demonstrate the accuracy of the simple method presented here.

Quadrature approximation properties of the spiral-phase quadrature transform

The notion of the 1-D analytic signal is well understood and has found many applications. At the heart of the analytic signal concept is the Hilbert transform. The problem in extending the concept of analytic signal to higher dimensions is that there is no unique multidimensional definition of the Hilbert transform. Also, the notion of analyticity is not so well under stood in higher dimensions. Of the several 2-D extensions of the Hilbert transform, the spiral-phase quadrature transform or the Riesz transform seems to be the natural extension and has attracted a lot of attention mainly due to its isotropic properties. From the Riesz transform, Larkin et al. constructed a vortex operator, which approximates the quadratures based on asymptotic stationary-phase analysis. In this paper, we show an alternative proof for the quadrature approximation property by invoking the quasi-eigenfunction property of linear, shift-invariant systems. We show that the vortex operator comes up as a natural consequence of applying this property. We also characterize the quadrature approximation error in terms of its energy as well as the peak spatial-domain error. Such results are available for 1-D signals, but their counter part for 2-D signals have not been provided. We also provide simulation results to supplement the analytical calculations.

Generalized eigenvalue decomposition of the field autocorrelation in correlation diffusion of photons in turbid media

We propose a novel numerical method based on a generalized eigenvalue decomposition for solving the diffusion equation governing the correlation diffusion of photons in turbid media. Medical imaging modalities such as diffuse correlation tomography and ultrasound-modulated optical tomography have the (elliptic) diffusion equation parameterized by a time variable as the forward model. Hitherto, for the computation of the correlation function, the diffusion equation is solved repeatedly over the time parameter. We show that the use of a certain time-independent generalized eigenfunction basis results in the decoupling of the spatial and time dependence of the correlation function, thus allowing greater computational efficiency in arriving at the forward solution. Besides presenting the mathematical analysis of the generalized eigenvalue problem on the basis of spectral theory, we put forth the numerical results that compare the proposed numerical method with the standard technique for solving the diffusion equation.

Binaural Signal Processing Motivated Generalized Analytic Signal Construction and AM-FM Demodulation

Binaural hearing studies show that the auditory system uses the phase-difference information in the auditory stimuli for localization of a sound source. Motivated by this finding, we present a method for demodulation of amplitude-modulated-frequency-modulated (AM-FM) signals using a ignal and its arbitrary phase-shifted version. The demodulation is achieved using two allpass filters, whose impulse responses are related through the fractional Hilbert transform (FrHT). The allpass filters are obtained by cosine-modulation of a zero-phase flat-top prototype halfband lowpass filter. The outputs of the filters are combined to construct an analytic signal (AS) from which the AM and FM are estimated. We show that, under certain assumptions on the signal and the filter structures, the AM and FM can be obtained exactly. The AM-FM calculations are based on the quasi-eigenfunction approximation. We then extend the concept to the demodulation of multicomponent signals using uniform and non-uniform cosine-modulated filterbank (FB) structures consisting of flat bandpass filters, including the uniform cosine-modulated, equivalent rectangular bandwidth (ERB), and constant-Q filterbanks. We validate the theoretical calculations by considering application on synthesized AM-FM signals and compare the performance in presence of noise with three other multiband demodulation techniques, namely, the Teager-energy-based approach, the Gabor's AS approach, and the linear transduction filter approach. We also show demodulation results for real signals.

Boundary perturbations and the Helmholtz equation in three dimensions

We propose an analytic perturbative scheme in the spirit of Lord Rayleigh's work for determining the eigenvalues of the Helmholtz equation in three dimensions inside an arbitrary boundary where the eigenfunction satisfies either the Dirichlet boundary condition or the Neumann boundary condition. Although numerous works are available in the literature for arbitrary boundaries in two dimensions, to the best of our knowledge the formulation in three dimensions is proposed for the first time. In this novel prescription, we have expanded the arbitrary boundary in terms of spherical harmonics about an equivalent sphere and obtained perturbative closed-form solutions at each order for the problem in terms of corrections to the equivalent spherical boundary for both the boundary conditions. This formulation is in parallel with the standard time-independent Rayleigh-Schrodinger perturbation theory. The efficacy of the method is tested by comparing the perturbative values against the numerically calculated eigenvalues for spheroidal, superegg and superquadric shaped boundaries. It is shown that this perturbation works quite well even for wide departure from spherical shape and for higher excited states too. We believe this formulation would find applications in the field of quantum dots and acoustical cavities.

Unusual eigenvalue spectrum and relaxation in the Levy-Ornstein-Uhlenbeck process

We consider the rates of relaxation of a particle in a harmonic well, subject to Levy noise characterized by its Levy index mu. Using the propagator for this Levy-Ornstein-Uhlenbeck process (LOUP), we show that the eigenvalue spectrum of the associated Fokker-Planck operator has the form (n + m mu)nu where nu is the force constant characterizing the well, and n, m is an element of N. If mu is irrational, the eigenvalues are all nondegenerate, but rational mu can lead to degeneracy. The maximum degeneracy is shown to be 2. The left eigenfunctions of the fractional Fokker-Planck operator are very simple while the right eigenfunctions may be obtained from the lowest eigenfunction by a combination of two different step-up operators. Further, we find that the acceptable eigenfunctions should have the asymptotic behavior vertical bar x vertical bar(-n1-n2 mu) as vertical bar x vertical bar -> infinity, with n(1) and n(2) being positive integers, though this condition alone is not enough to identify them uniquely. We also assert that the rates of relaxation of LOUP are determined by the eigenvalues of the associated fractional Fokker-Planck operator and do not depend on the initial state if the moments of the initial distribution are all finite. If the initial distribution has fat tails, for which the higher moments diverge, one can have nonspectral relaxation, as pointed out by Toenjes et al. Phys. Rev. Lett. 110, 150602 (2013)].

L-infinity norms of holomorphic modular forms in the case of compact quotient

We prove a sub-convex estimate for the sup-norm of L-2-normalized holomorphic modular forms of weight k on the upper half plane, with respect to the unit group of a quaternion division algebra over Q. More precisely we show that when the L-2 norm of an eigenfunction f is one, parallel to f parallel to(infinity) <<(epsilon) k(1/2-1/33+epsilon) for any epsilon > 0 and for all k sufficiently large.