Second-order Cumulant Expansion Research Articles

Zonostrophic instability driven by discrete particle noise

The consequences of discrete particle noise for a system possessing a possibly unstable collective mode are discussed. It is argued that a zonostrophic instability (of homogeneous turbulence to the formation of zonal flows) occurs just below the threshold for linear instability. The scenario provides a new interpretation of the random forcing that is ubiquitously invoked in stochastic models such as the second-order cumulant expansion or stochastic structural instability theory; neither intrinsic turbulence nor coupling to extrinsic turbulence is required. A representative calculation of the zonostrophic neutral curve is made for a simple two-field model of toroidal ion-temperature-gradient-driven modes. To the extent that the damping of zonal flows is controlled by the ion–ion collision rate, the point of zonostrophic instability is independent of that rate.

Zonal-flow dynamics from a phase-space perspective

The wave kinetic equation (WKE) describing drift-wave (DW) turbulence is widely used in the studies of zonal flows (ZFs) emerging from DW turbulence. However, this formulation neglects the exchange of enstrophy between DWs and ZFs and also ignores effects beyond the geometrical-optics limit. We derive a modified theory that takes both of these effects into account, while still treating DW quanta (“driftons”) as particles in phase space. The drifton dynamics is described by an equation of the Wigner–Moyal type, which is commonly known in the phase-space formulation of quantum mechanics. In the geometrical-optics limit, this formulation features additional terms missing in the traditional WKE that ensure exact conservation of the total enstrophy of the system, in addition to the total energy, which is the only conserved invariant in previous theories based on the WKE. Numerical simulations are presented to illustrate the importance of these additional terms. The proposed formulation can be considered as a phase-space representation of the second-order cumulant expansion, or CE2.

Stochastic simulation of anharmonic dissipation. I. Linear response regime.

Over decades, the theoretical study of the quantum dissipative dynamics was mainly based on the linear dissipation model. The study of the nonlinear dissipative dynamics in condensed phases, where there exist an infinite number of bath modes, is extremely difficult even if not impossible. This work put forward a stochastic scheme for the simulation of the nonlinear dissipative dynamics. In the linear response regime, the second-order cumulant expansion becomes exact to reproduce the effect of the bath on the evolution of the reduced system. Consequently, a Hermitian stochastic Liouville equation is derived without explicit treatment of the bath. Stochastic simulations for an anharmonic model illustrate that the dynamics dissipated by anharmonic bath exhibits substantial difference on temperature dependence compared to that with the Caldeira-Leggett model.

Evaluation of protein–ligand affinity prediction using steered molecular dynamics simulations

In computational drug design, ranking a series of compound analogs in a manner that is consistent with experimental affinities remains a challenge. In this study, we evaluated the prediction of protein–ligand binding affinities using steered molecular dynamics simulations. First, we investigated the appropriate conditions for accurate predictions in these simulations. A conic harmonic restraint was applied to the system for efficient sampling of work values on the ligand unbinding pathway. We found that pulling velocity significantly influenced affinity predictions, but that the number of collectable trajectories was less influential. We identified the appropriate pulling velocity and collectable trajectories for binding affinity predictions as 1.25 Å/ns and 100, respectively, and these parameters were used to evaluate three target proteins (FK506 binding protein, trypsin, and cyclin-dependent kinase 2). For these proteins using our parameters, the accuracy of affinity prediction was higher and more stable when Jarzynski’s equality was employed compared with the second-order cumulant expansion equation of Jarzynski’s equality. Our results showed that steered molecular dynamics simulations are effective for predicting the rank order of ligands; thus, they are a potential tool for compound selection in hit-to-lead and lead optimization processes.

A tutorial introduction to the statistical theory of turbulent plasmas, a half-century after Kadomtsev’sPlasma Turbulenceand the resonance-broadening theory of Dupree and Weinstock

In honour of the 50th anniversary of the influential review/monograph on plasma turbulence by B. B. Kadomtsev as well as the seminal works of T. H. Dupree and J. Weinstock on resonance-broadening theory, an introductory tutorial is given about some highlights of the statistical–dynamical description of turbulent plasmas and fluids, including the ideas of nonlinear incoherent noise, coherent damping, and self-consistent dielectric response. The statistical closure problem is introduced. Incoherent noise and coherent damping are illustrated with a solvable model of passive advection. Self-consistency introduces turbulent polarization effects that are described by the dielectric function${\mathcal{D}}$. Dupree’s method of using${\mathcal{D}}$to estimate the saturation level of turbulence is described; then it is explained why a more complete theory that includes nonlinear noise is required. The general theory is best formulated in terms of Dyson equations for the covariance$C$and an infinitesimal response function$R$, which subsumes${\mathcal{D}}$. An important example is the direct-interaction approximation (DIA). It is shown how to use Novikov’s theorem to develop an$\boldsymbol{x}$-space approach to the DIA that is complementary to the original$\boldsymbol{k}$-space approach of Kraichnan. A dielectric function is defined for arbitrary quadratically nonlinear systems, including the Navier–Stokes equation, and an algorithm for determining the form of${\mathcal{D}}$in the DIA is sketched. The independent insights of Kadomtsev and Kraichnan about the problem of the DIA with random Galilean invariance are described. The mixing-length formula for drift-wave saturation is discussed in the context of closures that include nonlinear noise (shielded by${\mathcal{D}}$). The role of$R$in the calculation of the symmetry-breaking (zonostrophic) instability of homogeneous turbulence to the generation of inhomogeneous mean flows is addressed. The second-order cumulant expansion and the stochastic structural stability theory are also discussed in that context. Various historical research threads are mentioned and representative entry points to the literature are given. Some outstanding conceptual issues are enumerated.

Exact stochastic unraveling of an optical coherence dynamics by cumulant expansion.

A numerically exact Monte Carlo scheme for calculation of open quantum system dynamics is proposed and implemented. The method consists of a Monte Carlo summation of a perturbation expansion in terms of trajectories in Liouville phase-space with respect to the coupling between the excited states of the molecule. The trajectories are weighted by a complex decoherence factor based on the second-order cumulant expansion of the environmental evolution. The method can be used with an arbitrary environment characterized by a general correlation function and arbitrary coupling strength. It is formally exact for harmonic environments, and it can be used with arbitrary temperature. Time evolution of an optically excited Frenkel exciton dimer representing a molecular exciton interacting with a charge transfer state is calculated by the proposed method. We calculate the evolution of the optical coherence elements of the density matrix and linear absorption spectrum, and compare them with the predictions of standard simulation methods.

The temperature dependence of vibronic lineshapes: linear electron-phonon coupling.

We calculate the effect of a linear electron-phonon coupling on vibronic transitions of dye molecules of arbitrary complexity. With the assumption of known vibronic frequencies (for instance from quantum-chemical calculations), we give expressions for the absorption or emission lineshapes in a second-order cumulant expansion. We show that the results coincide with those obtained from generalized Redfield theory if one uses the time-local version of the theory and applies the secular approximation. Furthermore, the theory allows to go beyond the Huang-Rhys approximation and can be used to incorporate Dushinsky effects in the treatment of the temperature dependence of optical spectra. We consider both, a pure electron-phonon coupling independent of the molecular vibrations and a coupling bilinear in the molecular vibrational modes and the phonon coordinates. We discuss the behavior of the vibronic density of states for various models for the spectral density representing the coupling of the vibronic system to the harmonic bath. We recover some of the results that have been derived earlier for the spin-boson model and we show that the behavior of the spectral density at low frequencies determines the dominant features of the spectra. In case of the bilinear coupling between the molecular vibrations and the phonons we give analytical expressions for different spectral densities. The spectra are reminiscent of those obtained from the well known Brownian oscillator model and one finds a zero-phonon line and phonon-side bands located at vibrational frequencies of the dye. The intensity of the phonon-side bands diminishes with increasing vibrational frequencies and with decreasing coupling strength (Huang-Rhys factor). It vanishes completely in the Markovian limit where only a Lorentzian zero-phonon line is observed.

On the Improvement of Free-Energy Calculation from Steered Molecular Dynamics Simulations Using Adaptive Stochastic Perturbation Protocols

The potential of mean force (PMF) calculation in single molecule manipulation experiments performed via the steered molecular dynamics (SMD) technique is a computationally very demanding task because the analyzed system has to be perturbed very slowly to be kept close to equilibrium. Faster perturbations, far from equilibrium, increase dissipation and move the average work away from the underlying free energy profile, and thus introduce a bias into the PMF estimate. The Jarzynski equality offers a way to overcome the bias problem by being able to produce an exact estimate of the free energy difference, regardless of the perturbation regime. However, with a limited number of samples and high dissipation the Jarzynski equality also introduces a bias. In our previous work, based on the Brownian motion formalism, we introduced three stochastic perturbation protocols aimed at improving the PMF calculation with the Jarzynski equality in single molecule manipulation experiments and analogous computer simulations. This paper describes the PMF reconstruction results based on full-atom molecular dynamics simulations, obtained with those three protocols. We also want to show that the protocols are applicable with the second-order cumulant expansion formula. Our protocols offer a very noticeable improvement over the simple constant velocity pulling. They are able to produce an acceptable estimate of PMF with a significantly reduced bias, even with very fast perturbation regimes. Therefore, the protocols can be adopted as practical and efficient tools for the analysis of mechanical properties of biological molecules.

Dynamics of radiationless transitions in large molecules: 4. An ensemble of molecules occurring in condensed medium at a finite temperature

Statistically averaged vibration-phonon (VP) states of large molecules occurring in a condensed medium are represented by superpositions of lower excited states of high-frequency intramolecular vibrations and phonons in the state of thermal equilibrium. The dynamics of amplitudes of these states is described by an ensemble of Hamiltonians, wherein the distribution of their parameters characterizes the configuration distribution of the medium and determines the broadening and shift of absorption bands and the time evolution of the ensemble population. The equations of motion for statistically averaged amplitudes, as found by second-order cumulant expansion, coincide with those found by solving the secular equation, which takes into account both intramolecular and VP interactions. The relative contribution of the decay and decoherence to the evolution of the amplitudes has been considered. Critical values at which the transition from regular recurrence cycles to stochastic dynamics takes place have been found for the VP coupling parameters. The ergodicity of VP systems and the conditions of separation of the intramolecular and phonon-stimulated dynamics have been considered.

Time-dependent approaches for the calculation of intersystem crossing rates

We present three formulas for calculating intersystem crossing rates in the Condon approximation to the golden rule by means of a time-dependent approach: an expression using the full time correlation function which is exact for harmonic oscillators, a second-order cumulant expansion, and a short-time approximation of this expression. While the exact expression and the cumulant expansion require numerical integration of the time correlation function, the integration of the short-time expansion can be performed analytically. To ensure convergence in the presence of large oscillations of the correlation function, we use a Gaussian damping function. The strengths and weaknesses of these approaches as well as the dependence of the results on the choice of the technical parameters of the time integration are assessed on four test examples, i.e., the nonradiative S(1) ⇝ T(1) transitions in thymine, phenalenone, flavone, and porphyrin. The obtained rate constants are compared with previous results of a time-independent approach. Very good agreement between the literature values and the integrals over the full time correlation functions are observed. Furthermore, the comparison suggests that the cumulant expansion approximates the exact expression very well while allowing the interval of the time integration to be significantly shorter. In cases with sufficiently high vibrational density of states also the short-time approximation yields rates in good agreement with the results of the exact formula. A great advantage of the time-dependent approach over the time-independent approach is its excellent computational efficiency making it the method of choice in cases of large energy gaps, large numbers of normal modes, and high densities of final vibrational states.

The role of Duschinsky rotation in intersystem crossing: A case study of uracil

The intersystem crossing rate for the transition between the lowest excited singlet and triplet electronic states of uracil was studied by means of ab initio methods. The rate was evaluated using the timedependent approach based on the correlation function and its two approximations: the second-order cumulant expansion and the short-time approximation. The normal modes of the singlet and triplet states are related by the Duschinsky transformation, i.e., by rotation and translation. It was found that for singlet-triplet adiabatic energy gaps below 6000 cm-1, the inclusion of the Duschinsky rotation is necessary for quantitative results. Above energy gaps of 6000 cm-1, the rates obtained with and without the Duschinsky rotation are similar. The cumulant expansion approximates well the correlation function. The short-time approximation, although crude, can be used as the first estimate of the rate.

Double-quantum two-dimensional electronic spectroscopy of a three-level system: Experiments and simulations

Double-quantum coherence two-dimensional (2Q2D) electronic spectroscopy is utilized to probe the dynamic fluctuations of electronic states in a solvated molecule at approximately twice the energy of the ground state bleach transition. The 2Q2D spectrum gives insight into the energetic position and spectral fluctuations (system-bath interaction) of the probed excited states. Combining it with single-quantum two-dimensional (1Q2D) electronic spectroscopy enables one to determine the strength of the excited state absorption transition and the relative detuning of electronic states, as well as the dynamics of the single-quantum coherence. To investigate the correlation of spectral fluctuations in different electronically excited states, we have carried out experiments on a solvated dye (Rhodamine 6G) with 23 fs pulses centered at the maximum of the linear absorption spectrum. The 2Q2D spectrum reveals three peaks of alternating signs with the major negative peak located at higher frequencies along the emission axis compared to the single positive peak. The 1Q2D spectrum, on the other hand, shows a negative peak stemming from excited state absorption at lower frequencies along the emission axis. Analysis of the signal in the homogeneous limit fails to account for this observation as well as the number of peaks in the 2Q2D spectrum. Employing a three-level model in which all time correlations of the third-order response function are accounted for via second-order cumulant expansion gives good agreement with both the 1Q2D and 2Q2D data. Furthermore, the analysis shows that the fluctuations of the probed electronic states are highly correlated, reflecting the modulation by a common nuclear bath and similarities in the nature of the electronic transitions.

Dynamical disorder in presence of exponential sink

In this Letter we show an elegant application of the path integral based method developed recently [J. Chem. Phys. 124 (2006) 204111] to calculate the lower bound to the survival probability of a particle diffusing in a harmonic well in presence of an exponential sink. Our lower bound is always finite and well behaved and does not blow up like the approximate survival probability when calculated using second-order cumulant expansion for some decorrelation time scales [J. Phys. Chem. B 110 (2006) 19061]. The problem can be mapped onto the problem of electron transfer dynamics in single protein molecule where the electron transfer rate shows an exponential dependence on the edge to edge distance between the donor and the acceptor.

Symmetry in Nonlinear Hydrologic Dynamics under Uncertainty: Ensemble Modeling of 2D Boussinesq Equation for Unsteady Flow in Heterogeneous Aquifers

In this paper, the probabilistic symmetry analysis approach proposed in the previous paper by Cayar and Kavvas in 2009 is adopted and applied to the analysis of nonlinear two-dimensional (2D) groundwater flow subject to random hydraulic conductivity field. The resulting model has the form of a two-dimensional Fokker-Planck equation (FPE). Following the methodology reported by Cayar and Kavvas in 2009, the 2D Boussinessq equation for unconfined groundwater flow in a heterogeneous aquifer is transformed to a system of ordinary differential equations (ODEs) through Lie group symmetry analysis in order to eliminate the spatial derivatives occurring in the governing equation. Next, these derived stochastic ODEs are ensemble averaged with second-order cumulant expansion and converted into a linear, deterministic partial differential equation. This new equation is called the FPE, and describes the evolution of the probability density function (PDF) of the hydraulic head, thereby, describing the ensemble behavior...

Combined Theoretical Modeling of Photoexcitation Spectrum of an Isolated Protonated Tyrosine

A photodepletion spectrum of gaseous protonated tyrosine molecules was obtained at 150 K by a UV laser spectroscopic technique in conjunction with mass spectrometry and interpreted by theoretical methods. The spectrum exhibits three distinct bands separated each other by about 800 cm(-1). Stable conformers of the molecular ion were determined by quantum mechanical density functional theory, and their electronic transition energies were obtained by a semiempirical quantum chemistry calculation. The whole pattern of the spectrum was reasonably reproduced by a combination of theoretical methods, the second-order cumulant expansion, a semiempirical quantum chemistry method, molecular dynamics simulation, and a semiclassical time-correlation function approach. The three spectral bands turned out to arise from the vibronic transition of two vibrational modes constituted by the "benzene breathing" mode and a torsional mode of the amino acid backbone. It is suggested that the major factor in spectral broadening is not conformational disorder or lifetime broadening but thermal fluctuation of the stable conformers. The good agreement between experimental and theoretical spectra exemplifies the validity of the theoretical methods applied for the present molecular system.

On the cumulant expansion up scaling of ground water contaminant transport equation with nonequilibrium sorption

The laboratory-scale ground water transport equation with nonequilibrium sorption reaction subjected to unsteady, nondivergence-free, and nonstationary velocity fields is up-scaled to the field-scale by using the ensemble-averaged equations obtained from the cumulant expansion ensemble-averaging method. It is found that existing ensemble-averaged equations obtained with the help of the cumulant expansion method for the system of linear partial differential equations are not second-order exact. Although the cumulant expansion methodology is designed for noncommuting operators, it is found that there are still commudativity requirements that need to be satisfied by the functions and constants exist in the coefficient matrix of the system of ordinary/partial differential equations. A reversibility requirement, which covers the commudativity requirements, is also proposed when applying the cumulant expansion method to a system of partial differential equations/a partial differential equation. The significance of the new velocity correction obtained in this study due to the applied second-order exact cumulant expansion is investigated on a numerical example with a linear trend in the distribution coefficient. It is found that the effect of the new velocity correction can be significant enough to affect the maximum concentration values and the plume center of mass in the case of a trending distribution coefficient in a physically heterogeneous environment.

Ligand Escape Pathways and (Un)Binding Free Energy Calculations for the Hexameric Insulin-Phenol Complex

Cooperative binding of phenolic species to insulin hexamers is known to stabilize pharmaceutical preparations of the hormone. Phenol dissociation is rapid on hexamer dissolution timescales, and phenol unbinding upon dilution is likely the first step in the conversion of (pharmaceutical) hexameric insulin to the active monomeric form upon injection. However, a clear understanding of the determinants of the rates of phenol unbinding remains obscure, chiefly because residues implicated in phenol exchange as determined by NMR are not all associated with likely unbinding routes suggested by the best-resolved hexamer structures. We apply random acceleration molecular dynamics simulation to identify potential escape routes of phenol from hydrophobic cavities in the hexameric insulin-phenol complex. We find three major pathways, which provide new insights into (un)binding mechanisms for phenol. We identify several residues directly participating in escape events that serve to resolve ambiguities from recent NMR experiments. Reaction coordinates for dissociation of phenol are developed based on these exit pathways. Potentials of mean force along the reaction coordinate for each pathway are resolved using multiple independent steered molecular dynamics simulations with second-order cumulant expansion of Jarzynski's equality. Our results for Δ F agree reasonably well within the range of known experimental and previous simulation magnitudes of this quantity. Based on structural analysis and energetic barriers for each pathway, we suggest a plausible preferred mechanism of phenolic exchange that differs from previous mechanisms. Several weakly-bound metastable states are also observed for the first time in the phenol dissociation reaction.

Statistics of an Unstable Barotropic Jet from a Cumulant Expansion

AbstractLow-order equal-time statistics of a barotropic flow on a rotating sphere are investigated. The flow is driven by linear relaxation toward an unstable zonal jet. For relatively short relaxation times, the flow is dominated by critical-layer waves. For sufficiently long relaxation times, the flow is turbulent. Statistics obtained from a second-order cumulant expansion are compared to those accumulated in direct numerical simulations, revealing the strengths and limitations of the expansion for different relaxation times.

The entropy of a constrained signal: A maximum entropy approach with applications

The information content of signals subject to constraints such as the total power at a set of frequencies is well understood. This paper illustrates a general approach to the computation of the entropy of a constrained signal. Although general solutions will be derived, exact solutions may be difficult and so three approximation techniques are discussed. One is based on the cumulant expansion. The lowest-order results from this expansion have a simple form. The use of an asymptotic expression to compute the signal entropy is explored when the constraints are such that the number of states available to the signal is small. The role of direct numerical calculations is discussed. Results are discussed in light of the problem of computing the entropy of a signal constrained by its moments. The lowest-order terms in the cumulant expansion of the entropy are used to find a generalized correlation filter to detect signals subject to different constraints. The critical role that entropy has in determining the sensitivity and specificity of such a detector is discussed. Another application is the computation of the entropy of a signal subject to polyspectral constraints. It is demonstrated that the entropy change is governed by the higher-order coherence. Applications of these findings to the detection of signals subject to polyspectral constraints is discussed as are the limitations of the second-order cumulant expansion. More general expressions for the entropy of such systems are derived.

Hybrid molecular dynamics‐quantum mechanics simulations of solute spectral properties in the condensed phase: Evaluation of simulation parameters

We have explored the impact of a number of basic simulation parameters on the results of a recently developed hybrid molecular dynamics-quantum mechanics (MD-QM) method (Mercer et al., J Phys Chem B 1999, 103, 7720). The method utilizes MD simulations to explore the ground-state configuration space of the system and QM evaluation of those structures to yield the time-dependent electronic transition energy, which is transformed into the optical line-broadening function using the second-order cumulant expansion. Both linear and nonlinear optical spectra can then be generated for comparison to experiment. The dependence of the resulting spectra on the length of the MD trajectory, the QM sampling rate, and the QM model chemistry have all been examined. In particular, for the system of oxazine-4 in methanol studied here, at least 20 ps of MD trajectory are needed for qualitative convergence of linear spectral properties, and >100 ps is needed for quantitative convergence. Surprisingly, little difference is found between the 3-21G and 6-31G(d) basis sets, and the CIS and TD-B3LYP methods yield remarkably similar spectra. The semiempirical INDO/s method yields the most accurate results, reproducing the experimental Stokes shift to within 5% and the FWHM to within 20%. Nonlinear 3-pulse photon echo peak shift (3PEPS) decays have also been simulated. Decays are generally poorly reproduced, though the initial peak shift which depends on the overall coupling of motions to the solute transition energy is within 15% of experiment for all model chemistries other than those using the STO-3G basis.