Normal Mode Space Research Articles

Phonon modal analysis of thermal transport in ThO2 with point defects using equilibrium molecular dynamics

Defects can significantly degrade the thermal conductivity of ThO2, an advanced nuclear fuel material as well as a surrogate for other fluorite-structured materials. We investigate how point defects in ThO2 impact phonon mode-resolved thermal transport. By incorporating phonon modes from lattice dynamics, we decompose the trajectory and heat flux to phonon normal mode space and extract key phonon properties, including phonon relaxation times and their contributions to thermal conductivity. We implement two methods. The first method is based on the Green Kubo formalism to resolve the contribution of each phonon mode to thermal conductivity. The second resolves the lifetime of individual phonon modes and the thermal conductivity is calculated using the Boltzmann transport equation within relaxation time approximation. Notably, a lower contribution of acoustic modes is revealed compared to perturbative approaches considering only three-phonon scattering processes. The effects of four types of point defects are evaluated. The strongest impact on a reduction in thermal conductivity is from Th interstitials, followed by Th vacancies. O interstitials/vacancies have a similar impact, albeit smaller than defects on the thorium sublattice. These observations are consistent with previous studies.

Random matrix theory for description of sound scattering on background internal waves in a shallow sea

The problem of propagation of low-frequency sound in a shallow waveguide with random hydrological inhomogeneity caused by background internal waves is considered. A new approach to statistical modeling of acoustic fields, based on the application of the random matrix theory and previously successfully used for deep-water acoustic waveguides, is used to the case of shallow-water waveguides. In this approach, sound scattering on random inhomogeneity is described using an ensemble of random propagator matrices which describe the transformation of the acoustic field in the space of normal waveguide modes. A study of the effect of sound “escaping” from a waveguide was carried out. The term “escaping” here means energy transfer to modes with stronger attenuation due to scattering on internal waves. A model of an underwater sound channel with an axis at a depth of about 45 meters is considered. It is shown that the first few modes propagating inside the water column are very little subject to losses due to the “escaping”. The strongest impact of the leakage scattering is experienced by the middle group of modes capable of reaching the sea surface. It is revealed as significant increasing of losses as compared to a horizontally homogeneous waveguide. On the other hand, the existence of linear mode combinations for which loss enhancement is practically absent has been revealed. These linear combinations correspond to the eigenfunctions of an inhomogeneous waveguide. Statistical analysis of propagator eigenfunctions indicates on qualitative differences of mechanisms of scattering for frequencies of 100 and 500 Hz.

Energy-recurrence breakdown and chaos in disordered Fermi–Pasta–Ulam–Tsingou lattices

In this paper, we consider the classic Fermi–Pasta–Ulam–Tsingou system as a model of interacting particles connected by harmonic springs with a quadratic nonlinear term (first system) and a set of second-order ordinary differential equations with variability (second system) that resembles Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system. In the absence of variability, the second system becomes Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system (first system). Variability is introduced to Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system to take into account inherent variations (for example, due to manufacturing processes), giving rise to heterogeneity in its parameters. We demonstrate that a percentage of variability smaller than a threshold can break the well-known energy recurrence phenomenon and induce localization in the energy normal-mode space. However, percentage of variability larger than the threshold may make the trajectories of the second system blow up in finite time. Using a multiple-scale expansion, we derive analytically a two normal-mode approximation that explains the mechanism for energy localization and blow up in the second system. We also investigate the chaotic behavior of the two systems as the percentage of variability is increased, utilizing the maximum Lyapunov exponent and Smaller Alignment Index. Our analysis shows that when there is almost energy localization in the second system, it is more probable to observe chaos, as the number of particles increases.

Multidimensional characterization of the conical intersection seam in the normal mode space.

Multidimensional conical intersection seam has been characterized by utilizing the dynamic resonances in the nonadiabatic transition probability experimentally observed in the predissociation of thioanisole isotopomers. The nonadiabatic bifurcation behavior of the reactive flux into either the Herzberg type-I (electronic) or type-II (vibrational) predissociation pathway is found to be strongly dependent on the quantum nature of the S1/S2 vibronic eigenstate, providing the essential information about structure and dynamic character of the conical intersection seam projected onto the normal mode space. By modifying the nature of the normal mode space through partial or full H/D substitution of the molecule, multiple aspects of the conical intersection seam could be characterized from different viewpoints set by the adjusted normal mode space. Theoretical calculations of potential energy curves along selected normal mode displacements support the experiment.

Enhanced sampling of protein conformational states for dynamic cross-docking within the protein-protein docking server SwarmDock.

The formation of specific protein‐protein interactions is often a key to a protein's function. During complex formation, each protein component will undergo a change in the conformational state, for some these changes are relatively small and reside primarily at the sidechain level; however, others may display notable backbone adjustments. One of the classic problems in the protein‐docking field is to be able to a priori predict the extent of such conformational changes. In this work, we investigated three protocols to find the most suitable input structure conformations for cross‐docking, including a robust sampling approach in normal mode space. Counterintuitively, knowledge of the theoretically best combination of normal modes for unbound‐bound transitions does not always lead to the best results. We used a novel spatial partitioning library, Aether Engine (see Supplementary Materials), to efficiently search the conformational states of 56 receptor/ligand pairs, including a recent CAPRI target, in a systematic manner and selected diverse conformations as input to our automated docking server, SwarmDock, a server that allows moderate conformational adjustments during the docking process. In essence, here we present a dynamic cross‐docking protocol, which when benchmarked against the simpler approach of just docking the unbound components shows a 10% uplift in the quality of the top docking pose.

Intermittent many-body dynamics at equilibrium

The equilibrium value of an observable defines a manifold in the phase space of an ergodic and equipartitioned many-body system. A typical trajectory pierces that manifold infinitely often as time goes to infinity. We use these piercings to measure both the relaxation time of the lowest frequency eigenmode of the Fermi-Pasta-Ulam chain, as well as the fluctuations of the subsequent dynamics in equilibrium. The dynamics in equilibrium is characterized by a power-law distribution of excursion times far off equilibrium, with diverging variance. Long excursions arise from sticky dynamics close to q-breathers localized in normal mode space. Measuring the exponent allows one to predict the transition into nonergodic dynamics. We generalize our method to Klein-Gordon lattices where the sticky dynamics is due to discrete breathers localized in real space.

Classical investigation of long-range coherence in biological systems.

Almost five decades ago, H. Fröhlich [H. Fröhlich, "Long-range coherence and energy storage in biological systems," Int. J. Quantum Chem. 2(5), 641-649 (1968)] reported, on a theoretical basis, that the excitation of quantum modes of vibration in contact with a thermal reservoir may lead to steady states, where under high enough rate of energy supply, only specific low-frequency modes of vibration are strongly excited. This nonlinear phenomenon was predicted to occur in biomolecular systems, which are known to exhibit complex vibrational spectral properties, especially in the terahertz frequency domain. However, since the effects of terahertz or lower-frequency modes are mainly classical at physiological temperatures, there are serious doubts that Fröhlich's quantum description can be applied to predict such a coherent behavior in a biological environment, as suggested by the author. In addition, a quantum formalism makes the phenomenon hard to investigate using realistic molecular dynamics simulations (MD) as they are usually based on the classical principles. In the current paper, we provide a general classical Hamiltonian description of a nonlinear open system composed of many degrees of freedom (biomolecular structure) excited by an external energy source. It is shown that a coherent behaviour similar to Fröhlich's effect is to be expected in the classical case for a given range of parameter values. Thus, the supplied energy is not completely thermalized but stored in a highly ordered fashion. The connection between our Hamiltonian description, carried out in the space of normal modes, and a more standard treatment in the physical space is emphasized in order to facilitate the prediction of the effect from MD simulations. It is shown how such a coherent phenomenon may induce long-range resonance effects that could be of critical importance at the biomolecular level. The present work is motivated by recent experimental evidences of long-lived excited low-frequency modes in protein structures, which were reported as a consequence of the Fröhlich's effect.

Low-dimensional [formula omitted]-tori in FPU lattices: Dynamics and localization properties

Recent studies on the Fermi–Pasta–Ulam (FPU) paradox, like the theory of q-breathers and the metastability scenario, dealing mostly with the energy localization properties in the FPU space of normal modes (q-space), motivated our first work on q-tori in the FPU problem (Christodoulidi et al., 2010) [19]. The q-tori are low-dimensional invariant tori hosting trajectories that present features relevant to the interpretation of FPU recurrences as well as the energy localization in q-space. The present paper is a continuation of our work in Christodoulidi et al. (2010) [19]. Our new results are: we extend a method of analytical computation of q-tori, using Poincaré–Lindstedt series, from the β to the α-FPU and we reach significantly higher expansion orders using an improved computer-algebraic program. We probe numerically the convergence properties as well as the level of precision of our computed series. We develop an additional algorithm in order to systematically locate values of the incommensurable frequencies used as an input in the PL series construction of q-tori corresponding to progressively higher values of the energy. We generalize a proposition proved in Christodoulidi et al. (2010) [19] regarding the so-called ‘sequence of propagation’ of an initial excitation in the PL series. We show by concrete examples how the latter interprets the localization patterns found in numerical simulations. We focus, in particular, on various types of extensive initial excitations that lead to q-tori solutions with exponentially localized profiles. Finally, we discuss the relation between q-tori, q-breathers (viewed as one-dimensional q-tori), and the so-called ‘FPU-trajectories’ invoked in the original study of the FPU problem.

Gaussian Binning of the Vibrational Distributions for the Cl + CH4(v4/2 = 0, 1) → H + CH3Cl(n1n2n3n4n5n6) Reactions

We test several binning techniques to obtain mode-specific final-state distributions for polyatomic reactions. Normal mode analysis is done after an exact transformation to the Eckart frame. Standard histogram binning (HB) and three different variants of the energy-based Gaussian binning (1GB) are employed to obtain the probabilities of the vibrational states. We consider the two major issues of the polyatomic quasiclassical product analysis, i.e., (1) rounding the classical action to the nearest integer can result in unphysical states and (2) the normal-mode analysis can break down for highly distorted geometries. We show that 1GB can handle issue 1 when the total vibrational energy is evaluated in the normal mode space using the harmonic approximation and both issues 1 and 2 can be solved when the total vibrational energy is calculated exactly in the Cartesian space. We found that anharmonicity in the quantized energy levels does not have a significant effect on the final-state distributions. Quasiclassical trajectory calculations are performed for the reactant ground-state and bending-excited Cl((2)P(3/2)) + CH(4)(v(4/2) = 0, 1) → H + CH(3)Cl reactions using an ab initio potential energy surface. The product analysis techniques are successfully applied to the CH(3)Cl product molecules and some qualitative features of the results are discussed.

One-loop partition function from normal modes for supergravity in AdS3

With the recently discovered formula, which relates the off-shell Euclidean one-loop determinant to the on-shell quantities such as normal and quasinormal modes in real spacetime, we work out the one-loop partition function for supergravity in AdS3 from scratch. In passing, we also provide the explicit expression for the one-loop determinant of a field with an arbitrary spin in AdS3. To achieve this, we first derive the determinant expression for the one-loop partition function in question using the powerful decomposition technique, and then we construct the normal modes in a purely algebraic way by demonstrating that the space of normal modes falls into the representation of SL(2,R) Lie algebra associated with AdS3. The whole procedure developed here turns out to be much simpler than the previous strategy.

Quantum breathers in a finite Heisenberg spin chain with antisymmetric interactions

A study of the likelihood of quantum breathers in a quantum Heisenberg spin system including a Dzyaloshinsky-Moriya interaction (DMI) is done through an extended Bose-Hubbard model while using the scheme of few body physics. The energy spectrum of the resulting Bose-Hubbard Hamiltonian, on a periodic one-dimensional lattice containing more than two quanta shows interesting detailed band structures. From a non degenerate, and a degenerate perturbation theory in addition to a numerical diagonalization, a careful investigation of these fine structures is set up. The attention is focussed on the effects of various interactions that are; the DMI, the Heisenberg in-plane (X, Y) as well as the out of plane exchange interaction on the energy spectrum of such a system. The outcome displays a possibility of an energy self-compensation in the system. We also computed the weight function of the eigenstates in direct space and in the space of normal modes. From a perturbation theory it is shown that the interaction between the quanta leads to an algebraic localization of the modified extended states in the normal-mode space of the non-interacting system that are coined quantum q-breathers excitations.

Spreading of wave packets in disordered systems with tunable nonlinearity

We study the spreading of single-site excitations in one-dimensional disordered Klein-Gordon chains with tunable nonlinearity |u(l)|(σ)u(l) for different values of σ. We perform extensive numerical simulations where wave packets are evolved (a) without and (b) with dephasing in normal-mode space. Subdiffusive spreading is observed with the second moment of wave packets growing as t(α). The dependence of the numerically computed exponent α on σ is in very good agreement with our theoretical predictions both for the evolution of the wave packet with and without dephasing (for σ≥2 in the latter case). We discuss evidence of the existence of a regime of strong chaos and observe destruction of Anderson localization in the packet tails for small values of σ.

Uncertainties of Estimates of Inertia–Gravity Energy in the Atmosphere. Part II: Large-Scale Equatorial Waves

Abstract This paper analyzes the spectra and spatiotemporal features of the large-scale inertia-gravity (IG) circulations in four analysis systems in the tropics. Of special interest is the Kelvin wave (KW), which represents between 7% and 25% of the total IG wave (zonal wavenumber k ≠ 0) energy. The mixed Rossby–gravity (MRG) mode comprises between 4% and 15% of the IG wave energy. At the longest scales, the KW spectra are fitted by a law while the MRG energy spectrum appears flat. At shorter scales both modes follow a −3 law. Energy spectra of the total IG wave motion at long zonal scales (zonal wavenumber smaller than 7) have slopes close to −1. The average circulation associated with KW is characterized by reverse flows in the upper and lower troposphere consistent with the ideas behind simple tropical models. The inverse projection is used to quantify the role of Kelvin and MRG waves in current analysis systems in the upper troposphere over the Indian Ocean. At these levels, easterlies between 10°S and 30°N are represented by the KW to a significant degree while the cross-equatorial flow toward the descending branch of the Hadley cell at 10°S is associated with the MRG waves. The transient structure of equatorial waves is presented in the space of normal modes defined by the zonal wavenumbers, meridional Hough functions, and the vertical eigenfunctions. The difference in the depth of the model domain in DART–CAM and NCEP–NCAR on one hand and ECMWF and NCEP on the other appears to be one reason for different wave propagation properties. In the latter case the vertical energy propagation is diagnosed by filtering the propagating KW modes back to physical space. The results agree with the linear theory of vertically propagating equatorial waves.

Boundary effects on quantum [formula omitted]-breathers in a Bose–Hubbard chain

We investigate the spectrum and eigenstates of a Bose–Hubbard chain containing two bosons with fixed boundary conditions. In the noninteracting case the eigenstates of the system define a two-dimensional normal-mode space. For the interacting case weight functions of the eigenstates are computed by perturbation theory and numerical diagonalization. We identify paths in the two-dimensional normal-mode space which are rims for the weight functions. The decay along and off the rims is algebraic. Intersection of two paths (rims) leads to a local enhancement of the weight functions. We analyze nonperturbative effects due to the degeneracies and the formation of two-boson bound states.

Q-breathers in discrete nonlinear Schrödinger lattices

q-breathers (QBs) are exact time-periodic solutions of extended nonlinear systems continued from the normal modes of the corresponding linearized system. They are localized in the space of normal modes. The existence of these solutions in a weakly anharmonic atomic chain explained essential features of the Fermi–Pasta–Ulam (FPU) paradox. We study QBs in one-, two- and three-dimensional discrete nonlinear Schrödinger (DNLS) lattices—theoretical playgrounds for light propagation in nonlinear optical waveguide networks, and the dynamics of cold atoms in optical lattices. We prove the existence of these solutions for weak nonlinearity. We find that the localization of QBs is controlled by a single parameter which depends on the norm density, nonlinearity strength and seed wave vector. At a critical value of that parameter QBs delocalize via resonances, signaling a breakdown of the normal mode picture and a transition into the strong mode–mode interaction regime. In particular, this breakdown takes place at one of the edges of the normal mode spectrum, and in a singular way also in the center of that spectrum. A stability analysis of QBs supplements these findings. We relate our findings to previous studies of time-periodic multibreather standing waves. For three-dimensional lattices, we find QB vortices which violate time reversal symmetry and generate a vortex ring flow of energy in normal mode space.

Periodic orbits, localization in normal mode space, and the Fermi–Pasta–Ulam problem

The Fermi–Pasta–Ulam problem was one of the first computational experiments. It has stirred the physics community since, and resisted a simple solution for half a century. The combination of straightforward simulations, efficient computational schemes for finding periodic orbits, and analytical estimates allows us to achieve significant progress. Recent results on q-breathers, which are time-periodic solutions that are localized in the space of normal modes of a lattice and maximize the energy at a certain mode number, are discussed, together with their relation to the Fermi–Pasta–Ulam problem. The localization properties of a q-breather are characterized by intensive parameters, that is, energy densities and wave numbers. By using scaling arguments, q-breather solutions are constructed in systems of arbitrarily large size. Frequency resonances in certain regions of wave number space lead to the complete delocalization of q-breathers. The relation of these features to the Fermi–Pasta–Ulam problem are discussed.

The Fermi–Pasta–Ulam problem: Periodic orbits, normal forms and resonance overlap criteria

Fermi, Pasta and Ulam observed that the excitation of a low frequency normal mode in a nonlinear acoustic chain leads to localization in normal mode space on large time scales. Fast equipartition (and thus complete delocalization) in the Fermi–Pasta–Ulam chain is restored if relevant intensive control parameters exceed certain threshold values. We compare recent results on periodic orbits (in the localization regime) and resonant normal forms (in a weak delocalization regime), and relate them to various resonance overlap criteria. We show that the approaches quantitatively agree in their estimate of the localization–delocalization threshold. A key ingredient for this transition are resonances of overtones.

Q-BREATHERS IN FPU-LATTICES — SCALING AND PROPERTIES FOR LARGE SYSTEMS

Recently q-breathers - time-periodic solutions which localize in the space of normal modes and maximize the energy density for some mode vector q0 - were obtained for finite nonlinear lattices. We scale these solutions to arbitrarily large lattices in various lattice dimensions. We study the scaling consequence for previously obtained analytical estimates of the localization length of q-breathers for β-FPU and α-FPU lattices. The first finding is that the degree of localization depends only on intensive quantities and is size independent. Secondly, a critical wave vector km is identified, which depends on one effective nonlinearity parameter, q-breathers minimize the localization length at k0 = km and completely delocalize in the limit k0 → 0, π.

Quantumq-breathers in a finite Bose-Hubbard chain: The case of two interacting bosons

We study the spectrum and eigenstates of the quantum discrete Bose-Hubbard Hamiltonian in a finite one-dimensional lattice containing two bosons. The interaction between the bosons leads to an algebraic localization of the modified extended states in the normal mode space of the noninteracting system. Weight functions of the eigenstates in the space of normal modes are computed by using numerical diagonalization and perturbation theory. We find that staggered states do not compactify in the dilute limit for large chains.

Modal and Nonmodal Symmetric Perturbations. Part II: Nonmodal Growths Measured by Total Perturbation Energy

Maximum nonmodal growths of total perturbation energy are computed for symmetric perturbations constructed from the normal modes presented in Part I. The results show that the maximum nonmodal growths are larger than the energy growth produced by any single normal mode for a give optimization time, and this is simply because the normal modes are nonorthogonal (measured by the inner product associated with the total perturbation energy norm). It is shown that the maximum nonmodal growths are produced mainly by paired modes, and this can be explained by the fact that the streamfunction component modes are partially orthogonal between different pairs and parallel within each pair in the streamfunction subspace. When the optimization time is very short (compared with the inverse Coriolis parameter), the nonmodal growth is produced mainly by the paired fastest propagating modes. When the optimization time is not short, the maximum nonmodal growth is produced almost solely by the paired slowest propagating modes and the growth can be very large for a wide range of optimization time if the parameter point is near the boundary and outside the unstable region. If the parameter point is near the boundary but inside the unstable region, the paired slowest propagating modes can contribute significantly to the energy growth before the fastest growing mode becomes the dominant component. The maximum nonmodal growths produced by paired modes are derived analytically. The analytical solutions compare well with the numerical results obtained in the truncated normal mode space. The analytical solutions reveal the basic mechanisms for four types of maximum nonmodal energy growths: the PP1 and PP2 nonmodal growths produced by paired propagating modes and the GD1 and GD2 nonmodal growths produced by paired growing and decaying modes. The PP1 growth is characterized by the increase of the cross-band kinetic energy that excessively offsets the decrease of the along-band kinetic and buoyancy energy. The situation is opposite for the PP2 growth. The GD1 (or GD2) growth is characterized by the reduction of the initial cross-band kinetic energy (or initial along-band kinetic and buoyancy energy) due to the inclusion of the decaying mode.