Multiscale theory of collective and quasiparticle modes in quantum nanosystems

Space warping order parameters and symmetry: application to multiscale simulation of macromolecular assemblies

Coarse-grained features of macromolecular assemblies are understood via a set of order parameters (OPs) constructed in terms of their all-atom configuration. OPs are shown to be slowly changing in time and capture the large-scale spatial features of macromolecular assemblies. The relationship of these variables to the classic notion of OPs based on symmetry breaking phase transitions is discussed. OPs based on space warping transformations are analyzed in detail as they naturally provide a connection between overall structure of an assembly and all-atom configuration. These OPs serve as the basis of a multiscale analysis that yields Langevin equations for OP dynamics. In this context, the characteristics of OPs and PCA modes are compared. The OPs enable efficient all-atom multiscale simulations of the dynamics of macromolecular assemblies in response to changes in microenvironmental conditions, as demonstrated on the structural transitions of cowpea chlorotic mottle virus capsid (CCMV) and RNA of the satellite tobacco mosaic virus (STMV).

MULTISCALE BORN-OPPENHEIMER THEORY OF COLLECTIVE ELECTRON-NUCLEAR DYNAMICS IN NANOSYSTEMS

Born-Oppenheimer theory is based on the separation in timescales between the nuclear and electron dynamics implied by the electron-to-nuclear mass ratio. This makes it naturally fit into a multiscale analysis. It is shown that a fully dynamical Born-Oppenheimer theory follows from a multiscale ansatz on the wave function and a Taylor expansion in the mass ratio. Allowing for a larger spatial scale of electron motion yields an understanding of boson, fermion, and more complex excitations that involve quasi-particles with an effective mass not equal to that of the electron. The theory involves a unified asymptotic expansion in a mass and length scale ratio, and preserves all many-body effects via accounting for the full strength of the interparticle forces. A novel mean-field theory emerges based on the fact that long-scale migration allows each electron to interact with many others on the space-time scale relevant to the coarse-grained equation. Implications for computational methods and applications to quantum nanosystems such as quantum dots, nanowires, superconducting nanoparticles, and liquid He droplets are discussed.

Scaling behavior of quantum nanosystems: emergence of quasi-particles, collective modes, and mixed exchange symmetry states

Examples of quantum nanosystems are graphene nanoribbons, molecular wires, and superconducting nanoparticles. The objective of the multiscale theory presented here is to provide a new perspective on the coupling of processes across scales in space and time underlying the dynamics of these systems. The long range objective for this multiscale approach is to serve as an efficient computational algorithm. Long space-time dynamics is derived using a perturbation expansion in the ratio ɛ of the nearest-neighbor distance to a nanometer-scale characteristic length and a theorem on the equivalence of long time-averages and expectation values. This dynamics is shown to satisfy a coarse-grained wave equation (CGWE) which takes a Schrödinger-like form with modified masses and interactions. The scaling of space and time is determined by the orders of magnitude of various contributions to the N-body potential. If the spatial scale of the coarse-graining is too large, the CGWE would imply an unbounded growth of gradients; if it is too short, the system's size would display uncontrolled growth inappropriate for the bound states of interest, i.e., collective motion or migration within a stable nanoassembly. The balance of these two extremes removes arbitrariness in the choice of the scaling of space-time. Since the long-scale dynamics of each Fermion involves its interaction with many others, we hypothesize that the solutions of the CGWE have mean-field character to good approximation, i.e., can be factorized into single-particle functions. This leads to a coarse-grained mean-field approximation that is distinct in character from traditional Hartree-Fock theory. A variational principle is used to derive equations for the single-particle functions. This theme is developed and used to derive an equation for low-lying disturbances from the ground state corresponding to long wavelength density disturbances or long-scale migration. An algorithm for the efficient simulation of quantum nanosystems is suggested.

Multiscale simulation of microbe structure and dynamics

A multiscale mathematical and computational approach is developed that captures the hierarchical organization of a microbe. It is found that a natural perspective for understanding a microbe is in terms of a hierarchy of variables at various levels of resolution. This hierarchy starts with the N -atom description and terminates with order parameters characterizing a whole microbe. This conceptual framework is used to guide the analysis of the Liouville equation for the probability density of the positions and momenta of the N atoms constituting the microbe and its environment. Using multiscale mathematical techniques, we derive equations for the co-evolution of the order parameters and the probability density of the N-atom state. This approach yields a rigorous way to transfer information between variables on different space-time scales. It elucidates the interplay between equilibrium and far-from-equilibrium processes underlying microbial behavior. It also provides framework for using coarse-grained nanocharacterization data to guide microbial simulation. It enables a methodical search for free-energy minimizing structures, many of which are typically supported by the set of macromolecules and membranes constituting a given microbe. This suite of capabilities provides a natural framework for arriving at a fundamental understanding of microbial behavior, the analysis of nanocharacterization data, and the computer-aided design of nanostructures for biotechnical and medical purposes. Selected features of the methodology are demonstrated using our multiscale bionanosystem simulator DeductiveMultiscaleSimulator. Systems used to demonstrate the approach are structural transitions in the cowpea chlorotic mosaic virus, RNA of satellite tobacco mosaic virus, virus-like particles related to human papillomavirus, and iron-binding protein lactoferrin.

Scaling behavior of electronic excitations in assemblies of molecules with degenerate ground states

The behavior of long space-time excitations in many-electron systems with ground state degeneracy is explored via multiscale analysis. The analysis starts with an ansatz for the wave function's dual dependence on the N-electron configuration (i.e., both by direct means and by indirect means via a set of order parameters). It is shown that a Dirac-like equation form of the wave equation emerges in the limit where the ratio epsilon (of the average nearest-neighbor distance to the characteristic length of the long-scale phenomenon of interest) is small. Examples of the long scale are the size of a quantum dot, nanotube, or wavelength of a density disturbance. The velocities in the Dirac-like equation are the transition moments of the single-particle momentum operator connecting degenerate ground states. While detailed band structure and the independent quasi-particle picture could underlie the behavior of some systems (as commonly suggested for graphene), the present scaling law results show it is not necessarily the only explanation. Rather, it can follow from the scaling properties of low-lying, long spatial scale excitations and ground state degeneracy, even in strongly interacting systems. The generality of our findings suggests graphene may be just one of many examples of Dirac-like equation behavior. A preliminary validation of our quantum scaling law for molecular arrays is presented. As our scaling law constitutes a coarse-grained wave equation, path integral or other methods derived from it hold great promise for calibration-free, long-time simulation of many-particle quantum systems.