Stephen Pankavich

Multiscaling for classical nanosystems: Derivation of Smoluchowski & Fokker–Planck equations

Using multiscale analysis and methods of statistical physics, we show that a solution to the N-atom Liouville equation can be decomposed via an expansion in terms of a smallness parameter ϵ, wherein the long scale time behavior depends upon a reduced probability density that is a function of slow-evolving order parameters. This reduced probability density is shown to satisfy the Smoluchowski equation up to O(ϵ2) for a given range of initial conditions. Furthermore, under the additional assumption that the nanoparticle momentum evolves on a slow time scale, we show that this reduced probability density satisfies a Fokker–Planck equation up to O(ϵ2). This approach has applications to a broad range of problems in the nanosciences.

Self-assembly of nanocomponents into composite structures: Derivation and simulation of Langevin equations

The kinetics of the self-assembly of nanocomponents into a virus, nanocapsule, or other composite structure is analyzed via a multiscale approach. The objective is to achieve predictability and to preserve key atomic-scale features that underlie the formation and stability of the composite structures. We start with an all-atom description, the Liouville equation, and the order parameters characterizing nanoscale features of the system. An equation of Smoluchowski type for the stochastic dynamics of the order parameters is derived from the Liouville equation via a multiscale perturbation technique. The self-assembly of composite structures from nanocomponents with internal atomic structure is analyzed and growth rates are derived. Applications include the assembly of a viral capsid from capsomers, a ribosome from its major subunits, and composite materials from fibers and nanoparticles. Our approach overcomes errors in other coarse-graining methods, which neglect the influence of the nanoscale configuration on the atomistic fluctuations. We account for the effect of order parameters on the statistics of the atomistic fluctuations, which contribute to the entropic and average forces driving order parameter evolution. This approach enables an efficient algorithm for computer simulation of self-assembly, whereas other methods severely limit the timestep due to the separation of diffusional and complexing characteristic times. Given that our approach does not require recalibration with each new application, it provides a way to estimate assembly rates and thereby facilitate the discovery of self-assembly pathways and kinetic dead-end structures.

Liquid-crystal transitions: a first-principles multiscale approach.

A rigorous theory of liquid-crystal transitions is developed starting from the Liouville equation. The starting point is an all-atom description and a set of order-parameter field variables that are shown to evolve slowly via Newtons equations. The separation of time scales between that of atomic collision or vibrations and the order-parameter fields enables the derivation of rigorous equations for stochastic order-parameter field dynamics. When the fields provide a measure of the spatial profile of the probability of molecular position, orientation, and internal structure, a theory of liquid-crystal transitions emerges. The theory uses the all-atom/continuum approach developed earlier to obtain a functional generalization of the Smoluchowski equation wherein key atomic details are embedded. The equivalent nonlocal Langevin equations are derived, and the computational aspects are discussed. The theory enables simulations that are much less computationally intensive than molecular dynamics and thus does not require oversimplification of the systems constituent components. The equations obtained do not include factors that require calibration and can thus be applicable to various phase transitions which overcomes the limitations of phenomenological field models. The relation of the theory to phenomenological descriptions of nematic and smectic phase transitions, and the possible existence of other types of transitions involving intermolecular structural parameters are discussed.

Multiscaling for systems with a broad continuum of characteristic lengths and times: Structural transitions in nanocomposites

The multiscale approach to N-body systems is generalized to address the broad continuum of long time and length scales associated with collective behaviors. A technique is developed based on the concept of an uncountable set of time variables and of order parameters (OPs) specifying major features of the system. We adopt this perspective as a natural extension of the commonly used discrete set of time scales and OPs which is practical when only a few, widely separated scales exist. The existence of a gap in the spectrum of time scales for such a system (under quasiequilibrium conditions) is used to introduce a continuous scaling and perform a multiscale analysis of the Liouville equation. A functional-differential Smoluchowski equation is derived for the stochastic dynamics of the continuum of Fourier component OPs. A continuum of spatially nonlocal Langevin equations for the OPs is also derived. The theory is demonstrated via the analysis of structural transitions in a composite material, as occurs for viral capsids and molecular circuits.

A Kernel-based Lagrangian method for imperfectly-mixed chemical reactions ☆

Abstract Current Lagrangian (particle-tracking) algorithms used to simulate diffusion–reaction equations must employ a certain number of particles to properly emulate the system dynamics—particularly for imperfectly-mixed systems. The number of particles is tied to the statistics of the initial concentration fields of the system at hand. Systems with shorter-range correlation and/or smaller concentration variance require more particles, potentially limiting the computational feasibility of the method. For the well-known problem of bimolecular reaction, we show that using kernel-based, rather than Dirac delta, particles can significantly reduce the required number of particles. We derive the fixed width of a Gaussian kernel for a given reduced number of particles that analytically eliminates the error between kernel and Dirac solutions at any specified time. We also show how to solve for the fixed kernel size by minimizing the squared differences between solutions over any given time interval. Numerical results show that the width of the kernel should be kept below about 12% of the domain size, and that the analytic equations used to derive kernel width suffer significantly from the neglect of higher-order moments. The simulations with a kernel width given by least squares minimization perform better than those made to match at one specific time. A heuristic time-variable kernel size, based on the previous results, performs on par with the least squares fixed kernel size.

The Effects of Latent Infection on the Dynamics of HIV

One way in which the human immunodeficiency virus (HIV-1) replicates within a host is by infecting activated CD

Global Existence and Increased Spatial Decay for the Radial Vlasov‐Poisson System with Steady Spatial Asymptotics

4+

Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity

4+ T-cells, which then produce additional copies of the virus. Even with the introduction of antiretroviral drug therapy, which has been very successful over the past decade, a large obstacle to the complete eradication of the virus is the presence of viral reservoirs in the form of latently infected CD