Mapping of diffusion in a channel with soft walls.

We study diffusion of pointlike particles biased toward the x axis by a quadratic potential U(x,y)=κ(x)y². This system mimics a channel with soft walls of some varying (effective) cross section A(x), depending on the stiffness κ(x). We show that diffusion in this geometry can also be mapped rigorously onto the longitudinal coordinate x by a procedure known for channels with hard walls [P.

Mapping of diffusion in a channel with abrupt change of diameter.

Mapping of the diffusion equation in a channel of varying cross section onto the longitudinal coordinate is already a well studied procedure for a slowly changing radius. We examine here the mapping of diffusion in a channel with abrupt change of diameter. In two dimensions, our considerations are based on solution of the exactly solvable geometry with abruptly doubled width at x=0.

Normal and anomalous diffusion in highly confined hard disk fluid mixtures.

Monte Carlo simulation is used to study binary mixtures of two-dimensional hard disks, confined to long, narrow, structureless pores with hard walls, in a regime of pore sizes where the large particles exhibit single file diffusion while the small particles diffuse normally. The dynamics of the small particles can be understood in the context of a hopping time, tau(21), that measures the time it takes for a small particle to escape the single file cage formed by its large particle neighbors, and can be linked to the long time diffusion coefficient. We find that tau(21) follows a power law as a function of the reduced pore radius for a wide range of particle size ratios with an exponent, alpha, that is independent of the size ratio, but linearly dependent on the Monte Carlo step size used in the dynamic scheme.

The amplitudes of viral blips in HIV-1 infected patients treated with antiretroviral therapy are power-law distributed.

We previously reported that in patients treated with highly active antiretroviral therapy (HAART) who achieve viral load (VL) suppression, low fluctuations of viral load over the threshold of detection (viral blips) more than 4 weeks apart occur at random, with a frequency that does not change with longer times of observation. The etiology of viral blips is currently unknown, but viral blip frequency inversely correlates with the decay of the latent reservoir, whose stability has been proposed as the major hurdle to HIV eradication. We show here that the distribution of viral blip amplitudes observed in a group of 272 patients successfully treated with highly active antiretroviral therapy appears to be power-law distributed.

Two definitions of the hopping time in a confined fluid of finite particles.

We consider a fluid of hard disks diffusing in a flat long narrow channel of width approaching from above the doubled diameter of the disks. In this limit, the disks can pass their neighbors only rarely, in a mean hopping time growing to infinity, so the disks start by diffusing anomalously. We study the hopping time, which is the crucial parameter of the theory describing the subsequent transition to normal diffusion.

Approximations of the generalized Fick-Jacobs equation.

We analyze the generalized Fick-Jacobs equation, obtained by a rigorous mapping of the diffusion equation in a quasi-one-dimensional (quasi-1D) (narrow 2D or 3D) channel with varying cross section A(x) onto the longitudinal coordinate x . We show that for constructing approximations and understanding their applicability in practice, it is crucial to study the 2D (3D) density inside the channel in the regime of stationary flow. We present algorithms enabling us to derive approximate formulas for the effective diffusion coefficient involving derivatives of A(x) higher than A'(x) and give examples for 2D channels.

Variational calculation of second-order reduced density matrices by strong N-representability conditions and an accurate semidefinite programming solver.

The reduced density matrix (RDM) method, which is a variational calculation based on the second-order reduced density matrix, is applied to the ground state energies and the dipole moments for 57 different states of atoms, molecules, and to the ground state energies and the elements of 2-RDM for the Hubbard model. We explore the well-known N-representability conditions (P, Q, and G) together with the more recent and much stronger T1 and T2(') conditions. T2(') condition was recently rederived and it implies T2 condition.

Stretched Markov nature of single-file self-dynamics.

We study the generalized diffusion of a tagged particle in a one-dimensional fluid of hard-point particles. The dynamics of a single particle in its nonuniform, nondeterministic environment is assumed known. On eliminating suitably defined transients from the exact solution, we find a universal form for the tagged particle dynamics when written in terms of stretched space and time, appearing as the classical telegrapher's equation.

Hopping time of a hard disk fluid in a narrow channel.

We use Monte Carlo (MC) and molecular dynamics (MD) methods to study the self-diffusion of hard disk fluids, confined within a narrow channel. The channels have a pore radius of Rp, above the passing limit of hard disk diameter (sigma(hd)). We focus on the average time (tau(hop)) needed for a hard disk to hop past a nearest neighbor in the longitudinal direction.

Hopping times of two hard disks diffusing in a channel.

A finite difference method was used to solve numerically the multidimensional diffusion equation describing the time evolution of two hard disks diffusing in a narrow hard channel. The authors extract an estimate for the average time tauhop needed for the disks to hop pass each other. For narrow channels near the hopping threshold, tauhop diverges and is consistent with the scaling prediction of the transition state theory.

Simple Hamiltonians which exhibit drastic failures by variational determination of the two-particle reduced density matrix with some well known N-representability conditions.

Calculations on small molecular systems indicate that the variational approach employing the two-particle reduced density matrix (2-RDM) as the basic unknown and applying the P, Q, G, T1, and T2 representability conditions provides an accuracy that is competitive with the best standard ab initio methods of quantum chemistry. However, in this paper we consider a simple class of Hamiltonians for which an exact ground state wave function can be written as a single Slater determinant and yet the same 2-RDM approach gives a drastically nonrepresentable result. This shows the need for stronger representability conditions than the mentioned ones.

Corrections to the Fick-Jacobs equation.

Diffusion in a quasi-one-dimensional channel, with cross section varying along the longitudinal coordinate, is considered. Using a rigorous mapping of the diffusion equation onto one dimension, eliminating transients in transverse direction(s), we derive an expansion of the effective diffusion coefficient D(x), which represents corrections to the Fick-Jacobs equation.

Mapping a homopolymer onto a model fluid.

We describe a linear homopolymer using a grand canonical ensemble formalism, a statistical representation that is very convenient for formal manipulations. We investigate the properties of a system where only next neighbor interactions and an external, confining, field are present and then show how a general pair interaction can be introduced perturbatively, making use of a Mayer expansion. Through a diagrammatic analysis, we shall show how constitutive equations derived for the polymeric system are equivalent to the Ornstein-Zernike and Percus-Yevick equations for a simple fluid and find the implications of such a mapping for the simple situation of Van der Waals mean field model for the fluid.

Extended Fick-Jacobs equation: variational approach.

We derive an extended Fick-Jacobs equation for the diffusion of noninteracting particles in a two- and symmetric three-dimensional channels of varying cross section A(x), using a variational approach. The result is a diffusion differential equation of second order in only one space (longitudinal) coordinate. This equation is tested on the task of calculating the stationary flux through a hyperboloidal tube, and its solution is compared with that of other methods.

Size-asymmetric primitive model at low temperature: description of ion pairing and location of the critical point.

We argue that Bjerrum's approach to ion pairing is inappropriate for the size-asymmetric primitive model in the neighborhood of its critical point, and propose a new approach based on the Stillinger-Lovett pairing procedure. The new approach recursively scales up the ion size until linear approximations are suitable for analyzing such a model. To locate the critical point, a residual van der Waals interaction between pairs is added, with an energy cutoff adjusted to match the critical temperature of the restricted primitive model.

At the boundary between reduced density-matrix and density-functional theories.

Here, we revisit the problem of finding the ground-state energy of an N-fermion fluid under an external field, with molecular structure as the ultimate target. Density-functional methods only have to deal with electron density, but require an empirical functional; reduced density-matrix methods involve a matrix on pair space and do give exact bounds, but require very complex linear programming to achieve their results. The polydensity representation that we introduce has the advantage of dealing only with densities, requires no empirical information, and also gives exact bounds; the major problem is that of accumulating and utilizing conditions on the densities that iteratively improve their realizability in the class of N-fermion systems.

Small population effects in stochastic population dynamics.

The discreteness of units of small populations can produce large fluctuations from a classical continuous representation, especially when null populations play a crucial role. These belong to what are here referred to as emergent and evanescent species. A few model biological systems are introduced in which this is the case, as well as a toy model that suggests a path to avoid the associated mathematical complexities.

Soluble stochastic dynamics of quasi-one-dimensional single-file fluid self-diffusion.

We solve a model of random-walk stochastic dynamics for hard single-file fluids in the experimentally important quasi-one-dimensional regime. This is a nontrivial extension of exact solution beyond one dimension. We point out that quasi-one-dimensional single-file self-diffusion of one-component hard fluids of diameter a under stochastic forces is equivalent at long time to a one-dimensional hard-rod fluid with the same linear density but a different diameter, a(eff).

Projection of two-dimensional diffusion in a narrow channel onto the longitudinal dimension.

Diffusion in a narrow two-dimensional channel of width A(x), depending on the longitudinal coordinate x, is the object of our study. We show how the 2+1 dimensional diffusion equation can be projected onto a 1+1 dimensional one, governing corresponding one-dimensional density, in a steady-state approximation. Then we demonstrate the method on a nontrivial exactly solvable case for A(x)=x and discuss projection of the initial condition.

Duration of an intermittent episode of viremia.

HIV-1 infected patients after being treated with potent combinations of antiretroviral drugs for 2-6 months typically reach a state in which virus can no longer be detected within their blood. These patients with undetectable virus occasionally have viral load measurements that are above the limit of detection of current assays. Such measurements are called blips.

Calculating the hopping times of confined fluids: two hard disks in a box.

The dynamical transition between the anomalous single file diffusion of highly confined fluids and bulk normal diffusion can be described by a phenomenological model involving a particle hopping time tau(hop). We suggest a theoretical formalism that will be useful for the calculation of tau(hop) for a variety of systems and test it using a simple model consisting of two hard disks confined to a rectangular box with hard walls. In the case where the particles are moving diffusively, we find the hopping time diverges as a power law in the threshold region with an exponent of -(3/2).

Equation of state for hard-sphere fluid in restricted geometry.

Many practical applications require the knowledge of the equation of state of fluids in restricted geometry. We study a hard-sphere fluid at equilibrium in a narrow cylindrical pore with hard walls for pore radii R<((square root 3)+2)/4 (in units of the hard sphere diameter). In this case each particle can interact only with its nearest neighbors, which makes possible the use of analytical methods to study the thermodynamics of the system.

The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions.

The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used P, Q, and G conditions. The additional conditions (called T1 and T2 here) are implicit in the work of Erdahl [Int. J.

Optimal viral production.

Viruses reproduce by multiplying within host cells. The reproductive fitness of a virus is proportional to the number of offspring it can produce during the lifetime of the cell it infects. If viral production rates are independent of cell death rate, then one expects natural selection will favor viruses that maximize their production rates.

The distribution of viral blips observed in HIV-1 infected patients treated with combination antiretroviral therapy.

Human immunodeficiency virus type 1 (HIV-1) infected patients treated with combination antiretroviral therapy frequently have the level of HIV-1 RNA detectable in plasma driven below the lower limit of detection of current assays, 50 copies ml(-1). Patients may continue to exhibit viral loads (VLs) below the assay limit for years, yet on some occasions the VL may be above the limit of detection. Whether these 'blips' in VL are simply assay errors or are indicative of intermittent episodes of increased viral replication is of great clinical concern.