James F. Lutsko

Stress and elastic constants in anisotropic solids: molecular dynamics techniques

The local stress tensor commonly used in statistical mechanics is cast in a form useful for molecular dynamics (MD) simulations. It is then used to derive fluctuation formulas for the local elastic constants of a system. The formulas are used in an MD measurement of the local elastic constants of an ideal crystal which are found to be in good agreement with the bulk elastic constants of the same system.

Theoretical Evidence for a Dense Fluid Precursor to Crystallization

We present classical density functional theory calculations of the free-energy landscape for fluids below their triple point as a function of density and crystallinity. We find that, both for a model globular protein and for a simple atomic fluid modeled with a Lennard-Jones interaction, it is free-energetically easier to crystallize by passing through a metastable dense fluid in accord with the Ostwald rule of stages but in contrast to the alternative of ordering and densifying at once as assumed in the classical picture of crystallization.

Globally optimal fuzzy decision trees for classification and regression

A fuzzy decision tree is constructed by allowing the possibility of partial membership of a point in the nodes that make up the tree structure. This extension of its expressive capabilities transforms the decision tree into a powerful functional approximant that incorporates features of connectionist methods, while remaining easily interpretable. Fuzzification is achieved by superimposing a fuzzy structure over the skeleton of a CART decision tree. A training rule for fuzzy trees, similar to backpropagation in neural networks, is designed. This rule corresponds to a global optimization algorithm that fixes the parameters of the fuzzy splits. The method developed for the automatic generation of fuzzy decision trees is applied to both classification and regression problems. In regression problems, it is seen that the continuity constraint imposed by the function representation of the fuzzy tree leads to substantial improvements in the quality of the regression and limits the tendency to overfitting. In classification, fuzzification provides a means of uncovering the structure of the probability distribution for the classification errors in attribute space. This allows the identification of regions for which the error rate of the tree is significantly lower than the average error rate, sometimes even below the Bayes misclassification rate.

Generalized expressions for the calculation of elastic constants by computer simulation

The general expressions, valid for any potential, for the calculation of elastic constants through computer simulation are given. At zero temperature, the elastic constants are found to be the sum of a generalization of the Born term and a term accounting for internal relaxations that arise when a system with more than one atom in the primitive unit cell is strained. The fluctuation formulae used in finite temperature simulations are found to be straightforward generalizations of those used for pair potentials. The connection between the finite‐temperature and zero‐temperature methods is also made.

Formalism for the calculation of local elastic constants at grain boundaries by means of atomistic simulation

A new formalism for use in atomistic simulations to calculate the full local elastic‐constant tensor in terms of local stresses and strains is presented. Results of simulations on a high‐angle (001) twist grain boundary are illustrated, using both a Lennard–Jones potential for Cu and an embedded‐atom potential for Au. The two conceptionally rather different potentials show similar anomalies in all elastic constants, confined to within a few lattice planes of the grain boundary, with an especially dramatic reduction in the resistance to shear parallel to the grain‐boundary plane. It is found that the primary cause of the anomalies is the atomic disorder near the grain boundary, as evidenced by the slice‐by‐slice radial distribution functions for the inhomogeneous interface system.

On the multifractal properties of the energy dissipation derived from turbulence data

Various difficulties can be eXpected in trying to eXtract from eXperimental data the distribution of singularities, the f(a) function, of the energy dissipation. One reason is that the multifractal model of turbulence implies a dependence of the viscous cutoff on the singularity eXponent. It is an open question if current hot-wire probes can resolve the scales implied by eXponents a significantly less than 1. Two eXactly soluble models are used to show how spurious scaling can occur, due to finite Reynolds number effects. In the Gaussian model the true velocity signal is replaced by independent Gaussian random variables. The dissipation, defined as the square of the difference of successive variables, has trivial scaling in so far as all the moments of spatial averages of the dissipation behave asymptotically as a uniform dissipation. Still, contamination by subdominant terms requires that scaling eXponents for high-order moments be identified over an increasingly large range ofb scales. If the available range is limited by the Reynolds number, scaling eXponents for high orders will be systematically underestimated and spurious intermittency will be inferred. Burgers’ model is used to highlight further problems. At finite Reynolds numbers, regions with no small-scale activity (away from shocks) have a residual dissipation which contributes a spurious point (a= l, f(a=) 1). In addition, when the f(a) function is obtained by Legendre transform techniques, conveX hull effects generate an entire spurious segment. Finally, Burgers’ model also indicates that the relation between eXponents of structure functions and eXponents of local dissipation moments, which goes back to Kolmogorovs (1962) work, leads to an inconsistency for structure functions of low positive order.

Dependence of the liquid-vapor surface tension on the range of interaction: A test of the law of corresponding states

The validity of the principle of corresponding states is investigated for the case of a potential with more than one intrinsic length scale. The planar surface tension of coexisting liquid and vapor phases of a fluid of Lennard-Jones atoms is studied as a function of the range of the potential using both Monte Carlo simulations and density functional theory (DFT). The interaction range is varied from r(c)(*) = 2.5 to r(c)(*) = 6 and the surface tension is determined for temperatures ranging from T(*) = 0.7 up to the critical temperature in each case. The simulation results are consistent with previous studies and are shown to obey the law of corresponding states even though the potential has two intrinsic length scales. It is further shown that the corresponding states principle can also be used to enhance the accuracy of some, but not all, DFT calculations of the surface tension. The results show that most of the cutoff dependence of the surface tension can be explained as a result of changes in the cutoff-dependent phase diagram and that corresponding states can be a useful tool for explaining differences between theory and simulation.

Observing classical nucleation theory at work by monitoring phase transitions with molecular precision.

It is widely accepted that many phase transitions do not follow nucleation pathways as envisaged by the classical nucleation theory. Many substances can traverse intermediate states before arriving at the stable phase. The apparent ubiquity of multi-step nucleation has made the inverse question relevant: does multistep nucleation always dominate single-step pathways? Here we provide an explicit example of the classical nucleation mechanism for a system known to exhibit the characteristics of multi-step nucleation. Molecular resolution atomic force microscopy imaging of the two-dimensional nucleation of the protein glucose isomerase demonstrates that the interior of subcritical clusters is in the same state as the crystalline bulk phase. Our data show that despite having all the characteristics typically associated with rich phase behaviour, glucose isomerase 2D crystals are formed classically. These observations illustrate the resurfacing importance of the classical nucleation theory by re-validating some of the key assumptions that have been recently questioned.

A dynamical theory of nucleation for colloids and macromolecules.

A general theory of nucleation for colloids and macromolecules in solution is formulated within the context of fluctuating hydrodynamics. A formalism for the determination of nucleation pathways is developed and stochastic differential equations for the evolution of order parameters are given. The conditions under which the elements of classical nucleation theory are recovered are determined. The theory provides a justification and extension of more heuristic equilibrium approaches based solely on the free energy. It is illustrated by application to the low-concentration/high-concentration transition in globular proteins, where a novel two-step mechanism is identified, where the first step involves the formation of long-wavelength density fluctuations, and the second step is the actual nucleation event occurring within the fluctuation.

Transport properties of dense dissipative hard-sphere fluids for arbitrary energy loss models

The revised Enskog approximation for a fluid of hard spheres which lose energy upon collision is discussed for the case that the energy is lost from the normal component of the velocity at collision but is otherwise arbitrary. Granular fluids with a velocity-dependent coefficient of restitution are an important special case covered by this model. A normal solution to the Enskog equation is developed using the Chapman-Enskog expansion. The lowest order solution describes the general homogeneous cooling state and a generating function formalism is introduced for the determination of the distribution function. The first order solution, evaluated in the lowest Sonine approximation, provides estimates for the transport coefficients for the Navier-Stokes hydrodynamic description. All calculations are performed in an arbitrary number of dimensions.