Advances in the Computational Sciences

The following sections are included:

• CONTENTS
• PREFACE

Good morning and welcome to Berni Alder’s 90^th Birthday Symposium. My name is Bruce Tarter and I’m representing Director Bill Goldstein, who could not be here this morning, but will be here for the afternoon session and will make comments of his own at the end of the Symposium…

Berni Alder profoundly influenced my research career at Lawrence Livermore National Laboratory and the Davis Campus’ Teller Tech, beginning in 1962 and lasting for over fifty years. I very much appreciate the opportunity provided by his Ninetieth Birthday Celebration to review some of the many high spots along the way.

The 1970 paper, “Decay of the Velocity Correlation Function” [Phys. Rev. A1, 18 (1970), see also Phys. Rev. Lett. 18, 988, (1967)] by Berni Alder and Tom Wainwright, demonstrated, by means of computer simulations, that the velocity autocorrelation function for a particle moving diffusively in a gas of hard disks decays algebraically in time as t⁻¹, and as t^−3/2 for a gas of hard spheres. These decays appear in non-equilibrium fluids and have no counterpart in fluids in thermodynamic equilibrium. The work of Alder and Wainwright stimulated theorists to find explanations for these “long time tails” using kinetic theory or a mesoscopic mode-coupling theory. This paper has had a profound influence on our understanding of the non-equilibrium properties of fluid systems. Here we discuss the kinetic origins of the long time tails, the microscopic foundations of modecoupling theory, and the implications of these results for the physics of fluids. We also mention applications of the long time tails and mode-coupling theory to other, seemingly unrelated, fields of physics. We are honored to dedicate this short review to Berni Alder on the occasion of his 90th birthday!

The fundamental problem of turbulent transition in wall-bounded hydrodynamic flow is still not well understood. Previous studies of channel and pipe flows do not include the role of wall roughness and only consider linear treatment with fluctuations to unsuccessfully destabilize the flow. We investigate if small amplitude distributed wall roughness with and without fluctuations could initiate the transition to turbulence by direct numerical simulation using a lattice-Boltzmann method that is equivalent to solving the nonlinear Navier-Stokes equation. The results show how a single roughness feature with microscopic amplitude on an otherwise smooth wall can generate the transition to turbulence in plane Poiseuille flow. The effect of fluctuations with smooth walls or the combined effect of fluctuations and wall perturbances predicts the onset of turbulence at about the same critical Reynolds number which is about twice the experimental one.

A model for diffusion in liquids that couples the dynamics of tracer particles to a fluctuating Stokes equation for the fluid is investigated in the limit of large Schmidt number. In this limit, the concentration of tracers is shown to satisfy a closed-form stochastic advection-diffusion equation that is used to investigate the collective diffusion of hydrodynamically-correlated tracers through a combination of Eulerian and Lagrangian numerical methods. This analysis indicates that transport in liquids is quite distinct from the traditional Fickian picture of diffusion. While the ensemble-averaged concentration follows Fick’s law with a diffusion coefficient that obeys the Stokes-Einstein relation, each instance of the diffusive mixing process exhibits long-ranged giant fluctuations around its average behavior. We construct a class of mesoscopic models for diffusion in liquids at different observation scales in which the renormalized diffusion coefficient depends on this scale. This indicates that the Fickian diffusion coefficient in liquids is not a material constant, but rather, changes with the scale at which experimental measurements are performed.

Hard sphere/disk systems are among the simplest models and have been used to address numerous fundamental problems in the field of statistical physics. The pioneering numerical works on the solid–fluid phase transition based on Monte Carlo (MC) and molecular dynamics (MD) methods published in 1957 represent historical milestones, which have had a significant influence on the development of computer algorithms and novel tools to obtain physical insights. This chapter addresses the works of Alder’s breakthrough regarding hard sphere/disk simulation: (i) event-driven molecular dynamics, (ii) long-time tail, (iii) molasses tail, and (iv) two-dimensional melting/crystallization. From a numerical viewpoint, there are serious issues that must be overcome for further breakthrough. Here, we present a brief review of recent progress in this area.

We present a brief overview of two key aspects of the structural glass transition (GT). The first part analyzes the structural slowing down in supercooled liquids, culminating in the kinetic GT predicted by mode-coupling theory (MCT); the theoretical predictions are confronted with the results of Molecular Dynamics simulations. The second part of the review examines the still controversial “random first order transition𠇍 (RFOT) from a quenched liquid to an “ideal glass𠇍 expected to occur at much lower temperatures, where equilibration times diverge, making the RFOT inaccessible to experiment and simulation. A “pedestrian𠇍 but robust version of the replica formalism (involving two weakly coupled replicae) is combined with integral equations for the pair structure of such a symmetric binary system, and leads to the prediction of a thermodynamic, weakly first-order RFOT both for the “soft sphere𠇍 and Lennard-Jones models, via qualitatively identical scenarios.

I do not know Berni Alder as a person, but I feel that I know him well through his seminal paper “Phase Transition in Elastic Disks𠇍 by B. J. Alder and T. E. Wainwright [1962], which was essential in motivating David Thouless and myself to think about phase transitions in two dimensional systems with a continuous symmetry. In the early 1970’s, the conventional wisdom was that a crystalline solid could not exist in a two dimensional world because of the rigorous Mermin-Wagner theorem prohibiting true long range translational order at any non-zero temperature. This contradiction was settled by the theory of dislocation mediated melting to an intermediate hexatic phase followed by a second transition to the isotropic fluid at a higher temperature. This scenario, with its associated sophisticated theory, seemed to settle the controversy of two dimensional melting once and for all. However, in our elation at understanding the fundamental physics and the essential excitations of melting in 2D, we had all forgotten that the early work of Berni Alder also showed that this melting involved a weak first order transition while theory now predicted melting by two successive continuous transitions with no discontinuity in area at the critical pressure. This discrepancy could be hand waved away by arguing that Berni’s system was far too small and his computers far too slow so that the areal discontinuity could be due to finite size effects or to failing to equilibrate the system. Experiments were not able to resolve the order of the transitions, but seemed to agree quantitatively with theory…

35 years ago Berni J. Alder showed the Boltzmann-Enskog kinetic theory failed to adequately account for the viscosity of fluids near solid density as determined by molecular dynamics simulation. This work, along with other notable simulation findings, provided great stimulus to the statistical mechanical studies of transport phenomena, particularly in dealing with collective effects in the time correlation functions of liquids. An extended theoretical challenge that remains partially resolved at best is the shear viscosity of supercooled liquids. How can one give a unified explanation of the so-called fragile and strong characteristic temperature behavior, with implications for the dynamics of glass transition? In this tribute on the occasion of his 90^th birthday symposium, we recount a recent study where simulation, combined with heuristic (transition-state) and first principles (linear response) theories, identifies the molecular mechanisms governing glassy-state relaxation. Such an interplay between simulation and theory is progress from the early days; instead of simulation challenging theory, now simulation and theory complement each other.

The empirically observed linear relationship between shock-wave velocity and particle velocity is compatible with the Eulerian finite-strain equation of state under tension as well as compression, identifying an ideal value of dynamic strength −P_H = K_0S/(K_0S’ + 1) and dynamic cohesive energy E_H − E₀ = 8V₀K_0S/(K_0S’ + 1)² (V, K and K’ are volume, bulk modulus and its pressure derivative; subscripts 0, S and H refer to zero-pressure, isentrope and Hugoniot states). The corresponding finitestrain estimate of the isentropic cohesive energy is E_S − E₀ = 9V₀K_0S(2 + 2n − K_0S’)/(2n³), with strain parameter n = 2 for the Eulerian (spatial) frame of reference.

The author has previously pointed out some similarities between selforganizing neural networks and quantum mechanics. These types of neural networks were originally conceived of as away of emulating the cognitive capabilities of the human brain. Recently extensions of these networks, collectively referred to as deep learning networks, have strengthened the connection between self-organizing neural networks and human cognitive capabilities. In this note we consider whether hardware quantum devices might be useful for emulating neural networks with human-like cognitive capabilities, or alternatively whether implementations of deep learning neural networks using conventional computers might lead to better algorithms for solving the many body Schrodinger equation.

I’m the young woman in the picture shown in Fig. 12.1 that appeared with the invitation to the Symposium to celebrate Berni Alder’s ninetieth birthday. I worked with Berni for over 25 years on the computer programs that provided the data he needed to write the fifteen papers published in scientific journals on Studies in Molecular Dynamics. My name appears at the end of each one thanking me for computer support. It has been interesting to look on the Internet to find my name in the middle of many foreign languages, including Japanese characters and Russian Cyrillic script. It shows how Berni’s work has been of interest to many scientists all over the world from the earliest years. Figure 12.1 was also included with articles written when he received the National Medal of Science from President Obama in 2009…

Explicit treatment of many-body Fermi statistics in path integral Monte Carlo (PIMC) results in exponentially scaling computational cost due to the near cancellation of contributions to observables from even and odd permutations. Through direct analysis of exchange statistics we find that individual exchange probabilities in homogeneous systems are, except for finite size effects, independent of the configuration of other permutations present. For two representative systems, ³He and the homogeneous electron gas, we show that this allows the entire antisymmetrized density matrix to be generated from a simple model depending on only a few parameters obtainable directly from a standard PIMC simulation. The result is a polynomial scaling algorithm and up to a 10 order of magnitude increase in efficiency in measuring fermionic observables for the systems considered.