13,817 research outputs found
Tagged particle diffusion in one-dimensional systems with Hamiltonian dynamics - II
We study various temporal correlation functions of a tagged particle in
one-dimensional systems of interacting point particles evolving with
Hamiltonian dynamics. Initial conditions of the particles are chosen from the
canonical thermal distribution. The correlation functions are studied in finite
systems, and their forms examined at short and long times. Various
one-dimensional systems are studied. Results of numerical simulations for the
Fermi-Pasta-Ulam chain are qualitatively similar to results for the harmonic
chain, and agree unexpectedly well with a simple description in terms of
linearized equations for damped fluctuating sound waves. Simulation results for
the alternate mass hard particle gas reveal that - in contradiction to our
earlier results [1] with smaller system sizes - the diffusion constant slowly
converges to a constant value, in a manner consistent with mode coupling
theories. Our simulations also show that the behaviour of the Lennard-Jones gas
depends on its density. At low densities, it behaves like a hard-particle gas,
and at high densities like an anharmonic chain. In all the systems studied, the
tagged particle was found to show normal diffusion asymptotically, with
convergence times depending on the system under study. Finite size effects show
up at time scales larger than sound traversal times, their nature being
system-specific.Comment: 15 pages, 12 figure
A volume-based hydrodynamic approach to sound wave propagation in a monatomic gas
We investigate sound wave propagation in a monatomic gas using a volume-based
hydrodynamic model. In Physica A vol 387(24) (2008) pp6079-6094, a microscopic
volume-based kinetic approach was proposed by analyzing molecular spatial
distributions; this led to a set of hydrodynamic equations incorporating a
mass-density diffusion component. Here we find that these new mass-density
diffusive flux and volume terms mean that our hydrodynamic model, uniquely,
reproduces sound wave phase speed and damping measurements with excellent
agreement over the full range of Knudsen number. In the high Knudsen number
(high frequency) regime, our volume-based model predictions agree with the
plane standing waves observed in the experiments, which existing kinetic and
continuum models have great difficulty in capturing. In that regime, our
results indicate that the "sound waves" presumed in the experiments may be
better thought of as "mass-density waves", rather than the pressure waves of
the continuum regime.Comment: Revised to aid clarification (no changes to presented model); typos
corrected, figures added, paper title change
Dominant Reaction Pathways in High Dimensional Systems
This paper is devoted to the development of a theoretical and computational
framework to efficiently sample the statistically significant thermally
activated reaction pathways, in multi-dimensional systems obeying Langevin
dynamics. We show how to obtain the set of most probable reaction pathways and
compute the corrections due to quadratic thermal fluctuations around such
trajectories. We discuss how to obtain predictions for the evolution of
arbitrary observables and how to generate conformations which are
representative of the transition state ensemble. We present an illustrative
implementation of our method by studying the diffusion of a point particle in a
2-dimensional funneled external potential.Comment: 18 pages, 7 figures. Improvement in the text and in the figures.
Version submitted for publicatio
Steady advection-diffusion around finite absorbers in two-dimensional potential flows
We perform an exhaustive study of the simplest, nontrivial problem in
advection-diffusion -- a finite absorber of arbitrary cross section in a steady
two-dimensional potential flow of concentrated fluid. This classical problem
has been studied extensively in the theory of solidification from a flowing
melt, and it also arises in Advection-Diffusion-Limited Aggregation. In both
cases, the fundamental object is the flux to a circular disk, obtained by
conformal mapping from more complicated shapes. We construct the first accurate
numerical solution using an efficient new method, which involves mapping to the
interior of the disk and using a spectral method in polar coordinates. Our
method also combines exact asymptotics and an adaptive mesh to handle boundary
layers. Starting from a well-known integral equation in streamline coordinates,
we also derive new, high-order asymptotic expansions for high and low P\'eclet
numbers (\Pe). Remarkably, the `high' \Pe expansion remains accurate even
for such low \Pe as . The two expansions overlap well near \Pe =
0.1, allowing the construction of an analytical connection formula that is
uniformly accurate for all \Pe and angles on the disk with a maximum relative
error of 1.75%. We also obtain an analytical formula for the Nusselt number
() as a function of the P\'eclet number with a maximum relative error of
0.53% for all possible geometries. Because our finite-plate problem can be
conformally mapped to other geometries, the general problem of two-dimensional
advection-diffusion past an arbitrary finite absorber in a potential flow can
be considered effectively solved.Comment: 29 pages, 12 figs (mostly in color
Effective transient behaviour of inclusions in diffusion problems
This paper is concerned with the effective transport properties of
heterogeneous media in which there is a high contrast between the phase
diffusivities. In this case the transient response of the slow phase induces a
memory effect at the macroscopic scale, which needs to be included in a
macroscopic continuum description. This paper focuses on the slow phase, which
we take as a dispersion of inclusions of arbitrary shape. We revisit the linear
diffusion problem in such inclusions in order to identify the structure of the
effective (average) inclusion response to a chemical load applied on the
inclusion boundary. We identify a chemical creep function (similar to the creep
function of viscoelasticity), from which we construct estimates with a reduced
number of relaxation modes. The proposed estimates admit an equivalent
representation based on a finite number of internal variables. These estimates
allow us to predict the average inclusion response under arbitrary time-varying
boundary conditions at very low computational cost. A heuristic generalisation
to concentration-dependent diffusion coefficient is also presented. The
proposed estimates for the effective transient response of an inclusion can
serve as a building block for the formulation of multi-inclusion homogenisation
schemes.Comment: 24 pages, 9 figures. Submitted to ZAMM (under review
Thermal conduction in classical low-dimensional lattices
Deriving macroscopic phenomenological laws of irreversible thermodynamics
from simple microscopic models is one of the tasks of non-equilibrium
statistical mechanics. We consider stationary energy transport in crystals with
reference to simple mathematical models consisting of coupled oscillators on a
lattice. The role of lattice dimensionality on the breakdown of the Fourier's
law is discussed and some universal quantitative aspects are emphasized: the
divergence of the finite-size thermal conductivity is characterized by
universal laws in one and two dimensions. Equilibrium and non-equilibrium
molecular dynamics methods are presented along with a critical survey of
previous numerical results. Analytical results for the non-equilibrium dynamics
can be obtained in the harmonic chain where the role of disorder and
localization can be also understood. The traditional kinetic approach, based on
the Boltzmann-Peierls equation is also briefly sketched with reference to
one-dimensional chains. Simple toy models can be defined in which the
conductivity is finite. Anomalous transport in integrable nonlinear systems is
briefly discussed. Finally, possible future research themes are outlined.Comment: 90 pages, revised versio
Semi-classical generalized Langevin equation for equilibrium and nonequilibrium molecular dynamics simulation
Molecular dynamics (MD) simulation based on Langevin equation has been widely
used in the study of structural, thermal properties of matters in difference
phases. Normally, the atomic dynamics are described by classical equations of
motion and the effect of the environment is taken into account through the
fluctuating and frictional forces. Generally, the nuclear quantum effects and
their coupling to other degrees of freedom are difficult to include in an
efficient way. This could be a serious limitation on its application to the
study of dynamical properties of materials made from light elements, in the
presence of external driving electrical or thermal fields. One example of such
system is single molecular dynamics on metal surface, an important system that
has received intense study in surface science. In this review, we summarize
recent effort in extending the Langevin MD to include nuclear quantum effect
and their coupling to flowing electrical current. We discuss its applications
in the study of adsorbate dynamics on metal surface, current-induced dynamics
in molecular junctions, and quantum thermal transport between different
reservoirs.Comment: 23 pages, 16 figur
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