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Register a new userNotes on configurational thermostat schemes
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We have linked together two contributions in the development of configurational thermostats, the BT thermostat and the SDC scheme. We have shown that recently proposed configurational thermostats are generalized and enriched in both understanding and content by the SDC scheme. We have presented the stochastic counterpart to the configurational thermostat.
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We implement the statistically sound G-JF thermostat for Langevin Dynamics simulations into the ESPREesSo molecular package for large-scale simulations of soft matter systems. The implemented integration method is tested against the integrator currently used by the molecular package in simulations of a fluid bilayer membrane. While the latter exhibits deviations in the sampling statistics that increase with the integration time step dt, the former reproduces near-correct configurational statistics for all dt within the stability range of the simulations. We conclude that, with very modest revisions to existing codes, one can significantly improve the performance of statistical sampling using Langevin thermostats.
Some properties of a Local discontinuous Galerkin (LDG) algorithm are demonstrated for the problem of evaluting a second derivative $g = f_{xx}$ for a given $f$. (This is a somewhat unusual problem, but it is useful for understanding the initial transient response of an algorithm for diffusion equations.) LDG uses an auxiliary variable to break this up into two first order equations and then applies techniques by analogy to DG algorithms for advection algorithms. This introduces an asymmetry into the solution that depends on the choice of upwind directions for these two first order equations. When using piecewise linear basis functions, this LDG solution $g_h$ is shown not to converge in an $L_2$ norm because the slopes in each cell diverge. However, when LDG is used in a time-dependent diffusion problem, this error in the second derivative term is transient and rapidly decays away, so that the overall error is bounded. I.e., the LDG approximation $f_h(x,t)$ for a diffusion equation $partial f / partial t = f_{xx}$ converges to the proper solution (as has been shown before), even though the initial rate of change $partial f_h / partial t$ does not converge. We also show results from the Recovery discontinuous Galerkin (RDG) approach, which gives symmetric solutions that can have higher rates of convergence for a stencil that couples the same number of cells.
We expand on the previously published Gr{o}nbech-Jensen Farago (GJF) thermostat, which is a thermodynamically sound variation on the St{o}rmer-Verlet algorithm for simulating discrete-time Langevin equations. The GJF method has been demonstrated to give robust and accurate configurational sampling of the phase space, and its applications to, e.g., Molecular Dynamics is well established. A new definition of the discrete-time velocity variable is proposed based on analytical calculations of the kinetic response of a harmonic oscillator subjected to friction and noise. The new companion velocity to the GJF method is demonstrated to yield correct and time-step-independent kinetic responses for, e.g., kinetic energy, its fluctuations, and Green-Kubo diffusion based on velocity autocorrelations. This observation allows for a new and convenient Leap-Frog algorithm, which efficiently and precisely represents statistical measures of both kinetic and configurational properties at any time step within the stability limit for the harmonic oscillator. We outline the simplicity of the algorithm and demonstrate its attractive time-step-independent features for nonlinear and complex systems through applications to a one-dimensional nonlinear oscillator and three-dimensional Molecular Dynamics.
A research and development (R&D) project related to the extension of the Geant4 toolkit has been recently launched to address fundamental methods in radiation transport simulation. The project focuses on simulation at different scales in the same experimental environment; this problem requires new methods across the current boundaries of condensed-random-walk and discrete transport schemes. The new developments have been motivated by experimental requirements in various domains, including nanodosimetry, astronomy and detector developments for high energy physics applications.
We develop numerical schemes for solving the isothermal compressible and incompressible equations of fluctuating hydrodynamics on a grid with staggered momenta. We develop a second-order accurate spatial discretization of the diffusive, advective and stochastic fluxes that satisfies a discrete fluctuation-dissipation balance, and construct temporal discretizations that are at least second-order accurate in time deterministically and in a weak sense. Specifically, the methods reproduce the correct equilibrium covariances of the fluctuating fields to third (compressible) and second (incompressible) order in the time step, as we verify numerically. We apply our techniques to model recent experimental measurements of giant fluctuations in diffusively mixing fluids in a micro-gravity environment [A. Vailati et. al., Nature Communications 2:290, 2011]. Numerical results for the static spectrum of non-equilibrium concentration fluctuations are in excellent agreement between the compressible and incompressible simulations, and in good agreement with experimental results for all measured wavenumbers.
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