41 research outputs found
Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics
A new method of deriving reduced models of Hamiltonian dynamical systems is
developed using techniques from optimization and statistical estimation. Given
a set of resolved variables that define a model reduction, the
quasi-equilibrium ensembles associated with the resolved variables are employed
as a family of trial probability densities on phase space. The residual that
results from submitting these trial densities to the Liouville equation is
quantified by an ensemble-averaged cost function related to the information
loss rate of the reduction. From an initial nonequilibrium state, the
statistical state of the system at any later time is estimated by minimizing
the time integral of the cost function over paths of trial densities.
Statistical closure of the underresolved dynamics is obtained at the level of
the value function, which equals the optimal cost of reduction with respect to
the resolved variables, and the evolution of the estimated statistical state is
deduced from the Hamilton-Jacobi equation satisfied by the value function. In
the near-equilibrium regime, or under a local quadratic approximation in the
far-from-equilibrium regime, this best-fit closure is governed by a
differential equation for the estimated state vector coupled to a Riccati
differential equation for the Hessian matrix of the value function. Since
memory effects are not explicitly included in the trial densities, a single
adjustable parameter is introduced into the cost function to capture a
time-scale ratio between resolved and unresolved motions. Apart from this
parameter, the closed equations for the resolved variables are completely
determined by the underlying deterministic dynamics
Spatial multi-level interacting particle simulations and information theory-based error quantification
We propose a hierarchy of multi-level kinetic Monte Carlo methods for
sampling high-dimensional, stochastic lattice particle dynamics with complex
interactions. The method is based on the efficient coupling of different
spatial resolution levels, taking advantage of the low sampling cost in a
coarse space and by developing local reconstruction strategies from
coarse-grained dynamics. Microscopic reconstruction corrects possibly
significant errors introduced through coarse-graining, leading to the
controlled-error approximation of the sampled stochastic process. In this
manner, the proposed multi-level algorithm overcomes known shortcomings of
coarse-graining of particle systems with complex interactions such as combined
long and short-range particle interactions and/or complex lattice geometries.
Specifically, we provide error analysis for the approximation of long-time
stationary dynamics in terms of relative entropy and prove that information
loss in the multi-level methods is growing linearly in time, which in turn
implies that an appropriate observable in the stationary regime is the
information loss of the path measures per unit time. We show that this
observable can be either estimated a priori, or it can be tracked
computationally a posteriori in the course of a simulation. The stationary
regime is of critical importance to molecular simulations as it is relevant to
long-time sampling, obtaining phase diagrams and in studying metastability
properties of high-dimensional complex systems. Finally, the multi-level nature
of the method provides flexibility in combining rejection-free and null-event
implementations, generating a hierarchy of algorithms with an adjustable number
of rejections that includes well-known rejection-free and null-event
algorithms.Comment: 34 page
Statistical equilibrium measures in micromagnetics
We derive an equilibrium statistical theory for the macroscopic description
of a ferromagnetic material at positive finite temperatures. Our formulation
describes the most-probable equilibrium macrostates that yield a coherent
deterministic large-scale picture varying at the size of the domain, as well as
it captures the effect of random spin fluctuations caused by the thermal noise.
We discuss connections of the proposed formulation to the Landau-Lifschitz
theory and to the studies of domain formation based on Monte Carlo lattice
simulations.Comment: 5 pages (2-column format
Error analysis of coarse-grained kinetic Monte Carlo method
In this paper we investigate the approximation properties of the
coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice
stochastic dynamics. We provide both analytical and numerical evidence that the
hierarchy of the coarse models is built in a systematic way that allows for
error control in both transient and long-time simulations. We demonstrate that
the numerical accuracy of the CGMC algorithm as an approximation of stochastic
lattice spin flip dynamics is of order two in terms of the coarse-graining
ratio and that the natural small parameter is the coarse-graining ratio over
the range of particle/particle interactions. The error estimate is shown to
hold in the weak convergence sense. We employ the derived analytical results to
guide CGMC algorithms and we demonstrate a CPU speed-up in demanding
computational regimes that involve nucleation, phase transitions and
metastability.Comment: 30 page