1,604 research outputs found
Event-Driven Monte Carlo: exact dynamics at all time-scales for discrete-variable models
We present an algorithm for the simulation of the exact real-time dynamics of
classical many-body systems with discrete energy levels. In the same spirit of
kinetic Monte Carlo methods, a stochastic solution of the master equation is
found, with no need to define any other phase-space construction. However,
unlike existing methods, the present algorithm does not assume any particular
statistical distribution to perform moves or to advance the time, and thus is a
unique tool for the numerical exploration of fast and ultra-fast dynamical
regimes. By decomposing the problem in a set of two-level subsystems, we find a
natural variable step size, that is well defined from the normalization
condition of the transition probabilities between the levels. We successfully
test the algorithm with known exact solutions for non-equilibrium dynamics and
equilibrium thermodynamical properties of Ising-spin models in one and two
dimensions, and compare to standard implementations of kinetic Monte Carlo
methods. The present algorithm is directly applicable to the study of the real
time dynamics of a large class of classical markovian chains, and particularly
to short-time situations where the exact evolution is relevant
Design of quasi-symplectic propagators for Langevin dynamics
A vector field splitting approach is discussed for the systematic derivation
of numerical propagators for deterministic dynamics. Based on the formalism, a
class of numerical integrators for Langevin dynamics are presented for single
and multiple timestep algorithms
Billion-atom Synchronous Parallel Kinetic Monte Carlo Simulations of Critical 3D Ising Systems
An extension of the synchronous parallel kinetic Monte Carlo (pkMC) algorithm
developed by Martinez {\it et al} [{\it J.\ Comp.\ Phys.} {\bf 227} (2008)
3804] to discrete lattices is presented. The method solves the master equation
synchronously by recourse to null events that keep all processors time clocks
current in a global sense. Boundary conflicts are rigorously solved by adopting
a chessboard decomposition into non-interacting sublattices. We find that the
bias introduced by the spatial correlations attendant to the sublattice
decomposition is within the standard deviation of the serial method, which
confirms the statistical validity of the method. We have assessed the parallel
efficiency of the method and find that our algorithm scales consistently with
problem size and sublattice partition. We apply the method to the calculation
of scale-dependent critical exponents in billion-atom 3D Ising systems, with
very good agreement with state-of-the-art multispin simulations
Langevin Simulations of a Long Range Electron Phonon Model
We present a Quantum Monte Carlo (QMC) study, based on the Langevin equation,
of a Hamiltonian describing electrons coupled to phonon degrees of freedom. The
bosonic part of the action helps control the variation of the field in
imaginary time. As a consequence, the iterative conjugate gradient solution of
the fermionic action, which depends on the boson coordinates, converges more
rapidly than in the case of electron-electron interactions, such as the Hubbard
Hamiltonian. Fourier Acceleration is shown to be a crucial ingredient in
reducing the equilibration and autocorrelation times. After describing and
benchmarking the method, we present results for the phase diagram focusing on
the range of the electron-phonon interaction. We delineate the regions of
charge density wave formation from those in which the fermion density is
inhomogeneous, caused by phase separation. We show that the Langevin approach
is more efficient than the Determinant QMC method for lattice sizes and that it therefore opens a potential path to problems including,
for example, charge order in the 3D Holstein model
Kinematics of Multigrid Monte Carlo
We study the kinematics of multigrid Monte Carlo algorithms by means of
acceptance rates for nonlocal Metropolis update proposals. An approximation
formula for acceptance rates is derived. We present a comparison of different
coarse-to-fine interpolation schemes in free field theory, where the formula is
exact. The predictions of the approximation formula for several interacting
models are well confirmed by Monte Carlo simulations. The following rule is
found: For a critical model with fundamental Hamiltonian H(phi), absence of
critical slowing down can only be expected if the expansion of
in terms of the shift psi contains no relevant (mass) term. We also introduce a
multigrid update procedure for nonabelian lattice gauge theory and study the
acceptance rates for gauge group SU(2) in four dimensions.Comment: 28 pages, 8 ps-figures, DESY 92-09
Random walks near Rokhsar-Kivelson points
There is a class of quantum Hamiltonians known as
Rokhsar-Kivelson(RK)-Hamiltonians for which static ground state properties can
be obtained by evaluating thermal expectation values for classical models. The
ground state of an RK-Hamiltonian is known explicitly, and its dynamical
properties can be obtained by performing a classical Monte Carlo simulation. We
discuss the details of a Diffusion Monte Carlo method that is a good tool for
studying statics and dynamics of perturbed RK-Hamiltonians without time
discretization errors. As a general result we point out that the relation
between the quantum dynamics and classical Monte Carlo simulations for
RK-Hamiltonians follows from the known fact that the imaginary-time evolution
operator that describes optimal importance sampling, in which the exact ground
state is used as guiding function, is Markovian. Thus quantum dynamics can be
studied by a classical Monte Carlo simulation for any Hamiltonian that is free
of the sign problem provided its ground state is known explicitly.Comment: 12 pages, 9 figures, RevTe
Computing quantum phase transitions
This article first gives a concise introduction to quantum phase transitions,
emphasizing similarities with and differences to classical thermal transitions.
After pointing out the computational challenges posed by quantum phase
transitions, a number of successful computational approaches is discussed. The
focus is on classical and quantum Monte Carlo methods, with the former being
based on the quantum-to classical mapping while the latter directly attack the
quantum problem. These methods are illustrated by several examples of quantum
phase transitions in clean and disordered systems.Comment: 99 pages, 15 figures, submitted to Reviews in Computational Chemistr
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