Malvin H. Kalos

A First-Passage Kinetic Monte Carlo algorithm for complex diffusion-reaction systems

We develop an asynchronous event-driven First-Passage Kinetic Monte Carlo (FPKMC) algorithm for continuous time and space systems involving multiple diffusing and reacting species of spherical particles in two and three dimensions. The FPKMC algorithm presented here is based on the method introduced in Oppelstrup et al. [10] and is implemented in a robust and flexible framework. Unlike standard KMC algorithms such as the n-fold algorithm, FPKMC is most efficient at low densities where it replaces the many small hops needed for reactants to find each other with large first-passage hops sampled from exact time-dependent Greens functions, without sacrificing accuracy. We describe in detail the key components of the algorithm, including the event-loop and the sampling of first-passage probability distributions, and demonstrate the accuracy of the new method. We apply the FPKMC algorithm to the challenging problem of simulation of long-term irradiation of metals, relevant to the performance and aging of nuclear materials in current and future nuclear power plants. The problem of radiation damage spans many decades of time-scales, from picosecond spikes caused by primary cascades, to years of slow damage annealing and microstructure evolution. Our implementation of the FPKMC algorithm has been able to simulate the irradiation of a metal sample for durations that are orders of magnitude longer than any previous simulations using the standard Object KMC or more recent asynchronous algorithms.

First-passage kinetic Monte Carlo method.

We present an efficient method for Monte Carlo simulations of diffusion-reaction processes. Introduced by us in a previous paper [Phys. Rev. Lett. 97, 230602 (2006)], our algorithm skips the traditional small diffusion hops and propagates the diffusing particles over long distances through a sequence of superhops, one particle at a time. By partitioning the simulation space into nonoverlapping protecting domains each containing only one or two particles, the algorithm factorizes the N -body problem of collisions among multiple Brownian particles into a set of much simpler single-body and two-body problems. Efficient propagation of particles inside their protective domains is enabled through the use of time-dependent Greens functions (propagators) obtained as solutions for the first-passage statistics of random walks. The resulting Monte Carlo algorithm is event-driven and asynchronous; each Brownian particle propagates inside its own protective domain and on its own time clock. The algorithm reproduces the statistics of the underlying Monte Carlo model exactly. Extensive numerical examples demonstrate that for an important class of diffusion-reaction models the algorithm is efficient at low particle densities, where other existing algorithms slow down severely.

Synchronous parallel kinetic Monte Carlo for continuum diffusion-reaction systems

A novel parallel kinetic Monte Carlo (kMC) algorithm formulated on the basis of perfect time synchronicity is presented. The algorithm is intended as a generalization of the standard n-fold kMC method, and is trivially implemented in parallel architectures. In its present form, the algorithm is not rigorous in the sense that boundary conflicts are ignored. We demonstrate, however, that, in their absence, or if they were correctly accounted for, our algorithm solves the same master equation as the serial method. We test the validity and parallel performance of the method by solving several pure diffusion problems (i.e. with no particle interactions) with known analytical solution. We also study diffusion-reaction systems with known asymptotic behavior and find that, for large systems with interaction radii smaller than the typical diffusion length, boundary conflicts are negligible and do not affect the global kinetic evolution, which is seen to agree with the expected analytical behavior. Our method is a controlled approximation in the sense that the error incurred by ignoring boundary conflicts can be quantified intrinsically, during the course of a simulation, and decreased arbitrarily (controlled) by modifying a few problem-dependent simulation parameters.

Quantum Many-Body Problems

We review methods used and results obtained in Monte Carlo calculations on quantum fluids and crystals. Available techniques are discussed for the computation of the energy and other expectation values by variational methods in which the absolute square of a trial function ψT is sampled by the Metropolis method. Recently developed methods for fermion systems are included. We give a more detailed exposition of the Green’s Function Monte Carlo method which permits exact numerical estimates of boson ground-state properties. Our survey of results comprises applications to 3He and 4He, hard-sphere fluids and crystals, spin-aligned hydrogen, the one-component plasma for bosons and fermions, and simple models of neutron and nuclear matter. The reliability of the product form of ψT in several applications is assessed. A selected set of related topics is also taken up: low temperature excitations, results obtained by the Wigner ℏ expansion, and evaluations of virial coefficients and pair correlations at finite temperatures.

Exact monte carlo method for continuum fermion systems

We offer a new proposal for the Monte Carlo treatment of many-fermion systems in continuous space. It is based upon diffusion Monte Carlo with significant modifications: correlated pairs of random walkers that carry opposite signs, different functions guide walkers of different signs, the Gaussians used for members of a pair are correlated, and walkers can cancel so as to conserve their expected future contributions. We report results for free-fermion systems and a fermion fluid with 14 3He atoms, where it proves stable and correct. Its computational complexity grows with particle number, but slowly enough to make interesting physics within the reach of contemporary computers.

Few-and Many-Fermion Problems

The success of Monte Carlo methods such as the Green’s function Monte Carlo method (GFMC) [4.1–8] in calculating the ground-state properties of many-body systems has led to their application to Fermi systems. For boson systems, ground-state properties may be calculated with only statistical errors. These Bose calculations have been done for large numbers of particles (N > 100) and in three spatial dimensions. Although some exact fermion calculations have been done [4.9–14], these calculations have generally been in one spatial dimension, where the fermion system can be mapped onto an equivalent Bose system, or with only a few particles (N ~ 3), where the difficulties associated with fermion systems can be controlled.

Sign problem of the fermionic shadow wave function

We present a whole series of methods to alleviate the sign problem of the fermionic shadow wave function in the context of variational Monte Carlo. The effectiveness of our techniques is demonstrated on liquid ^{3}He. We found that although the variance is reduced, the gain in efficiency is restricted by the increased computational cost. Yet, this development not only extends the scope of the fermionic shadow wave function, but also facilitates highly accurate quantum Monte Carlo simulations previously thought not feasible.

Monte Carlo methods in the physical sciences

I will review the role that Monte Carlo methods play in the physical sciences. They are very widely used for a number of reasons: they permit the rapid and faithful transformation of a natural or model stochastic process into a computer code. They are powerful numerical methods for treating the many-dimensional problems that derive from important physical systems. Finally, many of the methods naturally permit the use of modern parallel computers in efficient ways. In the presentation, I will emphasize four aspects of the computations: whether the computation derives from a natural or model stochastic process; whether the system under study is highly idealized or realistic; whether the Monte Carlo methodology is straightforward or mathematically sophisticated; and finally, the scientific role of the computation.

Adaptive importance sampling Monte Carlo simulation of rare transition events

We develop a general theoretical framework for the recently proposed importance sampling method for enhancing the efficiency of rare-event simulations [W. Cai, M. H. Kalos, M. de Koning, and V. V. Bulatov, Phys. Rev. E 66, 046703 (2002)], and discuss practical aspects of its application. We define the success/fail ensemble of all possible successful and failed transition paths of any duration and demonstrate that in this formulation the rare-event problem can be interpreted as a hit-or-miss Monte Carlo quadrature calculation of a path integral. The fact that the integrand contributes significantly only for a very tiny fraction of all possible paths then naturally leads to a standard importance sampling approach to Monte Carlo (MC) quadrature and the existence of an optimal importance function. In addition to showing that the approach is general and expected to be applicable beyond the realm of Markovian path simulations, for which the method was originally proposed, the formulation reveals a conceptual analogy with the variational MC (VMC) method. The search for the optimal importance function in the former is analogous to finding the ground-state wave function in the latter. In two model problems we discuss practical aspects of finding a suitable approximation for the optimal importance function. For this purpose we follow the strategy that is typically adopted in VMC calculations: the selection of a trial functional form for the optimal importance function, followed by the optimization of its adjustable parameters. The latter is accomplished by means of an adaptive optimization procedure based on a combination of steepest-descent and genetic algorithms.

Fermion Monte Carlo for continuum systems

We have been studying methods for the Monte Carlo treatment of many-fermion systems in continuous space. We use generalizations of diffusion Monte Carlo that involve ensembles of correlated pairs of random walkers that carry opposite signs. We have been able to exhibit stable long-term behavior of the random walks: the new method suppresses the usual decay of the signal-to-noise ratio of the integrals that appear in a quotient that estimates the energy. One way of checking this result is to calculate a bosonic system with the same Hamiltonian and to estimate separately the boson–fermion energy difference. Specifically, we performed a “fixed-node” calculation of 14 3He atoms, and then a “transient estimation” removing the fixed-node constraint. The rate of decay in imaginary time of the energy denominator gives a fairly accurate measure of the boson–fermion energy difference, in good agreement with the exact energy difference. This suggests a simple method of estimating fermion energies, viz., to use directly the bosonic energy and the energy difference measured in a transient calculation.