Strict Detailed Balance Research Articles

Spatial updating grand canonical Monte Carlo algorithms for fluid simulation: Generalization to continuous potentials and parallel implementation

Spatial updating grand canonical Monte Carlo algorithms are generalizations of random and sequential updating algorithms for lattice systems to continuum fluid models. The elementary steps, insertions or removals, are constructed by generating points in space either at random (random updating) or in a prescribed order (sequential updating). These algorithms have previously been developed only for systems of impenetrable spheres for which no particle overlap occurs. In this work, spatial updating grand canonical algorithms are generalized to continuous, soft-core potentials to account for overlapping configurations. Results on two- and three-dimensional Lennard-Jones fluids indicate that spatial updating grand canonical algorithms, both random and sequential, converge faster than standard grand canonical algorithms. Spatial algorithms based on sequential updating not only exhibit the fastest convergence but also are ideal for parallel implementation due to the absence of strict detailed balance and the nature of the updating that minimizes interprocessor communication. Parallel simulation results for three-dimensional Lennard-Jones fluids show a substantial reduction of simulation time for systems of moderate and large size. The efficiency improvement by parallel processing through domain decomposition is always in addition to the efficiency improvement by sequential updating.

Parallel Markov chain Monte Carlo simulations

With strict detailed balance, parallel Monte Carlo simulation through domain decomposition cannot be validated with conventional Markov chain theory, which describes an intrinsically serial stochastic process. In this work, the parallel version of Markov chain theory and its role in accelerating Monte Carlo simulations via cluster computing is explored. It is shown that sequential updating is the key to improving efficiency in parallel simulations through domain decomposition. A parallel scheme is proposed to reduce interprocessor communication or synchronization, which slows down parallel simulation with increasing number of processors. Parallel simulation results for the two-dimensional lattice gas model show substantial reduction of simulation time for systems of moderate and large size.

Sequential Metropolis Algorithms for Fluid Simulations

In this work, the implementation of our recently proposed sequential Metropolis algorithm in the grand canonical ensemble, a case particularly relevant for continuum fluids, is considered. By performing Monte Carlo simulations for the two-dimensional lattice gas, it is shown that our algorithm converges faster than all known grand canonical algorithms that satisfy strict detailed balance. The main advantages of the new algorithm are its simplicity, generality, and the possibility of parallel implementation.

Sequential updating algorithms for grand canonical Monte Carlo simulations

Strict detailed balance is essentially unnecessary for Markov chain Monte Carlo simulations to converge to the correct equilibrium distribution. Recently, we proposed a Monte Carlo algorithm based on sequential updating moves with partial randomness to guarantee correct sampling. The proposed algorithm only satisfies the weaker balance condition and converges faster than the Metropolis algorithm with strict detailed balance. In this work, we illustrate the efficiency of the algorithm for the two-dimensional lattice gas model in the grand canonical ensemble. Parallel implementation of the sequential algorithm on lattice systems indicates that parallel Monte Carlo simulations, if treated correctly, are not only as precise as serial implementation, but can also save significant computing time.

Acceleration of Markov chain Monte Carlo simulations through sequential updating

Strict detailed balance is not necessary for Markov chain Monte Carlo simulations to converge to the correct equilibrium distribution. In this work, we propose a new algorithm which only satisfies the weaker balance condition, and it is shown analytically to have better mobility over the phase space than the Metropolis algorithm satisfying strict detailed balance. The new algorithm employs sequential updating and yields better sampling statistics than the Metropolis algorithm with random updating. We illustrate the efficiency of the new algorithm on the two-dimensional Ising model. The algorithm is shown to identify the correct equilibrium distribution and to converge faster than the Metropolis algorithm with strict detailed balance. The main advantages of the new algorithm are its simplicity and the feasibility of parallel implementation through domain decomposition.

Strict detailed balance is unnecessary in Monte Carlo simulation

Detailed balance is an overly strict condition to ensure a valid Monte Carlo simulation. We show that, under fairly general assumptions, a Monte Carlo simulation need satisfy only the weaker balance condition. Not only does our proof show that sequential updating schemes are correct, but also it establishes the correctness of a whole class of new methods that simply leave the Boltzmann distribution invariant.

Improved Feynman’s path integral method with a large time step: Formalism and applications

We describe an efficient path integral scheme for calculating the propagator of an arbitrary quantum system, as well as that of a stochastic system in special cases where the Fokker–Planck equation obeys strict detailed balance. The basic idea is to split the respective Hamiltonian into two exactly solvable parts and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The resulting single step propagator allows rather large time steps in a path integral and leads to convergence with fewer time slices. Because it involves no system-specific reference system, the algorithm is amenable to all known numerical schemes available for evaluating quantum path integrals. In this way one obtains a highly accurate method, which is simultaneously fast, stable, and computationally simple. Numerical applications to the real time quantum dynamics in a double well and to the stochastic dynamics of a bistable system coupled to a harmonic mode show our method to be superior over the approach developed by the Makri group in their quasiadiabatic propagator representation, to say nothing about the propagation scheme based on the standard Trotter splitting.

The system-size expansion of the potential of a master equation without detailed balance

The potential of a one-dimensional master equation without strict detailed balance is expanded on the system-size. The general term in the expansion is given explicitly on the basis of the lower order terms. A computer simulation verifies the approach.

The Generalized Detailed Balance for Master Equation without the Strict Detailed Balance

The generalized detailed balance is revealed for one-dimensional many-step master equation without the strict detailed balance. To the leading order of the system-size the one and only one stationary solution is identified.

Master equation without strict detailed balance

The potentials of master equations without strict detailed balance are obtained in leading order of the system-size Ω by means of the Kramers-Moyal expansion. The approach of the new Fokker-Planck equation by Hanggi et al. is suggested to be extended to many-step Markov processes.