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Register a new userEfficiency of Rejection-free dynamic Monte Carlo methods for homogeneous spin models, hard disk systems, and hard sphere systems
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We construct asymptotic arguments for the relative efficiency of rejection-free Monte Carlo (MC) methods compared to the standard MC method. We find that the efficiency is proportional to $exp{({const} beta)}$ in the Ising, $sqrt{beta}$ in the classical XY, and $beta$ in the classical Heisenberg spin systems with inverse temperature $beta$, regardless of the dimension. The efficiency in hard particle systems is also obtained, and found to be proportional to $(rhoc -rho)^{-d}$ with the closest packing density $rhoc$, density $rho$, and dimension $d$ of the systems. We construct and implement a rejection-free Monte Carlo method for the hard-disk system. The RFMC has a greater computational efficiency at high densities, and the density dependence of the efficiency is as predicted by our arguments.
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We calculate the efficiency of a rejection-free dynamic Monte Carlo method for $d$-dimensional off-lattice homogeneous particles interacting through a repulsive power-law potential $r^{-p}$. Theoretically we find the algorithmic efficiency in the limit of low temperatures and/or high densities is asymptotically proportional to $rho^{tfrac{p+2}{2}}T^{-tfrac{d}{2}}$ with the particle density $rho$ and the temperature $T$. Dynamic Monte Carlo simulations are performed in 1-, 2- and 3-dimensional systems with different powers $p$, and the results agree with the theoretical predictions.
We show how to generalize the Lattice Switch Monte Carlo method to calculate the phase diagram of a binary system. A global coordinate transformation is combined with a modification of particle diameters, enabling the multi-component system in question to be explored and directly compared to a suitable reference state in a single Monte Carlo simulation. We use the method to evaluate the free energies of binary hard sphere crystals. Calculations at moderate size ratios, alpha=0.58 and alpha=0.73, are in agreement with previous results, and confirm AB2 and AB13 as stable structures. We also find that the AB(CsCl) structure is not entropically stable at the size ratio and volume at which it has been reported experimentally, and therefore that those observations cannot be explained by packing effects alone.
The hard-disk problem, the statics and the dynamics of equal two-dimensional hard spheres in a periodic box, has had a profound influence on statistical and computational physics. Markov-chain Monte Carlo and molecular dynamics were first discussed for this model. Here we reformulate hard-disk Monte Carlo algorithms in terms of another classic problem, namely the sampling from a polytope. Local Markov-chain Monte Carlo, as proposed by Metropolis et al. in 1953, appears as a sequence of random walks in high-dimensional polytopes, while the moves of the more powerful event-chain algorithm correspond to molecular dynamics evolution. We determine the convergence properties of Monte Carlo methods in a special invariant polytope associated with hard-disk configurations, and the implications for convergence of hard-disk sampling. Finally, we discuss parallelization strategies for event-chain Monte Carlo and present results for a multicore implementation.
We investigate Monte Carlo simulation strategies for determining the effective (depletion) potential between a pair of hard spheres immersed in a dense sea of much smaller hard spheres. Two routes to the depletion potential are considered. The first is based on estimates of the insertion probability of one big sphere in the presence of the other; we describe and compare three such methods. The second route exploits collective (cluster) updating to sample the depletion potential as a function of the separation of the big particles; we describe two such methods. For both routes we find that the sampling efficiency at high densities of small particles can be enhanced considerably by exploiting `geometrical shortcuts that focus the computational effort on a subset of small particles. All the methods we describe are readily extendable to particles interacting via arbitrary potentials.
We perform extensive MD simulations of two-dimensional systems of hard disks, focusing on the emph{on}-collision statistical properties. We analyze the distribution functions of velocity, free flight time and free path length for packing fractions ranging from the fluid to the solid phase. The behaviors of the mean free flight time and path length between subsequent collisions are found to drastically change in the coexistence phase. We show that single particle dynamical properties behave analogously in collisional and continuous time representations, exhibiting apparent crossovers between the fluid and the solid phase. We find that, both in collisional and continuous time representation, the mean square displacement, velocity autocorrelation functions, intermediate scattering functions and self part of the van Hove function (propagator), closely reproduce the same behavior exhibited by the corresponding quantities in granular media, colloids and supercooled liquids close to the glass or jamming transition.
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