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Register a new userCollisional Statistics and Dynamics of 2D Hard-Disk Systems: From Fluid to Solid
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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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We examine the question of the criteria of the relaxation to the equilibrium in the hard disk dynamics. In the Event-Chain Monte Carlo, we check the displacement distributions which follows to the exponential law.
We consider a modification of the well studied Hamiltonian Mean-Field model by introducing a hard-core point-like repulsive interaction and propose a numerical integration scheme to integrate numerically its dynamics. Our results show that the outcome of the initial violent relaxation is altered, and also that the phase-diagram is modified with a critical temperature at a higher value than in the non-collisional counterpart.
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.
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.
A system of hard spheres exhibits physics that is controlled only by their density. This comes about because the interaction energy is either infinite or zero, so all allowed configurations have exactly the same energy. The low density phase is liquid, while the high density phase is crystalline, an example of order by disorder as it is driven purely by entropic considerations. Here we study a family of hard spin models, which we call hardcore spin models, where we replace the translational degrees of freedom of hard spheres with the orientational degrees of freedom of lattice spins. Their hardcore interaction serves analogously to divide configurations of the many spin system into allowed and disallowed sectors. We present detailed results on the square lattice in $d=2$ for a set of models with $mathbb{Z}_n$ symmetry, which generalize Potts models, and their $U(1)$ limits, for ferromagnetic and antiferromagnetic senses of the interaction, which we refer to as exclusion and inclusion models. As the exclusion/inclusion angles are varied, we find a Kosterlitz-Thouless phase transition between a disordered phase and an ordered phase with quasi-long-ranged order, which is the form order by disorder takes in these systems. These results follow from a set of height representations, an ergodic cluster algorithm, and transfer matrix calculations.
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