Classical Monte Carlo Simulations Research Articles

Emergence of pseudo-time during optimal Monte Carlo sampling and temporal aspects of symmetry breaking and restoration.

We argue that one can associate a pseudo-time with sequences of configurations generated in the course of classical Monte Carlo simulations for a single-minimum bound state if the sampling is optimal. Hereby, the sampling rates can be, under special circumstances, calibrated against the relaxation rate and frequency of motion of an actual physical system. The latter possibility is linked to the optimal sampling regime being a universal crossover separating two distinct suboptimal sampling regimes analogous to the physical phenomena of diffusion and effusion, respectively. Bound states break symmetry; one may thus regard the pseudo-time as a quantity emerging together with the bound state. Conversely, when transport among distinct bound states takes place-thus restoring symmetry-a pseudo-time can no longer be defined. One can still quantify activation barriers if the latter barriers are smooth, but simulation becomes impractically slow and pertains to overdamped transport only. Specially designed Monte Carlo moves that bypass activation barriers-so as to accelerate sampling of the thermodynamics-amount to effusive transport and lead to severe under-sampling of transition-state configurations that separate distinct bound states while destroying the said universality. Implications of the present findings for simulations of glassy liquids are discussed.

Fragmentation dynamics of electron-impact double ionization of helium

We study the double ionization dynamics of a helium atom impacted by electrons with full-dimensional classical trajectory Monte Carlo simulation. The excess energy is chosen to cover a wide range of values from 5 eV to 1 keV for comparative study. At the lowest excess energy, i.e., close to the double-ionization threshold, it is found that the projectile momentum is totally transferred to the recoil-ion while the residual energy is randomly partitioned among the three outgoing electrons, which are then most probably emitted with an equilateral triangle configuration. Our results agree well with experiments as compared with early quantum-mechanical calculation as well as classical simulation based on a two-dimensional Bohr’s model. Furthermore, by mapping the final momentum vectors event by event into a Dalitz plot, we unambiguously demonstrate that the ergodicity has been reached and thus confirm a long-term scenario conceived by Wannier. The time scale for such few-body thermalization, from the initial nonequilibrium state to the final microcanonical distribution, is only about 100 attoseconds. Finally, we predict that, with the increase of the excess energy, the dominant emission configuration undergoes a transition from equilateral triangle to T-shape and finally to a co-linear mode. The associated signatures of such configuration transition in the electron-ion joint momentum spectrum and triple-electron angular distribution are also demonstrated.

Time-separated stochastic mechanics for the simulation of viscoelastic structures with local random material fluctuations

Simulating structures with history-dependent material models and uncertain material parameters is often computationally expensive. In contrast, the time-separated stochastic mechanics (TSM) has proven to provide an efficient yet accurate method for the computation of stochastic visco-elastic materials. In this work, the TSM is extended for viscoelastic structures with local random material fluctuations. The time-separated stochastic mechanics is based on a separation of the stochastic viscous evolution equation in time-dependent deterministic and time-independent stochastic terms enabling an efficient implementation. Here, the method is expanded by the Karhunen–Loéve expansion for the representation of the random fields of the material parameters. The method requires a low number of deterministic FEM simulations to approximate stress and reaction force, thus remarkably reducing the computational effort compared to classical Monte Carlo simulations. Multiple numerical simulations are presented to study accuracy, robustness and efficiency of the developed method.

Magnetic phase transition in a machine trained spin model: A study of hexagonal CrN monolayer

By employing the combination of non-collinear density functional theory, artificial neural network, and classical Monte Carlo simulation we study the ferromagnetic phase transition in CrN monolayers. The artificial neural network is successfully trained from different spin alignments to reproduce interaction energy at the DFT level in two dimensions. The training is performed solely based on spin spatial angles and second-order interactions energy without any speculation for the shape of spin Hamiltonian. The neural network model predicts the angle-dependent spin energy and mimics the magnetic anisotropy energy. The neural network is then validated for a larger supercell size against density functional theory results. Finally, the trained neural network spin model is used to study the finite temperature magnetization and phase transition in the two-dimensional CrN monolayer. The Curie temperature of the CrN monolayer is estimated equal to 600 K from the temperature-dependent magnetization and specific heat.

Seismic fragility analysis using stochastic polynomial chaos expansions

Within the performance-based earthquake engineering (PBEE) framework, the fragility model plays a pivotal role. Such a model represents the probability that the engineering demand parameter (EDP) exceeds a certain safety threshold given a set of selected intensity measures (IMs) that characterize the earthquake load. The-state-of-the art methods for fragility computation rely on full non-linear time–history analyses. Within this perimeter, there are two main approaches: the first relies on the selection and scaling of recorded ground motions; the second, based on random vibration theory, characterizes the seismic input with a parametric stochastic ground motion model (SGMM). The latter case has the great advantage that the problem of seismic risk analysis is framed as a forward uncertainty quantification problem. However, running classical full-scale Monte Carlo simulations is intractable because of the prohibitive computational cost of typical finite element models. Therefore, it is of great interest to define fragility models that link an EDP of interest with the SGMM parameters — which are regarded as IMs in this context. The computation of such fragility models is a challenge on its own and, despite a few recent studies, there is still an important research gap in this domain. This comes with no surprise as classical surrogate modeling techniques cannot be applied due to the stochastic nature of SGMM. This study tackles this computational challenge by using stochastic polynomial chaos expansions to represent the statistical dependence of EDP on IMs. More precisely, this surrogate model estimates the full conditional probability distribution of EDP conditioned on IMs. We compare the proposed approach with some state-of-the-art methods in two case studies. The numerical results show that the new method prevails over its competitors in estimating both the conditional distribution and the fragility functions.

DMRG and Monte Carlo studies of CrI3 magnetic phases and the phase transition

The monolayer of $\textrm{CrI}_3$ has been reported to exhibit the ferromagnetic order, with a Curie temperature of $45 K$ and off-plane easy axis, which has attracted much attention in the community of condensed matter physics. Using the Density Matrix Renormalization Group method, we investigate the role of correlation effects and using classical Monte Carlo simulations analyze the nature of phase transitions in the XXZ Hamiltonian on a honeycomb lattice, which can effectively model $\textrm{CrI}_3$. We show, that the magnetic ordering of the Hamiltonian's ground state can be well approximated by classical models in a wide range of an anisotropy parameter space. Using classical Monte Carlo simulations we estimate the Curie temperature of $49.7 K$ which is in good agreement with experimental result.

Satellite Dynamics Toolbox Library: a tool to model multi-body space systems for robust control synthesis and analysis

The level of maturity reached by robust control theory techniques nowadays contributes to a considerable minimization of the development time of an end-to-end control design of a spacecraft system. The advantage offered by this framework is twofold: all system uncertainties can be included from the very beginning of the design process; the validation and Verification (V&V) process is improved by fast detection of worst-case configurations that could escape to a classical sample-based Monte Carlo simulation campaign. Before proceeding to the control synthesis and analysis, a proper uncertain plant model has to be available in order to push these techniques to their limits of performance. In this spirit, the Satellite Dynamics Toolbox Library (SDTlib) offers many features to model a spacecraft system in a multi-body fashion on SIMULINK. Parametric models can be easily built in a Linear Fractional Transformation (LFT) form by including uncertainties and varying parameters with minimal number of repetitions. Uncertain Linear Time Invariant (LTI) and uncertain Linear Parameter-Varying (LPV) controllers can then be synthesized and analyzed in a straightforward way. The authors present in this article a tutorial, that can be downloaded at https://nextcloud.isae. fr/index.php/s/XDfRfHntejHTmmp, to show how to deal with an end-to-end robust design of a spacecraft mission and to provide to researchers a benchmark to test their own algorithms.

Quantify uncertainty by estimating the probability density function of the output of interest using MLMC based Bayes method

<p style='text-indent:20px;'>In uncertainty quantification, the quantity of interest is usually the statistics of the space and/or time integration of system solution. In order to reduce the computational cost, a Bayes estimator based on multilevel Monte Carlo (MLMC) is introduced in this paper. The cumulative distribution function of the output of interest, that is, the expectation of the indicator function, is estimated by MLMC method instead of the classic Monte Carlo simulation. Then, combined with the corresponding probability density function, the quantity of interest is obtained by using some specific quadrature rules. In addition, the smoothing of indicator function and Latin hypercube sampling are used to accelerate the reduction of variance. An elliptic stochastic partial differential equation is used to provide a research context for this model. Numerical experiments are performed to verify the advantage of computational reduction and accuracy improvement of our MLMC-Bayes method.</p>

An Application of Relative Entropy in Structural Safety Analysis of Elastoplastic Beam under Fire Conditions

The main aim of this work is to propose a new algorithm of reliability assessment for steel civil engineering structures subjected to fire temperatures. This new algorithm is based upon the relative probabilistic entropy concept elaborated by Bhattacharyya, and this probabilistic distance is sought in-between extreme and admissible deformations of some structural beam subjected to higher temperatures. Similar to the First Order Reliability Method, this approach uses the first two probabilistic characteristics of the structural response, when structural output may be modelled with the use of Gaussian distribution. The probabilistic structural response is found here using hybrid computational technique–the Finite Element Method system ABAQUS with its fully coupled thermo-elastic analysis with 3D solid elements and probabilistic modules implemented in the computer algebra system MAPLE. The probabilistic response is determined via a triple stochastic analysis, which is based on the classical Monte-Carlo simulation, iterative generalized stochastic perturbation technique, and also using semi-analytical probabilistic calculus. Final determination of the relative entropy in the Serviceability Limit State of the given structure and its comparison with the results obtained using the FORM analysis enables to calibrate this new technique to numerical values proposed in the engineering designing codes. Hence, a more accurate probabilistic method may use some experimental-based admissible values included in the existing design of legal provisions.

Improved pseudo-force approach for Monte Carlo Simulation of non-linear fractional oscillators under stochastic excitation

This paper presents a step-by-step procedure for the numerical integration of the fractional differential equation governing the response of single-degree-of-freedom (SDOF) non-linear systems endowed with fractional derivatives subjected to stochastic excitation. The procedure, labeled improved pseudo-force method (IPFM), is developed by extending a step-by-step integration scheme proposed by the second author for the numerical solution of classical differential equations. The IPFM relies on the following main steps: i) to use the Grünwald–Letnikov (GL) approximation of the fractional derivative; ii) to treat terms depending on the unknown values of the response, which result from the GL approximation as well as from the non-linear restoring forces, as pseudo-forces; iii) to handle non-linearities by performing iterations at each time step. The IPFM provides accurate solutions by using time steps of larger size compared to classical step-by-step integration schemes. In this paper, the IPFM is applied within the framework of classical Monte Carlo Simulation (MCS) to evaluate the time domain dynamic response of non-linear fractional systems subjected to the generic sample of a stochastic excitation.

Theoretical study of the magnetic properties of the CoCu2O3 compound

In this paper we present a theoretical study of the magnetic properties of the ${\mathrm{CoCu}}_{2}{\mathrm{O}}_{3}$ compound. The magnetic effective exchange interactions and zeroth-field splitting were computed using ab initio methods, then the magnetic order and transition temperature were determined using classical Monte Carlo simulations. We showed that, unlike other members of the $A{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ family, the presence of an additional magnetic atom, associated with a large folding of the puckered layers in the ($\stackrel{P\vec}{a}, \stackrel{P\vec}{b}$) directions, induces a magnetic pattern based on coupled three-leg ladders, quite different from the two-leg structural ladders. The propagation vector has been found to be $\stackrel{P\vec}{q}=(0,\frac{1}{2},\frac{1}{2})$. It is associated with a doubly degenerated ground state suggesting a doubling of the unit cell. The large ${\mathrm{Co}}^{2+}$-ion anisotropy was shown to be of crucial importance in the high transition temperature observed in this compound.

Eleven NanoHUB Simulation Tools Using RASPA Software To Demonstrate Classical Atomistic Simulations of Fluids and Nanoporous Materials.

Eleven interactive simulation tools were created on nanoHUB to help users learn how to perform classical atomistic simulations. These tools enable users to perform classical Monte Carlo and molecular dynamics simulations using RASPA software. These tools use comparatively small numbers of production cycles to keep the runtimes short, so that users will not be discouraged by long wait times to see results. Here, we show that these tools produce results of sufficient accuracy and reproducibility for learning purposes. The 11 tools developed were as follows: (1) calculation of the self-diffusion constant of gas molecules in metal-organic frameworks (MOFs), (2) gas adsorption in MOFs using the grand canonical ensemble, (3) Henry's coefficient calculator for gas molecules in MOFs and a zeolite, (4) adsorption of a gas mixture in a MOF, (5) self-diffusion of a gas mixture in a MOF, (6) void fraction calculation for several MOFs and zeolites, (7) surface area calculation for several MOFs and zeolites, (8) calculation of radial distribution function and self-diffusion constant for several pure gases, (9) energy distribution of adsorption sites using a probe molecule in MOFs, (10) molecular dynamics simulation of pure fluids in the NPT ensemble, and (11) gas adsorption in MOFs using the Gibbs ensemble.

Effect of layer charge density and cation concentration on sorption behaviors of heavy metal ions in the interlayer and nanopore of montmorillonite: A molecular dynamics simulation

Soil and ground water pollution by heavy metal is an identified global environment concern. Montmorillonite is widely adopted as an adsorbent used in retardation of heavy metals due to its excellent adsorption capacity, conveniency, and stability. However, the adsorption mechanism of heavy metal ions in the interlayer and nanopore of montmorillonite needs to be further studied. The classical grand canonical monte carlo (GCMC) and molecular dynamics (MD) simulations were performed to quantify the structural and dynamic properties of heavy metal ions in the interlayer and nanopore of montmorillonite (MMT). Two layer charge densities of MMT, two cation concentrations, and three typical heavy metal ions (Pb2+, Ba2+, and Cs+) were considered. With the increase of layer charge density, outer-sphere adsorption (OSA) was transformed into inner-sphere adsorption (ISA) for three heavy metal ions. The adsorption quantity increased with the increase of layer charge density and cation concentration. The calculated diffusion coefficients of three heavy metal caions in montmorillonite was bulk solution > nanopore > interlayer. The results and adsorption mechanism are helpful for understanding the natural processes of heavy metal ions in environment and developing more efficient remediation materials.

Quantum effects on the elastic properties of cubic boron nitride by path-integral Monte Carlo simulation

Using path-integral Monte Carlo simulation, we study the quantum effects on the elastic properties of cubic boron nitride (c-BN) crystal in a wide temperature range from 40 to 2000 K and a pressure range varying from 1 atm to 160 GPa. The Tersoff–Albe potential is employed to describe the interatomic interactions, and the elastic constants are determined by a direct method derived from the stress–strain curves. The calculations show that elastic constants C11, C12, and C44 are temperature and pressure dependent, and their values are in satisfactory agreement with available experimental results at room temperature. Furthermore, we estimate the Poisson ratio, anisotropic factor, longitudinal-wave elastic constant, Cauchy pressure, and Pugh’s ratio and their temperature and pressure dependence. In general, the quantum effect contributions to the elastic properties of the c-BN are extracted quantitatively from the differences between the path-integral and classical Monte Carlo simulation results. The quantum contribution is negligible at high temperatures but can be significant at low temperatures. It is responsible for variations of about 5∼7% of the elastic constants C11 and C44, and C12. The impacts of quantum contribution on the elastic properties of the c-BN crystal are similar to those found in the diamond.

Quantum simulations of dark sector showers

We consider dark sector scenarios where dark matter is accompanied by a dark photon and multiple-flavor dark fermions charged under the dark gauge group. We study quantum interference effects in dark sector jets, where multiple dark photons are emitted from high-energy dark fermions. We perform fully quantum simulations of dark sector showers and compare the results against those of the classical Monte-Carlo simulations. We find important differences in probability distributions of dark photon countings between quantum and classical computations. When the number of dark-fermion flavors is large, we find significant enhancements in large numbers of dark photon emissions. Such enhancements can provide distinguishing signals for our scenarios at particle colliders.

The Effect of Surface Composition on the Selective Capture of Atmospheric CO2 by ZIF Nanoparticles: The Case of ZIF-8.

We performed theoretical studies of CO2 capture in atmospheric conditions by the zeolitic imidazolate framework-8 (ZIF-8) via classical Monte Carlo (MC) simulations with Metropolis sampling and classical molecular dynamics (MD) simulations in the NVT and NPT ensembles and different thermodynamic conditions. The ZIF-8 framework was described by varying unit cell dimensions in the presence of pure gases of CO2, N2, O2, Ar, and H2O steam as well as binary mixtures of CO2:N2 and CO2:H2O in s 1:1 concentration. Different chemical compositions of the framework surface was considered to provide an accurate treatment of charge and charge distribution in the nanoparticle. Hence, surface groups were represented as unsaturated zinc atom (Zn+2), 2-methylimidazole (mImH), and deprotonated 2-methylimidazole (mIm-). Force field reparameterization of the surface sites was required to reproduce the interactions of the gas molecules with the ZIF-8 surface consistent with quantum mechanics (QM) calculations and Born-Oppenheimer molecular dynamics (BOMD). It was observed that ZIF-8 selectively captures CO2 due to the negligible concentrations of N2, O2, Ar, and H2O. These molecules spontaneously migrate to the inner pores of the framework. At the surface, there is a competitive interaction between H2O, CO2, and N2, for the positively charged ZIF-8 nanoparticle with a large binding energy advantage for water molecules (on average -62, -15, and -8 kcal/mol respectively). For the neutral ZIF-8 nanoparticle, the water molecules dominate the interactions due to the occurrence of hydrogen bond with the imidazolate groups at the surface. Simulations of binary mixtures of CO2/water steam and CO2/N2 were performed to investigate binding competition between these molecules for the framework positively charged and neutral surfaces. It was found that water molecules drastically block the interaction between CO2 molecules and the framework surface, decreasing CO2 capture in the central pore, and CO2 molecules fully block the interaction between N2 molecules and the framework. These findings show that CO2 capture by ZIF-8 is possible in atmospheric environments only upon dehydration of the atmospheric gas. It further shows that ZIF-8 capture of CO2 from the atmospheric environment is dependent on thermodynamic conditions and can be increased by decreasing temperature and/or increasing pressure.

Realistic magnetic thermodynamics by local quantization of a semiclassical Heisenberg model

Classical Monte Carlo simulation of the Heisenberg model poorly describes many thermodynamic phenomena due to its neglect of the quantum nature of spins. Alternatively, we discuss how to semiclassically approach the quantum problem and demonstrate a simple method for introducing a locally approximate form of spin quantization. While the procedure underestimates magnetic short-range order, our results suggest a simple correction for recovering realistic spin–spin correlations above the critical temperature. Moreover, ensemble fluctuations are found to provide reasonably accurate thermodynamics, largely reproducing quantum mechanically calculated heat capacities and experimental magnetometry for ferromagnetic Fe and antiferromagnetic RbMnF3. Extensions of the method are proposed to address remaining inaccuracies.

Charge glass in an extended dimer Hubbard model

The charge degrees of freedom in several different organic charge transfer salts display slow or glassy dynamics. In order to gain insight into this behaviour, we obtain the low energy theory for an extended dimer Hubbard model, taking into account the occupations of sites on neighbouring dimers. We take a classical limit of the resulting effective model of coupled spins and dimers and study it using classical Monte Carlo simulations. We find that frustration induced by intra- and inter-dimer interactions leads to glassiness in the charge degress of freedom in the absence of ordering of the spin degrees of freedom. Our results may have relevance to experimental observations of relaxor ferroelectric behaviour in the dynamics of organic charge transfer salts.

Exchange interactions and spin dynamics in the layered honeycomb ferromagnet CrI3

We derive the microscopic spin Hamiltonian for rhombohedral CrI$_3$ using extensive first-principles density functional theory (DFT) calculations which incorporate spin-orbit coupling and Hubbard U. Our calculations indicate a dominant nearest-neighbor ferromagnetic Heisenberg exchange with weaker further-neighbor Heisenberg terms. In addition, we find a Dzyaloshinskii-Moriya interaction which primarily drives a topological gap in the spin-wave spectrum at the Dirac point, and uncover a non-negligible antiferromagnetic Kitaev coupling between the S=3/2 Cr moments. The out-of-plane magnetic moment is stabilized by weak symmetric bond-dependent terms and a local single-ion anisotropy. Using linear spin wave theory, we find that our exchange parameters are in reasonably good agreement with inelastic neutron scattering (INS) experiments. Employing classical Monte Carlo simulations, we study the magnetic phase transition temperature $T_c$ and its evolution with an applied in-plane magnetic field. We further demonstrate how future high-resolution INS experiments on the magnon dispersion of single crystals in an in-plane magnetic field may be used to quantitatively extract the strength of the antiferromagnetic Kitaev exchange coupling.

Long-range order and quantum criticality in a dissipative spin chain

Environmental interaction is a fundamental consideration in any controlled quantum system. While interaction with a dissipative bath can lead to decoherence, it can also provide desirable emergent effects including induced spin-spin correlations. In this paper we show that under quite general conditions, a dissipative bosonic bath can induce a long-range ordered phase, without the inclusion of any additional direct spin-spin couplings. Through a quantum-to-classical mapping and classical Monte Carlo simulation, we investigate the $T=0$ quantum phase transition of an Ising chain embedded in a bosonic bath with Ohmic dissipation. We show that the quantum critical point is continuous, Lorentz invariant with a dynamical critical exponent $z=1.07(9)$, has correlation length exponent $\nu=0.80(5)$, and anomalous exponent $\eta=1.02(6)$, thus the universality class distinct from the previously studied limiting cases. The implications of our results on experiments in ultracold atomic mixtures and qubit chains in dissipative environments are discussed.