Multicanonical Monte Carlo Simulations Research Articles

Large deviation and anomalous fluctuations scaling in degree assortativity on configuration networks

By constructing a multicanonical Monte Carlo simulation, we obtain the full probability distribution ρ N (r) of the degree assortativity coefficient r on configuration networks of size N by using the multiple histogram reweighting method. We suggest that ρ N (r) obeys a large deviation principle, , where the rate function I is convex and possesses its unique minimum at , and ξ is an exponent that scales ρ N ’s with N. We show that ξ = 1 for Poisson random graphs, and ξ ⩾ 1 for scale-free networks in which ξ is a decreasing function of the degree distribution exponent γ. Our results reveal that the fluctuations of r exhibit an anomalous scaling with N in highly heterogeneous networks.

Microstructural Capture of Living Ultrathin Polymer Brush Evolution via Kinetic Simulation Studies.

Performing dynamic off-lattice multicanonical Monte Carlo simulations, we study the statics, dynamics, and scission-recombination kinetics of a self-assembled in situ-polymerized polydisperse living polymer brush (LPB), designed by surface-initiated living polymerization. The living brush is initially grown from a two-dimensional substrate by end-monomer polymerization-depolymerization reactions through seeding of initiator arrays on the grafting plane which come in contact with a solution of nonbonded monomers under good solvent conditions. The polydispersity is shown to significantly deviate from the Flory-Schulz type for low temperatures because of pronounced diffusion limitation effects on the rate of the equilibration reaction. The self-avoiding chains take up fairly compact structures of typical size Rg(N) ∼ Nν in rigorously two-dimensional (d = 2) melt, with ν being the inverse fractal dimension (ν = 1/d). The Kratky description of the intramolecular structure factor F(q), in keeping with the concept of generalized Porod scattering from compact particles with fractal contour, discloses a robust nonmonotonic fashion with qdF(q) ∼ (qRg)-3/4 in the intermediate-q regime. It is found that the kinetics of LPB growth, given by the variation of the mean chain length, follows a power law ⟨N(t)⟩ ∝ t1/3 with elapsed time after the onset of polymerization, whereby the instantaneous molecular weight distribution (MWD) of the chains c(N) retains its functional form. The variation of ⟨N(t)⟩ during quenches of the LPB to different temperatures T can be described by a single master curve in units of dimensionless time t/τ∞, where τ∞ is the typical (final temperature T∞-dependent) relaxation time which is found to scale as τ∞ ∝ ⟨N(t = ∞)⟩5 with the ultimate average length of the chains. The equilibrium monomer density profile ϕ(z) of the LPB varies as ϕ(z) ∝ ϕ-α with the concentration of segments ϕ in the system and the probability distribution c(N) of chain lengths N in the brush layer scales as c(N) ∝ N-τ. The computed exponents α ≈ 0.64 and τ ≈ 1.70 are in good agreement with those predicted within the context of the Diffusion-Limited Aggregation theory, α = 2/3 and τ = 7/4.

Microcanonical thermostatistics of aggregation transition in a system with anisotropically interacting molecules

Microcanonical thermostatistics analysis has been introduced as an important method in the study of phase transitions observed in intrinsically small systems, such as folding transitions in proteins and surface adsorption transitions of polymeric chains. Here we consider a lattice model and apply microcanonical analysis to investigate the aggregation transition of a system with anisotropically interacting molecules. By performing multicanonical Monte Carlo simulations we are able to obtain free-energy profiles from where we extract physical quantities related to the aggregation transition such as its transition temperature, latent heat, and free-energy barriers. Our results confirms that the aggregation transition is a first-order type of transition and that it is related to the nucleation of molecules into elongated aggregates. Also, our analysis revealed an unexpected non-monotonic behavior for the free-energy barrier as a function of the anisotropic ratio ξ between strong and weak interactions of the molecules, indicating that the nucleation kinetics might be also influenced by ξ.

Finite-size scaling of Monte Carlo simulations for the fcc Ising antiferromagnet: Effects of the low-temperature phase degeneracy

The Ising antiferromagnet on a face-centered cubic (fcc) lattice with nearest-neighbor interaction only is well known to exhibit a macroscopic (exponential in the system size $L$) ground-state degeneracy. With increasing temperature, this degeneracy is expected to be lifted and the model undergoes a first-order phase transition. For a model with an exponential degeneracy in the whole low-temperature phase, it was recently found that the finite-size scaling behavior is governed by leading correction terms $\sim L^{-2}$ instead of $\sim L^{-3}$ as usual. To test the conjecture that such a transmuted behavior may effectively persist also for the fcc antiferromagnet up to some crossover system size, we have performed parallel multicanonical Monte Carlo simulations for lattices of linear size $L \le 18$ with periodic boundary conditions and determined various inverse pseudo phase transition temperatures, as well as the extremal values of the specific heat and the energetic Binder parameter. We indeed find that, for the simulated lattice sizes, the conjectured transmuted finite-size scaling ansatz fits the data better than the standard ansatz. On this basis, we extrapolate for the transition temperature an estimate of $T_0 = 1.735047(46)$.

Phase Behavior of a Flexible Star Polymer Chain in Good Solvent near an Attractive Surface

AbstractThe phase behavior of a flexible star polymer chain in good solvent near an attractive surface is investigated by multicanonical Monte Carlo simulations where the specific heat, the parallel and the perpendicular components of the gyration tensor, and the fraction of adsorbed monomers are computed. Temperature–surface attraction strength pseudo‐phase diagrams are constructed for three chain lengths (N = 31, 41, 61) with three numbers of arms (f = 3, 4, 6). The star polymer chain adopts a desorbed expanded conformation for high temperatures and low surface attraction strength. With decreasing temperature and for sufficient surface attraction strength, the star polymer chain undergoes a transition from 3D to a 2D structure. At low temperatures, the chain is mostly adsorbed for low surface attraction strength and adopts a two‐layer arrangement at high values of the latter. In this context, it is also found that the phase behavior of flexible star polymer chains in good solvent near an attractive surface is affected by the number of arms f and that a star polymer chain with a high number of arms is necessary for a precise classification of all intermediate pseudo phases.

A subset multicanonical Monte Carlo method for simulating rare failure events

Estimating failure probabilities of engineering systems is an important problem in many engineering fields. In this work we consider such problems where the failure probability is extremely small (e.g. ≤10−10). In this case, standard Monte Carlo methods are not feasible due to the extraordinarily large number of samples required. To address these problems, we propose an algorithm that combines the main ideas of two very powerful failure probability estimation approaches: the subset simulation (SS) and the multicanonical Monte Carlo (MMC) methods. Unlike the standard MMC which samples in the entire domain of the input parameter in each iteration, the proposed subset MMC algorithm adaptively performs MMC simulations in a subset of the state space, which improves the sampling efficiency. With numerical examples we demonstrate that the proposed method is significantly more efficient than both of the SS and the MMC methods. Moreover, like the standard MMC, the proposed algorithm can reconstruct the complete distribution function of the parameter of interest and thus can provide more information than just the failure probabilities of the systems.

PHAST: Protein-like heteropolymer analysis by statistical thermodynamics

PHAST is a software package written in standard Fortran, with MPI and CUDA extensions, able to efficiently perform parallel multicanonical Monte Carlo simulations of single or multiple heteropolymeric chains, as coarse-grained models for proteins. The outcome data can be straightforwardly analyzed within its microcanonical Statistical Thermodynamics module, which allows for computing the entropy, caloric curve, specific heat and free energies. As a case study, we investigate the aggregation of heteropolymers bioinspired on Aβ25−33 fragments and their cross-seeding with IAPP20−29 isoforms. Excellent parallel scaling is observed, even under numerically difficult first-order like phase transitions, which are properly described by the built-in fully reconfigurable force fields. Still, the package is free and open source, this shall motivate users to readily adapt it to specific purposes.Program summaryProgram Title: PHASTProgram Files doi:http://dx.doi.org/10.17632/ggds2grzw9.1Licensing provisions: GNU GPL version 3Programming language: FORTRAN 90, MPICH 3.0.4, CUDA 8.0Nature of the problem: Nowadays powerful multicore processors (CPUs) and Graphical Processing Units (GPUs) became much popular and cost-effective, so enabling thermostatistical studies of complex molecular systems to be performed even in personal computers. PHAST provides not only an easily-reconfigurable parallel program for Monte Carlo simulations of linear heteropolymers, as coarse-grained models of proteins, but also permits the automatized microcanonical thermodynamic analysis of those systems.Solution method: PHAST has three main modules for complementary tasks as: to map any .pdb file-sequence to its inner lexicon by using a configurable hydrophobic scale, while the main simulational module performs parallel Monte Carlo simulations in the multicanonical (MUCA) ensemble and, the analysis module which extracts microcanonical observables from MUCA weights.External routines/libraries: cuRAND (CUDA), Grace-5.1.22 or higher.

The impact of bonded interactions on the ground-state geometries of a small flexible polymer

Bonded interactions in coarse-grained models of elastic polymers are commonly represented by the finitely extensible nonlinear elastic (FENE) potential. In this study, we perform parallel multicanonical Monte Carlo simulations to examine the impact of an additional Lennard-Jones term in the bonded potential on the geometry of ground-state structures of a short polymer. Employing microcanonical inflection point analysis and conformational analysis, we construct a hyper-phase diagram and identify ground-state structures with two distinct geometries.

Multicanonical Monte Carlo simulations of a de novo designed protein with end-to-end β-sheet

One of the smallest proteins with end-to-end β-sheet is the designed 36-residue protein DS119. We recently suggested that the rate-limiting step in the folding of the βαβ protein is the formation of the central helix that then provides a scaffold for the parallel β-sheet formed by the two chain ends. In the present report we investigate whether and how this folding mechanism depends on the energy function, and compare the efficiency of molecular dynamics and Monte Carlo implementations of multicanonical sampling. While we find the native structure with similar frequency as in our previous simulations, we observe that the folding mechanism differs for both force fields.

A new variant of multicanonical Monte Carlo algorithm with specifying the temperature range and its application to the hydration free energy change of fluorinated methane derivatives

The multicanonical Monte Carlo simulation is often carried out making use of a weight factor which is decided with specifying the energy range. Here we present a new variant of multicanonical Monte Carlo algorithm, ‘selected-multicanonical Monte Carlo’, in which the range of temperature is specified instead of the energy range and the configurations are selected which fit the desired temperature range. The new method has the advantage of adequately deciding the weight factor to create multicanonical ensemble. The new method is applied to free energy perturbation calculation to obtain the hydration free energy changes of methane derivatives.

Leading Fisher Partition Function Zeros as Indicators of Structural Transitions in Macromolecules

We show that leading zeros, i.e., isolated complex zeros of the canonical partition function near the positive real axis, are robust indicators of conformational transitions in finite macromolecular systems. For a flexible polymer with 250 monomers, we calcu- late the complex Fisher partition function zeros from a precise estimate of the density of states that was obtained via advanced multicanonical Monte Carlo simulations. Transition temperatures estimated from the leading zeros correspond very well to regions of enhanced thermal activity as signaled by canonical thermodynamic quantities such as the specific heat and fluctuations of the radius of gyration.

Polymer-attractive spherical cage system

We analyze the structural behavior of a single polymer chain inside an attractive sphere. Our model is composed of a coarse-grained polymer governed by Lennard-Jones interactions of the monomers and an attractive sphere potential which follows by integrating the monomer-monomer interaction over the (inner) surface of the sphere. By means of extensive multicanonical Monte Carlo simulations it is shown that the system exhibits a rich phase diagram in the adsorption strength-temperature (e − T) plane ranging from highly ordered and compact to extended and random coil structures and from desorbed to partially or even completely adsorbed conformations. These findings are identified with different energetic and structural observables. The resulting phase diagram in the e − T plane is compared with that for a polymer adsorbing to a plane, attractive substrate obtained previously by Moddel, Bachmann, and one of the authors.

Residual entropy of ordinary ice calculated from multicanonical Monte Carlo simulations

In this article, we calculated the residual entropy of ice I (ordinary ice) by multicanonical Monte Carlo simulations of two simple models with nearest neighbour interactions on 3D hexagonal lattices. Our estimate is found to be within 0.13% of an analytical approximation by Nagle, which is an improvement of Pauling's result from 1935. We pose it as a challenge to experimentalists to improve on the accuracy of a 1936 measurement by Giauque and Stout by about one order of magnitude, which would allow one to identify corrections to Pauling's value unambiguously. It is straightforward to transfer our methods to other crystal systems.

Ground-State Properties of a Polymer Chain in an Attractive Sphere

We analyze the structural formation of a polymer chain inside of an attractive sphere depending on the attraction strength. Our model is composed of a coarse-grained polymer and an attractive sphere potential. Within this frame, multicanonical Monte Carlo simulations are employed to identify the global minimum energies for a polymer chain interacting with the attractive inner wall of the sphere. Different compact structures are found with varying attraction strengths, among which are spherical, three/two, or monolayer. The conformational properties of these structures are presented.

Structural behavior of a polymer chain inside an attractive sphere

We analyze the structural behavior of a single polymer chain inside an attractive sphere. Our model is composed of a coarse-grained polymer and an attractive-sphere potential. By means of extensive multicanonical Monte Carlo simulations, it is shown that the system exhibits a rich phase diagram ranging from highly ordered compact to extended random coil structures and from desorbed to partially or even completely adsorbed conformations. These findings are identified with different structural observables.

Optical Filter-Based Mitigation of Group Delay Ripple- and PMD-Related Penalties for High-Capacity Metro Networks

In high-capacity metro networks, fiber Bragg gratings (FBGs) offer a potentially cost-effective solution for compensation of chromatic dispersion (CD). However, FBGs suffer from stochastic variations of their group delay, the so-called group delay ripple (GDR). We propose a novel statistical model to describe the effects of stochastic variations of GDR. The statistical properties of our model are verified by comparison to measurement data and Monte Carlo simulations as well as Multicanonical Monte Carlo (MMC) simulations. Results indicate that without further measures to counteract the GDR distortions, very large penalties (>; 10 dB) for the optical signal-to-noise ratio (OSNR) occur frequently at a bitrate of 112 Gbit/s. Thus, we investigated the performance of short and cost-effective optical finite and infinite impulse response equalizer structures to mitigate the GDR distortions and to enhance the signal quality. With the use of optical equalizers (which can be realized as planar lightwave circuits) we were able to reduce the mean OSNR penalty due to the GDR to less than 0.1 dB. We also demonstrate that the same filter structures can efficiently be used to mitigate all-order PMD distortions as well.

Temperature Effects on the Rovibrational Spectra of Pyrene-Based PAHs

Absorption infrared spectra have been computed for a variety of polycyclic aromatic hydrocarbon molecules of the pyrene family, taking into account anharmonicity and temperature effects, rovibrational quantization, and couplings. The energy levels are described by a second-order perturbative expansion of the rovibrational Hamiltonian in the vibrational and rotational quantum numbers, as relevant for a symmetric-top molecule, with ingredients obtained from quantum chemistry calculations. Multicanonical Monte Carlo simulations are carried out to compute bidimensional IR intensity histograms as a function of total energy and vibrational frequency, which then provide the absorption spectrum at arbitrary temperatures via a Laplace transformation. The main spectral features analyzed for neutral, anionic, and cationic pyrene indicate a strong dependence on temperature, in agreement with existing laboratory experiments, and a significant contribution of rotational degrees of freedom to the overall broadenings. The spectral shifts and broadenings reveal some sensitivity of anharmonicities to the charge and protonation states and, in the case of protonated pyrene and pyrenyl cation, on possible isomers and between aromatic and aliphatic C-H bands. Implications of the present work to the general issue of interstellar emission features are discussed.

Simulation of coexisting ferromagnetic order and disorder of geometrically frustrated Co 2Cl(OH) 3

The ground state and phase transition of Co 2Cl(OH) 3 were investigated by Monte Carlo simulation. This compound is a magnet, with a pyrochlore structure distorted along one axis. The magnetic structure at low temperatures consists of coexisting ferromagnetism and random spin, according to experiments. However, the formation mechanism of the coexistence and the interaction between the spins were unclear. We assumed an anisotropic Ising model and examined the ground state by multicanonical Monte Carlo simulation. In a nearest neighbor model, the ground states were highly degenerated. Almost all of the states were spin glass states, but a few of the states were ferromagnetic. The latter magnetic states were ferromagnetic at triangular layers and two in-one out random state at Kagome layers. The latter states should be stabilized if weak ferromagnetic interactions exist between second nearest neighbor spins and correspond to the states reported by the experiments. This expectation was confirmed by simulation.

Performance Evaluation of WDM Systems Through Multicanonical Monte Carlo Simulations

In wavelength-division-multiplexed (WDM) systems, performance is affected by the bit patterns on the various channels and, in order to account for this pattern dependence, one should perform a large number of simulations, each by using different patterns of suitable length in all channels and then average all results. However, since the number of patterns to be considered for averaging could be considerable, this approach would be very demanding from a computational point of view, even using multicanonical Monte Carlo (MMC) simulations. In single-channel systems, pattern dependence can be accounted for through the pattern perturbation method, a computationally efficient way for performing a random walk over an extended state space, which includes, in addition to noise components, also the transmitted bit pattern. In this way, not only the most relevant noise configurations but also the most relevant bit patterns (those having a major impact on the error events) are sampled more frequently by the MMC simulation, leading to enhanced efficiency in estimating the error probability. Here, the pattern perturbation method is extended to the WDM case, and simulations results are reported to show the effectiveness of the proposed approach.

The ferromagnetic–spin glass transition in PdMn alloys: symmetry breaking offerromagnetism and spin glass studied by a multicanonical method

The magnetism of Pd1 − xMnx is investigated theoretically. A localized spin model for Mn spins that interact withshort-range antiferromagnetic interactions and long-range ferromagnetic interactions viaitinerant d electrons is set up, with no adjustable parameters. A multicanonicalMonte Carlo simulation, combined with a procedure of symmetry breaking, isemployed to discriminate between the ferromagnetic and spin glass orders. Thetransition temperature and the low-temperature phase are determined from thetemperature variation of the specific heat and the probability distributions of theferromagnetic order parameter and the spin glass order parameter at differentconcentrations. The calculation results reveal that only the ferromagnetic phase exists atx < 0.02, that only the spinglass phase exists at x > 0.04, and that the two phases coexist at intermediate concentrations. This result agreessemi-quantitatively with experimental results.