Johannes Zierenberg

Exploring different regimes in finite-size scaling of the droplet condensation-evaporation transition.

We present a finite-size scaling analysis of the droplet condensation-evaporation transition of a lattice gas (in two and three dimensions) and a Lennard-Jones gas (in three dimensions) at fixed density. Parallel multicanonical simulations allow sampling of the required system sizes with precise equilibrium estimates. In the limit of large systems, we verify the theoretical leading-order scaling prediction for both the transition temperature and the finite-size rounding. In addition, we present an emerging intermediate scaling regime, consistent in all considered cases and with similar recent observations for polymer aggregation. While the intermediate regime locally may show a different effective scaling, we show that it is a gradual crossover to the large-system scaling behavior by including empirical higher-order corrections. This implies that care has to be taken when considering scaling ranges, possibly leading to completely wrong predictions for the thermodynamic limit. In this study, we consider a crossing of the phase boundary orthogonal to the usual fixed temperature studies. We show that this is an equivalent approach and, under certain conditions, may show smaller finite-size corrections.

Canonical free-energy barrier of particle and polymer cluster formation

A common approach to study nucleation rates is the estimation of free-energy barriers. This usually requires knowledge about the shape of the forming droplet, a task that becomes notoriously difficult in macromolecular setups starting with a proper definition of the cluster boundary. Here we demonstrate a shape-free determination of the free energy for temperature-driven cluster formation in particle as well as polymer systems. Combined with rigorous results on equilibrium droplet formation, this allows for a well-defined finite-size scaling analysis of the effective interfacial free energy at a fixed density. We first verify the theoretical predictions for the formation of a liquid droplet in a supersaturated particle gas by generalized-ensemble Monte Carlo simulations of a Lennard-Jones system. Going one step further, we then generalize this approach to cluster formation in a dilute polymer solution. Our results suggest an analogy with particle condensation, when the macromolecules are interpreted as extended particles.

Dilute Semiflexible Polymers with Attraction: Collapse, Folding and Aggregation

We review the current state on the thermodynamic behavior and structural phases of self- and mutually-attractive dilute semiflexible polymers that undergo temperature-driven transitions. In extreme dilution, polymers may be considered isolated, and this single polymer undergoes a collapse or folding transition depending on the internal structure. This may go as far as to stable knot phases. Adding polymers results in aggregation, where structural motifs again depend on the internal structure. We discuss in detail the effect of semiflexibility on the collapse and aggregation transition and provide perspectives for interesting future investigations.

Simulating flexible polymers in a potential of randomly distributed hard disks.

We perform equilibrium computer simulations of a two-dimensional pinned flexible polymer exposed to a quenched disorder potential consisting of hard disks. We are especially interested in the high-density regime of the disorder, where subtle structures such as cavities and channels play a central role. We apply an off-lattice growth algorithm proposed by Garel and Orland [J. Phys. A 23, L621 (1990)], where a distribution of polymers is constructed in parallel by growing each of them monomer by monomer. In addition we use a multicanonical Monte Carlo method in order to cross-check the results of the growth algorithm. We measure the end-to-end distribution and the tangent-tangent correlations. We also investigate the scaling behavior of the mean square end-to-end distance in dependence on the monomer number. While the influence of the potential in the low-density case is merely marginal, it dominates the configurational properties of the polymer for high densities.

Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects

We study long-range power-law correlated disorder on square and cubic lattices. In particular, we present high-precision results for the percolation thresholds and the fractal dimension of the largest clusters as a function of the correlation strength. The correlations are generated using a discrete version of the Fourier filtering method. We consider two different metrics to set the length scales over which the correlations decay, showing that the percolation thresholds are highly sensitive to such system details. By contrast, we verify that the fractal dimension d_{f} is a universal quantity and unaffected by the choice of metric. We also show that for weak correlations, its value coincides with that for the uncorrelated system. In two dimensions we observe a clear increase of the fractal dimension with increasing correlation strength, approaching d_{f}→2. The onset of this change does not seem to be determined by the extended Harris criterion.

Molecular Dynamics and Monte Carlo simulations in the microcanonical ensemble: Quantitative comparison and reweighting techniques

Molecular Dynamics (MD) and Monte Carlo (MC) simulations are the most popular simulation techniques for many-particle systems. Although they are often applied to similar systems, it is unclear to which extent one has to expect quantitative agreement of the two simulation techniques. In this work, we present a quantitative comparison of MD and MC simulations in the microcanonical ensemble. For three test examples, we study first- and second-order phase transitions with a focus on liquid-gas like transitions. We present MD analysis techniques to compensate for conservation law effects due to linear and angular momentum conservation. Additionally, we apply the weighted histogram analysis method to microcanonical histograms reweighted from MD simulations. By this means, we are able to estimate the density of states from many microcanonical simulations at various total energies. This further allows us to compute estimates of canonical expectation values.

Massively parallel multicanonical simulations

Abstract Generalized-ensemble Monte Carlo simulations such as the multicanonical method and similar techniques are among the most efficient approaches for simulations of systems undergoing discontinuous phase transitions or with rugged free-energy landscapes. As Markov chain methods, they are inherently serial computationally. It was demonstrated recently, however, that a combination of independent simulations that communicate weight updates at variable intervals allows for the efficient utilization of parallel computational resources for multicanonical simulations. Implementing this approach for the many-thread architecture provided by current generations of graphics processing units (GPUs), we show how it can be efficiently employed with of the order of 1 0 4 parallel walkers and beyond, thus constituting a versatile tool for Monte Carlo simulations in the era of massively parallel computing. We provide the fully documented source code for the approach applied to the paradigmatic example of the two-dimensional Ising model as starting point and reference for practitioners in the field. Program summary Program Title: cudamuca Program Files doi: http://dx.doi.org/10.17632/tzhfpdymv9.1 Licensing provisions: Creative Commons Attribution license (CC BY 4.0) Programming language: C, CUDA External routines/libraries: NVIDIA CUDA Toolkit 6.5 or newer Nature of problem: The program determines weights for a multicanonical simulation of the 2D Ising model to result in a flat energy histogram. A final production run with these weights provides an estimate of the density of states of the model. Solution method: The code uses a parallel variant of the multicanonical method employing many parallel walkers that accumulate a common histogram. The resulting histogram is used to determine the weight function for the next iteration. Once the iteration has converged, simulations visit all possible energies with the same probability. Additional comments including restrictions and unusual features: The system size and size of the population of replicas are limited depending on the memory of the GPU device used. Code repository at https://github.com/CQT-Leipzig/cudamuca .

Influence of lattice disorder on the structure of persistent polymer chains

We study the static properties of a semiflexible polymer exposed to a quenched random environment by means of computer simulations. The polymer is modeled as a two-dimensional Heisenberg chain. For the random environment we consider hard disks arranged on a square lattice. We apply an off-lattice growth algorithm as well as the multicanonical Monte Carlo method to investigate the influence of both disorder occupation probability and polymer stiffness on the equilibrium properties of the polymer. We show that the additional length scale induced by the stiffness of the polymer extends the well-known phenomenology considerably. The polymer?s response to the disorder is either contraction or extension depending on the ratio of polymer stiffness and void-space extension. Additionally, the periodic structure of the lattice is reflected in the observables that characterize the polymer.

Universality from disorder in the random-bond Blume-Capel model

Using high-precision Monte Carlo simulations and finite-size scaling we study the effect of quenched disorder in the exchange couplings on the Blume-Capel model on the square lattice. The first-order transition for large crystal-field coupling is softened to become continuous, with a divergent correlation length. An analysis of the scaling of the correlation length as well as the susceptibility and specific heat reveals that it belongs to the universality class of the Ising model with additional logarithmic corrections which is also observed for the Ising model itself if coupled to weak disorder. While the leading scaling behavior of the disordered system is therefore identical between the second-order and first-order segments of the phase diagram of the pure model, the finite-size scaling in the ex-first-order regime is affected by strong transient effects with a crossover length scale L^{*}≈32 for the chosen parameters.

Scaling laws for random walks in long-range correlated disordered media

We study the scaling laws of diffusion in two-dimensional media with long-range correlated disorder through exact enumeration of random walks. The disordered medium is modelled by percolation clusters with correlations decaying with the distance as a power law,