arXiv subjects

Peter Grassberger

Publications and source records attributed to Peter Grassberger.

At least 91 records · Page 5Linked to original sources

arXiv2005

The coil-globule transition of confined polymers

We study long polymer chains in a poor solvent, confined to the space between two parallel hard walls. The walls are energetically neutral and pose only a geometric constraint which changes the properties of the coil-globule (or "$θ$-") transition. We find that the $θ$ temperature increases monotonically with the width $D$ between the walls, in contrast to recent claims in the literature. Put in a wider context, the problem can be seen as a dimensional cross over in a tricritical point of a $ϕ^4$ model. We roughly verify the main scaling properties expected for such a phenomenon, but we find also somewhat unexpected very long transients before the asymptotic scaling regions are reached. In particular, instead of the expected scaling $R\sim N^{4/7}$ exactly at the ($D$-dependent) theta point we found that $R$ increases less fast than $N^{1/2}$, even for extremely long chains.

cond-mat.soft↗

arXiv2004

Simulations of grafted polymers in a good solvent

We present improved simulations of three-dimensional self avoiding walks with one end attached to an impenetrable surface on the simple cubic lattice. This surface can either be a-thermal, having thus only an entropic effect, or attractive. In the latter case we concentrate on the adsorption transition, We find clear evidence for the cross-over exponent to be smaller than 1/2, in contrast to all previous simulations but in agreement with a re-summed field theoretic $ε$-expansion. Since we use the pruned-enriched Rosenbluth method (PERM) which allows very precise estimates of the partition sum itself, we also obtain improved estimates for all entropic critical exponents.

cond-mat.soft↗

arXiv2004

Simulations of lattice animals and trees

The scaling behaviour of randomly branched polymers in a good solvent is studied in two to nine dimensions, using as microscopic models lattice animals and lattice trees on simple hypercubic lattices. As a stochastic sampling method we use a biased sequential sampling algorithm with re-sampling, similar to the pruned-enriched Rosenbluth method (PERM) used extensively for linear polymers. Essentially we start simulating percolation clusters (either site or bond), re-weigh them according to the animal (tree) ensemble, and prune or branch the further growth according to a heuristic fitness function. In contrast to previous applications of PERM, this fitness function is {\it not} the weight with which the actual configuration would contribute to the partition sum, but is closely related to it. We obtain high statistics of animals with up to several thousand sites in all dimension 2 <= d <= 9. In addition to the partition sum (number of different animals) we estimate gyration radii and numbers of perimeter sites. In all dimensions we verify the Parisi-Sourlas prediction, and we verify all exactly known critical exponents in dimensions 2, 3, 4, and >= 8. In addition, we present the hitherto most precise estimates for growth constants in d >= 3. For clusters with one site attached to an attractive surface, we verify the superuniversality of the cross-over exponent at the adsorption transition predicted by Janssen and Lyssy. Finally, we discuss the collapse of animals and trees, arguing that our present version of the algorithm is also efficient for some of the models studied in this context, but showing that it is {\it not} very efficient for the `classical' model for collapsing animals.

cond-mat.stat-mech↗

arXiv2004

Least Dependent Component Analysis Based on Mutual Information

We propose to use precise estimators of mutual information (MI) to find least dependent components in a linearly mixed signal. On the one hand this seems to lead to better blind source separation than with any other presently available algorithm. On the other hand it has the advantage, compared to other implementations of `independent' component analysis (ICA) some of which are based on crude approximations for MI, that the numerical values of the MI can be used for: (i) estimating residual dependencies between the output components; (ii) estimating the reliability of the output, by comparing the pairwise MIs with those of re-mixed components; (iii) clustering the output according to the residual interdependencies. For the MI estimator we use a recently proposed k-nearest neighbor based algorithm. For time sequences we combine this with delay embedding, in order to take into account non-trivial time correlations. After several tests with artificial data, we apply the resulting MILCA (Mutual Information based Least dependent Component Analysis) algorithm to a real-world dataset, the ECG of a pregnant woman. The software implementation of the MILCA algorithm is freely available at http://www.fz-juelich.de/nic/cs/software

physics.comp-ph↗

arXiv2004

Extracting Phases from Aperiodic Signals

We demonstrate by means of a simple example that the arbitrariness of defining a phase from an aperiodic signal is not just an academic problem, but is more serious and fundamental. Decomposition of the signal into components with positive phase velocities is proposed as an old solution to this new problem.

cond-mat.dis-nn↗

arXiv2004

Sequential Monte Carlo Methods for Protein Folding

We describe a class of growth algorithms for finding low energy states of heteropolymers. These polymers form toy models for proteins, and the hope is that similar methods will ultimately be useful for finding native states of real proteins from heuristic or a priori determined force fields. These algorithms share with standard Markov chain Monte Carlo methods that they generate Gibbs-Boltzmann distributions, but they are not based on the strategy that this distribution is obtained as stationary state of a suitably constructed Markov chain. Rather, they are based on growing the polymer by successively adding individual particles, guiding the growth towards configurations with lower energies, and using "population control" to eliminate bad configurations and increase the number of "good ones". This is not done via a breadth-first implementation as in genetic algorithms, but depth-first via recursive backtracking. As seen from various benchmark tests, the resulting algorithms are extremely efficient for lattice models, and are still competitive with other methods for simple off-lattice models.

cond-mat.soft↗

arXiv2004

Monte Carlo Protein Folding: Simulations of Met-Enkephalin with Solvent-Accessible Area Parameterizations

Treating realistically the ambient water is one of the main difficulties in applying Monte Carlo methods to protein folding. The solvent-accessible area method, a popular method for treating water implicitly, is investigated by means of Metropolis simulations of the brain peptide Met-Enkephalin. For the phenomenological energy function ECEPP/2 nine atomic solvation parameter (ASP) sets are studied that had been proposed by previous authors. The simulations are compared with each other, with simulations with a distance dependent electrostatic permittivity $ε(r)$, and with vacuum simulations ($ε=2$). Parallel tempering and a recently proposed biased Metropolis technique are employed and their performances are evaluated. The measured observables include energy and dihedral probability densities (pds), integrated autocorrelation times, and acceptance rates. Two of the ASP sets turn out to be unsuitable for these simulations. For all other sets, selected configurations are minimized in search of the global energy minima. Unique minima are found for the vacuum and the $ε(r)$ system, but for none of the ASP models. Other observables show a remarkable dependence on the ASPs. In particular, autocorrelation times vary dramatically with the ASP parameters. Three ASP sets have much smaller autocorrelations at 300 K than the vacuum simulations, opening the possibility that simulations can be speeded up vastly by judiciously chosing details of the force

cond-mat.soft↗

arXiv2004

Scaling of Star Polymers with one to 80 Arms

We present large statistics simulations of 3-dimensional star polymers with up to $f=80$ arms, and with up to 4000 monomers per arm for small values of $f$. They were done for the Domb-Joyce model on the simple cubic lattice. This is a model with soft core exclusion which allows multiple occupancy of sites but punishes each same-site pair of monomers with a Boltzmann factor $v<1$. We use this to allow all arms to be attached at the central site, and we use the `magic' value $v=0.6$ to minimize corrections to scaling. The simulations are made with a very efficient chain growth algorithm with resampling, PERM, modified to allow simultaneous growth of all arms. This allows us to measure not only the swelling (as observed from the center-to-end distances), but also the partition sum. The latter gives very precise estimates of the critical exponents $γ_f$. For completeness we made also extensive simulations of linear (unbranched) polymers which give the best estimates for the exponent $γ$.

cond-mat.soft↗

arXiv2004

Effective interactions between star polymers

We study numerically the effective pair potential between star polymers with equal arm lengths and equal number $f$ of arms. The simulations were done for the soft core Domb-Joyce model on the simple cubic lattice, to minimize corrections to scaling and to allow for an unlimited number of arms. For the sampling, we used the pruned-enriched Rosenbluth method (PERM). We find that the potential is much less soft than claimed in previous papers, in particular for $f\gg 1$. While we verify the logarithmic divergence of $V(r)$, with $r$ being the distance between the two cores, predicted by Witten and Pincus, we find for $f>20$ that the Mayer function is hardly distinguishable from that for a Gaussian potential.

cond-mat.soft↗

arXiv2004

Polymers Confined between Two Parallel Plane Walls

Single three dimensional polymers confined to a slab, i.e. to the region between two parallel plane walls, are studied by Monte Carlo simulations. They are described by $N$-step walks on a simple cubic lattice confined to the region $1 \le z \le D$. The simulations cover both regions $D << R_F$ and $D >> R_F$ (where $R_F \sim N^ν$ is the Flory radius, with $ν\approx 0.587$), as well as the cross-over region in between. Chain lengths are up to $N=80,000$, slab widths up to D=120. In order to test the analysis program and to check for finite size corrections, we actually studied three different models: (a) Ordinary random walks (mimicking $Θ$-polymers); (b) Self-avoiding walks (SAW); and (c) Domb-Joyce walks with the self-repulsion tuned to the point where finite size corrections for free (unrestricted) chains are minimal. For the simulations we employ the pruned-enriched-Rosenbluth method (PERM) with Markovian anticipation. In addition to the partition sum (which gives us a direct estimate of the forces exerted onto the walls), we measure the density profiles of monomers and of end points transverse to the slab, and the radial extent of the chain parallel to the walls. All scaling laws and some of the universal amplitude ratios are compared to theoretical predictions.

cond-mat.soft↗

arXiv2003

2-Dimensional Polymers Confined in a Strip

Single two dimensional polymers confined to a strip are studied by Monte Carlo simulations. They are described by N-step self-avoiding random walks on a square lattice between two parallel hard walls with distance 1 << D << N^ν(ν= 3/4 is the Flory exponent). For the simulations we employ the pruned-enriched-Rosenbluth method (PERM) with Markovian anticipation. We measure the densities of monomers and of end points as functions of the distance from the walls, the longitudinal extent of the chain, and the forces exerted on the walls. Their scaling with D and the universal ratio between force and monomer density at the wall are compared to theoretical predictions.

cond-mat.soft↗

arXiv2003

Hierarchical Clustering Based on Mutual Information

Motivation: Clustering is a frequently used concept in variety of bioinformatical applications. We present a new method for hierarchical clustering of data called mutual information clustering (MIC) algorithm. It uses mutual information (MI) as a similarity measure and exploits its grouping property: The MI between three objects X, Y, and Z is equal to the sum of the MI between X and Y, plus the MI between Z and the combined object (XY). Results: We use this both in the Shannon (probabilistic) version of information theory, where the "objects" are probability distributions represented by random samples, and in the Kolmogorov (algorithmic) version, where the "objects" are symbol sequences. We apply our method to the construction of mammal phylogenetic trees from mitochondrial DNA sequences and we reconstruct the fetal ECG from the output of independent components analysis (ICA) applied to the ECG of a pregnant woman. Availability: The programs for estimation of MI and for clustering (probabilistic version) are available at http://www.fz-juelich.de/nic/cs/software

q-bio.QM↗

arXiv2003

Hierarchical Clustering Using Mutual Information

We present a method for hierarchical clustering of data called {\it mutual information clustering} (MIC) algorithm. It uses mutual information (MI) as a similarity measure and exploits its grouping property: The MI between three objects $X, Y,$ and $Z$ is equal to the sum of the MI between $X$ and $Y$, plus the MI between $Z$ and the combined object $(XY)$. We use this both in the Shannon (probabilistic) version of information theory and in the Kolmogorov (algorithmic) version. We apply our method to the construction of phylogenetic trees from mitochondrial DNA sequences and to the output of independent components analysis (ICA) as illustrated with the ECG of a pregnant woman.

q-bio.QM↗

arXiv2003

Structure optimization in an off-lattice protein model

We study an off-lattice protein toy model with two species of monomers interacting through modified Lennard-Jones interactions. Low energy configurations are optimized using the pruned-enriched-Rosenbluth method (PERM), hitherto employed to native state searches only for off lattice models. For 2 dimensions we found states with lower energy than previously proposed putative ground states, for all chain lengths $\ge 13$. This indicates that PERM has the potential to produce native states also for more realistic protein models. For $d=3$, where no published ground states exist, we present some putative lowest energy states for future comparison with other methods.

cond-mat.stat-mech↗

arXiv2003

Are there really phase transitions in 1-d heat conduction models?

Recently, it has been claimed (O. V. Gendelman and A. V. Savin, Phys. Rev. Lett. {\bf 84}, 2381 (2000); A.V.Savin and O.V.Gendelman, arXiv: cond-mat/0204631 (2002)) that two nonlinear classical 1-d lattice models show transitions, at finite temperatures, where the heat conduction changes from being finite to being infinite. These are the well known Frenkel-Kontorova (FK) model and a model for coupled rotators. For the FK model we give strong theoretical arguments why such a phase transition is not to be expected. For both models we show numerically that the effects observed by Gendelman {\it et al.} are not true phase transitions but are rather the expected cross-overs associated to the conductivity divergence as $T\to 0$ and (for the FK model) $T\to\infty$.

cond-mat.stat-mech↗

arXiv2003

Estimating Mutual Information

We present two classes of improved estimators for mutual information $M(X,Y)$, from samples of random points distributed according to some joint probability density $μ(x,y)$. In contrast to conventional estimators based on binnings, they are based on entropy estimates from $k$-nearest neighbour distances. This means that they are data efficient (with $k=1$ we resolve structures down to the smallest possible scales), adaptive (the resolution is higher where data are more numerous), and have minimal bias. Indeed, the bias of the underlying entropy estimates is mainly due to non-uniformity of the density at the smallest resolved scale, giving typically systematic errors which scale as functions of $k/N$ for $N$ points. Numerically, we find that both families become {\it exact} for independent distributions, i.e. the estimator $\hat M(X,Y)$ vanishes (up to statistical fluctuations) if $μ(x,y) = μ(x) μ(y)$. This holds for all tested marginal distributions and for all dimensions of $x$ and $y$. In addition, we give estimators for redundancies between more than 2 random variables. We compare our algorithms in detail with existing algorithms. Finally, we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis (ICA), for improving ICA, and for estimating the reliability of blind source separation.

cond-mat.stat-mech↗

arXiv2003

Growth Algorithms for Lattice Heteropolymers at Low Temperatures

Two improved versions of the pruned-enriched-Rosenbluth method (PERM) are proposed and tested on simple models of lattice heteropolymers. Both are found to outperform not only the previous version of PERM, but also all other stochastic algorithms which have been employed on this problem, except for the core directed chain growth method (CG) of Beutler & Dill. In nearly all test cases they are faster in finding low-energy states, and in many cases they found new lowest energy states missed in previous papers. The CG method is superior to our method in some cases, but less efficient in others. On the other hand, the CG method uses heavily heuristics based on presumptions about the hydrophobic core and does not give thermodynamic properties, while the present method is a fully blind general purpose algorithm giving correct Boltzmann-Gibbs weights, and can be applied in principle to any stochastic sampling problem.

cond-mat.stat-mech↗

arXiv2003

A Growth-based Optimization Algorithm for Lattice Heteropolymers

An improved version of the pruned-enriched-Rosenbluth method (PERM) is proposed and tested on finding lowest energy states in simple models of lattice heteropolymers. It is found to outperform not only the previous version of PERM, but also all other fully blind general purpose stochastic algorithms which have been employed on this problem. In many cases it found new lowest energy states missed in previous papers. Limitations are discussed.

cond-mat.stat-mech↗