arXiv subjects

Peter Grassberger

Publications and source records attributed to Peter Grassberger.

At least 109 records · Page 6Linked to original sources

arXiv2002

Collapsed 2-Dimensional Polymers on a Cylinder

Single partially confined collapsed polymers are studied in two dimensions. They are described by self-avoiding random walks with nearest-neighbour attractions below the $Θ$-point, on the surface of an infinitely long cylinder. For the simulations we employ the pruned-enriched-Rosenbluth method (PERM). The same model had previously been studied for free polymers (infinite lattice, no boundaries) and for polymers on finite lattices with periodic boundary conditions. We verify the previous estimates of bulk densities, bulk free energies, and surface tensions. We find that the free energy of a polymer with fixed length $N$ has, for $N\to \infty$, a minimum at a finite cylinder radius $R^*$ which diverges as $T\to T_θ$. Furthermore, the surface tension vanishes roughly as $(T_θ-T)^α$ for $T\to T_θ$ with $α\approx 1.7$. The density in the interior of a globule scales as $(T_θ-T)^β$ with $β\approx 0.32$.

cond-mat.stat-mech↗

arXiv2002

Critical Percolation in High Dimensions

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4 to 13 dimensions. For d<6 they are preliminary, for d >= 6 they are between 20 to 10^4 times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing which allowed us to simulate clusters of millions of sites on computers with less than 500 MB memory; (ii) a histogram method which allowed us to obtain information for several p values from a single simulation; and (iii) a new variance reduction technique which is especially efficient at high dimensions where it reduces error bars by a factor up to approximately 30 and more. Based on these data we propose a new scaling law for finite cluster size corrections.

cond-mat.stat-mech↗

arXiv2002

Heat Conduction and Entropy Production in a One-Dimensional Hard-Particle Gas

We present large scale simulations for a one-dimensional chain of hard-point particles with alternating masses. We correct several claims in the recent literature based on much smaller simulations. Both for boundary conditions with two heat baths at different temperatures at both ends and from heat current autocorrelations in equilibrium we find heat conductivities kappa to diverge with the number N of particles. These depended very strongly on the mass ratios, and extrapolation to N -> infty resp. t -> infty is difficult due to very large finite-size and finite-time corrections. Nevertheless, our data seem compatible with a universal power law kappa ~ N^alpha with alpha approx 0.33. This suggests a relation to the Kardar-Parisi-Zhang model. We finally show that the hard-point gas with periodic boundary conditions is not chaotic in the usual sense and discuss why the system, when kept out of equilibrium, leads nevertheless to energy dissipation and entropy production.

nlin.CD↗

arXiv2002

Data Compression and Entropy Estimates by Non-sequential Recursive Pair Substitution

We argue that Non-sequential Recursive Pair Substitution (NSRPS) as suggested by Jiménez-Montaño and Ebeling can indeed be used as a basis for an optimal data compression algorithm. In particular, we prove for Markov sequences that NSRPS together with suitable codings of the substitutions and of the substitute series does not lead to a code length increase, in the limit of infinite sequence length. When applied to written English, NSRPS gives entropy estimates which are very close to those obtained by other methods. Using ca. 135 GB of input data from the project Gutenberg, we estimate the effective entropy to be $\approx 1.82$ bit/character. Extrapolating to infinitely long input, the true value of the entropy is estimated as $\approx 0.8$ bit/character.

physics.data-an↗

arXiv2002

Entropy estimation of symbol sequences

We discuss algorithms for estimating the Shannon entropy h of finite symbol sequences with long range correlations. In particular, we consider algorithms which estimate h from the code lengths produced by some compression algorithm. Our interest is in describing their convergence with sequence length, assuming no limits for the space and time complexities of the compression algorithms. A scaling law is proposed for extrapolation from finite sample lengths. This is applied to sequences of dynamical systems in non-trivial chaotic regimes, a 1-D cellular automaton, and to written English texts.

cond-mat.stat-mech↗

arXiv2002

Critical Behaviour of the Drossel-Schwabl Forest Fire Model

We present high statistics Monte Carlo results for the Drossel-Schwabl forest fire model in 2 dimensions. They extend to much larger lattices (up to $65536\times 65536$) than previous simulations and reach much closer to the critical point (up to $θ\equiv p/f = 256000$). They are incompatible with all previous conjectures for the (extrapolated) critical behaviour, although they in general agree well with previous simulations wherever they can be directly compared. Instead, they suggest that scaling laws observed in previous simulations are spurious, and that the density $ρ$ of trees in the critical state was grossly underestimated. While previous simulations gave $ρ\approx 0.408$, we conjecture that $ρ$ actually is equal to the critical threshold $p_c = 0.592...$ for site percolation in $d=2$. This is however still far from the densities reachable with present day computers, and we estimate that we would need many orders of magnitude higher CPU times and storage capacities to reach the true critical behaviour -- which might or might not be that of ordinary percolation.

cond-mat.stat-mech↗

arXiv2002

Stretched Polymers in a Poor Solvent

Stretched polymers with attractive interaction are studied in two and three dimensions. They are described by biased self-avoiding random walks with nearest neighbour attraction. The bias corresponds to opposite forces applied to the first and last monomers. We show that both in $d=2$ and $d=3$ a phase transition occurs as this force is increased beyond a critical value, where the polymer changes from a collapsed globule to a stretched configuration. This transition is second order in $d=2$ and first order in $d=3$. For $d=2$ we predict the transition point quantitatively from properties of the unstretched polymer. This is not possible in $d=3$, but even there we can estimate the transition point precisely, and we can study the scaling at temperatures slightly below the collapse temperature of the unstretched polymer. We find very large finite size corrections which would make very difficult the estimate of the transition point from straightforward simulations.

cond-mat.stat-mech↗

arXiv2001

On the trapping reaction with mobile traps

We present Monte Carlo results for the two-species trapping reaction $A+B \to B$ with diffusing $A$ and $B$ on lattices in one, two and three dimension. We use a novel algorithm which permits to simulate survival probabilities of $A$ partic les down to $<10^{-30}$ with high accuracy. The results for the survival probability agree much better with the exact asymptotic predictions of Bramson and Lebowitz (Phys. Rev. Lett. {\bf 61}, 2397 (1988)) than with the heuristics of Kang and Redner (J. Phys. A {\bf 17}, L451 (1984)). But there are very large deviations from either which show that even these simulations are far from asymptotia. This is supported by the rms. displacement of $A$ particles which clearly show that the asymptotic regime has not been reached, at least for $d=2$ and $d=3$.

cond-mat.stat-mech↗

arXiv2001

Transition to Localization of Biased Walkers in a Randomly Absorbing Enviroment

We study biased random walkers on lattices with randomly dispersed static traps in one, two and three dimensions. As the external bias is increased from zero the system undergoes a phase transition, most clearly manifested in the asymptotic drift velocity of survivors which is zero below a critical bias. This transition is second-order in one dimension but of first order in higher dimensions. The model can be mapped to a stretched polymer with attractive interaction between monomers, and this phase transition would then describe sudden unfolding of the polymer when the stretching force exceeds a critical value. We also present precise simulations of the zero bias case where we show unambiguously that the transition between the Rosenstock and Donsker-Varadhan regimes is first order in dimension $\ge 2$.

cond-mat.stat-mech↗

arXiv2000

Branched Polymers and Percolation

We study a supposed model for branched polymers which was shown in two dimensions to be in the universality class of ordinary percolation. We confirm this by high statistics simulations and show that it is in the percolation universality class also for three dimensions, in contrast to previous claims. These previous studies seem to have been mislead by huge corrections to scaling in this model.

cond-mat.stat-mech↗

arXiv2000

Opacity and Entanglement of Polymer Chains

We argue that the mean crossing number of a random polymer configuration is simply a measure of opacity, without being closely related to entanglement as claimed by several authors. We present an easy way of estimating its asymptotic behaviour numerically. These estimates agree for random walks (theta polymers), self-avoiding walks, and for compact globules with analytic estimates giving $\log N, a-b/N^{2ν-1},$ and $N^{1/3}$, respectively, for the average number of crossings per monomer in the limit $N\to \infty$. While the result for compact globules agrees with a rigorous previous estimate, the result for SAWs disagrees with previous numerical estimates.

cond-mat.stat-mech↗

arXiv2000

Slow Logarithmic Decay of Magnetization in the Zero Temperature Dynamics of an Ising Spin Chain: Analogy to Granular Compaction

We study the zero temperature coarsening dynamics in an Ising chain in presence of a dynamically induced field that favors locally the `-' phase compared to the `+' phase. At late times, while the `+' domains still coarsen as $t^{1/2}$, the `-' domains coarsen slightly faster as $t^{1/2}\log (t)$. As a result, at late times, the magnetization decays slowly as, $m(t)=-1 +{\rm const.}/{\log (t)}$. We establish this behavior both analytically within an independent interval approximation (IIA) and numerically. In the zero volume fraction limit of the `+' phase, we argue that the IIA becomes asymptotically exact. Our model can be alternately viewed as a simple Ising model for granular compaction. At late times in our model, the system decays into a fully compact state (where all spins are `-') in a slow logarithmic manner $\sim 1/{\log (t)}$, a fact that has been observed in recent experiments on granular systems.

cond-mat.stat-mech↗

arXiv2000

Exactness of the Annealed and the Replica Symmetric Approximations for Random Heteropolymers

We study a heteropolymer model with random contact interactions introduced some time ago as a simplified model for proteins. The model consists of self-avoiding walks on the simple cubic lattice, with contact interactions between nearest neighbor pairs. For each pair, the interaction energy is an independent Gaussian variable with mean value $B$ and variance $\D^2$. For this model the annealed approximation is expected to become exact for low disorder, at sufficiently high dimension and in the thermodynamic limit. We show that corrections to the annealed approximation in the 3-d high temperature phase are small, but do not vanish in the thermodynamic limit, and are in good agreement with our replica symmetric calculations. Such corrections derive from the fact that the overlap between two typical chains is nonzero. We explain why previous authors had come to the opposite conclusion, and discuss consequences for the thermodynamics of the model. Numerical results were obtained by simulating chains up to length 1400 by means of the recent PERM algorithm, in the coil and molten globular phases, well above the freezing temperature.

cond-mat.soft↗

arXiv1999

A Simple Model for the DNA Denaturation Transition

We study pairs of interacting self-avoiding walks on the 3d simple cubic lattice. They have a common origin and are allowed to overlap only at the same monomer position along the chain. The latter overlaps are indeed favored by an energetic gain. This is inspired by a model introduced long ago by Poland and Sheraga [J. Chem. Phys. {\bf 45}, 1464 (1966)] for the denaturation transition in DNA where, however, self avoidance was not fully taken into account. For both models, there exists a temperature T_m above which the entropic advantage to open up overcomes the energy gained by forming tightly bound two-stranded structures. Numerical simulations of our model indicate that the transition is of first order (the energy density is discontinuous), but the analog of the surface tension vanishes and the scaling laws near the transition point are exactly those of a second order transition with crossover exponent ϕ=1. Numerical and exact analytic results show that the transition is second order in modified models where the self-avoidance is partially or completely neglected.

cond-mat.soft↗

arXiv1999

Does mesoscopic disorder imply microscopic chaos?

We argue that Gaspard and coworkers do not give evidence for microscopic chaos in the sense in which they use the term. The effectively infinite number of molecules in a fluid can generate the same macroscopic disorder without any intrinsic instability. But we argue also that the notion of chaos in infinitely extended systems needs clarification: In a wider sense, even some systems without local instabilities can be considered chaotic.

chao-dyn↗

arXiv1998

Determination of the exponent gamma for SAWs on the two-dimensional Manhattan lattice

We present a high-statistics Monte Carlo determination of the exponent gamma for self-avoiding walks on a Manhattan lattice in two dimensions. A conservative estimate is $γ\gtapprox 1.3425(3)$, in agreement with the universal value 43/32 on regular lattices, but in conflict with predictions from conformal field theory and with a recent estimate from exact enumerations. We find strong corrections to scaling that seem to indicate the presence of a non-analytic exponent Delta < 1. If we assume Delta = 11/16 we find gamma = 1.3436(3), where the error is purely statistical.

cond-mat.stat-mech↗

arXiv1998

2-d Self-Avoiding Walks on a Cylinder

We present simulations of self-avoiding random walks on 2-d lattices with the topology of an infinitely long cylinder, in the limit where the cylinder circumference L is much smaller than the Flory radius. We study in particular the L-dependence of the size h parallel to the cylinder axis, the connectivity constant mu, the variance of the winding number around the cylinder, and the density of parallel contacts. While mu(L) and scale as as expected (in particular, \sim h/L), the number of parallel contacts decays as h/L^1.92, in striking contrast to recent predictions. These findings strongly speak against recent speculations that the critical exponent gamma of SAW's might be nonuniversal. Finally, we find that the amplitude for does not agree with naive expectations from conformal invariance.

cond-mat.stat-mech↗

arXiv1998

A New Monte Carlo Algorithm for Protein Folding

We demonstrate that the recently proposed pruned-enriched Rosenbluth method (P. Grassberger, Phys. Rev. E 56 (1997) 3682) leads to extremely efficient algorithms for the folding of simple model proteins. We test them on several models for lattice heteropolymers, and compare to published Monte Carlo studies. In all cases our algorithms are faster than all previous ones, and in several cases we find new minimal energy states. In addition to ground states, our algorithms give estimates for the partition sum at finite temperatures.

cond-mat.stat-mech↗