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T. Prellberg

Publications and source records attributed to T. Prellberg.

At least 19 recordsLinked to original sources
arXiv2018

Adsorption of neighbor-avoiding walks on the simple cubic lattice

We investigate neighbor-avoiding walks on the simple cubic lattice in the presence of an adsorbing surface. This class of lattice paths has been less studied using Monte Carlo simulations. Our investigation follows on from our previous results using self-avoiding walks and self-avoiding trails. The connection is that neighbor-avoiding walks are equivalent to the infinitely repulsive limit of self-avoiding walks with monomer-monomer interactions. Such repulsive interactions can be seen to enhance the excluded volume effect. We calculate the critical behavior of the adsorption transition for neighbor-avoiding walks, finding a critical temperature $T_{\text a}=3.274(9)$ and a crossover exponent $ϕ=0.482(13)$, which is consistent with the exponent for self-avoiding walks and trails, leading to an overall combined estimate for three dimensions of $ϕ_\text{3D}=0.484(7)$. While questions of universality have previously been raised regarding the value of adsorption exponents in three dimensions, our results indicate that the value of $ϕ$ in the strongly repulsive regime does not differ from its non-interacting value. However, it is clearly different from the mean-field value of $1/2$ and therefore not super-universal.

cond-mat.stat-mech↗
arXiv2017

Universality of crossover scaling for the adsorption transition of lattice polymers

Recently, it has been proposed that the adsorption transition for a single polymer in dilute solution, modeled by lattice walks in three dimensions, is not universal with respect to inter-monomer interactions. It has also been conjectured that key critical exponents $ϕ$, measuring the growth of the contacts with the surface at the adsorption point, and $1/δ$, which measures the finite-size shift of the critical temperature, are not the same. However, applying standard scaling arguments the two key critical exponents should be identical, thus pointing to a potential breakdown of these standard scaling arguments. This is in contrast to the well studied situation in two dimensions, where there are exact results from conformal field theory: these exponents are both accepted to be $1/2$ and universal. We use the flatPERM algorithm to simulate self-avoiding walks and trails on the hexagonal, square and simple cubic lattices up to length $1024$ to investigate these claims. Walks can be seen as a repulsive limit of inter-monomer interaction for trails, allowing us to probe the universality of adsorption. For each model we analyze several thermodynamic properties to produce different methods of estimating the critical temperature and the key exponents. We test our methodology on the two-dimensional cases and the resulting spread in values for $ϕ$ and $1/δ$ indicates that there is a systematic error that exceeds the statistical error usually reported. We further suggest a methodology for consistent estimation of the key adsorption exponents which gives $ϕ=1/δ=0.484(4)$ in three dimensions. We conclude that in three dimensions these critical exponents indeed differ from the mean-field value of $1/2$, but cannot find evidence that they differ from each other. Importantly, we also find no substantive evidence of any non-universality in the polymer adsorption transition.

cond-mat.stat-mech↗
arXiv2016

Grand-canonical solution of semi-flexible self-avoiding trails on the Bethe lattice

We consider a model of semi-flexible interacting self-avoiding trails (sISAT's) on a lattice, where the walks are constrained to visit each lattice edge at most once. Such models have been studied as an alternative to the self-attracting self-avoiding walks (SASAW) to investigate the collapse transition of polymers, with the attractive interactions being on site, as opposed to nearest-neighbor interactions in SASAW. The grand-canonical version of the sISAT model is solved on a four-coordinated Bethe lattice, and four phases appear: non-polymerized (NP), regular polymerized (P), dense polymerized (DP) and anisotropic nematic (AN), the last one present in the phase diagram only for sufficiently stiff chains. The last two phases are dense, in the sense that all lattice sites are visited once in AN phase and twice in DP phase. In general, critical NP-P and DP-P transition surfaces meet with a NP-DP coexistence surface at a line of bicritical points. The region in which the AN phase is stable is limited by a discontinuous critical transition to the P phase, and we study this somewhat unusual transition in some detail. In the limit of rods, where the chains are totally rigid, the P phase is absent and the three coexistence lines (NP-AN, AN-DP, and NP-DP) meet at a triple point, which is the endpoint of the bicritical line.

cond-mat.stat-mech↗
arXiv2016

Forces and pressures in adsorbing partially directed walks

Polymers in confined spaces lose conformational entropy. This induces a net repulsive entropic force on the walls of the confining space. A model for this phenomenon is a lattice walk between confining walls, and in this paper a model of an adsorbing partially directed walk is used. The walk is placed in a half square lattice $L^2_+$ with boundary $\partial L^2_+$, and confined between two vertical parallel walls, which are vertical lines in the lattice, a distance $w$ apart. The free energy of the walk is determined, as a function of $w$, for walks with endpoints in the confining walls and adsorbing in $\partial L^2_+$. This gives the entropic force on the confining walls as a function of $w$. It is shown that there are zero force points in this model and the locations of these points are determined, in some cases exactly, and in other cases asymptotically.

cond-mat.stat-mech↗
arXiv2014

A semi-flexible attracting-segment model of three-dimensional polymer collapse

Recently it has been shown that a two-dimensional model of self-attracting polymers based on attracting segments with the addition of stiffness displays three phases: a swollen phase, a globular, liquid-like phase, and an anisotropic crystal-like phase. Here, we consider the attracting segment model in three dimensions with the addition of stiffness. While we again identify a swollen and two distinct collapsed phases, we find that both collapsed phases are anisotropic, so that there is no phase in which the polymer resembles a disordered liquid drop. Moreover all the phase transitions are first order.

cond-mat.soft↗
arXiv2013

Numerical simulation of a lattice polymer model at its integrable point

We revisit an integrable lattice model of polymer collapse using numerical simulations. This model was first studied by Blöte and Nienhuis in J. Phys. A. {\bf 22}, 1415 (1989) and it describes polymers with some attraction, providing thus a model for the polymer collapse transition. At a particular set of Boltzmann weights the model is integrable and the exponents $ν=12/23\approx 0.522$ and $γ=53/46\approx 1.152$ have been computed via identification of the scaling dimensions $x_t=1/12$ and $x_h=-5/48$. We directly investigate the polymer scaling exponents via Monte Carlo simulations using the PERM algorithm. By simulating this polymer model for walks up to length 4096 we find $ν=0.576(6)$ and $γ=1.045(5)$, which are clearly different from the predicted values. Our estimate for the exponent $ν$ is compatible with the known $θ$-point value of 4/7 and in agreement with very recent numerical evaluation by Foster and Pinettes.

cond-mat.stat-mech↗
arXiv2013

The pressure exerted by adsorbing directed lattice paths and staircase polygons

A directed path in the vicinity of a hard wall exerts pressure on the wall because of loss of entropy. The pressure at a particular point may be estimated by estimating the loss of entropy if the point is excluded from the path. In this paper we determine asymptotic expressions for the pressure on the X-axis in models of adsorbing directed paths in the first quadrant. Our models show that the pressure vanishes in the limit of long paths in the desorbed phase, but there is a non-zero pressure in the adsorbed phase. We determine asymptotic approximations of the pressure for finite length Dyck paths and directed paths, as well as for a model of adsorbing staircase polygons with both ends grafted to the X-axis.

math-ph↗
arXiv2012

Self-avoiding trails with nearest neighbour interactions on the square lattice

Self-avoiding walks and self-avoiding trails, two models of a polymer coil in dilute solution, have been shown to be governed by the same universality class. On the other hand, self-avoiding walks interacting via nearest-neighbour contacts (ISAW) and self-avoiding trails interacting via multiply-visited sites (ISAT) are two models of the coil-globule, or collapse transition of a polymer in dilute solution. On the square lattice it has been established numerically that the collapse transition of each model lies in a different universality class. The models differ in two substantial ways. They differ in the types of subsets of random walk configurations utilised (site self-avoidance versus bond self-avoidance) and in the type of attractive interaction. It is therefore of some interest to consider self-avoiding trails interacting via nearest neighbour attraction (INNSAT) in order to ascertain the source for the difference in the collapse universality class. Using the flatPERM algorithm, we have performed computer simulations of this model. We present numerical evidence that the singularity in the free energy of INNSAT at the collapse transition has a similar exponent to that of the ISAW model rather than the ISAT model. This would indicate that the type of interaction used in ISAW and ISAT is the source of the difference in universality class.

cond-mat.stat-mech↗
arXiv2012

Weighting of topologically different interactions in a model of two-dimensional polymer collapse

We study by computer simulation a recently introduced generalised model of self-interacting self-avoiding trails on the square lattice that distinguishes two topologically different types of self-interaction: namely crossings where the trail passes across itself and collisions where the lattice path visits the same site without crossing. This model generalises the canonical interacting self-avoiding trail model of polymer collapse which has a strongly divergent specific heat at its transition point. We confirm the recent prediction that the asymmetry does not affect the universality class for a range of asymmetry. Certainly, where the weighting of collisions outweighs that of crossings this is well supported numerically. When crossings are weighted heavily relative to collisions the collapse transition reverts to the canonical theta-point-like behaviour found in interacting self-avoiding walks.

cond-mat.stat-mech↗
arXiv2012

Rare event sampling with stochastic growth algorithms

We discuss uniform sampling algorithms that are based on stochastic growth methods, using sampling of extreme configurations of polymers in simple lattice models as a motivation. We shall show how a series of clever enhancements to a fifty-odd year old algorithm, the Rosenbluth method, led to a cutting-edge algorithm capable of uniform sampling of equilibrium statistical mechanical systems of polymers in situations where competing algorithms failed to perform well. Examples range from collapsed homo-polymers near sticky surfaces to models of protein folding.

cond-mat.stat-mech↗
arXiv2009

Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point

We consider the Farey fraction spin chain in an external field $h$. Using ideas from dynamical systems and functional analysis, we show that the free energy $f$ in the vicinity of the second-order phase transition is given, exactly, by $$ f \sim \frac t{\log t}-\frac1{2} \frac{h^2}t \quad \text{for} \quad h^2\ll t \ll 1 . $$ Here $t=λ_{G}\log(2)(1-\fracβ{β_c})$ is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, $λ_G$. It follows that $λ_G$ determines the amplitude of both the specific heat and susceptibility singularities. To our knowledge, there is only one other microscopically defined interacting model for which the free energy near a phase transition is known as a function of two variables. Our results confirm what was found previously with a cluster approximation, and show that a clustering mechanism is in fact responsible for the transition. However, the results disagree in part with a renormalisation group treatment.

cond-mat.stat-mech↗
arXiv2008

A self-interacting partially directed walk subject to a force

We consider a directed walk model of a homopolymer (in two dimensions) which is self-interacting and can undergo a collapse transition, subject to an applied tensile force. We review and interpret all the results already in the literature concerning the case where this force is in the preferred direction of the walk. We consider the force extension curves at different temperatures as well as the critical-force temperature curve. We demonstrate that this model can be analysed rigorously for all key quantities of interest even when there may not be explicit expressions for these quantities available. We show which of the techniques available can be extended to the full model, where the force has components in the preferred direction and the direction perpendicular to this. Whilst the solution of the generating function is available, its analysis is far more complicated and not all the rigorous techniques are available. However, many results can be extracted including the location of the critical point which gives the general critical-force temperature curve. Lastly, we generalise the model to a three-dimensional analogue and show that several key properties can be analysed if the force is restricted to the plane of preferred directions.

cond-mat.stat-mech↗
arXiv2008

Polymers with self-attraction and stiffness: a generic phase structure

Recently it has been shown that a two-dimensional model of self-attracting polymers based on attracting segments displays two phase transitions, a theta-like collapse between swollen polymers and a globular state and another between the globular state and a polymer crystal. On the other hand, the canonical model based on attracting monomers on lattice sites displays only one: the standard tricritical theta collapse transition. Here we show that by considering both models with the addition of stiffness the two models display the same generic phase diagram. In fact we claim that any two-dimensional model of a self-attracting single polymer in solution based upon a fully volume-excluded backbone with isotropic short range attraction should show this same phase structure. We point to the model of hydrogen bonded polymers to demonstrate this observation. In three dimensions we note that more than one crystalline phase may occur.

cond-mat.soft↗
arXiv2008

Scaling of the atmosphere of self-avoiding walks

The number of free sites next to the end of a self-avoiding walk is known as the atmosphere. The average atmosphere can be related to the number of configurations. Here we study the distribution of atmospheres as a function of length and how the number of walks of fixed atmosphere scale. Certain bounds on these numbers can be proved. We use Monte Carlo estimates to verify our conjectures. Of particular interest are walks that have zero atmosphere, which are known as trapped. We demonstrate that these walks scale in the same way as the full set of self-avoiding walks, barring an overall constant factor.

cond-mat.stat-mech↗
arXiv2007

Exact Solution of Semi-Flexible and Super-Flexible Interacting Partially Directed Walks

We provide the exact generating function for semi-flexible and super-flexible interacting partially directed walks and also analyse the solution in detail. We demonstrate that while fully flexible walks have a collapse transition that is second order and obeys tricritical scaling, once positive stiffness is introduced the collapse transition becomes first order. This confirms a recent conjecture based on numerical results. We note that the addition of an horizontal force in either case does not affect the order of the transition. In the opposite case where stiffness is discouraged by the energy potential introduced, which we denote the super-flexible case, the transition also changes, though more subtly, with the crossover exponent remaining unmoved from the neutral case but the entropic exponents changing.

cond-mat.stat-mech↗
arXiv2007

On the location of the surface-attached globule phase in collapsing polymers

We investigate the existence and location of the surface phase known as the "Surface-Attached Globule" (SAG) conjectured previously to exist in lattice models of three-dimensional polymers when they are attached to a wall that has a short range potential. The bulk phase, where the attractive intra-polymer interactions are strong enough to cause a collapse of the polymer into a liquid-like globule and the wall either has weak attractive or repulsive interactions, is usually denoted Desorbed-Collapsed or DC. Recently this DC phase was conjectured to harbour two surface phases separated by a boundary where the bulk free energy is analytic while the surface free energy is singular. The surface phase for more attractive values of the wall interaction is the SAG phase. We discuss more fully the properties of this proposed surface phase and provide Monte Carlo evidence for self-avoiding walks up to length 256 that this surface phase most likely does exist. Importantly, we discuss alternatives for the surface phase boundary. In particular, we conclude that this boundary may lie along the zero wall interaction line and the bulk phase boundaries rather than any new phase boundary curve.

cond-mat.soft↗
arXiv2007

Finite-size scaling functions for directed polymers confined between attracting walls

The exact solution of directed self-avoiding walks confined to a slit of finite width and interacting with the walls of the slit via an attractive potential has been calculated recently. The walks can be considered to model the polymer-induced steric stabilisation and sensitised floculation of colloidal dispersions. The large width asymptotics led to a phase diagram different to that of a polymer attached to, and attracted to, a single wall. The question that arises is: can one interpolate between the single wall and two wall cases? In this paper we calculate the exact scaling functions for the partition function by considering the two variable asymptotics of the partition function for simultaneous large length and large width. Consequently, we find the scaling functions for the force induced by the polymer on the walls. We find that these scaling functions are given by elliptic theta-functions. In some parts of the phase diagram there is more a complex crossover between the single wall and two wall cases and we elucidate how this happens.

cond-mat.stat-mech↗