Thomas Prellberg

Critical exponents from nonlinear functional equations for partially directed cluster models

We present a method for the derivation of the generating function and computation of critical exponents for several cluster models (staircase, bar-graph, and directed column-convex polygons, as well as partially directed self-avoiding walks), starting with nonlinear functional equations for the generating function. By linearizing these equations, we first give a derivation of the generating functions. The nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz. Alternatively, taking the continuum limit leads to nonlinear differential equations, from which one can extract the scaling function. We find that all the above models are in the same universality class with exponents γu=-1/2, γi=-1/3, and ϕ=2/3. All models have as their scaling function the logarithmic derivative of the Airy function.

Uniform q-series asymptotics for staircase polygons

We present a uniform asymptotic expansion for the area-perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q-series 1 phi 1(0;y;q,x) as q to 1- from a new integral representation. This leads to a direct calculation of the scaling function for this model.

Scaling of self-avoiding walks and self-avoiding trails in three dimensions

Motivated by recent claims of a proof that the length scale exponent for the end-to-end distance scaling of self-avoiding walks is precisely 7/12 = 0.5833..., we present results of large-scale simulations of self-avoiding walks and self-avoiding trails with repulsive contact interactions on the simple cubic lattice. We find no evidence to support this claim; our estimate ν = 0.5874(2) is in accord with the best previous results from simulations.

Layering transitions for adsorbing polymers in poor solvents

An infinite hierarchy of layering transitions exists for model polymers in solution under poor solvent or low temperatures and near an attractive surface. A flat histogram stochastic growth algorithm known as FlatPERM has been used on a self- and surface interacting self-avoiding walk model for lengths up to 256. The associated phases exist as stable equilibria for large though not infinite length polymers and break the conjectured Surface Attached Globule phase into a series of phases where a polymer exists in specified layer close to a surface. We provide a scaling theory for these phases and the first-order transitions between them.

Towards a complete determination of the spectrum of a transfer operator associated with intermittency

The physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point. For an example of such an intermittent map, the Farey map, we give a simple proof that the transfer operator is self-adjoint on a suitably defined Hilbert space and characterize its spectrum. Using a suitable first-return map, we present a highly efficient numerical method for the determination of all the eigenvalues, including those embedded in the continuous spectrum.

A scaling theory of the collapse transition in geometric cluster models of polymers and vesicles

Much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk, polygon and animal models. The crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2- alpha =1/ phi with the alpha the canonical (thermodynamic) exponent that characterizes the divergence of the specific heat. The reasons for the validity of this relation are not widely known. The authors present a scaling theory of collapse transitions in such models. The free energy and canonical partition functions have finite-length scaling forms whilst the grand partition function has a tricritical scaling form. The link between the grand and canonical ensembles leads to the above scaling relation. They then comment on the validity of current estimates of the crossover exponent for interacting self-avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values.

Exact Solution of the Discrete (1 + 1)-Dimensional SOS Model with Field and Surface Interactions

We present the solution of a linear solid-on-solid (SOS) model. Configurations are partially directed walks on a two-dimensional square lattice and we include a linear surface tension, a magnetic field, and surface interaction terms in the Hamiltonian. There is a wetting transition at zero field and, as expected, the behavior is similar to a continuous model solved previously. The solution is in terms ofq-series most closely related to theq-hypergeometric functions1φ1.

Stretching of a chain polymer adsorbed at a surface

In this paper we present simulations of a surface-adsorbed polymer subject to an elongation force. The polymer is modelled by a self-avoiding walk on a regular lattice. It is confined to a half-space by an adsorbing surface with attractions for every vertex of the walk visiting the surface, and the last vertex is pulled perpendicular to the surface by a force. Using the recently proposed flatPERM algorithm, we calculate the phase diagram for a vast range of temperatures and forces. The strength of this algorithm is that it computes the complete density of states from one single simulation. We simulate systems of sizes up to 256 steps.

Stacking Models of Vesicles and Compact Clusters

We investigate three simple lattice models of two dimensional vesicles. These models differ in their behavior from the universality class of partially convex polygons, which has been recently established. They do not have the tricritical scaling of those models, and furthermore display a surprising feature: their (perimeter) free energy is discontinuous with an isolated value at zero pressure. We give the full asymptotic descriptions of the generating functions in area and perimeter variables from theq-series solutions and obtain the scaling functions where applicable.

Forcing Adsorption of a Tethered Polymer by Pulling

We present an analysis of a partially directed walk model of a polymer which at one end is tethered to a sticky surface and at the other end is subjected to a pulling force at fixed angle away from the point of tethering. Using the kernel method, we derive the full generating function for this model in two and three dimensions and obtain the respective phase diagrams. We observe adsorbed and desorbed phases with a thermodynamic phase transition in between. In the absence of a pulling force this model has a second-order thermal desorption transition which merely gets shifted by the presence of a lateral pulling force. On the other hand, if the pulling force contains a non-zero vertical component this transition becomes first order. Strikingly, we find that, if the angle between the pulling force and the surface is below a critical value, a sufficiently strong force will induce polymer adsorption, no matter how large the temperature of the system. Our findings are similar in two and three dimensions, an additional feature in three dimensions being the occurrence of a re-entrance transition at constant pulling force for low temperature, which has been observed previously for this model in the presence of pure vertical pulling. Interestingly, the re-entrance phenomenon vanishes under certain pulling angles, with details depending on how the three-dimensional polymer is modeled.