Jordan Brankov

Ground state of an infinite two-dimensional system of dipoles on a lattice with arbitrary rhombicity angle

A class of possible periodic orientational configurations of a system of dipole moments located at the sites of an infinite flat rhombic lattice with an arbitrary rhombicity angle α is studied by using the Luttinger and Tisza method. In the framework of the method it is obtained that: for α ⪅ 80° the ground state is ferromagnetic, at α = 60° being continuously degenerate in direction; for 80° ⪅ α ⩽ 90° the ground state is antiferromagnetic, at α = 90° being also continuously degenerate with respect to one parameter. It is shown that the restriction of the interaction range leads to a change in the type of the ground state: at α = 60° this takes place between the third and the fourth coordination spheres and at α = 30° at a distance of about 500 lattice constants. From comparison with known results of numerical experiments on finite systems it is established that the type of the ground state of a finite and an infinite system may be essentially different.

Finite-size scaling for systems with long-range interactions

Abstract The present review is devoted to the fundamental problems of finite-size scaling due to the presence of long-range interactions. The attention is focused on the precise formulation of critical finite-size scaling in the case that the bulk correlation length is identically infinite. The hypotheses, based on the notion of a finite-size scaling length, are formulated in a way which treats the cases of short- and long-range interactions on equal grounds, as well as the standard and modified (above the upper critical dimensionality) finite-size scaling. The general conjectures are tested on the exactly solvable mean spherical model with power-law interaction. Special attention is paid to the adequate mathematical techniques. The formulation of finite-size scaling at first-order phase transitions is also discussed, since the finite-size scaling length enters into the definition of the relevant scaled external field variable provided the system has d ′>0 infinite dimensions. The review closes with a summary and short discussion of some open problems.

The totally asymmetric exclusion process on a ring: Exact relaxation dynamics and associated model of clustering transition

The totally asymmetric simple exclusion process in discrete time is considered on finite rings with fixed number of particles. A translation-invariant version of the backward-ordered sequential update is defined for periodic boundary conditions. We prove that the so defined update leads to a stationary state in which all possible particle configurations have equal probabilities. Using the exact analytical expression for the propagator, we find the generating function for the conditional probabilities, average velocity and diffusion constant at all stages of evolution. An exact and explicit expression for the stationary velocity of TASEP on rings of arbitrary size and particle filling is derived. The evolution of small systems towards a steady state is clearly demonstrated. Considering the generating function as a partition function of a thermodynamic system, we study its zeros in planes of complex fugacities. At long enough times, the patterns of zeroes for rings with increasing size provide evidence for a transition of the associated two-dimensional lattice paths model into a clustered phase at low fugacities.

An investigation of finite-size scaling for systems with long-range interaction: The spherical model

A method is suggested for the derivation of finite-size corrections in the thermodynamic functions of systems with pair interaction potential decaying at large distancesr asr−d −σ, whered is the space dimensionality andσ>0. It allows for a unified treatment of short-range (σ=2) and long-range (σ<2) interaction. The asymptotic analysis is illustrated by the mean spherical model of general geometryLd−d′×∞d′ subject to periodic boundary conditions. The Fisher-Privman equation of state is generalized to arbitrary real values ofd⩾σ, 0⩽d′⩽σ. It is shown that theε-expansion may be used to study the breakdown of standard finite-size scaling at the borderline dimensionalities.

New surface critical exponents in the spherical model

The three-dimensional mean spherical model with a L-layer film geometry, under Neumann - Neumann and Neumann - Dirichlet boundary conditions is considered. Surafce fields and are supposed to act at the surfaces bounding the system. In the case of Neumann boundary conditions a new surface critical exponent is found. It is argued that this exponent corresponds to a special (surface - bulk) phase transition in the model. The Privman - Fisher scaling hypothesis for the free energy is verified and the corresponding scaling functions for both the Neumann - Neumann and Neumann - Dirichlet boundary conditions are explicitly derived. If the layer field is applied at some distance from the Dirichlet boundary, a family of critical exponents emerges: their values depend on the exponent defining how the distance scales with the finite size of the system, and interpolate continuously between the extreme cases and .

A five-vertex model interpretation of one-dimensional traffic flow

Here we solve a discrete one-dimensional traffic flow problem by mapping the allowed sets of car trajectories onto a line representation of the five-vertex model configurations. The fundamental flow diagram, obtained previously in a grand canonical ensemble, is rederived. Fluctuations of the flow are described quantitatively and two critical exponents are defined. The zero-density limit is studied by considering an ensemble of single directed self-avoiding loops on a finite torus.

Surface critical exponents for a three-dimensional modified spherical model

A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some � > 0, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility �1,1 has been evaluated exactly. For � = 1 we find that �1,1 is finite at the bulk critical temperature Tc, in contrast with the recently derived value 1,1 = 1 in the case of just one global spherical constraint. The result 1,1 = 1 is recovered only if � = �c = 2 − (12Kc) 1 , where Kc is the dimensionless critical coupling. When � > �c, �1,1 diverges exponentially as T → T + c . An effective hamiltonian which leads to an exactly solvable model with 1,1 = 2, the value for the n → ∞ limit of the corresponding O(n) model, is proposed too.

Asymptotic behavior of the solutions of the poisson-boltzmann equation and its physical consequences

Abstract Expressions are derived describing the asymptotic behavior of the solutions of the Poisson-Boltzmann equation (PBE) with increase of the surface potential or surface charge. In contrast to the solutions of the linearized PBE, the exact solutions remain finite everywhere in the aqueous medium at infinite surface charge density. This peculiarity gives rise to some notable physical effects: (a) the electrostatic disjoining pressure P el in a liquid film of fixed width cannot exceed a finite limiting value at any surface charge density; (b) at constant pressure, the width of a liquid film tends to a finite value at infinite increase of its surface charge; (c) the ζ-potential of an interface is limited from above and becomes insensitive to the value of the surface charge if the latter is sufficiently high. Numerical estimates show that the specific asymptotic behavior of the exact solutions of PBE manifests itself at surface charge densities which are readily accessible in experimental measurements. A comparison with the data available about foam films and lyotropic lamellar lipid phases provides evidence that P el and the width of a liquid film may become indistinguishably close to their asymptotic values. Thus, the asymptotic effects may be used also as a criterion for the extent of validity of the standard Gouy-Chapman theory.

Position dependence of the particle density in a double-chain section of a linear network in a totally asymmetric simple exclusion process.

We report here results on the study of the totally asymmetric simple exclusion processes (TASEP), defined on an open network, consisting of head and tail simple chain segments with a double-chain section inserted in-between. Results of numerical simulations for relatively short chains reveal an interesting new feature of the network. When the current through the system takes its maximum value, a simple translation of the double-chain section forward or backward along the network, leads to a sharp change in the shape of the density profiles in the parallel chains, thus affecting the total number of cars in that part of the network. In the symmetric case of equal injection and ejection rates α = β > 1/2 and equal lengths of the head and tail sections, the density profiles in the two parallel chains are almost linear, characteristic for the coexistence line (shock phase). Upon moving the section forward (backward), their shape changes to the one typical for the high (low) density phases of a simple chain. The total bulk density of cars in a section with a large number of parallel chains is evaluated too. The observed effect might have interesting implications for the traffic flow control as well as for biological transport processes in living cells. An explanation of this phenomenon is offered in terms of finite-size dependence of the effective injection and ejection rates at the ends of the double-chain section.

Exact density profiles for the fully asymmetric exclusion process with discrete-time dynamics on semi-infinite chains.

Exact density profiles in the steady state of the one-dimensional fully asymmetric simple-exclusion process on a semi-infinite chain are obtained in the case of forward-ordered sequential dynamics by taking the thermodynamic limit in our recent exact results for a finite chain with open boundaries. The corresponding results for sublattice-parallel dynamics follow from the relationship obtained by Rajewsky and Schreckenberg [Physica A 245, 139 (1997)], and for parallel dynamics from the mapping found by Evans, Rajewsky, and Speer [J. Stat. Phys. 95, 45 (1999)]. Our analytical expressions involve Laplace-type integrals, rather than complicated combinatorial expressions, which makes them convenient for taking the limit of a semi-infinite chain, and for deriving the asymptotic behavior of the density profiles at large distances from its end. By comparing the asymptotic results appropriate for parallel update with those published in the above cited paper by Evans, Rajewsky, and Speer, we find complete agreement except in two cases, in which we correct technical errors in the final results given there.