Jordan G Brankov

On the Finite-Size Scalling Equation for the Spherical Model

The mean spherical model with an arbitrary interaction potential, the Fourier transform of which has a long-wavelength exponent σ, 0<σ⩽2, is considered under periodic boundary conditions and fully finite geometry ind dimensions, when σ<d<2σ. A new form of the finite-size scaling equation for the spherical field in the critical region is derived, which relates the temperature shift to Madelung-type lattice constants. The method of derivation makes use of the Poisson summation formula and a Laplace transformation of the momentumspace correlation function.

Finite-size scaling for the mean spherical model with inverse power law interaction

A new analytical technique based on integral transformations with Mittag-Leffler-type kernels is used to derive the finite-size scaling function for the free energy per particle of the mean spherical model with inverse power law asymptotics of the interaction potential. The asymptotic formation of the singularities in the specific heat and magnetic susceptibility at the bulk critical point is studied.

An exactly solvable model for metal-insulator phase transition

A simple model of electrons interacting with photons which displays a metal-insulator phase transition in the case of s.c. and b.c.c. tight-binding bands is studied. A proof is given that the model is exactly solvable in the thermodynamic limit.

Finite-size effects in the approximating Hamiltonian method

The Husimi-Temperley mean spherical model, in which each two particles interact with equal strength, is considered. This model is shown to be equivalent to a d-dimensional model with periodic boundary conditions and interaction potential σJσ(r), where Jσ(r) ∼ r−d−σ as r→∞, σ > 0 being a parameter, in the limit σ→0. It is found that the approximating Hamiltonian method yields singular finite-size scaling functions both in the neighbourhood of the critical point and near a first-order phase transition. A modification of this method is suggested, which allows for all the essential configurations and reproduces the exact finite-size scaling near a first-order phase transition.

On finite-size scaling in the presence of dangerous irrelevant variables

A scaling hypothesis on finite-size scaling in the presence of a dangerous irrelevant variable is formulated for systems with long-range interaction and general geometryLd−d′×∞d′. A characteristic length which obeys a universal finite-size scaling relation is defined. The general conjectures are based on exact results for the mean spherical model with inverse power law interaction.

Effects of lattice distortion on the energy gap of a BCS superconductor

The model of Mattis and Langer for superconductors with a structural distortion associated with doubling of the lattice periodicity is studied. New results for the zero-temperature superconducting gap are found in the case of more than a half-filled band. The modification of both the electron density of states and the reduced BCS interaction is taken into account. A comparison with the results found for the additive type model is given.

On a class of exactly soluble statistical mechanical models with nonpolynomial interactions

The approximating Hamiltonian method of N. N. Bogolubov, Jr. is generalized to models with nonpolynomial intensive-observable interactions. The original Hamiltonian is proved to be thermodynamically equivalent to one linear in the intensive-observable trial Hamiltonian. We show that the exact expression for the free energy density in the thermodynamic limit can be obtained from a min-max principle for the system with trial Hamiltonian.

On the appearance of traffic jams in a long chain with a shortcut in the bulk

The Totally Asymmetric Simple Exclusion Process (TASEP) is studied on open long chains with a shunted section between two simple chain segments in the maximum current phase. The reference case, when the two branches are chosen with equal probability, is considered. The conditions for the occurrence of traffic jams and their properties are investigated both within the effective rates approximation and by extensive Monte Carlo simulations for arbitrary length of the shortcut. Our main results are: (1) For any length of the shortcut and any values of the external rates in the domain of the maximum current phase, there exists a position of the shortcut where the shunted segment is in a phase of coexistence with a completely delocalized domain wall; (2) The main features of the coexistence phase and the density profiles in the whole network are well described by the domain wall theory. Apart from the small inter-chain correlations, they depend only on the current through the shortcut; (3) The model displays unexpected features: (a) the current through the longer shunted segment is larger than the current through the shortcut, and (b) the delocalized domain wall in the coexistence phase of the long shunted segment induces similar behavior even in shortcuts containing a small number of sites; (4) From the viewpoint of vehicular traffic, most comfortable conditions for the drivers are provided when the shortcut is shifted downstream from the position of coexistence, when both the shunted segment and the shortcut exhibit low-density lamellar flow. Most unfavorable is the opposite case of upstream shifted shortcut, when both the shunted segment and the shortcut are in a high-density phase describing congested traffic of slowly moving cars. The above results are relevant also to phenomena like crowding of molecular motors moving along twisted protofilaments.