Exact results for the Kardar-Parisi-Zhang equation with spatially correlated noise

We investigate the Kardar-Parisi-Zhang (KPZ) equation in d spatial dimensions with Gaussian spatially long-range correlated noise -- characterized by its second moment -- by means of dynamic field theory and the renormalization group. Using a stochastic Cole-Hopf transformation we derive exact exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension . Below the lower critical dimension, there is a line marking the stability boundary between the short-range and long-range noise fixed points. For , the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above , one has to rely on some perturbational techniques. We discuss the location of this stability boundary in light of the exact results derived in this paper, and from results known in the literature. In particular, we conjecture that there might be two qualitatively different strong-coupling phases above and below the lower critical dimension, respectively. Received 5 August 1998     

Non-equilibrium behavior at a liquid-gas critical point

Second-order phase transitions in a non-equilibrium liquid-gas model with reversible mode couplings, i.e., model H for binary-fluid critical dynamics, are studied using dynamic field theory and the renormalization group. The system is driven out of equilibrium either by considering different values for the noise strengths in the Langevin equations describing the evolution of the dynamic variables (effectively placing these at different temperatures), or more generally by allowing for anisotropic noise strengths, i.e., by constraining the dynamics to be at different temperatures in d _|| - and d _⊥-dimensional subspaces, respectively. In the first, isotropic case, we find one infrared-stable and one unstable renormalization group fixed point. At the stable fixed point, detailed balance is dynamically restored, with the two noise strengths becoming asymptotically equal. The ensuing critical behavior is that of the standard equilibrium model H. At the novel unstable fixed point, the temperature ratio for the dynamic variables is renormalized to infinity, resulting in an effective decoupling between the two modes. We compute the critical exponents at this new fixed point to one-loop order. For model H with spatially anisotropic noise, we observe a critical softening only in the d _⊥-dimensional sector in wave vector space with lower noise temperature. The ensuing effective two-temperature model H does not have any stable fixed point in any physical dimension, at least to one-loop order. We obtain formal expressions for the novel critical exponents in a double expansion about the upper critical dimension d _c = 4 - d _|| and with respect to d _|| , i.e., about the equilibrium theory. Received 4 April 2002 Published online 13 August 2002     

Some remarks on pathologies of renormalization-group transformations for the Ising model

The results recently obtained by van Enter, Fernandez, and Sokal on non-Gibbsianness of the measurev =T _{b ,h} arising from the application of a single decimation transformationT _b , with spacingb, to the Gibbs measure _,h , of the Ising model, for suitably chosen large inverse temperature and nonzero external fieldh, are critically analyzed. In particular, we show that if, keeping fixed the same values of, h, andb, one iterates a sufficiently large number of timesn the transformationT _b , one obtains a new measurev = (T _b )_n,h which is Gibbsian and moreover very weakly coupled.     

Scaling behaviour in the fracture of fibrous materials

We study the existence of distinct failure regimes in a model for fracture in fibrous materials. We simulate a bundle of parallel fibers under uniaxial static load and observe two different failure regimes: a catastrophic and a slowly shredding. In the catastrophic regime the initial deformation produces a crack which percolates through the bundle. In the slowly shredding regime the initial deformations will produce small cracks which gradually weaken the bundle. The boundary between the catastrophic and the shredding regimes is studied by means of percolation theory and of finite-size scaling theory. In this boundary, the percolation density scales with the system size L, which implies the existence of a second-order phase transition with the same critical exponents as those of usual percolation. Received 24 June 1999     

An analytical method to compute an approximate value of the site percolation threshold P<Subscript>c</Subscript>

An analytical method to compute the site percolation threshold is introduced. This method yields an approximate value of larger or equal to the real value. As examples, the computation of is presented for 4 lattices in 2 dimensions: square, triangular, honeycomb and kagome. The results obtained are 0.592 871 6, 0.5, 0.765 069, 0.654 653 7, to be compared with the real values 0.592 746 0, 0.5, 0.697 043, 0.652 703 6. The method is not limited to 2 dimensions. Received 27 July 1999 and Received in final form 29 November 1999     

A lattice gas coupled to two thermal reservoirs: Monte Carlo and field theoretic studies

We investigate the collective behavior of an Ising lattice gas, driven to non-equilibrium steady states by being coupled to two thermal baths. Monte Carlo methods are applied to a two-dimensional system in which one of the baths is fixed at infinite temperature. Both generic long range correlations in the disordered state and critical properties near the second order transition are measured. Anisotropic scaling, a key feature near criticality, is used to extract and some critical exponents. On the theoretical front, a continuum theory, in the spirit of Landau-Ginzburg, is presented. Being a renormalizable theory, its predictions can be computed by standard methods of -expansions and found to be consistent with simulation data. In particular, the critical behavior of this system belongs to a universality class which is quite different from the uniformly driven Ising model. Received 4 October 2000     

Novel approaches to parallelism in soft-matter problems

Summary We outline a novel method of exploiting distributed computational resources to study problems in soft-matter science. We have applied our method to simulations of polymers, including hydrodynamic effects. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Planar cracks in the fuse model

We simulate the propagation of a planar crack in a quasi-two dimensional fuse model, confining the crack between two horizontal plates. We investigate the effect on the roughness of microcrack nucleation ahead of the main crack and study the structure of the damage zone. The two-dimensional geometry introduces a characteristic length in the problem, limiting the crack roughness. The damage ahead of the crack does not appear to change the scaling properties of the model, which are well described by gradient percolation. Received 29 March 2000     

Size and randomness effects on the temperature-dependent hopping conductivity of nanocrystalline chains

Developing a renormalization group approach, we study the hopping conductivity of nanocrystalline chains with different site energies. Exact calculations show that many parameters including nano-sizes, randomness of grain distributions, lattice distortions, site energies, transition rates, Fermi energy, and temperature influence the conductivity. Some new singular features, for example the frequency shift, the amplitude fluctuations, and the interchange between “peak” and “valley” behavior of the imaginary part of the conductivity can be caused by certain parameters mentioned above, while the interface distortions modulate mainly the overall amplitudes of the conductivity at the whole frequency region. Received 13 January 2000 and Received in final form 12 September 2000     

Adatom diffusion on a square lattice. Theoretical and numerical studies

A two-dimensional lattice-gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of the method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. The minimal relative error in determining the critical values of the interaction parameters is equal to . Using the RSRG method, we have explored phase diagrams of both a two-dimensional Ising spin model and of a square lattice gas with lateral interactions between adparticles. We also have investigated the influence of the attractive and repulsive interactions on both the thermodynamic properties of the lattice gas and the diffusion of adsorbed particles over surface. We have calculated adsorption isotherms and coverage dependences of the pair correlation function, isothermal susceptibility and the chemical diffusion coefficient. In addition, we have included in our analysis the interaction of the activated particle in the saddle point with its nearest neighbors. We have also used Monte Carlo (MC) technique to calculate these dependences. Despite the fact that both methods constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore, we conclude that the RSRG approach can be applied to characterize the thermodynamic and kinetic properties of systems of particles with strong lateral interactions. Received 1st September 1998 and Received in final form 8 March 2000