Prof John Cardy FRS

Abstract:

We investigate a lattice scalar field theory in the presence of a bias favouring the establishment of an energy current, as a model for stationary non-equilibrium processes at low temperature in a non-integrable system. There is a transition at a finite value of the bias to a gapless modulated phase which carries a classical current; however, unlike in similar, integrable, models, quantum effects also allow for a non-zero current at arbitrarily small bias. The transition is second-order in the magnetically disordered phase, but is pre-empted by a first-order transition in the ferromagnetic case, at least at the mean-field level. © 2000 Elsevier Science B.V. All rights reserved.

Abstract:

The high-temperature susceptibility of the q-state Potts model behaves as Γ|T - Tc|-y as T → Tc + , while for T → Tc - one may define both longitudinal and transverse susceptibilities, with the same power law but different amplitudes ΓL and ΓT. We extend a previous analytic calculation of the universal ratio Γ/ΓL in two dimensions to the low-temperature ratio ΓT/ΓL, and test both predictions with Monte Carlo simulations for q = 3 and 4. The data for q = 4 are inconclusive owing to large corrections to scaling, while for q = 3 they appear consistent with the prediction for ΓT/ΓL, but not with that for Γ/ΓL. A simple extrapolation of our analytic results to q → 1 indicates a similar discrepancy with the corresponding measured quantities in percolation. We point out that stronger assumptions were made in the derivation of the ratio Γ/ΓL, and our work suggests that these may be unjustified. © 2000 Elsevier Science B.V. All rights reserved.

Abstract:

The single-species reaction-diffusion process A + A → O is examined in the presence of an uncorrelated, quenched random velocity field. Utilizing a field-theoretic approach, we find that in two dimensions and below the density decay is altered from the case of purely diffusing reactants. In two dimensions the density amplitude is reduced in the presence of weak disorder, yielding the interesting result that Sinai disorder can cause reactions to occur at an increased rate. This is in contrast to the case of long-range correlated disorder, where it was shown that the reaction becomes sub-diffusion limited. However, when written in terms of the microscopic diffusion constant it is seen that increasing the disorder has the effect of reducing the rate of the reaction. Below two dimensions, the effect of Sinai disorder is much more severe and the reaction is shown to become sub-diffusion limited. Although there is no universal amplitude for the time-dependence of the density, it is universal when expressed in terms of the disorder-averaged diffusion length. The appropriate amplitude is calculated to one-loop order.

Abstract:

The reversible reactions A + A ⇌ C and A + B ⇌ C are investigated. From the exact Langevin equations describing our model, we set up a systematic approximation scheme to compute the approach of the density of C particles to its equilibrium value. We show that for a sufficiently long time t, this approach takes the form of a power law At-d/2, for any dimension d. The amplitude A is also computed exactly, but is expected to be model dependent. For uncorrelated initial conditions, the C density turns out to be a monotonic time function. The cases of correlated initial conditions and unequal diffusion constants are investigated as well. In the former, correlations may break the monotonicity of the density or in some special cases they may change the long time behaviour. For the latter, the power law remains valid, only the amplitude changes, even in the extreme case of immobile C particles. We also consider the case of segregated initial condition for which a reaction front is observed, and confirm that its width is governed by mean-field exponent in any dimension.