Invasion front dynamics in disordered environments

Invasion occurs in environments that are normally spatially disordered, however, the effect of such a randomness on the dynamics of the invasion front has remained less understood. Here, we study Fisher’s equation in disordered environments both analytically and numerically. Using the Effective Medium Approximation, we show that disorder slows down invasion velocity and for ensemble average of invasion velocity in disordered environment we have $$\bar{v}=v_0 (1-|\xi |^2/6)$$ v ¯ = v 0 ( 1 - | ξ | 2 / 6 ) where $$|\xi |$$ | ξ | is the amplitude of disorder and $$v_0$$ v 0 is the invasion velocity in the corresponding homogeneous environment given by $$v_0=2\sqrt{RD_0}$$ v 0 = 2 R D 0 . Additionally, disorder imposes fluctuations on the invasion front. Using a perturbative approach, we show that these fluctuations are Brownian with a diffusion constant of: $$D_{C}= \dfrac{1}{8} \xi ^2\sqrt{RD_0 (1-|\xi |^2/3)}$$ D C = 1 8 ξ 2 R D 0 ( 1 - | ξ | 2 / 3 ) . These findings were approved by numerical analysis. Alongside this continuum model, we use the Stepping Stone Model to check how our findings change when we move from the continuum approach to a discrete approach. Our analysis suggests that individual-based models exhibit inherent fluctuations and the effect of environmental disorder becomes apparent for large disorder intensity and/or high carrying capacities.

LOW TEMPERATURE NEUTRON DIFFRACTION ON CONGRUENT AND NEAR STOICHIOMETRIC LiNbO3

Neutron powder diffraction has been carried out on a congruent LiNbO 3 sample containing 7 Li isotope ( C 7 LN ) and a near stoichiometric Mg doped LiNbO 3 sample ( Mg : NSLN ) in the temperature range of 4 K and 90 K. Large anisotropic displacement parameters (ADPs) of the Li ions have shown evidence of large disorder along the c-axis for both samples. The results have shown no evidence for the existence of anomalous structural behavior for both samples at low temperatures, although abnormal structural features at 55 K and 100 K for a LiNbO 3 crystal having different Li content as the samples used in the present studies have been observed by Fernandez-Ruiz et al. [Phys. Rev. B72 (2005) 184108].

A COMPARISON OF HARMONIC CONFINEMENT AND DISORDER IN INDUCING LOCALIZATION EFFECTS IN A SUPERCONDUCTOR

We present a comparative study of the localization effects induced by harmonic trapping and random onsite disorder in a 2D s-wave superconductor. Performing a numeric computation of the Bogoliubov-de Gennes equations in the presence of parabolic trap and random disorder, we obtain the eigensolutions as a function of trap depth and disorder strength. While the wavefunctions demonstrate localization tendencies for moderate disorder strengths, with finally yielding an insulating behavior at large disorder, the harmonic confinement effects are seen to be vastly more drastic in inducing localization effects.

Faults self-organized by repeated earthquakes in a quasi-static antiplane crack model

We study a 2D quasi-static discrete crack anti-plane model of a tectonic plate with long range elastic forces and quenched disorder. The plate is driven at its border and the load is transferred to all elements through elastic forces. This model can be considered as belonging to the class of self-organized models which may exhibit spontaneous criticality, with four additional ingredients compared to sandpile models, namely quenched disorder, boundary driving, long range forces and fast time crack rules. In this "crack" model, as in the "dislocation" version previously studied, we find that the occurrence of repeated earthquakes organizes the activity on well-defined fault-like structures. In contrast with the "dislocation" model, after a transient, the time evolution becomes periodic with run-aways ending each cycle. This stems from the "crack" stress transfer rule preventing criticality to organize in favour of cyclic behaviour. For sufficiently large disorder and weak stress drop, these large events are preceded by a complex spacetime history of foreshock activity, characterized by a Gutenberg-Richter power law distribution with universal exponent B = 1±0.05. This is similar to a power law distribution of small nucleating droplets before the nucleation of the macroscopic phase in a first-order phase transition. For large disorder and large stress drop, and for certain specific initial disorder configurations, the stress field becomes frustrated in fast time: out-of-plane deformations (thrust and normal faulting) and/or a genuine dynamics must be introduced to resolve this frustration.