Convolution of multifractals and the local magnetization in a random field Ising chain

The local magnetization in the one-dimensional random-field Ising model is essentially the sum of two effective fields with multifractal probability measure. The probability measure of the local magnetization is thus the convolution of two multifractals. In this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution. The pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures. The generalized box dimensions of the convolution of Cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization. Returning to the study of this measure we apply the general inequalities and present numerical approximations of the D_q-spectrum. For the first time we are able to obtain results on multifractal properties of a physical quantity in the one-dimensional random-field Ising model which in principle could be measured experimentally. The numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h.Comment: An error in figure 1 was corrected, small additions were made to the introduction and the conclusions, some typos were corrected, 24 pages, LaTeX2e, 9 figure

Orbits and phase transitions in the multifractal spectrum

We consider the one dimensional classical Ising model in a symmetric dichotomous random field. The problem is reduced to a random iterated function system for an effective field. The D_q-spectrum of the invariant measure of this effective field exhibits a sharp drop of all D_q with q < 0 at some critical strength of the random field. We introduce the concept of orbits which naturally group the points of the support of the invariant measure. We then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so-called offshoots as well as for generic orbits in the non-overlapping case. The sharp drop in the D_q-spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field which is explicitly given. A similar drastic change near the points of a special family of periodic orbits explains a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a decisive role in this mechanism is played by a specific offshoot. We furthermore give rigorous upper and/or lower bounds on all D_q in a wide parameter range. In most cases the numerically obtained D_q coincide with either the upper or the lower bound. The results in this paper are relevant for the understanding of random iterated function systems in the case of moderate overlap in which periodic orbits with weak singularity can play a decisive role.Comment: The article has been completely rewritten; the title has changed; a section about the typical pointwise dimension as well as several references and remarks about more general systems have been added; to appear in J. Phys. A; 25 pages, 11 figures, LaTeX2

Phase diagram of the random field Ising model on the Bethe lattice

The phase diagram of the random field Ising model on the Bethe lattice with a symmetric dichotomous random field is closely investigated with respect to the transition between the ferromagnetic and paramagnetic regime. Refining arguments of Bleher, Ruiz and Zagrebnov [J. Stat. Phys. 93, 33 (1998)] an exact upper bound for the existence of a unique paramagnetic phase is found which considerably improves the earlier results. Several numerical estimates of transition lines between a ferromagnetic and a paramagnetic regime are presented. The obtained results do not coincide with a lower bound for the onset of ferromagnetism proposed by Bruinsma [Phys. Rev. B 30, 289 (1984)]. If the latter one proves correct this would hint to a region of coexistence of stable ferromagnetic phases and a stable paramagnetic phase.Comment: Article has been condensed and reorganized; Figs 3,5,6 merged; Fig 4 omitted; Some discussion added at end of Sec. III; 9 pages, 5 figs, RevTeX4, AMSTe

The randomly driven Ising ferromagnet, Part I: General formalism and mean field theory

We consider the behavior of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast switching, random external field. After introducing a general formalism for describing such systems, we consider here the mean-field theory. A novel type of first order phase transition related to spontaneous symmetry breaking and dynamic freezing is found. The non-equilibrium stationary state has a complex structure, which changes as a function of parameters from a singular-continuous distribution with Euclidean or fractal support to an absolutely continuous one.Comment: 12 pages REVTeX/LaTeX format, 12 eps/ps figures. Submitted to Journal of Physics

Evaluating Functional Diversity as Potential Early-Warning Indicator of Rangeland Degradation

Droughts and overgrazing play a crucial role in the degradation of semi-arid rangelands. This is evident in the loss of palatable long-lived grass species and bush encroachment. Early warning indicators are needed to mitigate long-term degradation and decline in essential forage provision. Functional diversity provides valuable information on ecosystem health. However, functional diversity indices have not yet been tested regarding their applicability as early warning indicators, revealing non-linear threshold behaviour. We therefore examined the following questions: (1) How do functional diversity indices respond to grazing pressure? (2) Does land tenure affect the relationship between functional diversity and grazing pressure? (3) Are functional diversity indices suitable early-warning indicators? To answer these questions, we conducted a space-for-time substitution of land use intensity of semi-arid rangelands in Namibia. Some 16 grazing gradients were selected, each starting at a cattle watering point where grazing pressure was highest. Gradients were located in four communal and four freehold farms. Communal farms were characterised by continuous grazing, while freehold farms by rotational grazing. In each transect we recorded plant species composition of the grass layer in 9 plots of 10 × 10 m each (N = 162 plots). Within each transect, these plots were logarithmically distributed. Various plant functional traits—all relating to plant life history or resource acquisition strategy—were measured for 142 dominant species, accounting for more than 80 % of the biomass, and indices of functional diversity were calculated. We found potential threshold behaviour in functional richness on freehold farms. Certain functional diversity indices revealed non-linear patterns in rangelands but are currently not a user-friendly early-warning indicator. To harness functional diversity, we need a more standardized method of calculation, and more functional trait databases for sub-Saharan species

Fracture propagation to the base of the Greenland Ice Sheet during supraglacial lake drainage

Author Posting. © The Authors, 2008. This is the author's version of the work. It is posted here by permission of American Association for the Advancement of Science for personal use, not for redistribution. The definitive version was published in Science 320 (2008): 778-781, doi:10.1126/science.1153360.Surface meltwater that reaches the base of an ice sheet creates a mechanism for the rapid response of ice flow to climate change. The process whereby such a pathway is created through thick, cold ice has not, however, been previously observed. We describe the rapid (<2 hours) drainage of a large supraglacial lake down 980 m through to the bed of the Greenland Ice Sheet initiated by water-driven fracture propagation evolving into moulin flow. Drainage coincided with increased seismicity, transient acceleration, ice sheet uplift and horizontal displacement. Subsidence and deceleration occurred over the following 24 hours. The short-lived dynamic response suggests an efficient drainage system dispersed the meltwater subglacially. The integrated effect of multiple lake drainages could explain the observed net regional summer ice speedup.Support was provided jointly by NSF and NASA through ARC-0520077 (S.B.D., M.P.B., I.M.H.) and ARC- 520382 (I.J.); The WHOI OCCI and Clark Arctic Research Initiative provided additional support to S.B.D., M.D.B., and D.L.; and a NERC (UK) Research Fellowship supported M.A.K

First-passage and first-exit times of a Bessel-like stochastic process

We study a stochastic process $X_t$ related to the Bessel and the Rayleigh processes, with various applications in physics, chemistry, biology, economics, finance and other fields. The stochastic differential equation is $dX_t = (nD/X_t) dt + \sqrt{2D} dW_t$, where $W_t$ is the Wiener process. Due to the singularity of the drift term for $X_t = 0$, different natures of boundary at the origin arise depending on the real parameter $n$: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behaviour is observed in the case of a regular boundary.Comment: 15 pages, 6 figures, submitted to Physical Review