J. M. Luck

Accumulation of arsenic from acidic mine waters by ferruginous bacterial accretions (stromatolites)

In the acidic stream (pH 2.2–4) of the Carnoules Pb-(Zn) mine, Gard, France, very high As contents (from 9 to 20%) can be accumulated as ferric arsenate and arsenate-sulphate precipitates in rapidly growing bacteria-made structures. The main bacterial forms are rod-shaped and sheathed, their sheath is made of Fe-As-rich material and is coated with ferric arsenate colloidal particles or may be partially included in authigenic crystals. Living forms ofThiobacillus-type bacteria have been recognized in the precipitates. The cyclic development of bacterial colonies alternating with sand deposition and erosive episodes results in the formation of As-rich ferruginous accretions. These laminated and dome-shaped bacterial constructions are similar to those of stromatolites. The extremely high contents of solute As in upstream flow (250 mg/1) are lowered by 2–3 order of magnitude downstream. Lead is also precipitated and concentrated in this FeAs-rich bacterial stromatolite (2500 ppm Pb). This accumulation and concentration of As and heavy metals via direct or induced microbial action limits pollution downflow. But seasonal storms could erode these FeAsPb-rich deposits and drastically increase pollution. The accumulation of ferric arsenate by bacterial stromatolites suggests that possible microbial remediation strategies may be used in acid mine drainage environments.

Localization of the Maximal Entropy Random Walk

We define a new class of random walk processes which maximize entropy. This maximal entropy random walk is equivalent to generic random walk if it takes place on a regular lattice, but it is not if the underlying lattice is irregular. In particular, we consider a lattice with weak dilution. We show that the stationary probability of finding a particle performing maximal entropy random walk localizes in the largest nearly spherical region of the lattice which is free of defects. This localization phenomenon, which is purely classical in nature, is explained in terms of the Lifshitz states of a certain random operator.

Nonequilibrium critical dynamics of ferromagnetic spin systems

We use simple models (the Ising model in one and two dimensions, and the spherical model in arbitrary dimension) to put to the test some recent ideas on the slow dynamics of nonequilibrium systems. In this review the focus is on the temporal evolution of two-time quantities and on the violation of the fluctuation-dissipation theorem, with special emphasis given to nonequilibrium critical dynamics.

Dynamics of the condensate in zero-range processes

For stochastic processes leading to condensation, the condensate, once it is formed, performs an ergodic stationary-state motion over the system. We analyse this motion, and especially its characteristic time, for zero-range processes. The characteristic time is found to grow with the system size much faster than the diffusive time scale, but not exponentially fast. This holds both in the mean-field geometry and on finite-dimensional lattices. In the generic situation where the critical mass distribution follows a power law, the characteristic time grows as a power of the system size.

Nonequilibrium dynamics of urn models

Dynamical urn models, such as the Ehrenfest model, played an important role in the early days of statistical mechanics. Dynamical many-urn models generalize the former models in two respects: the number of urns is macroscopic, and thermal effects are included. These many-urn models are exactly solvable in the mean-field geometry. They allow analytical investigations of the characteristic features of nonequilibrium dynamics referred to as aging, including the scaling of correlation and response functions in the two-time plane and the violation of the fluctuation-dissipation theorem. This paper contains a general presentation of these models, as well as a more detailed description of two dynamical urn models, the backgammon model and the zeta urn model.

Aging, Phase Ordering, and Conformal Invariance

In a variety of systems which exhibit aging, the two-time response function scales as R(t,s) approximately s(-1-a)f(t/s). We argue that dynamical scaling can be extended towards conformal invariance, thus obtaining the explicit form of the scaling function f. This quantitative prediction is confirmed in several spin systems, both for T<T(c) (phase ordering) and T = T(c) (nonequilibrium critical dynamics). The 2D and 3D Ising models with Glauber dynamics are studied numerically, while exact results are available for the spherical model with a nonconserved order parameter, both for short-ranged and long-ranged interactions, as well as for the mean-field spherical spin glass.

Geochemistry and water dynamics: II. Trace metals and Pb–Sr isotopes as tracers of water movements and erosion processes

Abstract A dynamic scheme for water movements over a flood in a small watershed was proposed in a previous paper [Ben Othman, D., Luck, J.M., Tournoud, M.G., 1997. Geochemistry and water dynamics: application to short-time scale flood phenomena in a small Mediterranean catchment: I. Alkalis, alkali-earths and Sr isotopes. Chem. Geol. [140] 9–28] based on major, alkali–alkali earth trace elements and Sr isotopes. In the present paper, metal contents and Pb isotope data are reported for the same samples, and compared to natural and anthropogenic local sources analysed for this purpose. Water treatment plants have stable 206 Pb/ 204 Pb but show relatively large 207 Pb/ 204 Pb variations, presumably related to yet unidentified (industrial?) Pb sources. Some Cu–Zn-rich chemicals used extensively on vineyards have unradiogenic 206 Pb/ 204 Pb around 17.7. Local rains sampled over two years show roughly similar low values. Pb from rocks is variable and more radiogenic ( 206 Pb/ 204 Pb ≈18.8–22.8). Except for the first hours, trace element concentrations in the dissolved load are similar to or slightly higher than those observed over the year, and similar to other moderately anthropogenic areas [Shiller, A.M., Boyle, E.A., 1987. Variability of dissolved trace metals in the Mississippi river. Geochim. Cosmochim. Acta [51] 3273–3277] for [Pb] (0.05–0.45 nM), [Zn] (10–75 nM) and [Cd] (0.03–0.18 nM). Dissolved [U] and [Co] show simple variations correlated to carbonate, related to local natural sources. Most trace elements in the particulate load are strongly correlated. Lead isotopes in the dissolved and particulate loads show ranges over the flood, again similar to those observed over the year ( 206 Pb/ 204 Pb ≈17.9–18.3), implying the same sources. The very good alignments observed in 207 Pb/ 204 Pb and 208 Pb/ 204 Pb vs. 206 Pb/ 204 Pb diagrams, especially for the particulate, are interpreted as mixing phenomena. Generally Pb isotopic signature of the dissolved load is less radiogenic than the particulate, indicating differences in sources or proportions and absence of isotopic equilibration with respect to the time of transfer. Pb isotopes shift regularly with time away from the road runoff endmember towards more radiogenic values. The strong negative correlation between Pb isotopes in the particulate load and Sr isotopes in the dissolved load, observed for the first time in a small watershed, probably reflects the local coupling between mechanical and chemical erosion, respectively.

Heterogeneities in granular materials

Different parts of a sandpile can exhibit very different dynamical behaviors ranging from jammed to fluid. To understand them, one needs to look at the networks of contacts between individual grains.

Fluctuation effects in metapopulation models: percolation and pandemic threshold.

Metapopulation models provide the theoretical framework for describing disease spread between different populations connected by a network. In particular, these models are at the basis of most simulations of pandemic spread. They are usually studied at the mean-field level by neglecting fluctuations. Here we include fluctuations in the models by adopting fully stochastic descriptions of the corresponding processes. This level of description allows to address analytically, in the SIS and SIR cases, problems such as the existence and the calculation of an effective threshold for the spread of a disease at a global level. We show that the possibility of the spread at the global level is described in terms of (bond) percolation on the network. This mapping enables us to give an estimate (lower bound) for the pandemic threshold in the SIR case for all values of the model parameters and for all possible networks.

A record-driven growth process

We introduce a novel stochastic growth process, the record-driven growth process, which originates from the analysis of a class of growing networks in a universal limiting regime. Nodes are added one by one to a network, each node possessing a quality. The new incoming node connects to the pre-existing node with best quality, that is, with record value for the quality. The emergent structure is that of a growing network, where groups are formed around record nodes (nodes endowed with the best intrinsic qualities). Special emphasis is put on the statistics of leaders (nodes whose degrees are the largest). The asymptotic probability for a node to be a leader is equal to the Golomb–Dickman constant ω = 0.624 329 ..., which arises in problems of combinatorial nature. This outcome solves the problem of the determination of the record breaking rate for the sequence of correlated inter-record intervals. The process exhibits temporal self-similarity in the late-time regime. Connections with the statistics of the cycles of random permutations, the statistical properties of randomly broken intervals, and the Kesten variable are given.