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We introduce and study a Lévy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.
We introduce and study a Lévy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.
Li and B in ascending magmas: an experimental study on their mobility and isotopic fractionation
(2022)
This research study focuses on the behaviour of Li and B during magmatic ascent, and decompression-driven degassing related to volcanic systems. The main objective of this dissertation is to determine whether it is possible to use the diffusion properties of the two trace elements as a tool to trace magmatic ascent rate. With this objective, diffusion-couple and decompression experiments have been performed in order to study Li and B mobility in intra-melt conditions first, and then in an evolving system during decompression-driven degassing.
Synthetic glasses were prepared with rhyolitic composition and an initial water content of 4.2 wt%, and all the experiments were performed using an internally heated pressure vessel, in order to ensure a precise control on the experimental parameters such as temperature and pressure.
Diffusion-couple experiments were performed with a fix pressure 300 MPa. The temperature was varied in the range of 700-1250 °C with durations between 0 seconds and 24 hours. The diffusion-couple results show that Li diffusivity is very fast and starts already at very low temperature. Significant isotopic fractionation occurs due to the faster mobility of 6Li compared to 7Li. Boron diffusion is also accelerated by the presence of water, but the results of the isotopic ratios are unclear, and further investigation would be necessary to well constrain the isotopic fractionation process of boron in hydrous silicate melts. The isotopic ratios results show that boron isotopic fractionation might be affected by the speciation of boron in the silicate melt structure, as 10B and 11B tend to have tetrahedral and trigonal coordination, respectively.
Several decompression experiments were performed at 900 °C and 1000 °C, with pressures going from 300 MPa to 71-77 MPa and durations of 30 minutes, two, five and ten hours, in order to trigger water exsolution and the formation of vesicles in the sample. Textural observations and the calculation of the bubble number density confirmed that the bubble size and distribution after decompression is directly proportional to the decompression rate.
The overall SIMS results of Li and B show that the two trace elements tend to progressively decrease their concentration with decreasing decompression rates. This is explained because for longer decompression times, the diffusion of Li and B into the bubbles has more time to progress and the melt continuously loses volatiles as the bubbles expand their volumes.
For fast decompression, Li and B results show a concentration increase with a δ7Li and δ11B decrease close to the bubble interface, related to the sudden formation of the gas bubble, and the occurrence of a diffusion process in the opposite direction, from the bubble meniscus to the unaltered melt. When the bubble growth becomes dominant and Li and B start to exsolve into the gas phase, the silicate melt close to the bubble gets depleted in Li and B, because of a stronger diffusion of the trace elements into the bubble.
Our data are being applied to different models, aiming to combine the dynamics of bubble nucleation and growth with the evolution of trace elements concentration and isotopic ratios. Here, first considerations on these models will be presented, giving concluding remarks on this research study. All in all, the final remarks constitute a good starting point for further investigations. These results are a promising base to continue to study this process, and Li and B can indeed show clear dependences on decompression-related magma ascent rates in volcanic systems.
Humankind and their environment need to be protected from the harmful effects of spent nuclear fuel, and therefore disposal in deep geological formations is favoured worldwide. Suitability of potential host rocks is evaluated, among others, by the retention capacity with respect to radionuclides. Safety assessments are based on the quantification of radionuclide migration lengths with numerical simulations as experiments cannot cover the required temporal (1 Ma) and spatial scales (>100 m).
Aim of the present thesis is to assess the migration of uranium, a geochemically complex radionuclide, in the potential host rock Opalinus Clay. Radionuclide migration in clay formations is governed by diffusion due to their low permeability and retarded by sorption. Both processes highly depend on pore water geochemistry and mineralogy that vary between different facies. Diffusion is quantified with the single-component (SC) approach using one diffusion coefficient for all species and the process-based multi-component (MC) option. With this, each species is assigned its own diffusion coefficient and the interaction with the diffuse double layer is taken into account. Sorption is integrated via a bottom-up approach using mechanistic surface complexation models and cation exchange. Therefore, reactive transport simulations are conducted with the geochemical code PHREEQC to quantify uranium migration, i.e. diffusion and sorption, as a function of mineralogical and geochemical heterogeneities on the host rock scale.
Sorption processes are facies dependent. Migration lengths vary between the Opalinus Clay facies by up to 10 m. Thereby, the geochemistry of the pore water, in particular the partial pressure of carbon dioxide (pCO2), is more decisive for the sorption capacity than the amount of clay minerals. Nevertheless, higher clay mineral quantities compensate geochemical variations. Consequently, sorption processes must be quantified as a function of pore water geochemistry in contact with the mineral assemblage.
Uranium diffusion in the Opalinus Clay is facies independent. Speciation is dominated by aqueous ternary complexes of U(VI) with calcium and carbonate. Differences in the migration lengths between SC and MC diffusion are with +/-5 m negligible. Further, the application of the MC approach highly depends on the quality and availability of the underlying data. Therefore, diffusion processes can be adequately quantified with the SC approach using experimentally determined diffusion coefficients.
The hydrogeological system governs pore water geochemistry within the formation rather than the mineralogy. Diffusive exchange with the adjacent aquifers established geochemical gradients over geological time scales that can enhance migration by up to 25 m. Consequently, uranium sorption processes must be quantified following the identified priority: pCO2 > hydrogeology > mineralogy.
The presented research provides a workflow and orientation for other potential disposal sites with similar pore water geochemistry due to the identified mechanisms and dependencies. With a maximum migration length of 70 m, the retention capacity of the Opalinus Clay with respect to uranium is sufficient to fulfill the German legal minimum requirement of a thickness of at least 100 m.
We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.
We analyse mobile-immobile transport of particles that switch between the mobile and immobile phases with finite rates. Despite this seemingly simple assumption of Poissonian switching, we unveil a rich transport dynamics including significant transient anomalous diffusion and non-Gaussian displacement distributions. Our discussion is based on experimental parameters for tau proteins in neuronal cells, but the results obtained here are expected to be of relevance for a broad class of processes in complex systems. Specifically, we obtain that, when the mean binding time is significantly longer than the mean mobile time, transient anomalous diffusion is observed at short and intermediate time scales, with a strong dependence on the fraction of initially mobile and immobile particles. We unveil a Laplace distribution of particle displacements at relevant intermediate time scales. For any initial fraction of mobile particles, the respective mean squared displacement (MSD) displays a plateau. Moreover, we demonstrate a short-time cubic time dependence of the MSD for immobile tracers when initially all particles are immobile.