A class of finite difference methods for solving fractional diffusion equations is considered. These methods are an extension of the weighted average methods for ordinary (non-fractional) diffusion ...equations. Their accuracy is of order (Δx)2 and Δt, except for the fractional version of the Crank–Nicholson method, where the accuracy with respect to the timestep is of order (Δt)2 if a second-order approximation to the fractional time-derivative is used. Their stability is analyzed by means of a recently proposed procedure akin to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, is found and checked numerically. Some examples are provided in which the new methods’ numerical solutions are obtained and compared against exact solutions.
A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the ...forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grünwald-Letnikov discretization of the Riemann-Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.
Exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature, and yet a finite lifetime is a frequent occurrence, ...and its effects on a number of random walk properties may be profound. For instance, whereas the average number of distinct sites visited by an immortal walker grows with time without bound, that of a mortal walker may, depending on dimensionality and rate of evanescence, remain finite or keep growing with the passage of time. This number can in turn be used to calculate other classic quantities such as the survival probability of a target surrounded by diffusing traps. If the traps are immortal, the survival probability will vanish with increasing time. However, if the traps are evanescent, the target may be spared a certain death. We analytically calculate a number of basic and broadly used quantities for evanescent random walkers.
We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples ...for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.
We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the ...motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.
The wine industry is extremely vulnerable to climate change given the impact of temperature, water or soil conditions on wine production. Transformation strategies toward more sustainable models ...would help the industry to overcome the weak points. The present study evaluates the environmental impact of the so-called CO2-alcoholic fermentation processes (CO2-AFP) strategy, a recently developed eco-innovative strategy, which offers a new pathway toward a greener wine-making production with a 16.79% reduction of the carbon footprint considering scopes 1 (direct emissions), 2 (emissions generated directly in the production of electricity), and 3 (indirect emissions embodied in the organization's value chain). This strategy is based on a novel Carbon Dioxide Utilization approach from a biogenic CO2 released by fermentation processes to produce a fully marketable and environmentally friendly chemical product at a global level, i.e., sodium carbonate.
This paper presents the strategy development corresponding to a real case: a medium-size winery and distillery in Spain, where the “CO2-AFP Strategy” has been successfully tested and scaled-up. Detailed and tested carbon capture and utilization schemes are used to evaluate the overall carbon footprint balance (carbon calculation, capture potential, and carbon balance) via an improved hybrid multiregional input-output-lifecycle assessment model (MRIO-LCA). The benefits go beyond the reduction of carbon footprint in the fermentation industry. The application of the “CO2-AFP Strategy” implies a revolution, in terms of Circular Economy, in the sodium carbonate industry, as the symbiotic process between the different stages of the value chain will allow downstream carbon footprint reductions, facilitating a greener sodium carbonate production.
•The AFP-Strategy proposes a method for carbon capture from alcoholic fermentation.•A real-case cost-effective carbon capture and utilization process is assessed.•The corporate carbon footprint is estimated through a tiered hybrid method.•Multiregional Input-Output and Life-cycle Assessment techniques are combined.•The hybrid model provide a footprint detailed into three scopes.
We address the problem of diffusion on a comb whose teeth display varying lengths. Specifically, the length ℓ of each tooth is drawn from a probability distribution displaying power law behavior at ...large ℓ,P(ℓ)∼ℓ^{-(1+α)} (α>0). To start with, we focus on the computation of the anomalous diffusion coefficient for the subdiffusive motion along the backbone. This quantity is subsequently used as an input to compute concentration recovery curves mimicking fluorescence recovery after photobleaching experiments in comblike geometries such as spiny dendrites. Our method is based on the mean-field description provided by the well-tested continuous time random-walk approach for the random-comb model, and the obtained analytical result for the diffusion coefficient is confirmed by numerical simulations of a random walk with finite steps in time and space along the backbone and the teeth. We subsequently incorporate retardation effects arising from binding-unbinding kinetics into our model and obtain a scaling law characterizing the corresponding change in the diffusion coefficient. Finally, we show that recovery curves obtained with the help of the analytical expression for the anomalous diffusion coefficient cannot be fitted perfectly by a model based on scaled Brownian motion, i.e., a standard diffusion equation with a time-dependent diffusion coefficient. However, differences between the exact curves and such fits are small, thereby providing justification for the practical use of models relying on scaled Brownian motion as a fitting procedure for recovery curves arising from particle diffusion in comblike systems.
We calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps (i.e., traps that disappear in the course of their motion). ...Our calculation is based on a fractional reaction-subdiffusion equation derived from a continuous time random walk model of the system. Contrary to an earlier method valid only in one dimension (d=1), the equation is applicable in any Euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport, the irreversible evanescence reaction, and the dimension in which both the traps and the target are embedded. Explicit results for the survival probability of the target are obtained for a density ρ(t) of traps which decays (i) exponentially and (ii) as a power law. In the former case, the target has a finite asymptotic survival probability in all integer dimensions, whereas in the latter case there are several regimes where the values of the decay exponent for ρ(t) and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one, two, and three dimensions.
A comparison of simulation results with the prediction of the structural properties of square-shoulder fluids is carried out to assess the performance of three theories: Tang-Lu's first-order mean ...spherical approximation, the simplified exponential approximation of the latter and the rational-function approximation. These three theoretical developments share the characteristic of being analytical in Laplace space and of reducing in the proper limit to the Percus-Yevick result for the hard-sphere fluid. Overall, the best agreement with the simulation data is obtained with the simplified exponential approximation.
The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the ...behavior of a two-particle system in two- and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability S(t) and the associated first-encounter time probability density H(t) over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time 〈T〉, as well as for the decay time T characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound t_{B} for the time at which S(t) starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to T depends only on the total diffusivity D=D_{1}+D_{2}, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity D. In two dimensions, the first subleading contribution to T is found to depend weakly on the ratio D_{1}/D_{2}. We also investigate the slow-diffusion limit when D_{2}≪D_{1}, and we discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when T can be expected to be a good approximation for 〈T〉.
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