Journal articles: 'HETEROGENOUS MEAN FIELD'

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Xu, Can, Yonggang Wu, Zhigang Zheng, and Longkun Tang. "Partial locking in phase-oscillator populations with heterogenous coupling." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 6 (June 2022): 063106. http://dx.doi.org/10.1063/5.0093318.

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We consider a variant of the mean-field model of coupled phase oscillators with uniform distribution of natural frequencies. By establishing correlations between the quenched disorder of intrinsic frequencies and coupling strength with both in- and out-coupling heterogeneities, we reveal a generic criterion for the onset of partial locking that takes place in a domain with the coexistence of phase-locked oscillators and drifters. The critical points manifesting the instability of the stationary states are obtained analytically. In particular, the bifurcation mechanism of the equilibrium states is uncovered by the use of frequency-dependent version of the Ott–Antonsen reduction consistently with the analysis based on the self-consistent approach. We demonstrate that both the manner of coupling heterogeneity and correlation exponent have influence on the emergent patterns of partial locking. Our research could find applicability in better understanding the phase transitions and related collective phenomena involving synchronization control in networked systems.

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Chicchi, Lorenzo, Gloria Cecchini, Ihusan Adam, Giuseppe de Vito, Roberto Livi, Francesco Saverio Pavone, Ludovico Silvestri, Lapo Turrini, Francesco Vanzi, and Duccio Fanelli. "Reconstruction scheme for excitatory and inhibitory dynamics with quenched disorder: application to zebrafish imaging." Journal of Computational Neuroscience 49, no. 2 (April 7, 2021): 159–74. http://dx.doi.org/10.1007/s10827-020-00774-1.

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AbstractAn inverse procedure is developed and tested to recover functional and structural information from global signals of brains activity. The method assumes a leaky-integrate and fire model with excitatory and inhibitory neurons, coupled via a directed network. Neurons are endowed with a heterogenous current value, which sets their associated dynamical regime. By making use of a heterogenous mean-field approximation, the method seeks to reconstructing from global activity patterns the distribution of in-coming degrees, for both excitatory and inhibitory neurons, as well as the distribution of the assigned currents. The proposed inverse scheme is first validated against synthetic data. Then, time-lapse acquisitions of a zebrafish larva recorded with a two-photon light sheet microscope are used as an input to the reconstruction algorithm. A power law distribution of the in-coming connectivity of the excitatory neurons is found. Local degree distributions are also computed by segmenting the whole brain in sub-regions traced from annotated atlas.

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Forbes, Emily, Matthew Back, Andrew Brooks, Natalia B. Petrovskaya, Sergei V. Petrovskii, Tom Pope, and Keith F. A. Walters. "Stability of Patches of Higher Population Density within the Heterogenous Distribution of the Gray Field Slug Deroceras reticulatum in Arable Fields in the UK." Insects 12, no. 1 (December 25, 2020): 9. http://dx.doi.org/10.3390/insects12010009.

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Exploitation of heterogenous distributions of Deroceras reticulatum, in arable fields by targeting molluscicide applications toward areas with higher slug densities, relies on these patches displaying sufficient spatio-temporal stability. Regular sampling of slug activity/distribution was undertaken using 1 ha rectangular grids of 100 refuge traps established in 22 commercial arable field crops. Activity varied significantly between the three years of the study, and the degree of aggregation (Taylor’s Power Law) was higher in fields with higher mean trap catches. Hot spot analysis detected statistically significant spatial clusters in all fields, and in 162 of the 167 individual assessment visits. The five assessment visits in which no clusters were detected coincided with low slug activity (≤0.07 per trap). Generalized Linear Models showed significant spatial stability of patches in 11 fields, with non-significant fields also characterized by low slug activity (≤1.2 per trap). Mantel’s permutation tests revealed a high degree of correlation between location of individual patches between sampling dates. It was concluded that patches of higher slug density were spatio-temporally stable, but detection using surface refuge traps (which rely on slug activity on the soil surface) was less reliable when adverse environmental conditions resulted in slugs retreating into the upper soil horizons.

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Hamdioui, S., and L. Vaivre-Douret. "Psycho-socio-emotional characteristics in high intellectual potential child regarding IQ profile (Homogenous/Heterogenous)." European Psychiatry 65, S1 (June 2022): S419. http://dx.doi.org/10.1192/j.eurpsy.2022.1064.

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Introduction Few studies have analyzed the psychometric profile (Homogenous/Heterogenous), established by the Wechsler scale in high intellectual potential children (HIP, IQ>130), regarding the psycho-socio-emotional characteristics. Objectives We aimed to look at the links between the IQ-profile and the psycho-socio-emotional characteristics in HIP. Methods Anamnestic questionnaire and Wechsler-Intelligence-Scale for children (WISC-V) were conducted and analyzed in 58 healthy children with HIP, aged 7-to-13 years-old (mean 10y; SD 1.8). It was possible to distinguished 27 Homogenousvs 30 Heterogenous IQ-profile. Results No significant difference between homogenous/heterogonous groups, FIQ was positively significantly correlated with “Reacting very little emotionally”, “Tendency to isolation”. Visual-Spatial-Index (VSI) with “Ability to adapt to new people” (r=-0.4, p=0.02), “few interests” (r=0.5, p=0.008). Verbal-Comprehension-Index (VCI) with “Reacting strongly to frustration”, “Difficulties to understand limits”, “Separation anxiety”. A significant difference between homogenous/heterogonous groups was shown regarding “few interests” with high rate in the heterogonous group (t= -2.34, p=0.023). Conclusions HIP seems to cover specific psycho-socio-emotional characteristics linked to IQ index distribution. Thus, it appears interesting to assess more the emotional and socio-cognitive field to understand these characteristics in HIP children. Disclosure No significant relationships.

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Allmeier, Sebastian, and Nicolas Gast. "Mean Field and Refined Mean Field Approximations for Heterogeneous Systems." ACM SIGMETRICS Performance Evaluation Review 50, no. 1 (June 20, 2022): 103–4. http://dx.doi.org/10.1145/3547353.3522653.

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Mean field approximation is a powerful technique to study the performance of large stochastic systems represented as n interacting objects. Applications include load balancing models, epidemic spreading, cache replacement policies, or large-scale data centers. Mean field approximation is asymptotically exact for systems composed of n homogeneous objects under mild conditions. In this paper, we study what happens when objects are heterogeneous. This can represent servers with different speeds or contents with different popularities. We define an interaction model that allows obtaining asymptotic convergence results for stochastic systems with heterogeneous object behavior and show that the error of the mean field approximation is of order O(1/n). More importantly, we show how to adapt the refined mean field approximation, developed by the authors of Gast et al. 2019, and show that the error of this approximation is reduced to O(1/n2). To illustrate the applicability of our result, we present two examples. The first addresses a list-based cache replacement model, RANDOM(m), which is an extension of the RANDOM policy. The second is a heterogeneous supermarket model. These examples show that the proposed approximations are computationally tractable and very accurate. For moderate system sizes (n ≈ 30) the refined mean field approximation tends to be more accurate than simulations for any reasonable simulation time.

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Allmeier, Sebastian, and Nicolas Gast. "Mean Field and Refined Mean Field Approximations for Heterogeneous Systems." Proceedings of the ACM on Measurement and Analysis of Computing Systems 6, no. 1 (February 24, 2022): 1–43. http://dx.doi.org/10.1145/3508033.

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Mean field approximation is a powerful technique to study the performance of large stochastic systems represented as n interacting objects. Applications include load balancing models, epidemic spreading, cache replacement policies, or large-scale data centers. Mean field approximation is asymptotically exact for systems composed of n homogeneous objects under mild conditions. In this paper, we study what happens when objects are heterogeneous. This can represent servers with different speeds or contents with different popularities. We define an interaction model that allows obtaining asymptotic convergence results for stochastic systems with heterogeneous object behavior, and show that the error of the mean field approximation is of order $O(1/n)$. More importantly, we show how to adapt the refined mean field approximation, developed by Gast et al., and show that the error of this approximation is reduced to O(1/n^2). To illustrate the applicability of our result, we present two examples. The first addresses a list-based cache replacement model, RANDOM(m), which is an extension of the RANDOM policy. The second is a heterogeneous supermarket model. These examples show that the proposed approximations are computationally tractable and very accurate. They also show that for moderate system sizes (30) the refined mean field approximation tends to be more accurate than simulations for any reasonable simulation time.

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Jumaah, Hayder A. "Modified Archie’s parameters for estimating water saturation for carbonate reservoir in north of Iraq." Journal of Petroleum Exploration and Production Technology 11, no. 10 (August 21, 2021): 3689–97. http://dx.doi.org/10.1007/s13202-021-01258-3.

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AbstractArchie’s parameters, cementation factor (m), saturation exponent (n) and tortuosity factor (a), are general factor that have effects on water saturation magnitude, due to their sensitivity to pores distribution, lithofacies properties and wettability, particularly in carbonate reservoirs. Water saturation magnitude has a direct effect in estimating initial oil-in-place values, and inaccuracy in its values will lead to huge impact errors in initial oil-in-place values, so it would affect the economics of field management and development plans. In this paper, the main objective was to investigate the impact of using conventional and modified Archie’s parameters in the determination of water saturation from well log interpretation for Tertiary reservoir in Khabaz oil field, a heterogenous carbonite reservoir in the north of Iraq which was affected by different digenesis processes that impacted the reservoir quality. Tertiary reservoir of Khabaz field consists of five geological units (A, B, C, D and E), and the selected well penetrated the top of the reservoir at 2200.5 m RTKB and passed through five geological units and reached total depth at 2348 m RTKB. The geothermal gradient of the field was 1.12 ℉ per 100 ft, and formation water resistivity (Rw) was about 0.029 Ω m. Water saturation was at first estimated from resistivity logs by Archie model with conventional known values of parameters (a, m and n) (1, 2 and 2), respectively, and then Archie’s parameters were modified and determined by graphical technique of Pickett plot for each geological unit to estimate water saturation. Finally, the results show the water saturation value was more sensitive for Archie’s parameter in low-porosity and high-clay-volume zone, but less sensitive in clean high-porosity zone, and water saturation values determined by modified Archie model were less about 18.5% at mean than their value by using conventional Archie’s parameters.

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Alabi, Tunrayo R., Julius Adewopo, Ojo Patrick Duke, and P. Lava Kumar. "Banana Mapping in Heterogenous Smallholder Farming Systems Using High-Resolution Remote Sensing Imagery and Machine Learning Models with Implications for Banana Bunchy Top Disease Surveillance." Remote Sensing 14, no. 20 (October 18, 2022): 5206. http://dx.doi.org/10.3390/rs14205206.

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Banana (and plantain, Musa spp.), in sub-Saharan Africa (SSA), is predominantly grown as a mixed crop by smallholder farmers in backyards and small farmlands, typically ranging from 0.2 ha to 3 ha. The crop is affected by several pests and diseases, including the invasive banana bunchy top virus (BBTV, genus Babuvirus), which is emerging as a major threat to banana production in SSA. The BBTV outbreak in West Africa was first recorded in the Benin Republic in 2010 and has spread to the adjoining territories of Nigeria and Togo. Regular surveillance, conducted as part of the containment efforts, requires the identification of banana fields for disease assessment. However, small and fragmented production spread across large areas poses complications for identifying all banana farms using conventional field survey methods, which is also time-consuming and expensive. In this study, we developed a remote sensing approach and machine learning (ML) models that can be used to identify banana fields for targeted BBTV surveillance. We used medium-resolution synthetic aperture radar (SAR), Sentinel 2A satellite imagery, and high-resolution RGB and multispectral aerial imagery from an unmanned aerial vehicle (UAV) to develop an operational banana mapping framework by combining the UAV, SAR, and Sentinel 2A data with the Support Vector Machine (SVM) and Random Forest (RF) machine learning algorithms. The ML algorithms performed comparatively well in classifying the land cover, with a mean overall accuracy (OA) of about 93% and a Kappa coefficient (KC) of 0.89 for the UAV data. The model using fused SAR and Sentinel 2A data gave an OA of 90% and KC of 0.86. The user accuracy (UA) and producer accuracy (PA) for the banana class were 83% and 78%, respectively. The BBTV surveillance teams used the banana mapping framework to identify banana fields in the BBTV-affected southwest Ogun state of Nigeria, which helped in detecting 17 sites with BBTV infection. These findings suggest that the prediction of banana and other crops in the heterogeneous smallholder farming systems is feasible, with the precision necessary to guide BBTV surveillance in large areas in SSA.

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Dias, C., and M. O. Hase. "Weakening connections in heterogeneous mean-field models." Journal of Statistical Mechanics: Theory and Experiment 2021, no. 1 (January 20, 2021): 013404. http://dx.doi.org/10.1088/1742-5468/abd4ca.

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Chong, Carsten, and Claudia Klüppelberg. "Partial mean field limits in heterogeneous networks." Stochastic Processes and their Applications 129, no. 12 (December 2019): 4998–5036. http://dx.doi.org/10.1016/j.spa.2018.12.018.

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Palakkool, Shamna Rajan, David J. Gower, and Ramachandran Kotharambath. "Osteology of the caecilian Gegeneophis carnosus (Beddome, 1870) (Amphibia: Gymnophiona: Grandisoniidae) from the Western Ghats of peninsular India." Vertebrate Zoology 72 (July 20, 2022): 561–76. http://dx.doi.org/10.3897/vz.72.e79911.

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Abstract The osteology of the poorly known grandisoniid caecilian Gegeneophis carnosus is described for the first time by applying high-resolution X-ray micro-computed tomography to some recently collected material. The ossified skeleton comprises a stegokrotaphic skull, lower jaw, and vertebral column. The braincase, composed of the sphenethmoid and os basale, is covered by eight other cranial elements viz. nasopremaxilla, frontal, parietal, squamosal, pterygoquadrate, maxillopalatine, vomer, and stapes. The eye is covered by the maxillopalatine, and an (open) orbit is absent. The sphenethmoid is not exposed and lacks a solum nasi or a ventral flange. The olfactory chamber lacks an olfactory eminence. Slight asymmetries were observed in the structure and/or size of the left and right frontals and parietals and in the number and size of some foramina. Except for pterygoquadrate and stapes, all bones are pierced by foramina for nerves and/or blood vessels. The lower jaw shows a typical caecilian pattern with dentigerous pseudodentary and edentulous pseudoangular. Numbers of vertebrae range from 123–130 (mean 126). The vertebrae are somewhat heterogenous, varying in size and proportions along the column. Comparisons are made with other caecilians, especially other grandisoniids. Aspects of the cranial osteology of Gegeneophis, such as the closed orbit, subterminal mouth, and stegokrotaphy are possible adaptations to dedicated fossoriality, but functional, behavioural, and field ecological data are not yet available to test this.

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Geva, Nadav, Valerie Vaissier, James Shepherd, and Troy Van Voorhis. "Mean field treatment of heterogeneous steady state kinetics." Chemical Physics Letters 685 (October 2017): 185–90. http://dx.doi.org/10.1016/j.cplett.2017.07.011.

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Ceyrolles, William J., Pascal Viot, and Julian Talbot. "Kinetics of Heterogeneous Adsorption: Mean-Field Theory and Simulations." Langmuir 18, no. 4 (February 2002): 1112–18. http://dx.doi.org/10.1021/la010101q.

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Mikhail, Christopher, Zach Pennington, Paul M. Arnold, Darrel S. Brodke, Jens R. Chapman, Norman Chutkan, Michael D. Daubs, et al. "Minimizing Blood Loss in Spine Surgery." Global Spine Journal 10, no. 1_suppl (January 2020): 71S—83S. http://dx.doi.org/10.1177/2192568219868475.

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Study Design: Broad narrative review. Objective: To review and summarize the current literature on guidelines, outcomes, techniques and indications surrounding multiple modalities of minimizing blood loss in spine surgery. Methods: A thorough review of peer-reviewed literature was performed on the guidelines, outcomes, techniques, and indications for multiple modalities of minimizing blood loss in spine surgery. Results: There is a large body of literature that provides a consensus on guidelines regarding the appropriate timing of discontinuation of anticoagulation, aspirin, nonsteroidal anti-inflammatory drugs (NSAIDs), and herbal supplements prior to surgery. Additionally, there is a more heterogenous discussion the utility of preoperative autologous blood donation facilitated by erythropoietin and iron supplementation for healthy patients slated for procedures with high anticipated blood loss and for whom allogeneic transfusion is likely. Intraoperative maneuvers available to minimize blood loss include positioning and maintaining normothermia. Tranexamic acid (TXA), bipolar sealer electrocautery, and topical hemostatic agents, and hypotensive anesthesia (mean arterial pressure (MAP) <65 mm Hg) should be strongly considered in cases with larger exposures and higher anticipated blood loss. There is strong level 1 evidence for the use of TXA in spine surgery as it reduces the overall blood loss and transfusion requirements. Conclusion: As the volume and complexity of spinal procedures rise, intraoperative blood loss management has become a pivotal topic of research within the field. There are many tools for minimizing blood loss in patients undergoing spine surgery. The current literature supports combining techniques to use a cost- effective multimodal approach to minimize blood loss in the perioperative period.

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Subramanian, Sriram Ganapathi, Matthew E. Taylor, Mark Crowley, and Pascal Poupart. "Decentralized Mean Field Games." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 9 (June 28, 2022): 9439–47. http://dx.doi.org/10.1609/aaai.v36i9.21176.

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Multiagent reinforcement learning algorithms have not been widely adopted in large scale environments with many agents as they often scale poorly with the number of agents. Using mean field theory to aggregate agents has been proposed as a solution to this problem. However, almost all previous methods in this area make a strong assumption of a centralized system where all the agents in the environment learn the same policy and are effectively indistinguishable from each other. In this paper, we relax this assumption about indistinguishable agents and propose a new mean field system known as Decentralized Mean Field Games, where each agent can be quite different from others. All agents learn independent policies in a decentralized fashion, based on their local observations. We define a theoretical solution concept for this system and provide a fixed point guarantee for a Q-learning based algorithm in this system. A practical consequence of our approach is that we can address a `chicken-and-egg' problem in empirical mean field reinforcement learning algorithms. Further, we provide Q-learning and actor-critic algorithms that use the decentralized mean field learning approach and give stronger performances compared to common baselines in this area. In our setting, agents do not need to be clones of each other and learn in a fully decentralized fashion. Hence, for the first time, we show the application of mean field learning methods in fully competitive environments, large-scale continuous action space environments, and other environments with heterogeneous agents. Importantly, we also apply the mean field method in a ride-sharing problem using a real-world dataset. We propose a decentralized solution to this problem, which is more practical than existing centralized training methods.

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Sun, Li-Hsien. "Mean Field Games with Heterogeneous Groups: Application to Banking Systems." Journal of Optimization Theory and Applications 192, no. 1 (October 19, 2021): 130–67. http://dx.doi.org/10.1007/s10957-021-01955-3.

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Persio, Luca Di, and Matteo Garbelli. "Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective." Symmetry 13, no. 1 (December 23, 2020): 14. http://dx.doi.org/10.3390/sym13010014.

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We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the supervised learning approach can be translated in terms of a (stochastic) mean-field optimal control problem by applying the Hamilton–Jacobi–Bellman (HJB) approach and the mean-field Pontryagin maximum principle. Our contribution sheds new light on a possible theoretical connection between mean-field problems and DL, melting heterogeneous approaches and reporting the state-of-the-art within such fields to show how the latter different perspectives can be indeed fruitfully unified.

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Kycko, Marlena, Bogdan Zagajewski, Marcin Kluczek, Anna Tardà, Lydia Pineda, Vicenç Palà, and Jordi Corbera. "Sentinel-2 and AISA Airborne Hyperspectral Images for Mediterranean Shrubland Mapping in Catalonia." Remote Sensing 14, no. 21 (November 2, 2022): 5531. http://dx.doi.org/10.3390/rs14215531.

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The Mediterranean ecosystem exhibits a particular geology and climate, which is characterized by mild, rainy winters and long, very hot summers with low precipitation; it has led to the emergence of resilient plant species. Such habitats contain a preponderance of shrubs, and collectively harbor 10% of the Earth's species, thus containing some of the most unique shrubby formations protecting against environmental natural degradation. Due to shrub species diversity, initial phases of forestland, heterogenous grasses, bare ground and stones, the monitoring of such areas is difficult. For this reason, the aim of this paper is to assess semi-automatic classifications of the shrubby formations based on multispectral Sentinel-2 and visible and near infrared (VINR) AISA-EAGLE II hyperspectral airborne images with a support of Canopy High Model (CHM) as a three-dimensional information and field-verified patterns, based on Match-T/DSM and aerial photos. Support Vector Machine (SVM) and Random Forest (RF) classifiers have been tested on a few scenarios featuring different combinations of spectral and Minimum Noise Fraction (MNF) transformed bands and vegetation indices. Referring to the results, the average overall accuracy for the SVM and AISA images (all tested data sets) was 78.23%, and for the RF: 79.85%. In the case of Sentinel-2, the SVM classifier obtained an average value of 83.63%, while RF: 85.32%; however, in the case of the shrubland, we would like to recommend the RF classifier, because the highest mean value of F1-score achieved was 91.86% (SVM offered few-percent-point worse results), and the required training time was quicker than SVM. Commonly available Sentinel-2 data offered higher accuracies for shrubland monitoring than did the airborne VNIR data.

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Zech, A., and S. Attinger. "Technical note: Analytical solution for the mean drawdown of steady state pumping tests in two-dimensional isotropic heterogeneous aquifers." Hydrology and Earth System Sciences Discussions 12, no. 7 (July 20, 2015): 6921–44. http://dx.doi.org/10.5194/hessd-12-6921-2015.

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Abstract. A new method is presented which allows to interpret steady state pumping test in heterogeneous isotropic transmissivity fields. In contrast to mean uniform flow, the pumping test drawdowns in heterogeneous media cannot be described by a single effective or equivalent value of hydraulic transmissivity. A radially depending description of transmissivity is required, including the parameters of aquifer heterogeneity: mean, variance and correlation length. Such a model is provided by the upscaling procedure Radial Coarse Graining, which describes the transition of near well to far field transmissivity effectively. Based on the Radial Coarse Graining Transmissivity, an analytical solution for a steady state pumping test drawdown is derived. The so-called effective well flow solution is derived for two cases: the ensemble mean of pumping tests and the drawdown at an individual heterogeneous transmissivity field. The analytical form of the solution allows to inversely estimate the parameters of aquifer heterogeneity from pumping test data. This is shown making use of virtual pumping test data, for both cases the ensemble mean drawdown and pumping tests at individual transmissivity fields. The effective well flow solution reproduces the drawdown for two-dimensional pumping tests in heterogeneous media and is a promising tool to estimate parameters of aquifer heterogeneity, in particular for the horizontal correlation length.

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Gkogkas, Marios Antonios, Benjamin Jüttner, Christian Kuehn, and Erik Andreas Martens. "Graphop mean-field limits and synchronization for the stochastic Kuramoto model." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 11 (November 2022): 113120. http://dx.doi.org/10.1063/5.0094009.

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Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator’s phase evolution and explains the transition from incoherent to coherent oscillations under simplifying assumptions, including all-to-all coupling with uniform strength. Real world networks, however, often display heterogeneous connectivity and coupling weights that influence the critical threshold for this transition. We formulate a general mean-field theory (Vlasov–Focker Planck equation) for stochastic Kuramoto-type phase oscillator models, valid for coupling graphs/networks with heterogeneous connectivity and coupling strengths, using graphop theory in the mean-field limit. Considering symmetric odd-valued coupling functions, we mathematically prove an exact formula for the critical threshold for the incoherence–coherence transition. We numerically test the predicted threshold using large finite-size representations of the network model. For a large class of graph models, we find that the numerical tests agree very well with the predicted threshold obtained from mean-field theory. However, the prediction is more difficult in practice for graph structures that are sufficiently sparse. Our findings open future research avenues toward a deeper understanding of mean-field theories for heterogeneous systems.

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Hu, Ying, Jianhui Huang, and Tianyang Nie. "Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints." SIAM Journal on Control and Optimization 56, no. 4 (January 2018): 2835–77. http://dx.doi.org/10.1137/17m1151420.

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Zech, Alraune, and Sabine Attinger. "Technical note: Analytical drawdown solution for steady-state pumping tests in two-dimensional isotropic heterogeneous aquifers." Hydrology and Earth System Sciences 20, no. 5 (May 3, 2016): 1655–67. http://dx.doi.org/10.5194/hess-20-1655-2016.

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Abstract. A new method is presented which allows interpreting steady-state pumping tests in heterogeneous isotropic transmissivity fields. In contrast to mean uniform flow, pumping test drawdowns in heterogeneous media cannot be described by a single effective or equivalent value of hydraulic transmissivity. An effective description of transmissivity is required, being a function of the radial distance to the well and including the parameters of log-transmissivity: mean, variance, and correlation length. Such a model is provided by the upscaling procedure radial coarse graining, which describes the transition of near-well to far-field transmissivity effectively. Based on this approach, an analytical solution for a steady-state pumping test drawdown is deduced. The so-called effective well flow solution is derived for two cases: the ensemble mean of pumping tests and the drawdown within an individual heterogeneous transmissivity field. The analytical form of the solution allows inversely estimating the parameters of aquifer heterogeneity. For comparison with the effective well flow solution, virtual pumping tests are performed and analysed for both cases, the ensemble mean drawdown and pumping tests at individual transmissivity fields. Interpretation of ensemble mean drawdowns showed proof of the upscaling method. The effective well flow solution reproduces the drawdown for two-dimensional pumping tests in heterogeneous media in contrast to Thiem's solution for homogeneous media. Multiple pumping tests conducted at different locations within an individual transmissivity field are analysed, making use of the effective well flow solution to show that all statistical parameters of aquifer heterogeneity can be inferred under field conditions. Thus, the presented method is a promising tool with which to estimate parameters of aquifer heterogeneity, in particular variance and horizontal correlation length of log-transmissivity fields from steady-state pumping test measurements.

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Wati, Kharisma Arby Ristiana, and Sulistiyawati. "Comparative Study of Homogeneous Class and Heterogeneous Class of Participation Level and Result Learning in System Reproduction Classes of SMA/MA." Proceeding International Conference on Science and Engineering 3 (April 30, 2020): 65–71. http://dx.doi.org/10.14421/icse.v3.470.

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The study is intended to determine how the differences between classes of homogeneous and class heterogeneous on the level of participation and the results of learning in class XI of the system of reproduction. The research is including field research with the approach of quantitative use study comparisons. Intake sample used technique purposive sampling and using three classes. the class is heterogeneous, class homogeneous male (M) and class heterogenous female (FM). Instruments retrieval of data using the data test multiple choice (posttest) and questionnaire level of participation. Mechanichal analysis of the data using a test statistic one way ANOVA and test Tukey as the test continued. The results of the study indicate that 1) the level of participation in class heterogeneous, class homogeneous male and class homogeneous female is not there differences in views of the value of the significant of 0,122 which means 0,122 > 0,05 and Fcount of 2,185 <Ftable. 2) The result of study on class of heterogenous, class homogeneous male, class homogeneous female there are differences in view on the value of significant of 0,000. Wich means that 0,000 <0.05 and Fcount amounted to 52,657> Ftable, based on test Tukey class that has the value of the results of studying the best is the class of homogeneous male.

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Mazich, K. A., P. C. Killgoar, and J. A. Ingram. "Mean-Field Calculations of the Dynamic Mechanical Properties of Heterogeneous Elastomer Blends." Rubber Chemistry and Technology 62, no. 2 (May 1, 1989): 305–14. http://dx.doi.org/10.5254/1.3536246.

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Abstract A method for calculating the dynamic mechanical properties of elastomer blends with co-continuous structures has been presented. The calculations are based on Kerner's packed-grain model for composite media. Comparisons of theoretical calculations with experimental data show that this model closely approximates the viscoelastic properties of blends with a co-continuous structure, at least in the glass-transition regions of the respective blend components. We have also shown that the storage modulus of co-continuous blends may be well-represented by a discrete-particle model. This result can be misleading, however, if the storage modulus alone is calculated and compared with experimental data. A comparison of viscoelastic data (log E′ and tan δ) with calculation clearly distinguishes the models and indicates that the packed-grain model is the correct representation of the structure of co-continuous blends. The agreement between theory and experiment reported above was principally found in the glass-transition regions of the respective components in the elastomer blend. We extended the comparison well into the rubbery region and found that the agreement between Kerner's mean-field theory and the experimental data was poor, particularly for the loss tangent. Different relaxation mechanisms (relaxations over greater periods of time) are available to the blend in the rubbery region of viscoelastic response, and these mechanisms are apparently not accounted for in the mean-field calculations.

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Mukhopadhyay, Arpan, A. Karthik, Ravi R. Mazumdar, and Fabrice Guillemin. "Mean field and propagation of chaos in multi-class heterogeneous loss models." Performance Evaluation 91 (September 2015): 117–31. http://dx.doi.org/10.1016/j.peva.2015.06.008.

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Yi, Zhenyan, Hongli Zhao, and Yunzhong Jiang. "Continuous Daily Evapotranspiration Estimation at the Field-Scale over Heterogeneous Agricultural Areas by Fusing ASTER and MODIS Data." Remote Sensing 10, no. 11 (October 26, 2018): 1694. http://dx.doi.org/10.3390/rs10111694.

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Continuous daily evapotranspiration (ET) monitoring at the field-scale is crucial for water resource management in irrigated agricultural areas in arid regions. Here, an integrated framework for daily ET, with the required spatiotemporal resolution, is described. Multi-scale surface energy balance algorithm evaluations and a data fusion algorithm are combined to optimally exploit the spatial and temporal characteristics of image datasets, collected by the advanced space-borne thermal emission reflectance radiometer (ASTER) and the moderate resolution imaging spectroradiometer (MODIS). Through combination with a linear unmixing-based method, the spatial and temporal adaptive reflectance fusion model (STARFM) is modified to generate high-resolution ET estimates for heterogeneous areas. The performance of this methodology was evaluated for irrigated agricultural fields in arid and semiarid areas of Northwest China. Compared with the original STARFM, a significant improvement in daily ET estimation accuracy was obtained by the modified STARFM (overall mean absolute percentage error (MAP): 12.9% vs. 17.2%; overall mean absolute percentage error (RMSE): 0.7 mm d−1 vs. 1.2 mm d−1). The modified STARFM additionally preserved more spatial details than the original STARFM for heterogeneous agricultural fields, and provided field-to-field variability in water use. Improvements were further evident in the continuous daily ET, where the day-to-day dynamics of ET estimates were captured. ET data fusion provides a unique means of monitoring continuous daily crop ET values at the field-scale in agricultural areas, and may have value in supporting operational water management decisions.

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Rudko, Igor M. "Unified field of detection probabilities for heterogeneous means of observations." Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitel'naya tekhnika i informatika, no. 53 (December 1, 2020): 93–101. http://dx.doi.org/10.17223/19988605/53/9.

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Vigil, R. Dennis, and Frank T. Willmore. "Oscillatory dynamics in a heterogeneous surface reaction: Breakdown of the mean-field approximation." Physical Review E 54, no. 2 (August 1, 1996): 1225–31. http://dx.doi.org/10.1103/physreve.54.1225.

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Huang, Feng, and Han-Shuang Chen. "An Improved Heterogeneous Mean-Field Theory for the Ising Model on Complex Networks." Communications in Theoretical Physics 71, no. 12 (December 2019): 1475. http://dx.doi.org/10.1088/0253-6102/71/12/1475.

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Li, Qian, Chenguang Song, Bin Wu, Yunpeng Xiao, and Bai Wang. "Social hotspot propagation dynamics model based on heterogeneous mean field and evolutionary games." Physica A: Statistical Mechanics and its Applications 508 (October 2018): 324–41. http://dx.doi.org/10.1016/j.physa.2018.05.100.

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Chen, Zheng, He Wang, Neil Qiang Su, Sai Duan, Tonghao Shen, and Xin Xu. "Beyond Mean-Field Microkinetics: Toward Accurate and Efficient Theoretical Modeling in Heterogeneous Catalysis." ACS Catalysis 8, no. 7 (April 11, 2018): 5816–26. http://dx.doi.org/10.1021/acscatal.8b00943.

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Vladimirski, Boris B., Joël Tabak, Michael J. O’Donovan, and John Rinzel. "Episodic activity in a heterogeneous excitatory network, from spiking neurons to mean field." Journal of Computational Neuroscience 25, no. 1 (March 6, 2008): 39–63. http://dx.doi.org/10.1007/s10827-007-0064-4.

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Ancel, Julien, Christine Fricker, and Hanene Mohamed. "Mean field analysis for bike and e-bike sharing systems." ACM SIGMETRICS Performance Evaluation Review 49, no. 2 (January 17, 2022): 12–14. http://dx.doi.org/10.1145/3512798.3512804.

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Electric bikes are deployed massively in preexisting bike sharing system in order to attract new users and replace cars on a larger scale (see [2]). But this causes interactions between the two populations of bikes. In this paper, we analyze a model of an homogeneous bike sharing system where two classes of bikes interact only through the finite capacity of stations. It models systems with both electric and normal bikes, these classes requiring different subscriptions. As far as we know (see [7]), it is the first stochastic large-scale analysis for integrated e-bike and bike sharing systems. The aim of the paper is to derive explicitly the limiting stationary distribution of the state of a station when the number of stations and the fleet size of each class increase at the same rate. Analysis for a spatially heterogeneous network is in preparation and discussed in Section 4.

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Dahmen, Karin, Deniz Ertaş, and Yehuda Ben-Zion. "Gutenberg-Richter and characteristic earthquake behavior in simple mean-field models of heterogeneous faults." Physical Review E 58, no. 2 (August 1, 1998): 1494–501. http://dx.doi.org/10.1103/physreve.58.1494.

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Chang, C. M., and H. D. Yeh. "Stochastic analysis of field-scale heat advection in heterogeneous aquifers." Hydrology and Earth System Sciences 16, no. 3 (March 1, 2012): 641–48. http://dx.doi.org/10.5194/hess-16-641-2012.

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Abstract. Owing to the analogy between the solute and heat transport processes, it can be expected that the rate of growth of the spatial second moments of the heat flux in a heterogeneous aquifer over relatively large space scales is greater than that predicted by applying the classical heat transport model. The motivation of stochastic analysis of heat transport at the field scale is therefore to quantify the enhanced growth of the field-scale second moments caused by the spatially varying specific discharge field. Within the framework of stochastic theory, an effective advection-dispersion equation containing effective parameters (namely, the macrodispersion coefficients) is developed to model the mean temperature field. The rate of growth of the field-scale spatial second moments of the mean temperature field in the principal coordinate directions is described by the macrodispersion coefficient. The variance of the temperature field is also developed to characterize the reliability to be anticipated in applying the mean heat transport model. It is found that the heterogeneity of the medium and the correlation length of the log hydraulic conductivity are important in enhancing the field-scale heat advection, while the effective thermal conductivity plays the role in reducing the field-scale heat advection.

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Chang, C. M., and H. D. Yeh. "Stochastic analysis of field-scale heat advection in heterogeneous aquifers." Hydrology and Earth System Sciences Discussions 8, no. 6 (November 22, 2011): 10311–31. http://dx.doi.org/10.5194/hessd-8-10311-2011.

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Abstract. Owing to the analogy between the solute and heat transport processes, it can be expected that the rate of growth of the spatial second moments of the heat flux in a heterogeneous aquifer over relatively large space scales is greater than that predicted by applying the classical heat transport model. The motivation of stochastic analysis of heat transport at the field scale is therefore to quantify the enhanced growth of the field-scale second moments caused by the spatially varying specific discharge field. Within the framework of stochastic theory, an effective advection-dispersion equation containing effective parameters (namely, the macrodispersion coefficients) is developed to model the mean temperature field. The rate of growth of the field-scale spatial second moments of the mean temperature field in the principal coordinate directions is described by the macrodispersion coefficient. The variance of the temperature field is also developed to characterize the reliability to be anticipated in applying the mean heat transport model. It is found that the heterogeneity of the medium and the correlation length of the log hydraulic conductivity are important in enhancing the field-scale heat advection, while the effective thermal conductivity plays the role in reducing the field-scale heat advection.

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Richter, F., K. Barfus, F. H. Berger, and U. Görsdorf. "The influence of cloud top variability from radar measurements on 3-D radiative transfer." Atmospheric Chemistry and Physics Discussions 7, no. 3 (June 11, 2007): 8087–111. http://dx.doi.org/10.5194/acpd-7-8087-2007.

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Abstract. In radiative transfer simulations the simplification of cloud top structure by homogenous assumptions can breed to mistakes in comparison to realistic heterogeneous cloud top structures. This paper examines the influence of cloud top heterogeneity on the radiation at the top of the atmosphere. The use of cloud top measurements with a high temporal resolution allows to analyse small spatial cloud top heterogeneities by using the frozen turbulence assumption for the time – space conversion. Radiative observations are often based on satellite measurements, whereas small spatial structures are not considered in such treatments. A spectral analysis of the cloud top measurements showed slopes of power spectra between –1.8 and –2.0, this values are larger then the spectra of –5/3 which is often applied to generate cloud field variability. The comparison of 3-D radiative transfer results from cloud fields with homogenous and heterogeneous tops have been done for a single wavelength of 0.6 μm. The radiative transfer calculations result in lower albedos for heterogeneous cloud tops. The differences of albedos between heterogeneous and homogeneous cloud top decrease with increasing solar zenith angle. The influence of cloud top variability on radiances is shown. Explicitly in forward direction the reflectances for heterogeneous tops are larger, in backward direction lower. The largest difference of the mean reflectances (mean over cloud field) between homogeneous and heterogeneous cloud top is approximately 0.3, which is 30% of illumination.

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Richter, F., K. Barfus, F. H. Berger, and U. Görsdorf. "The influence of cloud top variability from radar measurements on 3-D radiative transfer." Atmospheric Chemistry and Physics 7, no. 17 (September 14, 2007): 4699–708. http://dx.doi.org/10.5194/acp-7-4699-2007.

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Abstract. In radiative transfer simulations the simplification of cloud top structure by homogeneous assumptions can cause mistakes in comparison to realistic heterogeneous cloud top structures. This paper examines the influence of cloud top heterogeneity on the radiation at the top of the atmosphere. The use of cloud top measurements with a high temporal resolution allows the analysis of small spatial cloud top heterogeneities by using the frozen turbulence assumption for the time – space conversion. Radiative observations are often based on satellite measurements, whereas small spatial structures are not considered in such treatments. A spectral analysis of the cloud top measurements showed slopes of power spectra between –1.8 and –2.0, these values are larger than the spectra of –5/3 which is often applied to generate cloud field variability. The comparison of 3-D radiative transfer results from cloud fields with homogeneous and heterogeneous tops has been done for a single wavelength of 0.6 μm. The radiative transfer calculations result in lower albedos for heterogeneous cloud tops. The differences of albedos between heterogeneous and homogeneous cloud top decrease with increasing solar zenith angle. The influence of cloud top variability on radiances is shown. The reflectances for heterogeneous tops are explicitly larger in forward direction, in backward direction lower. The largest difference of the mean reflectances (mean over cloud field) between homogeneous and heterogeneous cloud top is approximately 0.3, which is 30% of illumination.

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Krause, Maximilian, and Thomas Böhlke. "Stochastic evaluation of stress and strain distributions in duplex steel." Archive of Applied Mechanics 91, no. 8 (April 8, 2021): 3527–40. http://dx.doi.org/10.1007/s00419-021-01925-1.

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AbstractAustenite–ferrite duplex steels generally consist of two differently textured polycrystalline phases with different glide mechanisms. For estimating the effective mechanical behavior of heterogeneous materials, there exist well established approaches, two of which are the classes of mean-field and full-field methods. In this work, the local fields resulting from these different approaches are compared using analytical calculations and full-field simulations. Duplex steels of various textures measured using X-ray diffraction are considered. Special emphasis is given to the influence of the crystallographic texture on the stress and strain distributions.

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Metz, Fernando L., and Thomas Peron. "Mean-field theory of vector spin models on networks with arbitrary degree distributions." Journal of Physics: Complexity 3, no. 1 (February 3, 2022): 015008. http://dx.doi.org/10.1088/2632-072x/ac4bed.

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Abstract Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie–Weiss, the Kuramoto, and the Sherrington–Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington–Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree c scales as a power c ∝ N b (b < 1) of the total number N ≫ 1 of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

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MOLA, E. E., I. M. IRURZUN, J. L. VICENTE, and D. A. KING. "MESOSCOPIC PATTERN FORMATION IN CATALYTIC PROCESSES BY AN EXTENSION OF THE MEAN FIELD APPROACH." Surface Review and Letters 10, no. 01 (February 2003): 23–38. http://dx.doi.org/10.1142/s0218625x03004585.

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For some years it has been known that a number of catalytic reactions, under specified steady operating conditions, exhibit oscillations, in the rate of product formation. These are often related to beautiful spatiotemporal patterns, including targets and spirals, on the metal surface. These examples of self-organizational phenomena have attracted considerable interest, because they are proving to be theoretically amenable. Here we review different approximations to model heterogeneous surface chemical reactions, which exhibit oscillatory behavior. A focal point is the use of a detailed knowledge of the dynamics of surface structural phase transition for modeling kinetic oscillations, which represent a severe test of our understanding of chemical processes at surfaces. Advantages and disadvantages of the Monte Carlo approach are presented to model heterogeneous oscillatory chemical reactions, with special emphasis if a Monte Carlo method is going to be applied to study the time evolution of a surface chemical reaction, as there should be a linear relationship between the time unit called the Monte Carlo step (MCS) and actual time. We conclude that special care must be taken when two or more processes are included in a simulation, because now overall MCS should be compatible with every individual process. The mean field approach (MFA) takes into account only reaction processes and completely neglects spatial correlation and fluctuations. Therefore, this approach is not adequate for describing the rich variety of spatial patterns that are experimentally observed. On the other hand, Monte Carlo approaches are severely limited by computational capabilities. To overcome MFA limitations we propose to extend the earlier work of King and coworkers [J. Chem. Phys.100, 14417 (1996)], which did not include spatial dependence, by adding diffusion processes and gas global coupling to the coupled reaction equations. The extended MFA can now be used as a new tool for the analysis of pattern formation in surface chemistry.

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Rao, Wei, Chao Yu, Juan Zhang, and Guozheng Kang. "New incremental secant linearization method for mean-field homogenization approach of elasto-viscoplastic microscopic heterogeneous materials." Composite Structures 271 (September 2021): 114125. http://dx.doi.org/10.1016/j.compstruct.2021.114125.

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Wang, Dezhi, Wei Wang, Zhu Han, and Zhaoyang Zhang. "Delay Optimal Random Access With Heterogeneous Device Capabilities in Energy Harvesting Networks Using Mean Field Game." IEEE Transactions on Wireless Communications 20, no. 9 (September 2021): 5543–57. http://dx.doi.org/10.1109/twc.2021.3068315.

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Cruz, Roberto de la, Pilar Guerrero, Fabian Spill, and Tomás Alarcón. "Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis." Journal of Theoretical Biology 407 (October 2016): 161–83. http://dx.doi.org/10.1016/j.jtbi.2016.07.028.

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Sánchez-Orgaz, E. M., F. D. Denia, J. Martínez-Casas, and L. Baeza. "3D Acoustic Modelling of Dissipative Silencers with Nonhomogeneous Properties and Mean Flow." Advances in Mechanical Engineering 6 (January 1, 2014): 537935. http://dx.doi.org/10.1155/2014/537935.

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A finite element approach is proposed for the acoustic analysis of automotive silencers including a perforated duct with uniform axial mean flow and an outer chamber with heterogeneous absorbent material. This material can be characterized by means of its equivalent acoustic properties, considered coordinate-dependent via the introduction of a heterogeneous bulk density, and the corresponding material airflow resistivity variations. An approach has been implemented to solve the pressure wave equation for a nonmoving heterogeneous medium, associated with the problem of sound propagation in the outer chamber. On the other hand, the governing equation in the central duct has been solved in terms of the acoustic velocity potential considering the presence of a moving medium. The coupling between both regions and the corresponding acoustic fields has been carried out by means of a perforated duct and its acoustic impedance, adapted here to include absorbent material heterogeneities and mean flow effects simultaneously. It has been found that bulk density heterogeneities have a considerable influence on the silencer transmission loss.

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Wu, Qingchu, and Shufang Chen. "Heterogeneous pair-approximation analysis for susceptible–infectious–susceptible epidemics on networks." Chaos: An Interdisciplinary Journal of Nonlinear Science 33, no. 1 (January 2023): 013113. http://dx.doi.org/10.1063/5.0112058.

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The pair heterogeneous mean-field (PHMF) model has been used extensively in previous studies to investigate the dynamics of susceptible–infectious–susceptible epidemics on complex networks. However, the approximate treatment of the classical or reduced PHMF models lacks a rigorous theoretical analysis. By means of the standard and full PHMF models, we first derived the equivalent conditions for the approximate model treatment. Furthermore, we analytically derived a novel epidemic threshold for the PHMF model, and we demonstrated via numerical simulations that this threshold condition differs from all those reported in earlier studies. Our findings indicate that both the reduced and full PHMF models agree well with continuous-time stochastic simulations, especially when infection is spreading at considerably higher rates.

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El Oula Frihi, Zahrate, Julian Barreiro-Gomez, Salah Eddine Choutri, and Hamidou Tembine. "Hierarchical Structures and Leadership Design in Mean-Field-Type Games with Polynomial Cost." Games 11, no. 3 (August 6, 2020): 30. http://dx.doi.org/10.3390/g11030030.

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This article presents a class of hierarchical mean-field-type games with multiple layers and non-quadratic polynomial costs. The decision-makers act in sequential order with informational differences. We first examine the single-layer case where each decision-maker does not have the information about the other control strategies. We derive the Nash mean-field-type equilibrium and cost in a linear state-and-mean-field feedback form by using a partial integro-differential system. Then, we examine the Stackelberg two-layer problem with multiple leaders and multiple followers. Numerical illustrations show that, in the symmetric case, having only one leader is not necessarily optimal for the total sum cost. Having too many leaders may also be suboptimal for the total sum cost. The methodology is extended to multi-level hierarchical systems. It is shown that the order of the play plays a key role in the total performance of the system. We also identify a specific range of parameters for which the Nash equilibrium coincides with the hierarchical solution independently of the number of layers and the order of play. In the heterogeneous case, it is shown that the total cost is significantly affected by the design of the hierarchical structure of the problem.

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Marín Radoszynski, Andrea, and Pierre Pinson. "Electric demand response and bounded rationality: mean-field control for large populations of heterogeneous bounded-rational agents." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, no. 2202 (June 7, 2021): 20190429. http://dx.doi.org/10.1098/rsta.2019.0429.

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The increased penetration of renewable energy sources into existing power systems induces challenges in supply–demand balancing. Demand-side flexibility is seen as an option to accommodate variability and limited predictability from renewable energy generation. Heat pumps at residential level, if well coordinated, can be one of those flexibility sources. The complexity involved is high though, since their coordinated operation combines control, population effects and the fact agents may actually not behave as rational decision-makers. We describe here a coordinated control framework that accounts for those aspects altogether. Decentralized model predictive control for large populations of heterogeneous agents is employed. As the cost to be minimized is affected by the population behaviour as a whole through the electricity price, the decentralized control is re-thought as a mean-field game. Existence and uniqueness of a Nash equilibrium are discussed while the Picard–Banach algorithm is used as a solution approach. It is extended to the case of bounded-rational agents. The impact on system dynamics of modelling agents as bounded rational is illustrated through numerical simulations. This article is part of the theme issue ‘The mathematics of energy systems’.

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Miklavic, S. J. "Mean-Field Potential for Heterogeneous Electrical Double Layers, with Application to the Surface Pressure of Charged Monolayers." Journal of Colloid and Interface Science 171, no. 2 (May 1995): 446–55. http://dx.doi.org/10.1006/jcis.1995.1201.

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Høgh Jensen, K., and J. C. Refsgaard. "Spatial Variability of Physical Parameters and Processes in Two Field Soils." Hydrology Research 22, no. 5 (October 1, 1991): 327–40. http://dx.doi.org/10.2166/nh.1991.0022.

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A numerical analysis of solute transport in two spatially heterogeneous fields is carried out assuming that the fields are composed of ensembles of one-dimensional non-interacting soil columns, each column representing a possible soil profile in statistical terms. The basis for the analysis is the flow simulation described in Part II (Jensen and Refsgaard, this issue), which serves as input to a transport model based on the convection-dispersion equation. The simulations of the average and variation in solute concentration in planes perpendicular to the flow direction are compared to measurements obtained from tracer experiments carried out at the two fields. Due to the limited amount of measurement data, it is difficult to draw conclusive evidence of the simulations, but reliable simulations are obtained of the mean behaviour within the two fields. The concept of equivalent soil properties is also tested for the transport problem in heterogeneous soils. Based on effective parameters for the retention and hydraulic conductivity functions it is possible to predict the mean transport in the two experimental fields.

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Journal articles on the topic 'HETEROGENOUS MEAN FIELD'

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