The percolation model of relative conductivities and phase permeabilities

The universal mathematical model of relative conductivities of percolation clusters and phase permeabilities of oil-water-saturated rocks is presented. It is obtained on the basis of percolation theory, porous body physics and statistics. The model takes into account the influence of change in pore space surface properties and the nature of fluid flow on the studied characteristics and may be applied for comprehensive analysis and modeling of technological processes of oil production.

Geometry of rare regions behind Griffiths singularities in random quantum magnets

In many-body systems with quenched disorder, dynamical observables can be singular not only at the critical point, but in an extended region of the paramagnetic phase as well. These Griffiths singularities are due to rare regions, which are locally in the ordered phase and contribute to a large susceptibility. Here, we study the geometrical properties of rare regions in the transverse Ising model with dilution or with random couplings and transverse fields. In diluted models, the rare regions are percolation clusters, while in random models the ground state consists of a set of spin clusters, which are calculated by the strong disorder renormalization method. We consider the so called energy cluster, which has the smallest excitation energy and calculate its mass and linear extension in one-, two-and three-dimensions. Both average quantities are found to grow logarithmically with the linear size of the sample. Consequently, the energy clusters are not compact: for the diluted model they are isotropic and tree-like, while for the random model they are quasi-one-dimensional.

Spreading of Infections on Network Models: Percolation Clusters and Random Trees

We discuss network models as a general and suitable framework for describing the spreading of an infectious disease within a population. We discuss two types of finite random structures as building blocks of the network, one based on percolation concepts and the second one on random tree structures. We study, as is done for the SIR model, the time evolution of the number of susceptible (S), infected (I) and recovered (R) individuals, in the presence of a spreading infectious disease, by incorporating a healing mechanism for infecteds. In addition, we discuss in detail the implementation of lockdowns and how to simulate them. For percolation clusters, we present numerical results based on site percolation on a square lattice, while for random trees we derive new analytical results, which are illustrated in detail with a few examples. It is argued that such hierarchical networks can complement the well-known SIR model in most circumstances. We illustrate these ideas by revisiting USA COVID-19 data.

A Size-Perimeter Discrete Growth Model for Percolation Clusters

Cluster growth models are utilized for a wide range of scientific and engineering applications, including modeling epidemics and the dynamics of liquid propagation in porous media. Invasion percolation is a stochastic branching process in which a network of sites is getting occupied that leads to the formation of clusters (group of interconnected, occupied sites). The occupation of sites is governed by their resistance distribution; the invasion annexes the sites with the least resistance. An iterative cluster growth model is considered for computing the expected size and perimeter of the growing cluster. A necessary ingredient of the model is the description of the mean perimeter as the function of the cluster size. We propose such a relationship for the site square lattice. The proposed model exhibits (by design) the expected phase transition of percolation models, i.e., it diverges at the percolation threshold p c . We describe an application for the porosimetry percolation model. The calculations of the cluster growth model compare well with simulation results.

Power law behavior of Elementary Cellular Automata

This paper shows that the percolation clusters from elementary cellular automata {30, 45, 60, 86, 99, 105, 129, 150, 153, 169, 182, 183, 184, 195 and 225 exhibit strong power law behavior, either under random initial conditions, a single occupied cell, or both. Most of the tail exponents are less than unity, implying diverging means and variances of cluster sizes. The analysis presented here is admittedly coarse in an effort to expedite its dissemination.

Percolation clusters of territorial inequality due to the spread of the coronavirus infection

Due to the influence of the coronavirus infection, the issues of regional governance and territorial planning have been included in the urgent agenda of territorial development for two years in a row. Within the framework of this issue, a number of challenges of territorial administration are already being investigated. One of them is maintaining the achieved level of economic development and replenishing problematic aspects due to the spread of the coronavirus infection, which seriously affected all economic processes in 2020. The article reveals the management structure of Sverdlovsk region (Russia) and analyses the main socio-economic indicators of the region’s development. The results of the analysis contribute to identifying the urgent problem of regional governance and territorial planning of Sverdlovsk region, i.e. the deepening inter-territorial inequality caused by an unfavourable epidemiological situation. The authors propose management solutions aimed at improving regional governance and territorial planning of Sverdlovsk region in the context of the identified problem.

Controllable Fabrication of Percolative Metal Nanoparticle Arrays Applied for Quantum Conductance-Based Strain Sensors

We use gas phase deposition of well-defined nanoparticles (NPs) to fabricate closely-spaced Pd NP arrays on flexible membranes prepatterned with interdigital electrodes (IDEs). The evolution of the morphology and electron conductance of the NP arrays during deposition is analyzed. The growth of two-dimensional percolation clusters of interconnected NPs, which correlate with the percolation pathway for electron conduction in the NP deposits, is demonstrated. The percolative nature of the NP arrays permits us to finely control the percolation geometries and conductance of the NP film by controlling the NP deposition time so as to realize a precise and reproducible fabrication of sensing materials. Electron transport measurements reveal that the electrical conductance of the NP films is dominated by electron tunneling or hopping across the NP percolating networks. Based on the percolative and quantum tunneling nature, the closely-spaced Pd NP films on PET membranes are used as flexible strain sensors. The sensor demonstrates an excellent response ability to distinguish tiny deformations down to 5×10−4 strain and a high sensitivity with a large gauge factor of 200 up to 4% applied strain.

The first step from molecules to life: Formation of large random molecules acting as micro-environments

<p>The mathematician John von Neumann, through his work on universal constructors, discovered<br />a generalized version of the central dogma of molecular biology biology in the 1940s, long  <br />before the biological version had been discovered. While his discovery played no role in the  <br />development of molecular biology, we may benefit from a similar mathematical approach to find  <br />clues on the origin of life. This then involves addressing those problems in the field that  <br />do not depend on the details of organic chemistry. We can then consider a general set of  <br />models that describe machines capable of self-maintenance and self-replication formulated in  <br />terms of a set of building blocks and their interactions. </p> <p>The analogue of the origin of life problem is then to explain how one can get to such  <br />machines starting from a set of only building blocks. A fundamental obstacle one then faces  <br />is the limit on the complexity of low fidelity replicating systems, preventing building  <br />blocks from getting assembled randomly into low fidelity machines which can then improve due  <br />to natural selection [1]. A generic way out of this problem is for the entire ecosystem of  <br />machines to have been encapsulated in a micro-structure with fixed inner surface features  <br />that would have boosted the fidelity [2]. Such micro-structures could have formed as a result  <br />of the random assembly of building blocks, leading to so-called percolation clusters [2].</p> <p>This then leads us to consider how in the real world a percolation process involving the  <br />random assembly of organic molecules can be realized. A well studied process in the  <br />literature is the assembly of organic compounds in ice grains due to UV radiation and heating  <br />events [3,4,5]. This same process will also lead to the percolation process if it proceeds  <br />for a sufficiently long period [2].</p> <p>In this talk I will discuss the percolation process in more detail than has been done in [2],  <br />explaining how it leads to the necessary symmetry breakings such as the origin of chiral  <br />molecules needed to explain the origin of life.   </p> <p> </p> <p>[1] Eigen, M., 1971. Self-organization of matter and the evolution of biological  <br />macromolecules. Naturwissenschaften 58, 465-523.</p> <p>[2] Mitra, S., 2019. Percolation clusters of organics in interstellar ice grains as the  <br />incubators of life, Progress in Biophysics and Molecular Biology 149, 33-38.</p> <p>[3] Ciesla, F., and Sandford.,S., 2012. Organic Synthesis via Irradiation and Warming of Ice  <br />Grains in the Solar Nebula. Science 336, 452-454.</p> <p>[4] Muñoz Caro, G., et al., 2002. Amino acids from ultraviolet irradiation of interstellar ice  <br />analogues. Nature 416, 403-406.</p> <p>[5]  Meinert, C,., et al., 2016. Ribose and related sugars from ultraviolet irradiation of  <br />interstellar ice analogs. Science 352, 208-212.</p>