arXiv subjects

Michał Łepek

Publications and source records attributed to Michał Łepek.

9 recordsLinked to original sources

arXiv2026

A mini-review on combinatorial solutions to the Marcus-Lushnikov irreversible aggregation

Over the past decade, a combinatorial framework for discrete, finite, and irreversibly aggregating systems has emerged. This work reviews its progress, practical applications, and limitations. We outline the approach's assumptions and foundations, based on direct enumeration of system states, contrasting with classical Smoluchowski and Marcus-Lushnikov methods. Using the constant kernel as an example, we derive combinatorial expressions for the average number of clusters of a given size and their standard deviation, and present the complete probability distribution for cluster counts. The method is then extended to several kernels (additive, product, linear-chain, condensation) by explicitly enumerating ways to form clusters of a given size. For general kernels, approximate solutions are obtained via recursive expressions, enabling predictions without explicit solutions. Applications to aerosol growth and planetesimal formation are demonstrated, with comparisons to numerical results. We summarize issues of validity and precision and propose open problems. The appendix includes partial Bell polynomials, generating functions, Lagrange inversion, potential applications, and links between combinatorial and scaling solutions of the Smoluchowski equation.

cond-mat.stat-mech↗

arXiv2023

Towards fractal origins of the community structure in complex networks: a model-based approach

In this paper, we pose a hypothesis that the structure of communities in complex networks may result from their latent fractal properties. This hypothesis is based not only on the general observation that many real networks have multilevel organization, which is reminiscent of the geometric self-similarity of classical fractals. Quantitative arguments supporting this hypothesis are: first, many non-fractal real complex networks that have a well-defined community structure reveal fractal properties when suitably diluted; second, the scale-free community size distributions observed in many real networks directly relate to scale-invariant box mass distributions, which have recently been described as a fundamental feature of fractal complex networks. We test this hypothesis in a general model of evolving network with community structure that exhibits dual scale invariance: at the level of node degrees and community sizes, respectively. We show that, at least in this model, the proposed hypothesis cannot be rejected. The argument for this is that a kind of fractal core can be identified in the networks studied, which appears as a macroscopic connected component when the edges between modules identified by the community detection algorithm are removed in a supervised manner.

physics.soc-ph↗

arXiv2021

How initial condition impacts aggregation -- a systematic numerical study

In a process of aggregation, a finite number of particles merge irreversibly to create growing clusters. In this work, impact of particular initial conditions: monodisperse, power-law, exponential, and inspired by condensation nuclei was tested against several aggregation processes: constant, additive, product, electrorheological, anti-social, Berry's, Brownian, shear, and gravitational kernels. Coagulating systems consisting of a thousand monomers were observed in the late time of the evolution, for the moment when only fifty clusters were left in the system. In this way, the impact of particular initial condition was revealed in relation to other initial conditions. Several similarities between different kernels were observed, among others, strong correspondence between the constant and the Brownian kernel was confirmed. In case of the product kernel, unexpected behaviour related to the condensation nuclei initial conditions was revealed and explained. According to the results, it is doubtful that a phase transition can be triggered by appropriate setting of initial condition.

cond-mat.stat-mech↗

arXiv2021

Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework

For this work, we studied a finite system of discreet-size aggregating particles for two types of kernels with arbitrary parameters, a condensation (or branched-chain polymerization) kernel, $K(i,j)=(A+i)(A+j)$, and a linear combination of the constant and additive kernels, $K(i,j)=A+i+j$. They were solved under monodisperse initial conditions in the combinatorial approach where discreet time is counted as subsequent states of the system. A generating function method and Lagrange inversion were used for derivations. Expressions for an average number of clusters of a given size and its corresponding standard deviation were obtained and tested against numerical simulation. High precision of the theoretical predictions can be observed for a wide range of $A$ and coagulation stages, excepting post-gel phase in the case of the condensation kernel (a giant cluster presence is preserved). For appropriate $A$, these two kernels reproduced known results of the constant, additive and product kernels. Beside a previously solved linear-chain kernel, they extend the number of arbitrary-parameter kernels solved in the combinatorial approach.

cond-mat.stat-mech↗

arXiv2021

Exact combinatorial approach to finite coagulating systems through recursive equations

This work outlines an exact combinatorial approach to finite coagulating systems through recursive equations and use of generating function method. In the classic approach the mean-field Smoluchowski coagulation is used. However, the assumptions of the mean-field theory are rarely met in real systems which limits the accuracy of the solution. In our approach, cluster sizes and time are discrete, and the binary aggregation alone governs the time evolution of the systems. By considering the growth histories of all possible clusters and applying monodisperse initial conditions, the exact expression for the probability of finding a coagulating system with an arbitrary kernel in a given cluster configuration is derived. Then, the average number of such clusters and the standard deviation of these solutions can be calculated. In this work, recursive equations for all possible growth histories of clusters are introduced. The correctness of our expressions was proved based on the comparison with numerical results obtained for systems with constant, multiplicative and additive kernels. For the first time the exact solutions for the multiplicative and additive kernels were obtained with this framework. In addition, our results were compared with the results arising from the solutions to the mean-field Smoluchowski equation. Our theoretical predictions outperform the classic approach.

cond-mat.stat-mech↗

arXiv2020

Combinatorial solutions to generalized electrorheological kernel aggregation

For this paper, we studied the time evolution of a system of coagulating particles under a generalized electrorheological (ER) kernel with real power, $K\left(i,j\right) = \left( \frac{1}{i}+\frac{1}{j} \right)^α$, and monodisperse initial conditions. We used a combinatorial framework in which time and cluster sizes were discrete and the binary aggregation governed the time evolution of the system. We modified a previously-known solution for the constant kernel to cover the generalized ER kernel and used it in the framework to obtain the exact expression for the cluster size distribution (the average number of particles of a given size) and the standard deviation. Our theoretical solution is validated by a comparison to numerically simulated results for several values of $α$ and to the experimental data of coagulating polystyrene particles. Theoretical predictions were accurate for any time of the aggregation process and for a wide range of $α$.

cond-mat.stat-mech↗

arXiv2020

Automatic cough detection based on airflow signals for portable spirometry system

We give a short introduction to cough detection efforts that were undertaken during the last decade and we describe the solution for automatic cough detection developed for the AioCare portable spirometry system. In contrast to more popular analysis of sound and audio recordings, we fully based our approach on airflow signals only. As the system is intended to be used in a large variety of environments and different patients, we trained and validated the algorithm using AioCare-collected data and the large database of spirometry curves from the NHANES database by the American National Center for Health Statistics. We trained different classifiers, such as logistic regression, feed-forward artificial neural network, support vector machine, and random forest to choose the one with the best performance. The ANN solution was selected as the final classifier. The classification results on the test set (AioCare data) are: 0.86 (sensitivity), 0.91 (specificity), 0.91 (accuracy) and 0.88 (F1 score). The classification methodology developed in this study is robust for detecting cough events during spirometry measurements. As far as we know, the solution presented in this work is the first fully reproducible description of the automatic cough detection algorithm based totally on airflow signals and the first cough detection implemented in a commercial spirometry system that is to be published.

q-bio.QM↗

arXiv2018

Spatial evolution of Hindmarsh-Rose neural network with time delays

Spatial relations between neurons in the network with time delays play a crucial role in determining dynamics of the system. During the development of the nervous system different types of neurons group together to enable specific functions of the network. Right spatial distances, thus right time delays between cells are crucial for an appropriate functioning of the system. To model the process of neural migration we proposed simple but effective model of network spatial evolution based on Hindmarsh-Rose neurons and Metropoli-Hastings Monte Carlo algorithm. Under the specific assumptions and using appropriate parameters of the neural evolution the network can converge to the desirable state giving the opportunity of achieving large variety of spectra. We show that there is a specific range of network size in space which allows it to generate assumed output. A network or generally speaking a system with time delays (corresponding to the arrangement in the physical space) of specific output properties has a specific spatial dimension that allows it to function properly.

nlin.AO↗

arXiv2018

Coagulation with product kernel and arbitrary initial conditions: Exact kinetics within the Marcus-Lushnikov framework

The time evolution of a system of coagulating particles under the product kernel and arbitrary initial conditions is studied. Using the improved Marcus-Lushnikov approach, the master equation is solved for the probability $W(Q,t)$ to find the system in a given mass spectrum $Q=\{n_1,n_2,\dots,n_g\dots\}$, with $n_g$ being the number of particles of size $g$. The exact expression for the average number of particles, $\langle n_g(t)\rangle$, at arbitrary time $t$ is derived and its validity is confirmed in numerical simulations of several selected initial mass spectra.

cond-mat.stat-mech↗