1
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Gryczak DW, Lenzi EK, Rosseto MP, Evangelista LR, Zola RS. Non-Markovian Diffusion and Adsorption-Desorption Dynamics: Analytical and Numerical Results. Entropy (Basel) 2024; 26:294. [PMID: 38667848 PMCID: PMC11048754 DOI: 10.3390/e26040294] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2024] [Revised: 03/21/2024] [Accepted: 03/23/2024] [Indexed: 04/28/2024]
Abstract
The interplay of diffusion with phenomena like stochastic adsorption-desorption, absorption, and reaction-diffusion is essential for life and manifests in diverse natural contexts. Many factors must be considered, including geometry, dimensionality, and the interplay of diffusion across bulk and surfaces. To address this complexity, we investigate the diffusion process in heterogeneous media, focusing on non-Markovian diffusion. This process is limited by a surface interaction with the bulk, described by a specific boundary condition relevant to systems such as living cells and biomaterials. The surface can adsorb and desorb particles, and the adsorbed particles may undergo lateral diffusion before returning to the bulk. Different behaviors of the system are identified through analytical and numerical approaches.
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Affiliation(s)
| | - Ervin K. Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil;
| | - Michely P. Rosseto
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil;
| | - Luiz R. Evangelista
- Departamento de Física, Universidade Estadual de Maringá, Maringá 87020-900, PR, Brazil;
- Istituto dei Sistemi Complessi (ISC–CNR), Via dei Taurini, 19, 00185 Rome, Italy
- Department of Molecular Science and Nanosystems, Ca’ Foscari University of Venice, Via Torino 155, 30175 Mestre (VE), Italy
| | - Rafael S. Zola
- Department of Physics, Universidade Tecnológica Federal do Paraná, Apucarana 86812-460, PR, Brazil;
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2
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Souza DLM, Gabrick EC, Protachevicz PR, Borges FS, Trobia J, Iarosz KC, Batista AM, Caldas IL, Lenzi EK. Adaptive exponential integrate-and-fire model with fractal extension. Chaos 2024; 34:023107. [PMID: 38341761 DOI: 10.1063/5.0176455] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2023] [Accepted: 01/08/2024] [Indexed: 02/13/2024]
Abstract
The description of neuronal activity has been of great importance in neuroscience. In this field, mathematical models are useful to describe the electrophysical behavior of neurons. One successful model used for this purpose is the Adaptive Exponential Integrate-and-Fire (Adex), which is composed of two ordinary differential equations. Usually, this model is considered in the standard formulation, i.e., with integer order derivatives. In this work, we propose and study the fractal extension of Adex model, which in simple terms corresponds to replacing the integer derivative by non-integer. As non-integer operators, we choose the fractal derivatives. We explore the effects of equal and different orders of fractal derivatives in the firing patterns and mean frequency of the neuron described by the Adex model. Previous results suggest that fractal derivatives can provide a more realistic representation due to the fact that the standard operators are generalized. Our findings show that the fractal order influences the inter-spike intervals and changes the mean firing frequency. In addition, the firing patterns depend not only on the neuronal parameters but also on the order of respective fractal operators. As our main conclusion, the fractal order below the unit value increases the influence of the adaptation mechanism in the spike firing patterns.
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Affiliation(s)
- Diogo L M Souza
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Enrique C Gabrick
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Department of Physics, Humboldt University Berlin, Newtonstraße 15, 12489 Berlin, Germany
- Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | | | - Fernando S Borges
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, New York 11203, USA
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, 09606-045 São Bernardo do Campo, SP, Brazil
| | - José Trobia
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, Brazil
| | - Kelly C Iarosz
- University Center UNIFATEB, 84266-010 Telêmaco Borba, PR, Brazil
| | - Antonio M Batista
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
| | - Ervin K Lenzi
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Departament of Physics, State University of Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, Ponta Grossa 84030-900, PR, Brazil
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3
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Lenzi EK, Zola RS, Rosseto MP, Mendes RS, Ribeiro HV, da Silva LR, Evangelista LR. Results for Nonlinear Diffusion Equations with Stochastic Resetting. Entropy (Basel) 2023; 25:1647. [PMID: 38136527 PMCID: PMC10742535 DOI: 10.3390/e25121647] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2023] [Revised: 12/03/2023] [Accepted: 12/08/2023] [Indexed: 12/24/2023]
Abstract
In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate.
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Affiliation(s)
- Ervin K. Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil;
- National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro 22290-180, RJ, Brazil;
| | - Rafael S. Zola
- Departamento de Física, Universidade Tecnológica Federal do Paraná, Apucarana 86812-460, PR, Brazil;
| | - Michely P. Rosseto
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil;
| | - Renio S. Mendes
- Departamento de Física, Universidade Estadual de Maringá, Maringa 87020-900, PR, Brazil; (R.S.M.); (H.V.R.); (L.R.E.)
| | - Haroldo V. Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Maringa 87020-900, PR, Brazil; (R.S.M.); (H.V.R.); (L.R.E.)
| | - Luciano R. da Silva
- National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro 22290-180, RJ, Brazil;
- Departamento de Física, Universidade Federal do Rio Grande do Norte, Natal 59078-900, RN, Brazil
| | - Luiz R. Evangelista
- Departamento de Física, Universidade Estadual de Maringá, Maringa 87020-900, PR, Brazil; (R.S.M.); (H.V.R.); (L.R.E.)
- Istituto dei Sistemi Complessi (ISC–CNR), Via dei Taurini, 19, 00185 Rome, Italy
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4
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Zola RS, Lenzi EK, da Silva LR, Lenzi MK. Entropy Production in a Fractal System with Diffusive Dynamics. Entropy (Basel) 2023; 25:1578. [PMID: 38136458 PMCID: PMC10742906 DOI: 10.3390/e25121578] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2023] [Revised: 11/09/2023] [Accepted: 11/22/2023] [Indexed: 12/24/2023]
Abstract
We study the entropy production in a fractal system composed of two subsystems, each of which is subjected to an external force. This is achieved by using the H-theorem on the nonlinear Fokker-Planck equations (NFEs) characterizing the diffusing dynamics of each subsystem. In particular, we write a general NFE in terms of Hausdorff derivatives to take into account the metric of each system. We have also investigated some solutions from the analytical and numerical point of view. We demonstrate that each subsystem affects the total entropy and how the diffusive process is anomalous when the fractal nature of the system is considered.
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Affiliation(s)
- Rafael S. Zola
- Departmento de Física, Universidade Tecnológica Federal do Paraná—Campus de Apucarana, Apucarana 86812-460, PR, Brazil
| | - Ervin K. Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil;
| | - Luciano R. da Silva
- Departamento de Física, Universidade Federal do Rio Grande do Norte, Natal 59078-900, RN, Brazil;
| | - Marcelo K. Lenzi
- Departamento de Engenharia Química, Universidade Federal do Paraná, Curitiba 81531-980, PR, Brazil;
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5
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Evangelista LR, Lenzi EK. Nonlinear Fokker-Planck Equations, H-Theorem and Generalized Entropy of a Composed System. Entropy (Basel) 2023; 25:1357. [PMID: 37761656 PMCID: PMC10528951 DOI: 10.3390/e25091357] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2023] [Revised: 08/11/2023] [Accepted: 08/13/2023] [Indexed: 09/29/2023]
Abstract
We investigate the dynamics of a system composed of two different subsystems when subjected to different nonlinear Fokker-Planck equations by considering the H-theorem. We use the H-theorem to obtain the conditions required to establish a suitable dependence for the system's interaction that agrees with the thermodynamics law when the nonlinearity in these equations is the same. In this framework, we also consider different dynamical aspects of each subsystem and investigate a possible expression for the entropy of the composite system.
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Affiliation(s)
- Luiz R. Evangelista
- Dipartimento di Scienza Applicata e Tecnologia (DISAT) del Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
- Istituto dei Sistemi Complessi del Consiglio Nazionale delle Ricerche (ISC-CNR) co Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790, Maringá 87020-900, Paraná, Brazil
| | - Ervin K. Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Avenida General Carlos Cavalcanti, 4748, Ponta Grossa 84030-900, Paraná, Brazil;
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6
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Gabrick EC, Sayari E, Souza DLM, Borges FS, Trobia J, Lenzi EK, Batista AM. Fractal and fractional SIS model for syphilis data. Chaos 2023; 33:093124. [PMID: 37712917 DOI: 10.1063/5.0153122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2023] [Accepted: 08/25/2023] [Indexed: 09/16/2023]
Abstract
This work studies the SIS model extended by fractional and fractal derivatives. We obtain explicit solutions for the standard and fractal formulations; for the fractional case, we study numerical solutions. As a real data example, we consider the Brazilian syphilis data from 2011 to 2021. We fit the data by considering the three variations of the model. Our fit suggests a recovery period of 11.6 days and a reproduction ratio (R0) equal to 6.5. By calculating the correlation coefficient (r) between the real data and the theoretical points, our results suggest that the fractal model presents a higher r compared to the standard or fractional case. The fractal formulation is improved when two different fractal orders with distinguishing weights are considered. This modification in the model provides a better description of the data and improves the correlation coefficient.
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Affiliation(s)
- Enrique C Gabrick
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Elaheh Sayari
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Diogo L M Souza
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Fernando S Borges
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, New York 11203, USA
| | - José Trobia
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, 09606-045 São Bernardo do Campo, SP, Brazil
| | - Ervin K Lenzi
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Department of Physics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Antonio M Batista
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, 09606-045 São Bernardo do Campo, SP, Brazil
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7
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Lopes DD, Cunha BRD, Martins AF, Gonçalves S, Lenzi EK, Hanley QS, Perc M, Ribeiro HV. Machine learning partners in criminal networks. Sci Rep 2022; 12:15746. [PMID: 36130960 PMCID: PMC9492767 DOI: 10.1038/s41598-022-20025-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2022] [Accepted: 09/07/2022] [Indexed: 12/03/2022] Open
Abstract
Recent research has shown that criminal networks have complex organizational structures, but whether this can be used to predict static and dynamic properties of criminal networks remains little explored. Here, by combining graph representation learning and machine learning methods, we show that structural properties of political corruption, police intelligence, and money laundering networks can be used to recover missing criminal partnerships, distinguish among different types of criminal and legal associations, as well as predict the total amount of money exchanged among criminal agents, all with outstanding accuracy. We also show that our approach can anticipate future criminal associations during the dynamic growth of corruption networks with significant accuracy. Thus, similar to evidence found at crime scenes, we conclude that structural patterns of criminal networks carry crucial information about illegal activities, which allows machine learning methods to predict missing information and even anticipate future criminal behavior.
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Affiliation(s)
- Diego D Lopes
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR, 87020-900, Brazil
| | - Bruno R da Cunha
- Rio Grande do Sul Superintendency, Brazilian Federal Police, Porto Alegre, RS, 90160-093, Brazil.,National Police Academy, Brazilian Federal Police, Brasília, DF, 71559-900, Brazil
| | - Alvaro F Martins
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR, 87020-900, Brazil
| | - Sebastián Gonçalves
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, 91501-970, Brazil
| | - Ervin K Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, 84030-900, Brazil
| | - Quentin S Hanley
- School of Science and Technology, Nottingham Trent University, Clifton Lane, Nottingham, NG11 8NS, UK
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000, Maribor, Slovenia. .,Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan. .,Alma Mater Europaea, Slovenska ulica 17, 2000, Maribor, Slovenia. .,Complexity Science Hub Vienna, Josefstädterstraße 39, 1080, Vienna, Austria.
| | - Haroldo V Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR, 87020-900, Brazil.
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8
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Rosseto MP, Evangelista LR, Lenzi EK, Zola RS, Ribeiro de Almeida RR. Frequency-Dependent Dielectric Permittivity in Poisson-Nernst-Planck Model. J Phys Chem B 2022; 126:6446-6453. [PMID: 35984722 DOI: 10.1021/acs.jpcb.2c03663] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The approaches developed for studying the polarization of molecules and the dynamics of ions in dielectric materials are usually considered separately. The two effects are often believed to take place in different frequency ranges. The low frequency response is usually dominated by ionic migration, whereas the high frequency response is played by molecular polarization. The goal here is to clarify the interplay between free and bound charge densities and their influences on permittivity and impedance profiles by proposing a version of the Poisson-Nernst-Planck (PNP) model that allows to include the effect of a frequency-dependent (and thus not instantaneous) polarizability.
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Affiliation(s)
- M P Rosseto
- Department of Physics, Maringá State University, Maringá, Paraná 87020-900, Brazil
| | - L R Evangelista
- Department of Physics, Maringá State University, Maringá, Paraná 87020-900, Brazil
| | - E K Lenzi
- Department of Physics, Ponta Grossa State University, Ponta Grossa, Paraná 84010-330, Brazil
| | - R S Zola
- Department of Physics, Maringá State University, Maringá, Paraná 87020-900, Brazil.,Department of Physics, Federal University of Technology - Paraná, Apucarana, Paraná 86812-460, Brazil
| | - R R Ribeiro de Almeida
- Department of Physics, Federal University of Technology - Paraná, Apucarana, Paraná 86812-460, Brazil
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9
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Górska K, Horzela A, Lenzi EK, Pagnini G, Sandev T. Generalized Cattaneo (telegrapher's) equations in modeling anomalous diffusion phenomena. Phys Rev E 2020; 102:022128. [PMID: 32942420 DOI: 10.1103/physreve.102.022128] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2020] [Accepted: 07/20/2020] [Indexed: 11/07/2022]
Abstract
We study generalized Cattaneo (telegrapher's) equations involving memory effects introduced by smearing the time derivatives. Consistency conditions where the smearing functions obey restrict freedom in their choice but the proposed scheme goes beyond the approach based on using fractional derivatives. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, i.e., are normalizable and nonnegative on their domains. Nonnegativity of solutions is demonstrated by methods of positive definite and completely monotonic functions with the Bernstein theorem being the cornerstone of the ongoing proofs. Analysis of exactly solvable examples and relevant mean-squared displacements enables us to classify diffusion processes described by such got solutions and to identify them with either ordinary or anomalous diffusion which character may change over time. To complete the present research we compare our results with those obtained using the continuous-time random-walk and the continuous-time persistent random-walk approaches.
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Affiliation(s)
- K Górska
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Kraków, Poland
| | - A Horzela
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Kraków, Poland
| | - E K Lenzi
- Departamento de Fisica, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil
| | - G Pagnini
- BCAM-Basque Centre for Applied Mathematics, 48009 Bilbao, Basque Country Spain and Ikerbasque-Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain
| | - T Sandev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, 1000 Skopje, Macedonia, Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany and Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, 1000 Skopje, Macedonia
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10
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Barbero G, Evangelista LR, Lenzi EK. Frequency dispersion in the fractional Langmuir approximation for the adsorption–desorption phenomena. Proc Math Phys Eng Sci 2020. [DOI: 10.1098/rspa.2019.0570] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We propose that a kinetic equation of fractional order may be used to account for the frequency dispersion of the effective parameters in the framework of the Langmuir approximation for the adsorption–desorption phenomena. A frequency dependence of these parameters naturally arises in this formalism, indicating that it may play a similar role of a generalization of the Langmuir isotherm obtained from a homogeneous distribution of relaxation times. The fractional approach formalism opens the possibility to consider different relaxation regimes characterizing the interfacial behaviour of electrolytic cells, and may be a powerful tool to interpret complex impedance spectroscopy data.
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Affiliation(s)
- Giovanni Barbero
- Dipartimento di Scienza Applicata del Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
- National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye shosse 31, 115409 Moscow, Russian Federation
| | - Luiz Roberto Evangelista
- Dipartimento di Scienza Applicata del Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790, 87020-900 Maringá, Paraná, Brazil
| | - Ervin K. Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790, 87020-900 Maringá, Paraná, Brazil
- Departamento de Física, Universidade Estadual de Ponta Grossa, 87030-900, Ponta Grossa, Paraná, Brazil
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11
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Sigaki HYD, Lenzi EK, Zola RS, Perc M, Ribeiro HV. Learning physical properties of liquid crystals with deep convolutional neural networks. Sci Rep 2020; 10:7664. [PMID: 32376993 PMCID: PMC7203147 DOI: 10.1038/s41598-020-63662-9] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2019] [Accepted: 04/03/2020] [Indexed: 02/03/2023] Open
Abstract
Machine learning algorithms have been available since the 1990s, but it is much more recently that they have come into use also in the physical sciences. While these algorithms have already proven to be useful in uncovering new properties of materials and in simplifying experimental protocols, their usage in liquid crystals research is still limited. This is surprising because optical imaging techniques are often applied in this line of research, and it is precisely with images that machine learning algorithms have achieved major breakthroughs in recent years. Here we use convolutional neural networks to probe several properties of liquid crystals directly from their optical images and without using manual feature engineering. By optimizing simple architectures, we find that convolutional neural networks can predict physical properties of liquid crystals with exceptional accuracy. We show that these deep neural networks identify liquid crystal phases and predict the order parameter of simulated nematic liquid crystals almost perfectly. We also show that convolutional neural networks identify the pitch length of simulated samples of cholesteric liquid crystals and the sample temperature of an experimental liquid crystal with very high precision.
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Affiliation(s)
- Higor Y D Sigaki
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR, 87020-900, Brazil
| | - Ervin K Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, 84030-900, Brazil
| | - Rafael S Zola
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR, 87020-900, Brazil
- Departamento de Física, Universidade Tecnológica Federal do Paraná, Apucarana, PR, 86812-460, Brazil
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000, Maribor, Slovenia
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
- Complexity Science Hub Vienna, Josefstädterstraße 39, 1080, Vienna, Austria
| | - Haroldo V Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR, 87020-900, Brazil.
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12
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Tateishi AA, Ribeiro HV, Sandev T, Petreska I, Lenzi EK. Quenched and annealed disorder mechanisms in comb models with fractional operators. Phys Rev E 2020; 101:022135. [PMID: 32168676 DOI: 10.1103/physreve.101.022135] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2019] [Accepted: 02/11/2020] [Indexed: 06/10/2023]
Abstract
Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb model is a simplified description of diffusion on percolation clusters, where the comblike structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comblike structure by a generalized fractal structure. Our hybrid comb models thus represent a diffusion where different comblike structures describe different quenched disorder mechanisms, and the fractional operators account for various annealed disorder mechanisms. We find exact solutions for the diffusion propagator and mean square displacement in terms of different memory kernels used for defining the fractional operators. Among other findings, we show that these models describe crossovers from subdiffusion to Brownian or confined diffusions, situations emerging in empirical results. These results reveal the critical role of interactions between geometrical restrictions and memory effects on modeling anomalous diffusion.
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Affiliation(s)
- A A Tateishi
- Departamento de Física, Universidade Tecnologica Federal de Pato Branco, Pato Branco, Paraná 85503-390, Brazil
| | - H V Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Maringá, Paraná 87020-900, Brazil
| | - T Sandev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
- Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
| | - I Petreska
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
| | - E K Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Av. Carlos Cavalcanti 4748, 84030-900 Ponta Grossa, Paraná, Brazil
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Lenzi EK, Lenzi MK, Ribeiro HV, Evangelista LR. Extensions and solutions for nonlinear diffusion equations and random walks. Proc Math Phys Eng Sci 2019; 475:20190432. [PMID: 31824219 DOI: 10.1098/rspa.2019.0432] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2019] [Accepted: 10/03/2019] [Indexed: 11/12/2022] Open
Abstract
We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify that framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, 84030-900, Ponta Grossa, PR, Brazil.,National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, 22290-180, Rio de Janeiro, RJ, Brazil
| | - M K Lenzi
- Departamento de Engenharia Química, Universidade Federal do Paraná, Av. Cel. Francisco H. dos Santos 210, 81531-980, Curitiba, PR, Brazil
| | - H V Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900, Maringá PR, Brazil
| | - L R Evangelista
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900, Maringá PR, Brazil.,National Institute of Science and Technology on Complex Fluids, Instituto de Física-USP, São Paulo, SP 05508-090, Brazil.,Dipartimento di Scienza Applicata del Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
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de Souza RF, Granado SA, Evangelista LR, Lenzi EK, Li Q, Zola RS. Effect of dynamically changing the substrate's easy axis on the response time of nematic samples. J Phys Condens Matter 2018; 30:505401. [PMID: 30457123 DOI: 10.1088/1361-648x/aaea51] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Recent discoveries of advanced photocontrolled materials have kindled a great deal of interest on their use as command surfaces that switch easy axis under light radiation. One noticeable point when using switchable surfaces on any application is how the dynamical process propagates to the bulk directors. In this paper, we theoretically study the effect of a relaxing easy axis over time on a nematic sample when finite anchoring energy and surface viscosity are included. We first consider the case where just one of the substrates decay over time in an initially distorted director organization. Next, we assume that both substrates can be switched simultaneously. From the calculated director we obtained the optical profile and finally the molecular response time of the material. The response time depends on both the materials and the surfaces properties including its decay time. Our results might be used for understanding and engineering liquid crystal displays and other electro-optical devices with photocontrolled alignment layers.
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Affiliation(s)
- R F de Souza
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790-87020-900 Maringá, Paraná, Brazil
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de Souza RF, Lenzi EK, de Souza RT, Evangelista LR, Li Q, Zola RS. Surface induced twist in nematic and chiral nematic liquid crystals: stick-slip-like and constrained motion. Soft Matter 2018; 14:2084-2093. [PMID: 29485156 DOI: 10.1039/c7sm02122d] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Surface driven pattern formation is an intriguing phenomenon in the liquid crystal field. Owing to its ability to transmit torque, one can generate different patterns by propagating distortions on the optical wavelength scale in the sample from the surface. Here, we theoretically investigate (from the elasticity point of view) twist deformations induced by a rotating easy axis at one surface, by considering the anchoring energy and surface viscosity of nematic and chiral nematic samples. The model is solved analytically in the limit of strong anchoring and numerically for a low anchoring strength situation. Such rotation could be induced, in principle, by light-controlling the orientation of an azobenzene monolayer coated at one of the glass substrates or by an in-plane rotating field. We discuss the role of the surface parameters and the different distortions, and calculate light transmission using the Jones method. Three different regimes are identified: free twist, stick-slip twist, and constrained twist. The results obtained here may be relevant for liquid crystal active waveplates and for determining surface viscosity and the azimuthal anchoring energy.
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Affiliation(s)
- R F de Souza
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil
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Abstract
We use the H theorem to establish the entropy and the entropic additivity law for a system composed of subsystems, with the dynamics governed by the Klein-Kramers equations, by considering relations among the dynamics of these subsystems and their entropies. We start considering the subsystems governed by linear Klein-Kramers equations and verify that the Boltzmann-Gibbs entropy is appropriated to this dynamics, leading us to the standard entropic additivity, S_{BG}^{(1∪2)}=S_{BG}^{1}+S_{BG}^{2}, consistent with the fact that the distributions of the subsystem are independent. We then extend the dynamics of these subsystems to independent nonlinear Klein-Kramers equations. For this case, the results show that the H theorem is verified for a generalized entropy, which does not preserve the standard entropic additivity for independent distributions. In this scenario, consistent results are obtained when a suitable coupling among the nonlinear Klein-Kramers equations is considered, in which each subsystem modifies the other until an equilibrium state is reached. This dynamics, for the subsystems, results in the Tsallis entropy for the system and, consequently, verifies the relation S_{q}^{(1∪2)}=S_{q}^{1}+S_{q}^{2}+(1-q)S_{q}^{1}S_{q}^{2}/k, which is a nonadditive entropic relation.
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Affiliation(s)
- M A F Dos Santos
- Departamento de Física, Universidade Estadual de Ponta Grossa, Av. General Carlos Cavalcanti, 4748, Ponta Grossa, PR 87030-900, Brazil
| | - E K Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Av. General Carlos Cavalcanti, 4748, Ponta Grossa, PR 87030-900, Brazil
- National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ 22290-180, Brazil
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Abstract
The search for patterns in time series is a very common task when dealing with complex systems. This is usually accomplished by employing a complexity measure such as entropies and fractal dimensions. However, such measures usually only capture a single aspect of the system dynamics. Here, we propose a family of complexity measures for time series based on a generalization of the complexity-entropy causality plane. By replacing the Shannon entropy by a monoparametric entropy (Tsallis q entropy) and after considering the proper generalization of the statistical complexity (q complexity), we build up a parametric curve (the q-complexity-entropy curve) that is used for characterizing and classifying time series. Based on simple exact results and numerical simulations of stochastic processes, we show that these curves can distinguish among different long-range, short-range, and oscillating correlated behaviors. Also, we verify that simulated chaotic and stochastic time series can be distinguished based on whether these curves are open or closed. We further test this technique in experimental scenarios related to chaotic laser intensity, stock price, sunspot, and geomagnetic dynamics, confirming its usefulness. Finally, we prove that these curves enhance the automatic classification of time series with long-range correlations and interbeat intervals of healthy subjects and patients with heart disease.
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Affiliation(s)
- Haroldo V Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil
| | - Max Jauregui
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil
| | - Luciano Zunino
- Centro de Investigaciones Ópticas (CONICET La Plata - CIC), C.C. 3, 1897 Gonnet, Argentina
- Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata (UNLP), 1900 La Plata, Argentina
| | - Ervin K Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR 84030-900, Brazil
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Lenzi EK, Zola RS, Ribeiro HV, Vieira DS, Ciuchi F, Mazzulla A, Scaramuzza N, Evangelista LR. Ion Motion in Electrolytic Cells: Anomalous Diffusion Evidences. J Phys Chem B 2017; 121:2882-2886. [PMID: 28293957 DOI: 10.1021/acs.jpcb.7b01097] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
In this study, we argue that ion motion in electrolytic cells containing Milli-Q water, weak electrolytes, or liquid crystals may exhibit unusual diffusive regimes that deviate from the expected behavior, leading the system to present an anomalous diffusion. Our arguments lie on the investigation of the electrical conductivity and its relationship with the mean square displacement, which may be used to characterize the ionic motion. In our analysis, the Poisson-Nernst-Planck diffusional model is used with extended boundary conditions to simulate the charge transfer, accumulation, and/or adsorption-desorption at the electrode surfaces.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa , 87030-900 Ponta Grossa, Paraná, Brazil
| | - R S Zola
- Departamento de Física, Universidade Estadual de Maringá , Avenida Colombo, 5790, 87020-900 Maringá, Paraná, Brazil.,Departamento de Física, Universidade Tecnológica Federal do Paraná , Rua Marcílio Dias, 635, 86812-460 Apucarana, Paraná, Brazil
| | - H V Ribeiro
- Departamento de Física, Universidade Estadual de Maringá , Avenida Colombo, 5790, 87020-900 Maringá, Paraná, Brazil
| | - Denner S Vieira
- Departamento de Física, Universidade Estadual de Maringá , Avenida Colombo, 5790, 87020-900 Maringá, Paraná, Brazil
| | | | | | | | - L R Evangelista
- Departamento de Física, Universidade Estadual de Maringá , Avenida Colombo, 5790, 87020-900 Maringá, Paraná, Brazil
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Guimarães VG, Lenzi EK, Evangelista LR, Zola FC, de Souza RT, Zola RS. Symmetry breaking in an electrolytic cell under AC field and non-identical adsorbing electrodes. J Electroanal Chem (Lausanne) 2017. [DOI: 10.1016/j.jelechem.2017.02.019] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Lenzi EK, Ribeiro HV, Tateishi AA, Zola RS, Evangelista LR. Anomalous diffusion and transport in heterogeneous systems separated by a membrane. Proc Math Phys Eng Sci 2016; 472:20160502. [PMID: 27956877 DOI: 10.1098/rspa.2016.0502] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Diffusion of particles in a heterogeneous system separated by a semipermeable membrane is investigated. The particle dynamics is governed by fractional diffusion equations in the bulk and by kinetic equations on the membrane, which characterizes an interface between two different media. The kinetic equations are solved by incorporating memory effects to account for anomalous diffusion and, consequently, non-Debye relaxations. A rich variety of behaviours for the particle distribution at the interface and in the bulk may be found, depending on the choice of characteristic times in the boundary conditions and on the fractional index of the modelling equations.
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Affiliation(s)
- E K Lenzi
- Departamento de Física , Universidade Estadual de Ponta Grossa , Ponta Grossa, Paraná 87030-900, Brazil
| | - H V Ribeiro
- Departamento de Física , Universidade Estadual de Maringá , Maringá, Paraná 87020-900, Brazil
| | - A A Tateishi
- Departamento de Física , Universidade Tecnológica Federal do Paraná , Pato Branco, Paraná 85503-390, Brazil
| | - R S Zola
- Departamento de Física, Universidade Estadual de Maringá, Maringá, Paraná 87020-900, Brazil; Departamento de Física, Universidade Tecnológica Federal do Paraná, Apucarana, Paraná 86812-460, Brazil
| | - L R Evangelista
- Departamento de Física , Universidade Estadual de Maringá , Maringá, Paraná 87020-900, Brazil
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Alves LGA, Mendes RS, Lenzi EK, Ribeiro HV. Scale-Adjusted Metrics for Predicting the Evolution of Urban Indicators and Quantifying the Performance of Cities. PLoS One 2015; 10:e0134862. [PMID: 26356081 PMCID: PMC4565645 DOI: 10.1371/journal.pone.0134862] [Citation(s) in RCA: 39] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2015] [Accepted: 07/14/2015] [Indexed: 11/18/2022] Open
Abstract
More than a half of world population is now living in cities and this number is expected to be two-thirds by 2050. Fostered by the relevancy of a scientific characterization of cities and for the availability of an unprecedented amount of data, academics have recently immersed in this topic and one of the most striking and universal finding was the discovery of robust allometric scaling laws between several urban indicators and the population size. Despite that, most governmental reports and several academic works still ignore these nonlinearities by often analyzing the raw or the per capita value of urban indicators, a practice that actually makes the urban metrics biased towards small or large cities depending on whether we have super or sublinear allometries. By following the ideas of Bettencourt et al. [PLoS ONE 5 (2010) e13541], we account for this bias by evaluating the difference between the actual value of an urban indicator and the value expected by the allometry with the population size. We show that this scale-adjusted metric provides a more appropriate/informative summary of the evolution of urban indicators and reveals patterns that do not appear in the evolution of per capita values of indicators obtained from Brazilian cities. We also show that these scale-adjusted metrics are strongly correlated with their past values by a linear correspondence and that they also display crosscorrelations among themselves. Simple linear models account for 31%-97% of the observed variance in data and correctly reproduce the average of the scale-adjusted metric when grouping the cities in above and below the allometric laws. We further employ these models to forecast future values of urban indicators and, by visualizing the predicted changes, we verify the emergence of spatial clusters characterized by regions of the Brazilian territory where we expect an increase or a decrease in the values of urban indicators.
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Affiliation(s)
- Luiz G. A. Alves
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil
- National Institute of Science and Technology for Complex Systems, CNPq, Rio de Janeiro, RJ 22290-180, Brazil
| | - Renio S. Mendes
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil
- National Institute of Science and Technology for Complex Systems, CNPq, Rio de Janeiro, RJ 22290-180, Brazil
| | - Ervin K. Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil
- National Institute of Science and Technology for Complex Systems, CNPq, Rio de Janeiro, RJ 22290-180, Brazil
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR 84030-900, Brazil
| | - Haroldo V. Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil
- Departamento de Física, Universidade Tecnológica Federal do Paraná, Apucarana, PR 86812-460, Brazil
- * E-mail:
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Mendes GA, Ribeiro MS, Mendes RS, Lenzi EK, Nobre FD. Nonlinear Kramers equation associated with nonextensive statistical mechanics. Phys Rev E Stat Nonlin Soft Matter Phys 2015; 91:052106. [PMID: 26066118 DOI: 10.1103/physreve.91.052106] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2015] [Indexed: 06/04/2023]
Abstract
Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified q-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results.
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Affiliation(s)
- G A Mendes
- Departamento de Física, Universidade Federal do Maranhão, Avenida dos Portugueses 1966, 65080-805 São Luís-MA, Brazil
| | - M S Ribeiro
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
| | - R S Mendes
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá-PR, Brazil
- National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
| | - E K Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Avenida Carlos Cavalcanti 4748, 84030-900 Ponta Grossa-PR, Brazil
- National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
| | - F D Nobre
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
- National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
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Guimarães VG, Ribeiro HV, Li Q, Evangelista LR, Lenzi EK, Zola RS. Unusual diffusing regimes caused by different adsorbing surfaces. Soft Matter 2015; 11:1658-1666. [PMID: 25633342 DOI: 10.1039/c5sm00151j] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
A confined liquid with dispersed neutral particles is theoretically studied when the limiting surfaces present different dynamics for the adsorption-desorption phenomena. The investigation considers different non-singular kernels in the kinetic equations at the walls, where the suitable choice of the kernel can account for the relative importance of physisorption or chemisorption. We find that even a small difference in the adsorption-desorption rate of one surface (relative to the other) can drastically affect the behavior of the whole system. The surface and bulk densities and the dispersion are calculated when several scenarios are considered and anomalous-like behaviors are found. The approach described here is closely related to experimental situations, and can be applied in several contexts such as dielectric relaxation, diffusion-controlled relaxation in liquids, liquid crystals, and amorphous polymers.
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Affiliation(s)
- Veridiana G Guimarães
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil.
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Malacarne LC, Savi EL, Baesso ML, Lenzi EK, Astrath NGC. Role of photophysics processes in thermal lens spectroscopy of fluids: a theoretical study. J Phys Chem A 2014; 118:5983-8. [PMID: 25017719 DOI: 10.1021/jp505255a] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Photophysics processes are ubiquitous in nature and difficult to be quantitatively characterized by conventional spectroscopy. Alternatively, pump-probe methods have been widely applied to study these complex processes. In this context, the thermal lens technique is a precise spectroscopic tool for material characterization and presents a wide range of applications in chemical analysis. Here, we present an all numerical approach to analyze the dynamics of photophysics processes and to identify the role of individual contributions of photoreaction and mass diffusion in the thermal lens experiments. The results are essential for a proper understanding of the dominant physical mechanisms in laser-induced photodegradation, which allow precise data analysis of the effects in photosensitive fluids.
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Affiliation(s)
- L C Malacarne
- Departamento de Física, Universidade Estadual de Maringá , Maringá, Paraná 87020-900, Brazil
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Picoli S, del Castillo-Mussot M, Ribeiro HV, Lenzi EK, Mendes RS. Universal bursty behaviour in human violent conflicts. Sci Rep 2014; 4:4773. [PMID: 24758839 PMCID: PMC3998014 DOI: 10.1038/srep04773] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2013] [Accepted: 04/02/2014] [Indexed: 12/03/2022] Open
Abstract
Understanding the mechanisms and processes underlying the dynamics of collective violence is of considerable current interest. Recent studies indicated the presence of robust patterns characterizing the size and timing of violent events in human conflicts. Since the size and timing of violent events arises as the result of a dynamical process, we explore the possibility of unifying these observations. By analyzing available catalogs on violent events in Iraq (2003–2005), Afghanistan (2008–2010) and Northern Ireland (1969–2001), we show that the inter-event time distributions (calculated for a range of minimum sizes) obeys approximately a simple scaling law which holds for more than three orders of magnitude. This robust pattern suggests a hierarchical organization in size and time providing a unified picture of the dynamics of violent conflicts.
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Affiliation(s)
- S Picoli
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil
| | - M del Castillo-Mussot
- Instituto de Física, Universidad Nacional Autónoma de México, 04510 México D.F., Mexico
| | - H V Ribeiro
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil
| | - E K Lenzi
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil
| | - R S Mendes
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil
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Alves LGA, Ribeiro HV, Lenzi EK, Mendes RS. Distance to the scaling law: a useful approach for unveiling relationships between crime and urban metrics. PLoS One 2013; 8:e69580. [PMID: 23940525 PMCID: PMC3734155 DOI: 10.1371/journal.pone.0069580] [Citation(s) in RCA: 59] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2013] [Accepted: 06/13/2013] [Indexed: 11/24/2022] Open
Abstract
We report on a quantitative analysis of relationships between the number of homicides, population size and ten other urban metrics. By using data from Brazilian cities, we show that well-defined average scaling laws with the population size emerge when investigating the relations between population and number of homicides as well as population and urban metrics. We also show that the fluctuations around the scaling laws are log-normally distributed, which enabled us to model these scaling laws by a stochastic-like equation driven by a multiplicative and log-normally distributed noise. Because of the scaling laws, we argue that it is better to employ logarithms in order to describe the number of homicides in function of the urban metrics via regression analysis. In addition to the regression analysis, we propose an approach to correlate crime and urban metrics via the evaluation of the distance between the actual value of the number of homicides (as well as the value of the urban metrics) and the value that is expected by the scaling law with the population size. This approach has proved to be robust and useful for unveiling relationships/behaviors that were not properly carried out by the regression analysis, such as [Formula: see text] the non-explanatory potential of the elderly population when the number of homicides is much above or much below the scaling law, [Formula: see text] the fact that unemployment has explanatory potential only when the number of homicides is considerably larger than the expected by the power law, and [Formula: see text] a gender difference in number of homicides, where cities with female population below the scaling law are characterized by a number of homicides above the power law.
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Affiliation(s)
- Luiz G. A. Alves
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Maringá, Brazil
| | - Haroldo V. Ribeiro
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Maringá, Brazil
| | - Ervin K. Lenzi
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Maringá, Brazil
| | - Renio S. Mendes
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Maringá, Brazil
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Mantovani MC, Ribeiro HV, Lenzi EK, Picoli S, Mendes RS. Engagement in the electoral processes: scaling laws and the role of political positions. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 88:024802. [PMID: 24032971 DOI: 10.1103/physreve.88.024802] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2013] [Revised: 07/12/2013] [Indexed: 06/02/2023]
Abstract
We report on a statistical analysis of the engagement in the electoral processes of all Brazilian cities by considering the number of party memberships and the number of candidates for mayor and councillor. By investigating the relationships between the number of party members and the population of voters, we have found that the functional forms of these relationships are well described by sublinear power laws (allometric scaling) surrounded by a multiplicative log-normal noise. We have observed that this pattern is quite similar to those we previously reported for the relationships between the number of candidates (mayor and councillor) and population of voters [Europhys. Lett. 96, 48001 (2011)], suggesting that similar universal laws may be ruling the engagement in the electoral processes. We also note that the power-law exponents display a clear hierarchy, where the more influential is the political position the smaller is the value of the exponent. We have also investigated the probability distributions of the number of candidates (mayor and councillor), party memberships, and voters. The results indicate that the most influential positions are characterized by distributions with very short tails, while less influential positions display an intermediate power-law decay before showing an exponential-like cutoff. We discuss the possibility that, in addition to the political power of the position, limitations in the number of available seats can also be connected with this changing of behavior. We further believe that our empirical findings point out to an under-representation effect, where the larger the city is, the larger are the obstacles for more individuals to become directly engaged in the electoral process.
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Affiliation(s)
- M C Mantovani
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil and Universidade Tecnológica Federal do Paraná, Campus Campo Mourão, 87301-006 Campo Mourão, Paraná, Brazil
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Ribeiro HV, Guimarães RR, Teixeira-Souza RT, Mukai H, Fernandes PRG, Lenzi EK, Mendes RS. Antipersistent behavior of defects in a lyotropic liquid crystal during annihilation. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 87:054501. [PMID: 23767661 DOI: 10.1103/physreve.87.054501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2013] [Indexed: 06/02/2023]
Abstract
We report on the dynamical behavior of defects of strength s=±1/2 in a lyotropic liquid crystal during the annihilation process. By following their positions using time-resolved polarizing microscopy technique, we present statistically significant evidence that the relative velocity between defect pairs is Gaussian distributed, antipersistent, and long-range correlated. We further show that simulations of the Lebwohl-Lasher model reproduce quite well our experimental findings.
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Affiliation(s)
- H V Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, 87020-900, Maringá, Paraná, Brazil.
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29
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Lenzi EK, da Silva LR, Tateishi AA, Lenzi MK, Ribeiro HV. Diffusive process on a backbone structure with drift terms. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 87:012121. [PMID: 23410297 DOI: 10.1103/physreve.87.012121] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2012] [Revised: 12/09/2012] [Indexed: 06/01/2023]
Abstract
The effects of an external force on a diffusive process subjected to a backbone structure are investigated by considering the system governed by a Fokker-Planck equation with drift terms. Our results show an anomalous spreading which may present different diffusive regimes connected to anomalous diffusion and stationary states.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, PR, Brazil.
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30
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de Paula JL, Santoro PA, Zola RS, Lenzi EK, Evangelista LR, Ciuchi F, Mazzulla A, Scaramuzza N. Non-Debye relaxation in the dielectric response of nematic liquid crystals: surface and memory effects in the adsorption-desorption process of ionic impurities. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 86:051705. [PMID: 23214803 DOI: 10.1103/physreve.86.051705] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2012] [Indexed: 06/01/2023]
Abstract
We demonstrate theoretically that the presence of ions in insulating materials such as nematic liquid crystals may be responsible for the dielectric spectroscopy behavior observed experimentally. It is shown that, at low frequencies, an essentially non-Debye relaxation process takes place due to surface effects. This is accomplished by investigating the effects of the adsorption-desorption process on the electrical response of an electrolytic cell when the generation and recombination of ions is present. The adsorption-desorption is governed by a non-usual kinetic equation in order to incorporate memory effects related to a non-Debye relaxation and the roughness of the surface. The analysis is carried out by searching for solutions to the drift-diffusion equation that satisfy the Poisson equation relating the effective electric field to the net charge density. We also discuss the effect of the mobility of the ions, i.e., situations with equal and different diffusion coefficients for positive and negative ions, on the impedance and obtain an exact expression for the admittance. The model is compared with experimental results measured for the impedance of a nematic liquid crystal sample and a very good agreement is obtained.
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Affiliation(s)
- J L de Paula
- Departamento di Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil
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31
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Ribeiro HV, Zunino L, Lenzi EK, Santoro PA, Mendes RS. Complexity-entropy causality plane as a complexity measure for two-dimensional patterns. PLoS One 2012; 7:e40689. [PMID: 22916097 PMCID: PMC3419253 DOI: 10.1371/journal.pone.0040689] [Citation(s) in RCA: 48] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2012] [Accepted: 06/11/2012] [Indexed: 11/18/2022] Open
Abstract
Complexity measures are essential to understand complex systems and there are numerous definitions to analyze one-dimensional data. However, extensions of these approaches to two or higher-dimensional data, such as images, are much less common. Here, we reduce this gap by applying the ideas of the permutation entropy combined with a relative entropic index. We build up a numerical procedure that can be easily implemented to evaluate the complexity of two or higher-dimensional patterns. We work out this method in different scenarios where numerical experiments and empirical data were taken into account. Specifically, we have applied the method to fractal landscapes generated numerically where we compare our measures with the Hurst exponent; liquid crystal textures where nematic-isotropic-nematic phase transitions were properly identified; 12 characteristic textures of liquid crystals where the different values show that the method can distinguish different phases; and Ising surfaces where our method identified the critical temperature and also proved to be stable.
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Affiliation(s)
- Haroldo V Ribeiro
- Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Maringá, Brazil.
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Abstract
A wide range of physical and biological systems exhibit complex behaviours characterised by a scale-invariant structure of the fluctuations in their output signals. In the context of plant populations, scaling relationships are typically allometric. In this study, we analysed spatial variation in the size of maize plants (Zea Mays L.) grown in agricultural plots at constant densities and found evidence of scaling in the size fluctuations of plants. The findings indicate that the scaling of the probability distribution of spatial size fluctuation exhibits non-Gaussian behaviour compatible with a Lévy stable process. The scaling relationships were observed for spatial scales spanning three orders of magnitude. These findings should provide additional information for the selection and development of empirically accurate models of pattern formation in plant populations.
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Affiliation(s)
- S Picoli
- Departamento de Física, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil.
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33
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Tateishi AA, Lenzi EK, da Silva LR, Ribeiro HV, Picoli S, Mendes RS. Different diffusive regimes, generalized Langevin and diffusion equations. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 85:011147. [PMID: 22400552 DOI: 10.1103/physreve.85.011147] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2011] [Revised: 10/28/2011] [Indexed: 05/31/2023]
Abstract
We investigate a generalized Langevin equation (GLE) in the presence of an additive noise characterized by the mixture of the usual white noise and an arbitrary one. This scenario lead us to a wide class of diffusive processes, in particular the ones whose noise correlation functions are governed by power laws, exponentials, and Mittag-Leffler functions. The results show the presence of different diffusive regimes related to the spreading of the system. In addition, we obtain a fractional diffusionlike equation from the GLE, confirming the results for long time.
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Affiliation(s)
- A A Tateishi
- Departamento de Física, Universidade Estadual de Maringá Avenida Colombo, 5790-87020-900 Maringá-PR, Brazil
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34
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Evangelista LR, Lenzi EK, Barbero G, Macdonald JR. Anomalous diffusion and memory effects on the impedance spectroscopy for finite-length situations. J Phys Condens Matter 2011; 23:485005. [PMID: 22082531 DOI: 10.1088/0953-8984/23/48/485005] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
The contribution of ions to the electrical impedance of an electrolytic cell limited by perfect blocking electrodes is determined by considering the role of the anomalous diffusion process and memory effects. Analytical solutions for fractional diffusion equations together with Poisson's equation relating the effective electric field to the net charge density are found. This procedure allows the construction of general expressions for the electrochemical impedance satisfying the Kramers-Kronig relations when the diffusion of ions in the cell is characterized by the usual, as well as by anomalous, behavior.
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Affiliation(s)
- L R Evangelista
- Departamento di Física, Universidade Estadual de Maringá, Maringá, Paraná, Brazil.
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35
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Lenzi EK, Fernandes PRG, Petrucci T, Mukai H, Ribeiro HV. Anomalous-diffusion approach applied to the electrical response of water. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 84:041128. [PMID: 22181108 DOI: 10.1103/physreve.84.041128] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2011] [Indexed: 05/31/2023]
Abstract
We investigate the electrical response of Milli-Q deionized water by using a fractional diffusion equation of distributed order with the interfaces (i.e., the boundary conditions at the electrodes limiting the sample) governed by integrodifferential equations. We also consider that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies Poisson's equation. In addition, the good agreement between the experimental data and this approach evidences the presence of anomalous diffusion due to the surface effects in this system.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil
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36
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Santoro PA, de Paula JL, Lenzi EK, Evangelista LR. Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell. J Chem Phys 2011; 135:114704. [PMID: 21950880 DOI: 10.1063/1.3637944] [Citation(s) in RCA: 51] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023] Open
Affiliation(s)
- P A Santoro
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil
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37
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Mendes RS, Ribeiro HV, Freire FCM, Tateishi AA, Lenzi EK. Universal patterns in sound amplitudes of songs and music genres. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 83:017101. [PMID: 21405792 DOI: 10.1103/physreve.83.017101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/20/2010] [Revised: 11/10/2010] [Indexed: 05/30/2023]
Abstract
We report a statistical analysis of more than eight thousand songs. Specifically, we investigated the probability distribution of the normalized sound amplitudes. Our findings suggest a universal form of distribution that agrees well with a one-parameter stretched Gaussian. We also argue that this parameter can give information on music complexity, and consequently it helps classify songs as well as music genres. Additionally, we present statistical evidence that correlation aspects of the songs are directly related to the non-Gaussian nature of their sound amplitude distributions.
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Affiliation(s)
- R S Mendes
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900, Maringá, Paraná, Brazil
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38
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Malacarne LC, Astrath NGC, Lukasievicz GVB, Lenzi EK, Baesso ML, Bialkowski SE. Time-resolved thermal lens and thermal mirror spectroscopy with sample-fluid heat coupling: a complete model for material characterization. Appl Spectrosc 2011; 65:99-104. [PMID: 21211159 DOI: 10.1366/10-06096] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
This work presents a theoretical study of a heat transfer effect, taking into account the heat transfer within the heated sample and out to the surrounding medium. The analytical solution is used to model the thermal lens and thermal mirror effects and the results are compared with the finite element analysis (FEA) software solution. The FEA modeling results were found to be in excellent agreement with the analytical solutions. Our results also show that the heat transfer between the sample surface and the air coupling fluid does not introduce an important effect over the induced phase shift in the sample when compared to the solution obtained without considering axial heat flux. On the other hand, the thermal lens created in the air coupling fluid has a significant effect on the predicted time-dependent photothermal signals. When water is used as fluid, the heat coupling leads to a more significant effect in both sample and fluid phase shift. Our results could be used to obtain physical properties of low optical absorption fluids by using a reference solid sample in both thermal lens and thermal mirror experiments.
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Affiliation(s)
- Luis C Malacarne
- Departamento de Física, Universidade Estadual de Maringá, Maringá-PR, 87020-900, Brazil.
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39
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Lenzi EK, Yednak CAR, Evangelista LR. Non-Markovian diffusion and the adsorption-desorption process. Phys Rev E Stat Nonlin Soft Matter Phys 2010; 81:011116. [PMID: 20365332 DOI: 10.1103/physreve.81.011116] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2009] [Indexed: 05/29/2023]
Abstract
The non-Markovian diffusion of dispersed particles in a semi-infinite cell of an isotropic fluid limited by an adsorbing-desorbing surface is theoretically investigated. The density of dispersed particles in the bulk is a time dependent function and the time dependent density of surface particles is governed by a modified kinetic equation with a time dependent kernel. In this framework, the densities of bulk and surface particles are analytically determined, taking into account the conservation of the number of particles immersed in the sample. This system exhibits anomalous diffusion behavior as well as memory effects in the adsorption-desorption process. The results obtained here are expected to be useful to investigate the adsorption-desorption phenomena of neutral as well as charged particles in an isotropic fluid in contact with a solid substrate when the anomalous diffusion is present.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR, Brazil
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40
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Lenzi EK, Evangelista LR, Lenzi MK, da Silva LR. Fokker-Planck equation in a wedge domain: anomalous diffusion and survival probability. Phys Rev E Stat Nonlin Soft Matter Phys 2009; 80:021131. [PMID: 19792101 DOI: 10.1103/physreve.80.021131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2009] [Revised: 06/27/2009] [Indexed: 05/28/2023]
Abstract
We obtain exact solutions and the survival probability for a Fokker-Planck equation subjected to the two-dimensional wedge domain. We consider a spatial dependence in the diffusion coefficient and the presence of external forces. The results show an anomalous spreading of the solution and, consequently, a nonusual behavior of the survival probability which can be connected to anomalous diffusion.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, 87020-900 Maringá, PR, Brazil
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41
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Rossato R, Lenzi MK, Evangelista LR, Lenzi EK. Fractional diffusion equation in a confined region: surface effects and exact solutions. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 76:032102. [PMID: 17930291 DOI: 10.1103/physreve.76.032102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2007] [Indexed: 05/25/2023]
Abstract
Surface effects on a diffusion process governed by a fractional diffusion equation in a confined region with spatial and time dependent boundary conditions are investigated. First, we consider the one-dimensional case with the boundary conditions rho(0,t)=Phi0(t) and rho(a,t)=Phia(t). Subsequently, the two-dimensional case in the cylindrical symmetry with rho(a,theta,t)=Phia(theta,t) and rho(b,theta,t)=Phib(theta,t) is investigated. For these cases, we also obtain exact solutions for an arbitrary initial condition by using the Green's function approach.
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Affiliation(s)
- R Rossato
- Universidade Estadual de Maringá, Departamento de Física, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil
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42
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Zola RS, Lenzi EK, Evangelista LR, Barbero G. Memory effect in the adsorption phenomena of neutral particles. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 75:042601. [PMID: 17500940 DOI: 10.1103/physreve.75.042601] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2006] [Revised: 12/21/2006] [Indexed: 05/15/2023]
Abstract
The adsorption-desorption phenomenon of neutral particles dissolved in an isotropic fluid is investigated by using a nonsingular kernel in the kinetic equation at the limiting surfaces. To account for the relevance of a memory effect, three types of kernels in the kinetic equation are considered. Similar kernels have been used to investigate nonexponential relaxation including several contexts such as dielectric relaxation, diffusion-controlled relaxation in liquids, liquid crystals, and amorphous polymers. A suitable choice for a temporal kernel can account for the relative importance of physisorption or chemisorption, according to the time scale governing the adsorption phenomena, and can be the key mechanism to understand the specific roles of both processes. By using a general procedure, the time evolution of the density of particles is determined in closed analytical form. The analysis is relevant in the description of the adsorption phenomena in general.
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Affiliation(s)
- R S Zola
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil
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43
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Malacarne LC, Mendes RS, Lenzi EK, Lenzi MK. General solution of the diffusion equation with a nonlocal diffusive term and a linear force term. Phys Rev E Stat Nonlin Soft Matter Phys 2006; 74:042101. [PMID: 17155110 DOI: 10.1103/physreve.74.042101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2006] [Indexed: 05/12/2023]
Abstract
We obtain a formal solution for a large class of diffusion equations with a spatial kernel dependence in the diffusive term. The presence of this kernel represents a nonlocal dependence of the diffusive process and, by a suitable choice, it has the spatial fractional diffusion equations as a particular case. We also consider the presence of a linear external force and source terms. In addition, we show that a rich class of anomalous diffusion, e.g., the Lévy superdiffusion, can be obtained by an appropriated choice of kernel.
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Affiliation(s)
- L C Malacarne
- Universidade Estadual de Maringá, Departamento de Física, 87020-900 Maringá, Paraná, Brazil
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44
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Malacarne LC, Mendes RS, Lenzi EK, Picoli S, Dal Molin JP. A non-Gaussian model in polymeric network. Eur Phys J E Soft Matter 2006; 20:395-9. [PMID: 16953341 DOI: 10.1140/epje/i2006-10028-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2006] [Accepted: 07/28/2006] [Indexed: 05/11/2023]
Abstract
We investigate a finite chain approximation, the non-Gaussian Tsallis distribution, to the polymeric network, which gives an improvement to the Gaussian model. This distribution presents some necessary characteristics, like a cutoff to the maximum chain length and a continuous limit to the Gaussian one for a large number of monomers. It also presents a simple quadratic structure that allows to generalize the Gaussian properties such as exact-moments calculation and Wick theorem. We obtain the free-energy density in its full tensorial structure.
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Affiliation(s)
- L C Malacarne
- Departamento de Fisica, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900, Maringá, Paraná, Brazil.
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45
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Assis PC, de Souza RP, da Silva PC, da Silva LR, Lucena LS, Lenzi EK. Non-Markovian Fokker-Planck equation: solutions and first passage time distribution. Phys Rev E Stat Nonlin Soft Matter Phys 2006; 73:032101. [PMID: 16605577 DOI: 10.1103/physreve.73.032101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2005] [Indexed: 05/08/2023]
Abstract
We investigate the solutions and first passage time distribution for an anomalous diffusion process governed by a generalized non-Markovian Fokker-Planck equation. In our analysis, we also consider the presence of external forces and absorbent (source) terms. In addition, we show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained from the solutions found here.
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Affiliation(s)
- P C Assis
- International Center for Complex Systems and Departamento de Física, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil
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46
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Yednak CAR, Freire FCM, Lenzi EK, Evangelista LR. Dynamical behavior of the director field for splay-bend deformations in nematic liquid crystals. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 72:042701. [PMID: 16383447 DOI: 10.1103/physreve.72.042701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2005] [Indexed: 05/05/2023]
Abstract
The exact dynamical evolution of the director field for splay-bend deformations, in nematic liquid crystal samples limited by inhomogeneous surfaces, is determined in the one-constant approximation. The initial conditions and boundary-value problem concerning the situation of strong anchoring at the surfaces of a sample of slab shape of thickness d is analytically solved in the presence of a time dependent external electric field, and taking into account the viscous torque. The results are used to analytically obtain the time dependence of the phase shift between the two components of a linearly polarized beam impinging perpendicularly on the sample. The analysis can be relevant to investigate the phase retardation of a nematic cell submitted to an external voltage which is lower than or in the order of the Féedericksz threshold to induce deformations in the sample.
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Affiliation(s)
- C A R Yednak
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790-87020-900 Maringá (PR), Brazil
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47
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Pedron IT, Mendes RS, Buratta TJ, Malacarne LC, Lenzi EK. Logarithmic diffusion and porous media equations: a unified description. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 72:031106. [PMID: 16241410 DOI: 10.1103/physreve.72.031106] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2005] [Indexed: 05/04/2023]
Abstract
In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a Lorentzian form, consequently this equation characterizes a superdiffusion like a Lévy kind. In addition an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension, is obtained. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.
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Affiliation(s)
- I T Pedron
- Universidade Estadual do Oeste do Paraná, Rua Pernambuco, 1777, 85960-000, Marechal Cândido Rondon, Paraná, Brazil
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48
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Fa KS, Lenzi EK. Time-fractional diffusion equation with time dependent diffusion coefficient. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 72:011107. [PMID: 16089937 DOI: 10.1103/physreve.72.011107] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2005] [Indexed: 05/03/2023]
Abstract
We consider the time-fractional diffusion equation with time dependent diffusion coefficient given by (O)O(alpha)(C)(t) W (x,t) = D(alpha,gamma)(t)(gamma) [theta(2) W (x,t) /theta x(2)], where O is the Caputo operator. We investigate its solutions in the infinite and the finite domains. The mean squared displacement and the mean first passage time are also considered. In particular, for alpha = 0 , the mean squared displacement is given by <x(2)> approximately t(gamma) and we verify that the mean first passage time is finite for superdiffusive regimes.
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Affiliation(s)
- Kwok Sau Fa
- Departamento de Física, Universidade Estadual de Maringá, Brazil
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Lenzi EK, Mendes RS, Andrade JS, da Silva LR, Lucena LS. N-dimensional fractional diffusion equation and Green function approach: spatially dependent diffusion coefficient and external force. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 71:052101. [PMID: 16089577 DOI: 10.1103/physreve.71.052101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2004] [Revised: 12/08/2004] [Indexed: 05/03/2023]
Abstract
We investigate an N-dimensional fractional diffusion equation with radial symmetry by using the Green function approach. We consider, in our analysis, the spatial dependence on the diffusion coefficient and the presence of an external force. In particular, we employ boundary conditions in a finite interval and after we extend it to a semi-infinite interval. We also show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained from the solutions found here.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá PR, Brazil
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Fa KS, Lenzi EK. Anomalous diffusion, solutions, and first passage time: Influence of diffusion coefficient. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 71:012101. [PMID: 15697637 DOI: 10.1103/physreve.71.012101] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2004] [Indexed: 05/24/2023]
Abstract
We investigate the solutions and the first passage time for anomalous diffusion processes governed by the usual diffusion equation. We consider a space- and time-dependent diffusion coefficient and the presence of absorbing boundaries. We obtain analytical results for the probability distribution and the first passage time distribution for finite and semi-infinite intervals. In addition, we compare our results for the first passage time distribution with the one obtained by the usual diffusion equation with constant diffusion coefficient.
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Affiliation(s)
- Kwok Sau Fa
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900, Maringá-PR, Brazil
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