Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos
Tallennettuna:
| Päätekijä: | |
|---|---|
| Julkaisupäivä: | 2018 |
| Aineistotyyppi: | Doctoral thesis |
| Kieli: | por |
| Lähde: | Repositório Comum do Brasil - Deposita |
| Download full: | https://deposita.ibict.br/handle/deposita/479 |
Yhteenveto: | Anomalous diffusion is an ubiquitous phenomenon which have been studied by several approaches, including simulation, analytical methods and by experiments. Either fractional partial differential as non-linear partial differential equations were able to describe the phenomenon, in particular, the heavy tails observed in the PDFs. Besides, one can observe some lack in controlling anomalous diffusion in many attempts of simulation. Consequently, the relationship to adequate models becomes somewhat impaired. To perform a systematic calibration between simulations and models, in this work, we explore the relationship among the diffusion coefficients and diffusion exponents with the order of fractional derivatives, the q-Gaussian parameter and model diffusion constant by means of a systematic fitting procedure of the simulation data using linear and non-linear approaches. Using CTRW with a criterion to control mean waiting time and step length variance, a full range of well controlled cases from subdiffusion to superdiffusion were generated. Theoretical models are expressed by means of generalised diffusion equations with fractional derivatives in space, in time and by the non-linearity of the porous medium equation. To decide how to assess the diffusion constant, the order of fractional derivatives and the q-Gaussian parameter from the simulation data in each case of anomalous diffusion, we compare the accuracy of two methods: (1) by analysis of the dispersion of the variance over time and, (2) by the optimisation of the solutions of the theoretical models to the histogram of positions. The relative accuracies of the models are also analysed for each regimen of anomalous diffusion. We highlight relations between the simulation parameters and model parameters. Among those, Tsallis-Buckman scaling law identity is verified. The study discusses methods to link model parameters to simulation parameters. |
| _version_ | 1872168167146520576 |
|---|---|
| author | Pereira, Ana |
| author_browse | Pereira, Ana |
| author_facet | Pereira, Ana |
| author_role | author |
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| collection | Repositório Comum do Brasil - Deposita |
| contributor_str_mv | Fernandes, José Faria, Allbens |
| dc.contributor.advisor-co1.fl_str_mv | Faria, Allbens |
| dc.contributor.advisor-co1Lattes.fl_str_mv | http://lattes.cnpq.br/4216801992845696 |
| dc.contributor.advisor1.fl_str_mv | Fernandes, José |
| dc.contributor.advisor1Lattes.fl_str_mv | http://lattes.cnpq.br/0824886862509279 |
| dc.contributor.author.fl_str_mv | Pereira, Ana |
| dc.contributor.authorLattes.fl_str_mv | http://lattes.cnpq.br/5199859493258720 |
| dc.date.accessioned.fl_str_mv | 2024-01-10T18:42:35Z |
| dc.date.issued.fl_str_mv | 2018 |
| dc.format.none.fl_str_mv | application/pdf application/pdf |
| dc.identifier.uri.fl_str_mv | https://deposita.ibict.br/handle/deposita/479 |
| dc.language.iso.fl_str_mv | por |
| dc.publisher.country.fl_str_mv | Brasil |
| dc.publisher.department.fl_str_mv | Centro Federal de Educação Tecnológica de Minas Gerais - CEFET-MG |
| dc.publisher.none.fl_str_mv | Centro Federal de Educação Tecnológica de Minas Gerais |
| dc.publisher.program.fl_str_mv | Programa de Pós-Graduação em Modelagem Matemática e Computacional |
| dc.rights.driver.fl_str_mv | info:eu-repo/semantics/openAccess |
| dc.source.none.fl_str_mv | reponame:Repositório Comum do Brasil - Deposita instname:Instituto Brasileiro de Informação em Ciência e Tecnologia (Ibict) instacron:IBICT |
| dc.subject.cnpq.fl_str_mv | Matemática Aplicada |
| dc.subject.por.fl_str_mv | Difusão anômala. Equação de difusão fracionária. q-gaussiana. Calibração. |
| dc.title.por.fl_str_mv | Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos |
| dc.type.driver.fl_str_mv | info:eu-repo/semantics/doctoralThesis |
| dc.type.status.fl_str_mv | info:eu-repo/semantics/publishedVersion |
| description | Anomalous diffusion is an ubiquitous phenomenon which have been studied by several approaches, including simulation, analytical methods and by experiments. Either fractional partial differential as non-linear partial differential equations were able to describe the phenomenon, in particular, the heavy tails observed in the PDFs. Besides, one can observe some lack in controlling anomalous diffusion in many attempts of simulation. Consequently, the relationship to adequate models becomes somewhat impaired. To perform a systematic calibration between simulations and models, in this work, we explore the relationship among the diffusion coefficients and diffusion exponents with the order of fractional derivatives, the q-Gaussian parameter and model diffusion constant by means of a systematic fitting procedure of the simulation data using linear and non-linear approaches. Using CTRW with a criterion to control mean waiting time and step length variance, a full range of well controlled cases from subdiffusion to superdiffusion were generated. Theoretical models are expressed by means of generalised diffusion equations with fractional derivatives in space, in time and by the non-linearity of the porous medium equation. To decide how to assess the diffusion constant, the order of fractional derivatives and the q-Gaussian parameter from the simulation data in each case of anomalous diffusion, we compare the accuracy of two methods: (1) by analysis of the dispersion of the variance over time and, (2) by the optimisation of the solutions of the theoretical models to the histogram of positions. The relative accuracies of the models are also analysed for each regimen of anomalous diffusion. We highlight relations between the simulation parameters and model parameters. Among those, Tsallis-Buckman scaling law identity is verified. The study discusses methods to link model parameters to simulation parameters. |
| eu_rights_str_mv | openAccess |
| format | doctoralThesis |
| id | IBICT-1_bfa2300907f655625ccb0abc94af9fa0 |
| instacron_str | IBICT |
| institution | IBICT |
| instname_str | Instituto Brasileiro de Informação em Ciência e Tecnologia (Ibict) |
| language | por |
| network_acronym_str | IBICT-1 |
| network_name_str | Repositório Comum do Brasil - Deposita |
| oai_identifier_str | oai:deposita.ibict.br:deposita/479 |
| publishDate | 2018 |
| publishDateSort | 2018 |
| publisher.none.fl_str_mv | Centro Federal de Educação Tecnológica de Minas Gerais |
| reponame_str | Repositório Comum do Brasil - Deposita |
| repository.mail.fl_str_mv | deposita@ibict.br |
| repository.name.fl_str_mv | Repositório Comum do Brasil - Deposita - Instituto Brasileiro de Informação em Ciência e Tecnologia (Ibict) |
| repository_id_str | 4658 |
| spelling | Fernandes, Joséhttp://lattes.cnpq.br/0824886862509279Faria, Allbenshttp://lattes.cnpq.br/4216801992845696http://lattes.cnpq.br/5199859493258720Pereira, Ana2024-01-10T18:42:35Z2018https://deposita.ibict.br/handle/deposita/479Anomalous diffusion is an ubiquitous phenomenon which have been studied by several approaches, including simulation, analytical methods and by experiments. Either fractional partial differential as non-linear partial differential equations were able to describe the phenomenon, in particular, the heavy tails observed in the PDFs. Besides, one can observe some lack in controlling anomalous diffusion in many attempts of simulation. Consequently, the relationship to adequate models becomes somewhat impaired. To perform a systematic calibration between simulations and models, in this work, we explore the relationship among the diffusion coefficients and diffusion exponents with the order of fractional derivatives, the q-Gaussian parameter and model diffusion constant by means of a systematic fitting procedure of the simulation data using linear and non-linear approaches. Using CTRW with a criterion to control mean waiting time and step length variance, a full range of well controlled cases from subdiffusion to superdiffusion were generated. Theoretical models are expressed by means of generalised diffusion equations with fractional derivatives in space, in time and by the non-linearity of the porous medium equation. To decide how to assess the diffusion constant, the order of fractional derivatives and the q-Gaussian parameter from the simulation data in each case of anomalous diffusion, we compare the accuracy of two methods: (1) by analysis of the dispersion of the variance over time and, (2) by the optimisation of the solutions of the theoretical models to the histogram of positions. The relative accuracies of the models are also analysed for each regimen of anomalous diffusion. We highlight relations between the simulation parameters and model parameters. Among those, Tsallis-Buckman scaling law identity is verified. The study discusses methods to link model parameters to simulation parameters.A difusão anômala é um fenômeno onipresente que tem sido estudado por meio de diferentes abordagens, incluindo simulações, análise de métodos e experimentos. Tanto a equação de difusão fracionária quanto a equação de difusão não linear foram capazes de descrever tal fenômeno, em particular, os casos onde aparecem funções densidade de probabilidade com caudas pesadas. Observa-se, no entanto, uma falta de um controle sistemático nas simulações de processos de difusão anômala. Consequentemente, a relação de quais os modelos mais adequados para a descrição destes processos fica prejudicada. Neste trabalho, exploramos as relações os parâmetros coeficientes de difusão e entre os expoentes de difusão, ordem da derivada fracionária e o parâmetro q-gaussiano, através de um procedimento de ajuste sistemático dos dados das simulações usando abordagens lineares e não lineares. Os dados simulados são gerados por um random walk de tempo contínuo com média do tempo de espera entre os saltos e variância do comprimento dos saltos controlados para gerar casos de subdifusão, superdifusão e difusão normal. Os modelos teóricos são expressos por meio de equações de difusão generalizadas ora por derivadas fracionárias no tempo ou espaço, ora pela não linearidade da equação de meios porosos. A avaliação dos parâmetros coeficientes de difusão e dos expoentes de difusão obtidos a partir das simulações é feita utilizando duas diferentes abordagens: (1) estudando a evolução temporal da variância amostral dos deslocamentos e, (2) otimizando as soluções dos modelos teóricos aos histogramas de soluções. A precisão dos modelos também é estudada para cada regime de difusão anômala. São analisadas as relações entre os parâmetros obtidos por meio das simulações e os parâmetros teóricos. Dentre os quais, a relação de Tsallis-Buckman é verificada. Faz-se, ainda, uma discussão sobre os métodos para relacionar os parâmetros das simulações com os parâmetros dos modelos.Sudeste-1application/pdfapplication/pdfporCentro Federal de Educação Tecnológica de Minas GeraisPrograma de Pós-Graduação em Modelagem Matemática e ComputacionalBrasilCentro Federal de Educação Tecnológica de Minas Gerais - CEFET-MGDifusão anômala. Equação de difusão fracionária. q-gaussiana. 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| spellingShingle | Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos Pereira, Ana Difusão anômala. Equação de difusão fracionária. q-gaussiana. Calibração. Matemática Aplicada |
| status_str | publishedVersion |
| title | Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos |
| title_full | Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos |
| title_fullStr | Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos |
| title_full_unstemmed | Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos |
| title_short | Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos |
| title_sort | Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos |
| topic | Difusão anômala. Equação de difusão fracionária. q-gaussiana. Calibração. Matemática Aplicada |
| url | https://deposita.ibict.br/handle/deposita/479 |