{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/38840"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/38840","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"A stochastic treatment of reaction and diffusion","abstract":"We develop a theory for the analysis of chemical reactions in \"isolated\" containers. The main tool for this analysis consists of Boltzmann maps, which are discrete time dynamical systems that describe the time evolution of the normalized concentrations of the chemicals in the reactions. Moreover, the use of these maps allows us to draw conclusions about the continuous dynamical systems that the law of mass action associates with the different reactions. The theorems we prove show that entropy is a strict Liapunov function and that no complex evolution is expected out of the discrete dynamical systems. In fact, we prove convergence to a fixed point for most of the possible cases, and we give solid arguments for the convergence of the remaining ones. The analysis of the continuous systems is more complicated, and fewer results have been proven. However, the conclusions we draw are similar to those relative to the Boltzmann maps. Therefore, we suggest that no chaos is to be found in systems that do not exchange energy nor matter with the outer environment, both for the discrete and for the continuous cases. Such a phenomenon is more likely to occur in \"closed\" or in \"open\" reactors. Finally, we argue that the discrete dynamical systems have more physical content than the continuous ones, and that Boltzmann maps may be useful in the analysis of the non chaotic regions of many other kinds of finite dimensional maps.","abstract_html":"We develop a theory for the analysis of chemical reactions in "isolated" containers. The main tool for this analysis consists of Boltzmann maps, which are discrete time dynamical systems that describe the time evolution of the normalized concentrations of the chemicals in the reactions. Moreover, the use of these maps allows us to draw conclusions about the continuous dynamical systems that the law of mass action associates with the different reactions. The theorems we prove show that entropy is a strict Liapunov function and that no complex evolution is expected out of the discrete dynamical systems. In fact, we prove convergence to a fixed point for most of the possible cases, and we give solid arguments for the convergence of the remaining ones. The analysis of the continuous systems is more complicated, and fewer results have been proven. However, the conclusions we draw are similar to those relative to the Boltzmann maps. Therefore, we suggest that no chaos is to be found in systems that do not exchange energy nor matter with the outer environment, both for the discrete and for the continuous cases. Such a phenomenon is more likely to occur in "closed" or in "open" reactors. Finally, we argue that the discrete dynamical systems have more physical content than the continuous ones, and that Boltzmann maps may be useful in the analysis of the non chaotic regions of many other kinds of finite dimensional maps.","abstract_has_math":false,"creators":["Rondoni, Lamberto"],"institution":"Virginia Tech","degree_name":"Ph. 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The main tool for this analysis consists of Boltzmann maps, which are discrete time dynamical systems that describe the time evolution of the normalized concentrations of the chemicals in the reactions. Moreover, the use of these maps allows us to draw conclusions about the continuous dynamical systems that the law of mass action associates with the different reactions. The theorems we prove show that entropy is a strict Liapunov function and that no complex evolution is expected out of the discrete dynamical systems. In fact, we prove convergence to a fixed point for most of the possible cases, and we give solid arguments for the convergence of the remaining ones. The analysis of the continuous systems is more complicated, and fewer results have been proven. However, the conclusions we draw are similar to those relative to the Boltzmann maps. Therefore, we suggest that no chaos is to be found in systems that do not exchange energy nor matter with the outer environment, both for the discrete and for the continuous cases. Such a phenomenon is more likely to occur in \"closed\" or in \"open\" reactors. Finally, we argue that the discrete dynamical systems have more physical content than the continuous ones, and that Boltzmann maps may be useful in the analysis of the non chaotic regions of many other kinds of finite dimensional maps."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A stochastic treatment of reaction and diffusion"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Zweifel, Paul F.","Streater, R.F."],"dc:contributor.committeemember":["Rossi, John","Bowden, Robert L.","Hagedorn, George","Hannsgen, Kenneth B."],"dc:contributor.department":["Mathematical Physics"],"dc:creator":["Rondoni, Lamberto"],"dc:date.accessioned":["2014-03-14T21:16:17Z"],"dc:date.available":["2014-03-14T21:16:17Z","2008-07-28"],"dc:date.issued":["1991"],"dc:description.abstract":["We develop a theory for the analysis of chemical reactions in \"isolated\" containers. The main tool for this analysis consists of Boltzmann maps, which are discrete time dynamical systems that describe the time evolution of the normalized concentrations of the chemicals in the reactions. Moreover, the use of these maps allows us to draw conclusions about the continuous dynamical systems that the law of mass action associates with the different reactions. The theorems we prove show that entropy is a strict Liapunov function and that no complex evolution is expected out of the discrete dynamical systems. In fact, we prove convergence to a fixed point for most of the possible cases, and we give solid arguments for the convergence of the remaining ones. The analysis of the continuous systems is more complicated, and fewer results have been proven. However, the conclusions we draw are similar to those relative to the Boltzmann maps. 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