Keywords

modeling and simulation, statistical mechanics, materials science, phase transitions, stochastic processes

Select the category the research project fits.

Mathematical/Computational Sciences

Is this submission part of ICaP/PW (Introductory Composition at Purdue/Professional Writing)?

No

Abstract

Phase transitions within large-scale systems may be modeled by using partial differential equations, in which system dynamics are captured by appropriate polynomial potentials. The ability to simulate and predict phase transition behavior has many applications, from material behaviors (e.g., liquid crystal phase transformations, coherent movement of granular materials) to traffic congestion. Coherent structures in these systems evolve along a single spatial dimension randomly through time; thus, the statistical behavior of these fields is of greater interest than particular system results. Past research focused on deriving solutions to the system probability density function (PDF) and verifying solutions for fourth-order and other simple potentials. Until recently, the extent to which these solutions could be verified was limited by computing power. This work focused on verifying solutions for PDFs of sixth-order and tenth-order potentials, which describe more complex phase transition behaviors, and determining their respective correlation functions. Large-scale MATLAB simulations were used to model the evolution of fields at certain system “temperatures”, for which statistical PDFs and correlation functions were computed. Once fully validated, this approach will enable a better understanding of successive phase transitions in complex materials, and allow for accurate modeling of these system behaviors based on material properties. In the future it would be beneficial to evaluate the field dynamics of higher-order potentials at a smaller scale to gain further insight on the behavior of stochastic processes in large-scale systems.

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Thermodynamics of Coherent Structures near Phase Transitions

Phase transitions within large-scale systems may be modeled by using partial differential equations, in which system dynamics are captured by appropriate polynomial potentials. The ability to simulate and predict phase transition behavior has many applications, from material behaviors (e.g., liquid crystal phase transformations, coherent movement of granular materials) to traffic congestion. Coherent structures in these systems evolve along a single spatial dimension randomly through time; thus, the statistical behavior of these fields is of greater interest than particular system results. Past research focused on deriving solutions to the system probability density function (PDF) and verifying solutions for fourth-order and other simple potentials. Until recently, the extent to which these solutions could be verified was limited by computing power. This work focused on verifying solutions for PDFs of sixth-order and tenth-order potentials, which describe more complex phase transition behaviors, and determining their respective correlation functions. Large-scale MATLAB simulations were used to model the evolution of fields at certain system “temperatures”, for which statistical PDFs and correlation functions were computed. Once fully validated, this approach will enable a better understanding of successive phase transitions in complex materials, and allow for accurate modeling of these system behaviors based on material properties. In the future it would be beneficial to evaluate the field dynamics of higher-order potentials at a smaller scale to gain further insight on the behavior of stochastic processes in large-scale systems.