We present a Gibbs random field model for the microscopic interactions in a viscoplastic fluid. The energy function is derived from the Gibbs potential in terms of the external stress and internal energy. The resulting Gibbs distribution, over a configuration space of microscopic interactions, can mimic experimentally observed macroscopic behavioral phenomena that depend on the externally applied stress. A simulation algorithm that can be used to approximate samples from the Gibbs distribution is given and it is used to gain several insights about the model. The model has two parameters for the internal energy of the material in the absence of external stress and a third parameter for a constant externally applied stress. An approximating differential equation for the expected proportion of the material in the solid phase is derived by a spatio-temporal rescaling of the toroidal square lattice upon which the Gibbs random field model is defined. The asymptotic dynamics of this tri-parametric family of differential equation matches with those of the rescaled simulations from the Gibbs field model and can account for the macroscopic behaviors, including solid-fluid phase transitions in the presence of constant as well as varying external stress and the associated hysteresis.
Raaz Sainudiin is Senior Lecturer in the School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand. He finished his Ph.D. in statistics at Cornell in 2005 and was a Post-Doctoral Research Fellow at the Department of Statistics, University of Oxford, UK from 2005-2007.
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