The Boltzmann equation near and far from equilibrium
If an ideal gas is in thermodynamic equilibrium, the distribution of particle velocities is given explicitly by the Maxwellian (a.k.a. Maxwell-Boltzmann) distribution. If the gas is out of equilibrium, modeling the dynamics is naturally much more difficult. The evolution equation satisfied by the particle density, known as the Boltzmann equation, lacks a suitable well-posedness theory despite its long history and widespread use in statistical physics. So far, global solutions have only been constructed for initial data that is sufficiently close to an equilibrium state. In this talk, I will describe progress from the last few years on the "large-data" or "far-from-equilibrium" regime, including joint work with C. Henderson and A. Tarfulea on local existence, instantaneous filling of vacuum regions, and continuation criteria.
Boundary Element Methods for Acoustic Wave Propagation
Simulating acoustic wave propagation has important applications in many engineering problems. In this talk, we will consider modelling focused ultrasound techniques used for the non-invasive treatment of liver cancer. This application requires the simulation of high-frequency acoustic waves propagating through soft tissue and scattering at the rib bones. The boundary element method (BEM) is an efficient numerical method to solve the Helmholtz transmission system. We will survey the design of boundary integral equations and preconditioners for the iterative linear solver. The computational results of a benchmarking exercise involving hundreds of different preconditioned formulations uncover the dependencies of BEM’s efficiency on the numerical and physical parameters. Finally, we will consider acoustic transmission at high-contrast media and apply the BEM to resonances of water-entrained air bubbles.
Image-based, organ-scale computational modeling of prostate cancer growth
The current clinical management of prostate cancer (PCa) enables its detection at early organ-confined stages by combining regular screening and patient classification in risk groups. Although these newly diagnosed tumors do not usually pose a threat to the patient, most PCa cases are prescribed a radical treatment immediately after diagnosis (e.g., surgery or radiotherapy). However, the limited individualization of the clinical management beyond risk-group definition has led to significant overtreatment and undertreatment rates, which might adversely impact the patients’ lives and life expectancy, respectively. Thus, PCa is a paradigmatic disease in which an individualized predictive technology could make a crucial difference in clinical practice, thereby separating less aggressive tumors that could be safely monitored from lethal tumors that require immediate treatment. To address this critical need, we can use routine clinical and imaging data to construct and parametrize personalized mathematical models of PCa growth including the key mechanisms involved in this pathology. Then, we can run computer simulations with these models to forecast the growth of a patient’s tumor, which may assist physicians in clinical-decision making. In this seminar, I will show that these models can reproduce tumor growth over the local anatomy of a patient’s prostate extracted from imaging data, along with the dynamics of the Prostate Specific Antigen (PSA, a ubiquitous biomarker in PCa clinical management). Additionally, I will discuss the importance of the inhibitive effect of growth-induced mechanical stress on PCa and how the compression exerted by concomitant benign prostatic hyperplasia dramatically impedes tumor growth. Finally, I will argue that these imaging-based models could constitute a promising computational technology to assist physicians to provide a personalized clinical management of PCa.
Plasma Instabilities in the Young Solar Wind: Thermodynamics far from Equilibrium
understanding how the young solar wind expands and is accelerated and provides a natural laboratory for studying plasma physics processes that arise throughout the Universe. One key feature of the young solar wind, due to its diffuse and hot nature, is that the collision frequency is low compared to other dynamic timescales, enabling the plasma to maintain significant deviations from local thermodynamic equilibrium. In such weakly collisional systems, these departures can drive unstable wave growth and serve as signatures of wave-particle interactions that act to transfer energy between the charged particles and electromagnetic fields. In this talk, we discuss measurements of non-equilibrium solar wind structure made by spacecraft throughout the inner heliosphere, and the application of mathematical tools to infer what mechanisms, in particular instabilities, may be driven by these structures. Understanding this energy transfer will shed new light on the processes driving solar and stellar winds, heating accretion disks, and stirring the interplanetary and interstellar medium.
The 2021 Los Alamos - Arizona Days Conference sponsored by CNLS and The UA Program in Applied Mathematics will be held entirely virtual this year on Monday, May 17th and Tuesday, May 18th. More details coming soon!