Jeremiah Birrell

My research interests revolve around mathematical physics and numerical analysis. Recent research topics include:

• Nonequilibrium statistical mechanics and entropy production.
• Singular limits of stochastic differential equations, including Hamiltonian systems and randomly perturbed geodesic flow on a Riemannian manifold.
• Numerical methods for the relativistic Boltzmann equation that can efficiently capture the emergence of chemical non-equilibrium and the decoupling process for systems of massive bosons and fermions. My focus has been on moving frame type methods utilizing orthogonal polynomials.
• A posteriori error bounds and validated numerics for dynamical systems, especially singularly perturbed or stiff problems.

I obtained my PhD through the Program in Applied Mathematics at the University of Arizona in August of 2014 under Dr. Johann Rafelski of the Department of Physics and with support from the National Defense Science and Engineering (NDSEG) Graduate Fellowship. During my PhD, I worked on several topics in astroparticle physics, as applied to quark gluon plasma and neutrino transport in the early universe. See my CV for a list of papers, or use the INSPIRE or arXiv links to the left.

Most recently, I have been working in the area of nonequilibrium statistical mechanics, studying singular limits of stochastic differential equations with Dr. Janek Wehr and collaborators. In our paper Small Mass Limit of a Langevin Equation on a Manifold, we derived the limiting equation for randomly perturbed geodesic motion with forcing. In this system, an anomalous drift term can arise in the limit, as a result of inhomogeneities in the noise coefficients, even if the original system. We are currently studying this noise induced drift phenomenon in randomly perturbed Hamiltonian systems, as well as investigating its effect on entropy production.

In my dissertation, I studied the evolution and properties of the relic (or cosmic) neutrino distribution from neutrino freeze-out at T=O(1) MeV through the free-streaming era up to today, characterizing the deviation of the neutrino spectrum from equilibrium and the dependence of the freeze-out process on natural constants.

To perform this characterization, I developed a Boltzmann equation solver based on a dynamical basis of orthogonal polynomials (moving frame) that is capable of efficiently capturing the emergence of chemical non-equilibrium and the reheating process. I also developed an improved procedure for analytically simplifying the scattering integrals that describe the two-body neutrino-neutrino and neutrino-electron interactions governing the freeze-out process. This resulted in a paper published in the Journal of Computational Physics.

This method was applied to the neutrino freeze-out process. In particular, it was used to investigate potential mechanisms of producing the measured value of the cosmological effective number of neutrinos.