The McKean-Vlasov Equation in Finite Volume

We study the McKean--Vlasov equation on the finite tori of length scale $L$ in $d$--dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in \cite{GP} and \cite{KM}. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, $\theta^{\sharp}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $\theta < \theta^{\sharp}$ and prove, abstractly, that a {\it critical} transition must occur at $\theta = \theta^{\sharp}$. However for this system we show that under generic conditions -- $L$ large, $d \geq 2$ and isotropic interactions -- the phase transition is in fact discontinuous and occurs at some \theta\t < \theta^{\sharp}. Finally, for H--stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the \theta\t(L) tend to a definitive non--trivial limit as $L\to\infty$

Singular Cucker-Smale Dynamics

The existing state of the art for singular models of flocking is overviewed, starting from microscopic model of Cucker and Smale with singular communication weight, through its mesoscopic mean-filed limit, up to the corresponding macroscopic regime. For the microscopic Cucker-Smale (CS) model, the collision-avoidance phenomenon is discussed, also in the presence of bonding forces and the decentralized control. For the kinetic mean-field model, the existence of global-in-time measure-valued solutions, with a special emphasis on a weak atomic uniqueness of solutions is sketched. Ultimately, for the macroscopic singular model, the summary of the existence results for the Euler-type alignment system is provided, including existence of strong solutions on one-dimensional torus, and the extension of this result to higher dimensions upon restriction on the smallness of initial data. Additionally, the pressureless Navier-Stokes-type system corresponding to particular choice of alignment kernel is presented, and compared - analytically and numerically - to the porous medium equation

A silicon avalanche photodiode detector circuit for Nd:YAG laser scattering

A silicon avalanche photodiode with an internal gain of about 50 to 100 is used in a temperature controlled environment to measure the Nd:YAG laser Thomson scattered spectrum in the wavelength range from 700 to 1150 nm. A charge sensitive preamplifier has been developed for minimizing the noise contribution from the detector electronics. Signal levels as low as 20 photoelectrons (S/N = 1) can be detected. Measurements show that both the signal and the variance of the signal vary linearly with the input light level over the range of interest, indicating Poisson statistics. The signal is processed using a 100 ns delay line and a differential amplifier which subtracts the low frequency background light component. The background signal is amplified with a computer controlled variable gain amplifier and is used for an estimate of the measurement error, calibration, and Z{sub eff} measurements of the plasma. The signal processing has been analyzed using a theoretical model to aid the system design and establish the procedure for data error analysis. 4 refs., 5 figs