Brownian motion with restoring drift: The petit and micro-canonical ensembles

Letf(Q) be odd and positive near +∞. Then the non-linear wave equation ?² Q/?t ²??² Q/?x ²?f(Q)=0, considered on the circle 0≤x<L, can be written in Hamiltonian formQ ^⊙ =?H/?P, P ^⊙ =??H/?Q with $$P = Q^\cdot and H = \tfrac{1}{2}\mathop \smallint \limits_0^L (Q')^2 + \mathop \smallint \limits_0^L F(Q) + \tfrac{1}{2}\mathop \smallint \limits_0^L P^2 ;$$ the corresponding flow preserves the (suitably interpreted) “petit ensemble”e ^?H d ^∞ Qd ^∞ P; and forL↓∞,Q settles down to the stationary diffusion with infinitesimal operator 1/2 ?²/?Q ²+m(Q)?/?Q,m being the logarithmic derivative of the ground state of ?d ²/dQ ² ‖F(Q). This diffusion is the “Brownian motion with restoring drift”; see McKean-Vaninsky [1993(1)]. For reasons suggested by the paper of Lebowitz-Rose-Speer [1988] on NLS, it is interesting to study the “micro-canonical ensemble” obtained by restricting to the sphere $\int\limits_0^L {Q^2 } = N$ and makingL↓∞ with fixedD=N/L. Now, forF(Q)/Q ²→∞, the same type of diffusion appears, but with drift arising from the modified potentialF(Q)+cQ ²,c being chosen so that the mean ofQ ² is the assigned numberD. The proof employs Döblin's method of “loops” [1937] and steepest descent. The same is true forF(Q)=m ² Q ², only now the proof is elementary. The outcome is also the same ifF(Q)/Q ²→0, providedD is smaller than the petit canonical mean ofQ ²; forD larger than this mean, the matter is more subtle and the outcome is unknown.     

AES-depth-profiling of thin annealed Pt-films on Si(100)

Thin PtSi films can be grown by evaporating Pt on Si(100) at RT and subsequent annealing of the system at 600–700 K. Contaminants like oxygen are known to have a strong influence on this reaction. In the present study we concentrate on the effect of oxygen partial pressure during the annealing on the silicide growth process. Under proper vacuum conditions annealing at 500 K leads to a homogeneous Pt₂Si film which reacts around 600 K completely to PtSi. A substantial oxygen partial pressure ( 0.1 mbar) in contrast results in an incomplete reaction in the same temperature range: unreacted platinum remains at the surface separated from the silicide by an oxygen enriched layer.Presented at the Seminar on Secondary Electrons in Electron Spectroscopy, Microscopy, and Microanalysis, Chlum (The Czech Republic), 21–24 September, 1993.This work was supported by Deutsche Forschungsgemeinschaft (DFG) through Sonderforschungsbereich 292.We thank Dr. W. Platz (Deutsche Aerospace AG, Ottobrunn) for providing us with Pt evaporated Si-wafers and Th. Hierl (Lehrstuhl für Angewandte Physik, University of Erlangen-Nürnberg), who performed the RBS measurements for AES calibration.