On the Mixing of Diffusing Particles

We study how the order of N independent random walks in one dimension evolves with time. Our focus is statistical properties of the inversion number m, defined as the number of pairs that are out of sort with respect to the initial configuration. In the steady-state, the distribution of the inversion number is Gaussian with the average ~N^2/4 and the standard deviation sigma N^{3/2}/6. The survival probability, S_m(t), which measures the likelihood that the inversion number remains below m until time t, decays algebraically in the long-time limit, S_m t^{-beta_m}. Interestingly, there is a spectrum of N(N-1)/2 distinct exponents beta_m(N). We also find that the kinetics of first-passage in a circular cone provides a good approximation for these exponents. When N is large, the first-passage exponents are a universal function of a single scaling variable, beta_m(N)--> beta(z) with z=(m-)/sigma. In the cone approximation, the scaling function is a root of a transcendental equation involving the parabolic cylinder equation, D_{2 beta}(-z)=0, and surprisingly, numerical simulations show this prediction to be exact.Comment: 9 pages, 6 figures, 2 table

Complete positivity of nonlinear evolution: A case study

Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown on a concrete example.Comment: extended version, 3 appendices added (on mixed states, projection postulate, nonlocality), to be published in Phys. Rev.

Measurement of the quasi-elastic axial vector mass in neutrino-oxygen interactions

The weak nucleon axial-vector form factor for quasi-elastic interactions is determined using neutrino interaction data from the K2K Scintillating Fiber detector in the neutrino beam at KEK. More than 12,000 events are analyzed, of which half are charged-current quasi-elastic interactions nu-mu n to mu- p occurring primarily in oxygen nuclei. We use a relativistic Fermi gas model for oxygen and assume the form factor is approximately a dipole with one parameter, the axial vector mass M_A, and fit to the shape of the distribution of the square of the momentum transfer from the nucleon to the nucleus. Our best fit result for M_A = 1.20 \pm 0.12 GeV. Furthermore, this analysis includes updated vector form factors from recent electron scattering experiments and a discussion of the effects of the nucleon momentum on the shape of the fitted distributions.Comment: 14 pages, 10 figures, 6 table