On the Mixing of Diffusing Particles
E. Ben-Naim
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 $\langle m\rangle \simeq N^2/4$ and the standard
deviation $\sigma\simeq N^{3/2}/6$. 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, $S_m\sim t^{-\beta_m}$. 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, $\beta_m(N)\to \beta(z)$ with \hbox{$z=(m-\langle
m\rangle)/\sigma$}. 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.
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