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 $⟨m⟩\simeq N^2/4$ and the standard deviation $\sigma \simeq N^{3/2}/6$. The survival probability, $S_m\left(t\right)$, 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 $N\left(N-1\right)/2$ distinct exponents ${\beta }_m\left(N\right)$. 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, ${\beta }_m\left(N\right)\to \beta \left(z\right)$, with $z=\left(m-⟨m⟩\right)/\sigma$. In the cone approximation, the scaling function is a root of a transcendental equation involving the parabolic cylinder equation, $D_{2\beta }\left(-z\right)=0$, and surprisingly, numerical simulations show this prediction to be exact.

Figure 1

Space-time diagram of a four-particle system. The circled $+s$ and $-s$ indicate whether the inversion number increases or decreases when two trajectories cross. Four out of five crossings increase the inversion number, and accordingly, the inversion number increases from $m=0$ to $m=3$.

Figure 2

(Color online) The normalized average $⟨m\left(t\right)⟩/\left[N\left(N-1\right)/4\right]$ versus time $t$. The results correspond to an average over ${10}^2$ independent realizations of a system with $N={10}^3$ random walks. Also shown for reference is a line with slope of 1/2.

Figure 3

(Color online) The distribution of the inversion number in the intermediate time regime $1⪡t⪡N^2$. Shown is the distribution $p_m\equiv p_m\left(N,t\right)$ versus the variable $\left[m-⟨m\left(t\right)⟩\right]/\sigma \left(t\right)$. The results are from ${10}^5$ independent realizations of a system with $N={10}^3$ random walks. The distribution is shown at times $t={10}^2$ (diamonds), $t={10}^3$ (squares), and $t={10}^4$ (circles). Also shown for reference is normal distribution (13).

Figure 4

(Color online) The survival probability $S_m\left(t\right)$ versus $t$ for a four-particle system. Shown are the quantities $S_2$ (bottom curve) and $S_3,\dots ,S_6$ (top curve). The number of independent Monte Carlo runs varies from ${10}^6$ for the slowest decay to ${10}^{12}$ for the fastest decay.

Figure 5

(Color online) The first-passage exponent ${\beta }_m$ versus $m$ for $N=4,5,6,7$. Shown are simulation results (circles) and the outcome of the cone approximation (squares).

Figure 6

(Color online) The exponent $\beta$ versus the scaling variable $z$, shown using (a) a linear-linear plot and (b) a linear-logarithmic plot. The simulation results are from Monte Carlo runs with $N=50$ (diamonds), $N=100$ (squares), and $N=200$ (circles) particles. The number of independent realizations varies from ${10}^4$ for slow first-passage processes to ${10}^8$ for fast one. The solid line shows theoretical prediction (28).