a way that the mean particle density $\langle n(t,x)\rangle$ obeys the equation \_t n(t,x)=\_x\^2 n(t,x)+n(t,x). \[eq:meannBRW\] A particular realization of this BRW is represented in Fig. \[fig:plotBRW\].\

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   ![\[fig:plotBRW\][(a)]{} One realization of the continuous BRW up to time $t=8$. To guide the eye, we also plot the theoretical (truncated) mean position of the boundaries of the BRW, namely $\pm\bar X_t=\pm 2t\mp\frac32 \ln t$ (dashed lines). [(b)]{} Distribution of the particles at times $t=2,6$ and $8$ for this particular realization in bins of size 1 ($\log_{10}$ scale on the vertical axis). We see the bulk building up a smoother (more “deterministic”) distribution as time elapses, while the low-density tails remain noisy. Also, for this realization, the distribution is skewed towards negative values of $x$, due to an accidentally large drift in the initial stages, whose memory is kept throughout the evolution. ](plotBRW.eps "fig:"){width=".45\textwidth"}   ![\[fig:plotBRW\][(a)]{} One realization of the continuous BRW up to time $t=8$. To guide the eye, we also plot the theoretical (truncated) mean position of the boundaries of the BRW, namely $\pm\bar X_t=\pm 2t\mp\frac32 \ln t$ (dashed lines). [(b)]{} Distribution of the particles at times $t=2,6$ and $8$ for this particular realization in bins of size 1 ($\log_{10}$ 