The Laplacian of a function $u$ at a point $x$ measures the average extent to which the value of $u$ at $x$ deviates from the value of $u$ at nearby points to $x$ (cf. the mean value theorem for harmonic functions). As such, it naturally occurs in any system in which some quantity at a point is influenced by the value of the same quantity at nearby points. (This also explains the link between Laplacians the Laplacian and Brownian motion (or random walks), in which one repeatedly travels from a point $x$ to a randomly selected nearby point to $x$.)
The Laplacian of a function $u$ at a point $x$ measures the average extent to which the value of $u$ at $x$ deviates from the value of $u$ at nearby points to $x$ (cf. the mean value theorem for harmonic functions). As such, it naturally occurs in any system in which some quantity at a point is influenced by the value of the same quantity at nearby points. (This also explains the link between Laplacians and Brownian motion, in which one repeatedly travels from a point $x$ to a randomly selected nearby point to $x$.)