# Brownian dimension

One way to determine an objects' dimension is to answer the following question

How does the number $$\mathbf{N}$$ of circles of radius $$r$$ that are required to cover the object change with $$r$$?

For instance, a line of length $$L$$ can be covered by $$L/r$$ circles of radius $$r$$, and $$\mathbf{N}\sim r^{-1}$$. A square of side length $$L$$ can be covered by $$\sim L^2/r^2$$ circles of radius $$r$$, and $$\mathbf{N}\sim r^{-2}$$. It is for this reason that we call a line 1 dimensional, and we call the square 2 dimensional. Here we'd like to ask, what is the dimensionality of a particle undergoing Brownian motion (diffusion)?

A particle undergoes Brownian motion through the action of a random force so that its displacement in a time interval $$\Delta t$$ behaves as $$\sim\sqrt{2D\Delta t}$$. Because the particle is 0d, we might expect that its trajectory is 1d, as for particles in projectile motion.

Suppose we have a perfectly sampled trajectory of a particle, $$\gamma_B$$. We want to lay down a set of circles, $$C$$, of radius $$r$$ to cover this path, i.e. so that every section of the particle's path in the 2d plane is overlaid by a portion of a circle $$C_i \in C$$. $$\mathbf{N}_\textrm{Brownian}$$ changes with $$r$$ according to $$\mathbf{N}_\textrm{Brownian} \sim r^d$$. What is $$\lvert d \rvert$$, i.e. what is the dimensionality of Brownian motion?

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