Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution ρ ρ ( x , t ) = N 4 π π D t exp ⁡ ⁡ ( − − x 2 4 D t ) . This expression (which is a normal distribution with the mean μ μ = 0 and variance σ σ 2 = 2 D t usually called Brownian motion B t ) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by E [ x 2 ] = 2 D t . This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root. His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.