He regarded the increment of particle positions in time τ τ in a one-dimensional (x) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a random variable ( q ) with some probability density function φ φ ( q ) (i.e., φ φ ( q ) is the probability density for a jump of magnitude q , i.e., the probability density of the particle incrementing its position from x to x + q in the time interval τ τ ). Further, assuming conservation of particle number, he expanded the number density ρ ρ ( x , t + τ τ ) (number of particles per unit volume around x ) at time t + τ τ in a Taylor series, ρ ρ ( x , t + τ τ ) = ρ ρ ( x , t ) + τ τ ∂ ∂ ρ ρ ( x , t ) ∂ ∂ t + ⋯ ⋯ = ∫ ∫ − − ∞ ∞ ∞ ∞ ρ ρ ( x − − q , t ) φ φ ( q ) d q = E q [ ρ ρ ( x − − q , t ) ] = ρ ρ ( x , t ) ∫ ∫ − − ∞ ∞ ∞ ∞ φ φ ( q ) d q − − ∂ ∂ ρ ρ ∂ ∂ x ∫ ∫ − − ∞ ∞ ∞ ∞ q φ φ ( q ) d q + ∂ ∂ 2 ρ ρ ∂ ∂ x 2 ∫ ∫ − − ∞ ∞ ∞ ∞ q 2 2 φ φ ( q ) d q + ⋯ ⋯ = ρ ρ ( x , t ) ⋅ ⋅ 1 − − 0 + ∂ ∂ 2 ρ ρ ∂ ∂ x 2 ∫ ∫ − − ∞ ∞ ∞ ∞ q 2 2 φ φ ( q ) d q + ⋯ ⋯ where the second equality is by definition of φ φ . The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd moments) vanish because of space symmetry. What is left gives rise to the following relation: ∂ ∂ ρ ρ ∂ ∂ t = ∂ ∂ 2 ρ ρ ∂ ∂ x 2 ⋅ ⋅ ∫ ∫ − − ∞ ∞ ∞ ∞ q 2 2 τ τ φ φ ( q ) d q + higher-order even moments. Where the coefficient after the Laplacian, the second moment of probability of displacement q , is interpreted as mass diffusivity D : D = ∫ ∫ − − ∞ ∞ ∞ ∞ q 2 2 τ τ φ φ ( q ) d q . Then the density of Brownian particles ρ at point x at time t satisfies the diffusion equation : ∂ ∂ ρ ρ ∂ ∂ t = D ⋅ ⋅ ∂ ∂ 2 ρ ρ ∂ ∂ x 2 ,