Brownian motion apart once the particle is in motion. Also, there would be a distribution of different possible Δ"V"s instead of always just one in a realistic situation. The diffusion equation yields an approximation of the time evolution of the probability density function associated to the position of the particle going under a Brownian movement under the physical definition. The approximation is valid on short timescales. The time evolution of the position of the Brownian particle itself is best described using Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on
Brownian surface Brownian motion, Brownian surfaces are named after 19th-century biologist Robert Brown. Efficient generation of fractional Brownian surfaces poses significant challenges. Since the Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cholesky decomposition method. A more efficient method is Stein's method, which generates an auxiliary stationary Gaussian process using the circulant embedding approach and then adjusts this auxiliary process to obtain the desired nonstationary Gaussian process. The figure below shows three typical realizations of fractional Brownian surfaces for different values of the roughness or Hurst parameter. The Hurst parameter is always between zero and