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
Geometric Brownian motion above: formula_27. import numpy as np import pandas as pd import matplotlib.pyplot as plt mu=1 n=50 dt=0.1 x0=100 x=pd.DataFrame() np.random.seed(1) for sigma in np.arange(0.8,2,0.2): x.columns=np.arange(0.8,2,0.2) plt.plot(x) plt.legend(x.columns) plt.xlabel('t') plt.ylabel('X') plt.title('Realizations of Geometric Brownian Motion with different variances\n mu=1') plt.show() GBM can be extended to the case where there are multiple correlated price paths. Each price path follows the underlying process where the Wiener processes are correlated such that formula_31 where formula_32. For the multivariate case, this implies that Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock