where δ is the Dirac function and <> is average. These properties indicate that a value at t is not completely correlated to a value at t ′ , that is, discontinuous. Typically, this white noise is used to calculate Brownian dynamics simulations. However, it is dangerous to use this random function for Brown motion at a molecular time scale. This is because recent a study shows that Brown motion is continuous in a short time scale 8 , which is also supported by our MD simulations (Fig. ). Instead, we considered a smooth random function (Fig.