We describe the hydrodynamic model in Section 4, which further includes the inertia of the fluid and takes into account the transient hydrodynamic interactions between the particle and the fluid. Subsequently, we describe the Langevin model in Section 3, which considers explicitly the inertia of the Brownian particle. At first in Section 2 we introduce the pure diffusion model corresponding to Einstein’s microscopic picture. In an order of progressively more accurate hydrodynamic interactions between the particle and the fluid, we organize various theoretical models as follows. For a demonstrative purpose their values are a = 1, c s = 50, ρ = ρ B = 1, v = 1, and k B T = 1 in reduced units. The definitions of variables are in the text. Τ c s = a/ c s, viscous time τ v = a 2/ v, Brownian relaxation time τ B = m/ ξ, and diffusive time τ D = a 2/ D ∞. This can be clearly seen by the following relation: 31, 32Ĭ(t), D(t), and 〈Δ x 2( t)〉 of a Brownian particle (1 D) according to the Langevin model, incompressible viscous hydrodynamics, and its correction due to compressible effects at the short time scale. Compared to the well-known mean-squared displacement (MSD), which is denoted by 〈Δ r 2( t)〉 with the displacement Δ r( t) = r( t) − r(0), the VACF contains equivalent dynamical information. 30 In general, due to its interaction with the surrounding fluid, the particle’s velocity becomes randomized and the magnitude of 〈 v(0) It measures how similar the velocity v after time t is to the initial velocity.
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