Brownian dynamics Fluctuation Dissipation Theorem

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

According to the fluctuation-dissipation theorem [1], the electrical polarizability of polyelectrolytes is related to the fluctuations of the dipole moment generated in the counterion atmosphere around the polyions in the absence of an applied electric field [2-4], Here we calculate the fluctuations by computer simulation to determine anisotropy of the electrical polarizability Aa of model DNA fragments in salt-free aqueous solutions [5-7]. The Metropolis Monte Carlo (MC) Brownian dynamics method [8-12] is applied to calculate counterion distributions, electric potentials, and fluctuations of counterion polarization. 

What physical forces affect colloid dynamics Three forces acting on neutral colloids are readily identified, namely random thermal forces, hydrodynamic interactions, and direct interactions. The random thermal forces are created by fluctuations in the surrounding medium they cause polymers and colloids to perform Brownian motion. As shown by fluctuation-dissipation theorems, the random forces on different colloid particles are not independent they have cross-correlations. The cross-correlations are described by the hydrodynamic interaction tensors, which determine how the Brownian displacements of nearby colloidal particles are correlated. The hydrodynamic drag experienced by a moving particle, as modified by hydrodynamic interactions with other nearby particles, is also described by a hydrodynamic interaction tensor. 

A second catch is the noise. If one observes the movements of a colloidal particle, the Brownian motion will be evident. There may be a constant drift in the dynamics, but the movement will be irregular. Likewise, if one observes a phase-separating liquid mixture on the mesoscale, the concentration levels would not be steady, but fluctuating. The thermodynamic mean-field model neglects all fluctuations, but they can be restored in the dynamical equations, similar to added noise in particle Brownian dynamics models. The result is a set of stochastic diffusion equations, with an additional random noise source tj [20]. In principle, the value and spectrum of the noise is dictated by a fluctuation dissipation theorem, but usually one simply takes a white noise source. 

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