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Brownian dynamics (BD) is an accurate and computationally efficient mesoscale simulation technique used to study the dynamics and material properties of synthetic and biological polymeric solutions with different architectures, under both equilibrium and nonequilibrium (with the presence of an external field) conditions. A common model used to describe the physics of a polymer is the bead-spring model, where beads resemble the centers of hydrodynamic resistance and they are connected by a network of springs, where the spring tension is generally related nonlinearly to the spring extension. In comparison to a fully resolved molecular dynamics model, this bead-spring model abstracts away the fine (unnecessary) details of real macromoleculaes.
The stochastic differential equation governing the positions of
$$ \begin{aligned} \text{d}\boldsymbol{r}{\nu} & = \ & \left[Pe\boldsymbol{\kappa}\cdot\boldsymbol{r}{\nu}+\frac{1}{4}\sum_{\mu=1}^{N_\mathrm{b}}\nabla_{\mu}\cdot\mathbf{D_{\mu\nu}}+\frac{1}{4}\sum_{\mu=1}^{N_\mathrm{b}}\mathbf{D}{\nu\mu}\cdot\boldsymbol{F}{\mu}\right]\mathrm{d}t \ & + \frac{1}{\sqrt{2}}\sum_{\mu=1}^{N_\mathrm{b}}\mathbf{C}{\nu\mu}\cdot\mathrm{d}\boldsymbol{W}{\mu} \end{aligned} $$
where
where
$$\boldsymbol{v}^{\prime}(\boldsymbol{x}{\nu})=\mathbf{D}{\nu\mu}\cdot\boldsymbol{F}(\boldsymbol{x}_{\mu}).$$
The form of
The diffusivity matrix