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In the Langevin equation with hydrodynamic interactions the stochastic force on particle $a$ is: $$ \sqrt{2k_BT} A^{ab}_{ij} \xi^{b}_j(t)$$ where $\xi$ is a unit white noise. Here $ A^{ab}_{ij} $ is the square root of the mobility matrix (and spare a multiplicative constant the diffusion matrix) in the sense that: $$M_{ij}^{ab}=A_{ik}^{ac}A^{bc}_{jk}\tag{1}$$ My question is: does (1) uniquely define $A$ and if not how do we choose the 'correct' $A$?

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The answer to this question is no it is not uniquely defined - but it doesn't matter since any solution to (1) will work (remember that it is multiplied by something that it is intrinsically random).

That being said there are some typical ways in which the matrix $A$ (in the literature more commonly called $B$) is found. These include things like Cholesky decomposition.

I found a useful source to be: http://leonardo.inf.um.es/macromol/publications/212JCP11.pdf

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