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# Uniqueness of the square-root of the diffusion matrix?

2018-02-23 08:05:22

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' $A$?