Why is Diffusion Called a Second Rank Tensor Variable?

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Diffusion is classified as a second rank tensor variable because it describes the relationship between input and output vectors, particularly in anisotropic materials where diffusivity varies by location and direction. The diffusion tensor, initially developed for colloidal solutions and now utilized in MRI, captures this complexity. Diagonal components of the diffusion tensor represent translational diffusivities, while off-diagonal components indicate cross-directional flux due to concentration gradients. Each component of the diffusion tensor matrix corresponds to an apparent diffusion coefficient, which can be transformed into true diffusion coefficients through eigenvalue calculations. Understanding these components is crucial for accurately modeling diffusion processes in various materials.
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Hi all:
Does anyone explain why we call diffusion as the second rank tensor variable? I understand Based on the tensor definition, we know tensor expresses the relationship between input vector and output vector. How about diffusion? what is its input vector? what is its output vector? thanks in advance

xf
 
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The diffusion tensor (or the related Oseen mobility tensor) are used to allow the material to be anisotropic. The diffusivity varies with location and direction.

The current use of diffusion tensor comes from MRI (diffusion tensor imaging), but the concept was developed for colloidal solutions.
 
Andy Resnick said:
The diffusion tensor (or the related Oseen mobility tensor) are used to allow the material to be anisotropic. The diffusivity varies with location and direction.

The current use of diffusion tensor comes from MRI (diffusion tensor imaging), but the concept was developed for colloidal solutions.

Thank you. Could you tell me what the meaning of each component of diffusion tensor matrix is ? For example, what do off-diagonal terms represent? What do diagonal terms represent?
thanks
 
The diagonal components are translational diffusivities/mobilities. I think the off-diagonal components correspond to rotational mobility/diffusion.
 
Thanks. I do not really understand your answer. After I search the internet and get some explanation, for diffusion tensor matrix, each component of matrix represents an apparent diffusion coefficient. These diffusion coefficient are not real diffusion coefficient. After we calculate the eigenvalue of the matrix, we can get true and objective diffusion coefficient which is independent of coordinate frame selection. I am not sure these explanations are correct or not. Anyway thank you. Andy Resnick

xf
 
The diagonal components represent flux in some direction caused by a concentration gradient in that same direction. The off-diagonal components represent flux in some direction caused by a concentration gradient in another direction (which is less intuitive).
 
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