Why is Diffusion Called a Second Rank Tensor Variable?

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    Diffusion Tensor
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Discussion Overview

The discussion centers around the classification of diffusion as a second rank tensor variable, exploring the definitions and implications of this classification in the context of anisotropic materials and diffusion tensor imaging. Participants seek to clarify the meaning of the components of the diffusion tensor matrix and their physical interpretations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the input and output vectors associated with diffusion in the context of tensor definitions.
  • Another participant explains that the diffusion tensor allows for anisotropic behavior, with diffusivity varying by location and direction.
  • There is a request for clarification on the meaning of each component of the diffusion tensor matrix, specifically regarding off-diagonal and diagonal terms.
  • One participant asserts that diagonal components represent translational diffusivities, while off-diagonal components correspond to rotational mobility or diffusion.
  • Another participant expresses uncertainty about the interpretation of the diffusion tensor matrix components, suggesting that they represent apparent diffusion coefficients and that true diffusion coefficients can be derived from the eigenvalues of the matrix.
  • A later reply states that diagonal components represent flux in a direction caused by a concentration gradient in the same direction, while off-diagonal components represent flux in one direction caused by a concentration gradient in another direction.

Areas of Agreement / Disagreement

Participants express differing interpretations of the diffusion tensor components, with some agreeing on the roles of diagonal and off-diagonal terms while others remain uncertain or propose alternative explanations. The discussion does not reach a consensus on the interpretations presented.

Contextual Notes

There are unresolved questions regarding the definitions and interpretations of the components of the diffusion tensor matrix, as well as the relationship between apparent and true diffusion coefficients. The discussion reflects varying levels of understanding and interpretation among participants.

xfshi2000
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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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