Scaling law: Difference between revisions

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* make educated guesses at local transport mechanisms
* make educated guesses at local transport mechanisms


The typical scaling law expression for an (output) variable ''y'' as a function of some system variables ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is:
The typical scaling law expression for a (dependent) variable ''y'' as a function of some (independent) system variables ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is:
:<math>y = \alpha_0 x_1^{\alpha_1} x_2^{\alpha_1} ...</math>
:<math>y = \alpha_0 x_1^{\alpha_1} x_2^{\alpha_1} ...</math>
Here, the &alpha;<sub>i</sub> are the scaling parameters.
Here, the &alpha;<sub>i</sub> are the scaling parameters.
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However, a proper analysis requires:
However, a proper analysis requires:
* using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately)
* using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately)
* using (linearly) ''statistically independent'' variables (applying, e.g., Principal Component Analysis)
* guaranteeing the (linear) ''statistical independence'' of the independent variables (applying, e.g., Principal Component Analysis)


== Confinement time scaling ==
== Confinement time scaling ==