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Can physical quantities such as displacement and velocity be defined for a body moving in fractional dimensions?

For a point particle, the distance/displacement between any two points would be infinite for an ideal fractal. The situation would be a bit different if we consider real world examples, say in biology, the self-similarity would stop at some scale. How do we still use concepts from fractional calculus to define such quantities (if at all)?

References supporting the answers would be highly appreciated.

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  • $\begingroup$ "Moving in fractional dimensions" doesn't match what you describe. The idealised fractal is an infinitely sized surface of m dimensions embedded in a finite region of n (>m) domensional space. "Motion" is defined on either the m or n dimensional spaces. $\endgroup$
    – Paul Childs
    Commented Jul 1, 2019 at 5:33
  • $\begingroup$ Actually, consider a fractal set $F$ embedded in some suitable D-dimensional space. We can look at a function $x: t\rightarrow f\in F$ as a motion of a point in the fractal. While the velocity $x'(t)$ in the embedding space might be badly defined, it is not obvious to me that fractional derivatives have to diverge. $\endgroup$
    – Anders Sandberg
    Commented Jul 1, 2019 at 11:50

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