Could anybody tell me what does scale invariance means?

Is there any book or article that describes [ and gives examples ] about it.

  • $\begingroup$ Dear Poli we can't answer this question other than with a Wikipedia answer, which you've already linked to. Scale invariance, as the linked article shows, shows up a great deal in science. A great deal of work has already gone into that article: could you please ask a question about something you don't understand there, then it will be a question we can chew on. Or maybe your question is a resource recommendation? $\endgroup$ Dec 20, 2013 at 21:40

1 Answer 1


Scale invariance can be thought of as 'self-similarity'. What this really means is that regardless of how much you zoom into or out of an object (be it a function, or a physical object, or the like) it looks exactly the same.

Fractals are good examples of self-similarity. Shown below is an animation of the self-similarity of the Mandelbrot set:


Another well known example of scale invariance in physics is the Wiener process, which is a continuous-time stochastic (random) process. It's also often called standard Brownian motion. No matter how much you zoom into a Wiener process you still get quantitatively the same thing:


Also, taken from the page on scale invariance:

In quantum field theory, scale invariance has an interpretation in terms of particle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.

My last example (though scale invariance shows up in many places in physics and astronomy) is something in which I personally study as a researcher, and that is the self-similarity of the dark matter halo profile. The dark matter halos surrounding dwarf galaxies up to the largest bound structures in the universe (galaxy clusters), all seem to follow a fairly simple density profile. This density profile is called the Navarro-Frenk-White (NFW) profile, and has the following form:

$$ \rho_{NFW}(r) = \frac{\delta_{c} \rho_{crit}}{\frac{r}{r_{s}} \left(1 + \frac{r}{r_{s}} \right)^{2}}$$

As far as I know, people still don't completely understand why, physically speaking, this profile is what you observe in large gravity only simulations over many decades of mass. The underlying physics is simple, it's just gravitational interactions between particles, but the intuition as to why they form these structures in bulk has not been properly explained.


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