I would like to know the mathematical relation and the physical meaning in terms of real example.
I'm not quite clear to what you are referring based on the question. I have a suspicion that you might be talking about relating critical exponents to fractal structure, but could you be more specific?
I see researchers use the power-law index values as a measure of fragmentation processes (e.g., if we plot fragment size distribution data in double logarithmic scale and determine the slope based on power law fit). How is fractal dimension better than the power-law exponent value or power-law index? To clarrify, I put one figre as attachemnt file. Is it possible to directly say that this power-law exponent (1.752) is the fractal dimension of the fragment size distribution?
Fractal concepts in surface growth 1995
By-- A. L. Barabasi & H. E. Stanley
May be this book can help you, what you are searching for...
In the field of fractal physiology (perhaps not close to your area) and allometric relationships the power-law index or the scaling exponent h of a fractal time series (e.g. heart beat intervals) is related to its fractal dimension : FD = 2-h/2. An excellent book about that was written by Bruce West: "Where medicine went wrong" (section 5.2., pp.182-191).
Fractal dimension is closely related to geometry. When you observe larger and larger part of a fractal object then its observed volume ((or "mass") increases as some power of the size of "field of view" (think, for example, about the mass of a tree). But is this reversible, i.e. is power law an indicator of "fractalness"? It is tempting to think so, but there is no proof.
A study done by Korcak in 1938 on the surface areas of a group of islands had an important inspiration on the fractal theory of Mandelbrot. Korcak discovered that when the logarithm of the number of Aegean islands greater than or equal to a stated size A is plotted against the logarithm of A yields a straight line.
The thing which makes fractal dimension different than a simple power law equation is the the way we use the independent parameter. It goes into the horizontal axis as an increasing "cumulative value", such as the number of islands greater than a certain size. Mandelbrot used this concept to measure the length of a coast (i.e. to calculate the fractal dimension) by changing the measure which is the distance between two legs of a compass.
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