Miao et al. (2011) show a correlation between the bulk density of biomass and the screen size as a power law expression. Why is the correlation characterized by a negative sign in the power of the characteristic size?

A power-law relationship can be represented as Y = kX^n, where Y and X are the two variables (in this case, Y is the bulk density of biomass and X is the screen size), k is a proportionality constant, and n is the exponent (the power of the characteristic size).

A negative exponent (n) in the power-law relationship means that as the value of X (screen size) increases, the value of Y (bulk density of biomass) decreases, and vice versa. This inverse relationship between the two variables is characterized by the negative exponent.

Bulk density refers to the mass of biomass per unit of volume, whereas the screen size represents the characteristic particle size of the biomass. A higher screen size means larger particles in the biomass.

The negative sign in the power of the characteristic size represents an inverse relationship between the bulk density of biomass and the screen size. One possible reason for this inverse correlation could be that as the screen size increases (larger particles), the biomass particles may have more void spaces among them which result in lower bulk density. Conversely, as the screen size decreases (smaller particles), the biomass particles may pack more closely, leading to higher bulk density. In summary, the negative sign in the power of the characteristic size in the power-law expression describes an inverse relationship between the bulk density of biomass and screen size, possibly due to the increased void spaces in larger particles affecting the overall mass per unit volume of biomass.