Answer to the Question 09/01
THE POWER OF DIMENSIONThe question was:
Physical quantities always have dimensions that are products of powers of basic units. E.g., the energy is measured in joules and
1 J = kg*m2/sec2
Why aren't there any quantities which are NOT powers of elementary units, but rather are more complicated functions?
(12/01) Most of the answers submitted to us concentrated on "non-power-law" examples which might lead to absurd result. A more constructive approach to the question has been submitted by Jared Daniel Kaplan (9/2001) from Stanford (e-mail jaredk@stanford.edu). However, we feel that we did not receive a really general proof of the case that details all the assumptions involved.
An excellent exposition and in depth consideration of this question can be found in the book Scaling, self-similarity, and intermediate asymptotics by G.I. Barenblatt (Cambridge U. Press, 1996). (We are grateful to Jerome Gosset (e-mail jerome.gosset1@libertysurf.fr) for bringing this book to our attention.) Our discussion below largely follows that book.
Answer: Only power laws (and products of power laws) satisfy the principle underlying the concept of "unit" as defined below.
Detailed explanation
A set of fundamental units that is sufficient to measure properties of a class of phenomena is called a system of units. E.g., gram, centimeter and second can form a system for a broad range of phenomena. Similarly, kilogram, meter and second can form another system of units. Both systems rely on mass M, length L and time T units, and we shall say that they belong to one class of sytems of units. On the other hand, kilogram-force, speed-of-light and minute, also form a system. However, they belong to a different class (force, velocity, time).
The function which determines factor by which the numerical value of a physical quantity changes upon passage between two systems of units within the same class is called dimension function. E.g., the dimension function for "mass density" is M/L3. It tells us that if mass unit has been increased by a factor of 2 and length unit has been increased by a factor of 4, the number of the same quantity will decrease by factor of 32.
Can there be a dimension function sinM*logT? No! We are going to show that the dimension function must be a power-law monomial. This is a consequence of a simple principle:
All systems within a single class are equivalent, and there is no single distinguished or somehow preferred system of units
Without loss of generality, let us consider some specific system of units, e.g. the system built on L, M and T. Consider a mechanical quantity A that depends on all these units and the dimension function is [A]=F(L,M,T). Now let us consider systems of units 1 and 2 which have been obtained from an original system L,M,T by decreasing the units by factors L1,M1,T1 and L2,M2,T2, respectively. If the value of our mechanical quantity in the original units was A, then in the new units it will be A1=A*F(L1,M1,T1) and A2=A*F(L2,M2,T2), respectively, and therefore:
A1/A2=F(L1,M1,T1)/F(L2,M2,T2).
However, we could have used system 1 as our basic system, and consequently treated system 2, as modification (decrease) of system 1 units by factors L2/L1, M2/M1, T2/T1, and therefore
A2=A1F(L2/L1, M2/M1, T2/T1).
By comparing the above expressions we see that
F(L1,M1,T1)/F(L2,M2,T2)= F(L2/L1, M2/M1, T2/T1)
If we now differentiate both sides with respect to L2, and then set L2=L1=L, M2=M1=M, T2=T1=T, then we get:
DLF(L,M,T)/F(L,M,T)=(1/L)DLF(1,1,1)=c/L,
where c simply denotes DLF(1,1,1), and DL denotes partial derivative with respect to L. Integrating the last equation with respect to L, we find that
F(L,M,T)=LcG(M,T),
where G is some (as yet) undetermined function of two variables. By repeating the above procedure with M and with T we will arrive to the conclusion that the remaining variables must also appear as power laws, and consequently,
F=g Lc Md Tf,
where g, c, d, f are constants. This proves that dimension function must be a power-law monomial.
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