Up to this point, I thought that when doing a dimensional analysis using the Buckingham-Pi theorem the exponents are generally quite "simple". For example 0.5, 1 or 2 and so on.
However, I now have found a paper where the exponents of the dimensionless numbers that are formulated by a data-driven approach have "strange" exponents like 0.07 or 0.304. This seems a bit odd to me and brings me to the questions: are such exponents (still) physically meaningful? If so, in which cases does such a type of exponents occur (and why)?
Thank you very much!
When performing dimensional analysis, exponents are used to express the relationship between different physical quantities. The exponents used in dimensional analysis are determined by the units of the quantities being analyzed. Here are some guidelines regarding the use of exponents in dimensional analysis:
The exponents must be integers or fractions with integer numerators and denominators. This is because the exponents represent the powers to which the units are raised.
The exponents can be positive, negative, or zero. A positive exponent indicates that the unit is raised to that power, a negative exponent indicates that the unit is in the denominator, and a zero exponent indicates that the unit is not present in the expression.
The exponents are determined by the dimensions of the physical quantities involved. For example, if you are analyzing a distance (L), time (T), and mass (M), the dimensions can be represented as [L^a * T^b * M^c], where a, b, and c are the exponents.
The exponents should be chosen in a way that ensures dimensional consistency. This means that the dimensions of all terms in an equation or expression should be the same. For example, if you are analyzing a formula like F = ma (force equals mass times acceleration), the dimensions on both sides of the equation must match. If the dimension of mass is [M^1], and the dimension of acceleration is [L^1 * T^-2], then the dimension of force should be [M^1 * L^1 * T^-2] to maintain dimensional consistency.
It's important to note that the specific values of the exponents are determined based on the specific equation or problem you are working with. Different equations and problems may require different sets of exponents.
Some of my thought:
In physics, theories about phase transition could give various non-integer exponents (see e.g. https://en.wikipedia.org/wiki/Ising_critical_exponents).
Sometimes, exponents could arise from specific type of ODE (e.g. https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation). For example, I am working on one simple physics model and got a second-order ODE. One coefficient is an definite integral of some special function (I can only calculate it numerically). Because of this coefficient, the solution of ODE has non-integer exponents.
In dimensional analysis using Buckingham pi method, the exponents have a high significance in describing nature of a specific model or it's characteristic behavior e.g. fluid flow property in a mechanical system.
Usually integer exponents represent a homogeneous model in which n dimensional variables are related to m fundamental quantities and require a total of n-m quantities to make the model dimensionless and independent.
However in the case you've referred to, the exponents are non-integers and therefore represent a dimensional inhomogeneity and/or nonlinearities. In such cases , methods involving the reduction of higher order models to an equivalent simplified order may be necessary to investigate.
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