Minimal Functional Forms in Statisical Mechanics Part 2

In Part 1, we considered minimal functional forms in a variable such as ei (single particle energy) leading to the derivation of statistical mechanical distributions. Our argument was that more complicated functional forms carry more information and that there must be a reason for such extra information. For example, the simplest rotatinoally invariant form using a vector v is v dot v, yet any f(v dot v) manifests rotatioanl invarariance just as well. As a result, one is not only concerned with rotational invariance, but a functional form which matches reality and hopefully (due to underlying symmetries) is as simple as possible. We suggested seeking minimal functional forms in some variable such as single particle energy when trying to create statistical distributions. This is a math procedure and so here we try to examine any underlying physics linked to this idea. This physics may even appear in deterministic Newtonian mechanics. In particular, dp/dt = F, Newton’s second law, can be made rotationally invariant through: v dot dp = F dot dr. Integrating on the LHS yields .5 mo v dot v (nonrelativistically) which is the simplest rotationally invariant form. The RHS which is work, however, contains the notion that W1+W2 = W3+W4+W5 etc. In other words, a given work number may be written as a sum of different work terms in many ways and each is equally legitimate, i.e. there is a statistical notion of equal probability for each scenario. We suggest that this is linked to the idea of p(ei) = exp(- a v dot v). It is not just the v dot v piece which was the focus of Part 1 that is important, but also the notion of p(e1)p(e2)=p(e1+e2) which is already apparent in the notion of work. The variable e (single particle energy) may be viewed in terms of a quantity which adds to another single particle e. We suggest, however, that the value e may also have a different physical interpretation in terms of information. It may represent internal information. This viewpoint is also contained in v dot dp = F dot dr. In particular, given an ei, it may be created through Sum i=1,N ei/N pieces and all of these should have the same weight. This is not a new idea because in statistical mechanics texts, the number of states is given by E power N1, where N1 is the degrees of freedom. This is a power law in E and so exp(-E/T) is not the only distribution in nature. exp(-e/T) is a distribution associated with e1+e2=e3+e4, while an ei power law is linked with ei*ej=ek*el. In previous notes, we argued that these represent minimal functionals in e, but there is a deterministic Newtonian reason for both being considered. The sum leads to the Maxwell-Boltzmann distribution and the produce for e/T>>1 to the Tsallis and Kaniadakis distributions. In other words, the minimal ei functional solutions to p(ei)p(ej)=p(ek)p(el), namely ei+ej=ek+el and ei*ej=ek*el for e/T>>1 both follow from the Newtonian notion of work, we argue and one should not selectively consider one without the other, i.e. the Maxwell-Boltzman solution (ei+ej=ek+el) is directly linked with the Tsallis and Kaniadakis ones ei*ej=ek*el, e/T>>1.