On the Derivation of the Maxwell Boltzmann Distribution
| Main Author: | Ruggeri, Francesco R. |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2018
|
| Subjects: | |
| Online Access: |
https://zenodo.org/record/1220890 |
Daftar Isi:
- It is known that the statistical factor exp(-e/T), or at least exp(-Ae), appears very naturally from Boltzmann ́s equilibrium criterion f(e1)f(e2)-f(e3)f(e4) and holds in both relativistic and nonrelativistic cases. If one lets e1=e and e2 =e+de, one is able to obtain an equation for f(e) alone, namely 1/f df/de=-A where A is constant. This leads to an exp(-eA) factor and suggests that this factor may arise in more general cases than just the collisions of two particles. The purpose of this note is to put forward a physical picture in which there is the ̈appearance ̈ of an increasing difficulty of obtaining high energy states. Ultimately, however, it seems that the high ̈decay ̈ rate of high energy or momentum particles may be the cause of f(e+de) being smaller than f(e). We argue that high energy particles tend to smash into nearby particles doing work and losing their energy and so have a higher tendency of ̈decaying ̈ or ̈shedding energy ̈ than lower energy particles. Here it should be noted that ̈high energy ̈ is relative to an average energy or temperature. We argue that this is the dynamical reason for a decrease in f(e). Such a dynamical picture does not depend on counting states or the idea of entropy. This is not to say that these are not ̈in the picture ̈, but rather suggests that there should be a dynamical driver of equilibrium. We argue that it is important to have a physical picture based on dynamics to describe the Maxwell-Boltzmann distribution, and ultimately the factor exp(-E/T) which is used throughout statistical mechanics.