Abstract
In this chapter, we present some elements of Maxwell’s formalism and derivation that builds the kinetic theory of the ideal gas. It derives some rather useful results from first principles. While the main results that we shall see in this section can be viewed as a special case of the more general concepts and principles that will be provided later on, the purpose here is to give a quick snapshot on the taste of this matter and to demonstrate how the statistical approach to physics, which is based on very few reasonable assumptions, gives rise to rather far–reaching results and conclusions.
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Notes
- 1.
Rudolf Julius Emanuel Clausius (1822–1888) was a German physicist and mathematician who is considered to be one of the central pioneers of thermodynamics.
- 2.
James Clerk Maxwell (1831–1879) was a Scottish physicist and mathematician, whose other prominent achievement was formulating classical electromagnetic theory.
- 3.
Ludwig Eduard Boltzmann (1844–1906) was an Austrian physicist, who has founded contributions in statistical mechanics and thermodynamics. He was one of the advocators of the atomic theory when it was still very controversial.
- 4.
Average – over time as well.
- 5.
Consider this as an experimental fact.
- 6.
Another example of the Boltzmann form \(e^{-\epsilon /(kT)}\) is the barometric formula: considering gravity, the pressure increment \(\text {d}P\) between height h and height \(h+\text {d}h\) in an ideal–gas atmosphere, must be equal to \(-\mu g\text {d}h\), which is the pressure contributed by a layer of thickness \(\text {d}h\), where \(\mu \) is the mass density. Thus, \(\text {d}P/\text {d}h=-\mu g\). But by the equation of state, \(\mu =Nm/V=mP/kT\), which gives the simple differential equation \(\text {d}P/\text {d}h=-Pmg/kT\), whose solution is \(P=P_0e^{-mgh/kT}\), and so, \(\mu =-P'/g=\rho _0e^{-mgh/kT}\), which is proportional to the probability density. Then, here we have \(\epsilon =mgh\), the gravitational potential energy of one particle.
- 7.
Similar idea to the one of the earlier derivation of the Gaussian pdf of the ideal gas.
- 8.
Note that for \(v_y=v_z=0\), the factor \(v_x\tau A/V\), in the forthcoming equation, is clearly the relative volume (and hence the probability) of being in the ‘box’ in which a particle must be found in order to pass the hole within \(\tau \) seconds. When \(v_y\) and \(v_z\) are non–zero, instead of a rectangular box, this region becomes a parallelepiped, but the relative volume remains \(v_x\tau A/V\) independently of \(v_y\) and \(v_z\).
References
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Merhav, N. (2018). Kinetic Theory and the Maxwell Distribution. In: Statistical Physics for Electrical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-62063-3_1
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DOI: https://doi.org/10.1007/978-3-319-62063-3_1
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