Maxwell - Boltzmann Distribution

In 1859, after reading a paper about the diffusion of molecules by Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.23 This was the first-ever statistical law in physics.24 Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.25 In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases."26 In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution

https://en.wikipedia.org/wiki/Kinetic_theory_of_gases#:~:text=In 1859%2C after,Maxwell–Boltzmann distribution

The distribution can be derived on the ground that it maximizes the entropy of the system. A list of derivations are:

Maximum entropy probability distribution in the phase space, with the constraint of conservation of average energy
Canonical ensemble.

For a system containing a large number of identical non-interacting, non-relativistic classical particles in thermodynamic equilibrium, the fraction of thAle particles within an infinitesimal element of the three-dimensional velocity space $d^{3} v$ , centered on a velocity vector v of magnitude $v$ , is given by

f (v) d^{3} v = [\frac{m}{2 π k_{B} T}]^{3 / 2} \exp (- \frac{m v^{2}}{2 k_{B} T}) d^{3} v,

m is the particle mass;
_k_B is the Boltzmann constant;
T is thermodynamic temperature;
$f (v)$ is a probability distribution function, properly normalized so that $\int f (v) d^{3} v$ over all velocities is unity.

One can write the element of velocity space as $d^{3} v = d v_{x} d v_{y} d v_{z}$ , for velocities in a standard Cartesian coordinate system, or as $d^{3} v = v^{2} d v d Ω$

In the one dimensional case the Maxwellian reduces to a Normal - Gaussian Distribution with a standard deviation of $\sqrt{k_{B} T / m} .$ (from Kinetic Theory of Gases)

f (v_{x}) d v_{x} = \sqrt{\frac{m}{2 π k_{B} T}} \exp (- \frac{m v_{x}^{2}}{2 k_{B} T}) d v_{x},

To go from the distribution of velocity vectors to the speed distribution we can integrate in Spherical coordinates over all solid angles giving $4 π$ and using the element of velocity space $d^{3} v = v^{2} d v d Ω$ giving

f (v) = 4 π v^{2} [\frac{m}{2 π k_{B} T}]^{3 / 2} \exp (- \frac{m v^{2}}{2 k_{B} T})

or we can stay in a cartesian frame where :

u = \sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}}

where the Maxwellian is just a reduction of the Chi distribition - the distribution of the positive square root of a sum of squared independent Gaussian random variables:
If $Z_{1}, \dots, Z_{k}$ are k independent, normally distributed random variables with mean 0 and standard deviation 1, then the statistic

Y = \sqrt{\sum_{i = 1}^{k} Z_{i}^{2}}

The probability density function (pdf) of the chi-distribution is

f (x; k) = {\begin{cases} \frac{x^{k - 1} e^{- x^{2} / 2}}{2^{k / 2 - 1} Γ (\frac{k}{2})}, & x \geq 0; \\ 0, & otherwise . \end{cases}

where Γ(z) is the gamma function.

Then the Maxwellian is equivalent to the chi distribution with three degrees of freedom and scale parameter $a = \sqrt{k_{B} T / m} .$

f (v) = [\frac{m}{2 π k_{B} T}]^{3 / 2} 4 π v^{2} \exp (- \frac{m v^{2}}{2 k_{B} T}) .

Generalized quantum Boltzmann distribution

Δίνει την πιθανότητα ένα σύστημα να βρεθεί σε μια κατάσταση όταν τοποθετηθεί σε δεξαμενή θερμότητας Τ:

p (E r) = \frac{1}{Z} g (E_{r}) e^{- β E_{r}}

Where $$Z=\sum_r e^{\beta e_r}$$

β = \frac{1}{k T}

This distribution is derived in the Closed system - Heat Bath - Canonical ensemble. It gives the probability that a system when placed in a heat bath at temperature T (i.e. with a temperature parameter β = 1/ kT) should be in aparticular state. We see that this probability depends on the energy of the state. The only property of the heat bath on which it depends is the temperature of the heat bath. The quantity Z, defined by (2.23), is known as the partition function of the system. We shall see that it plays a central role in studying systems at a fixed temperature.

Qualitively, g(Er) increases rapidly with E while the propability of occupation of hgher states decreases rapidly, therefore their multiplication is a delta function.

Mandl's detailed derivation of the density of states f(E): [[Mandl_7.7.pdf]]

Another derivation can be performed using Lagrange Multipliers :

Maximization of Entropy with 2 contrains

[[Hill_5p4_MaxwellBoltzmannDistribution.pdf]]