Maxwell - 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]]