Maxwell boltzmann

In statistical mechanics, Maxwell–Boltzmann statistics describes the average distribution of non-interacting material particles over various energy states in thermal equilibrium, and is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible.
The expected number of particles with energy




ε

i




{\displaystyle \varepsilon _{i}}
for Maxwell–Boltzmann statistics is




⟨

N

i


⟩
=



g

i



e

(

ε

i


−
μ
)

/

k
T




=


N
Z




g

i



e

−

ε

i



/

k
T


,


{\displaystyle \langle N_{i}\rangle ={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/kT}}}={\frac {N}{Z}}\,g_{i}e^{-\varepsilon _{i}/kT},}
where:





ε

i




{\displaystyle \varepsilon _{i}}
is the energy of the i-th energy level,




⟨

N

i


⟩


{\displaystyle \langle N_{i}\rangle }
is the average number of particles in the set of states with energy




ε

i




{\displaystyle \varepsilon _{i}}
,





g

i




{\displaystyle g_{i}}
is the degeneracy of energy level i, that is, the number of states with energy




ε

i




{\displaystyle \varepsilon _{i}}
which may nevertheless be distinguished from each other by some other means,
μ is the chemical potential,
k is Boltzmann's constant,
T is absolute temperature,
N is the total number of particles:
Z is the partition function:
e is Euler's numberEquivalently, the number of particles is sometimes expressed as




⟨

N

i


⟩
=


1

e

(

ε

i


−
μ
)

/

k
T




=


N
Z




e

−

ε

i



/

k
T


,


{\displaystyle \langle N_{i}\rangle ={\frac {1}{e^{(\varepsilon _{i}-\mu )/kT}}}={\frac {N}{Z}}\,e^{-\varepsilon _{i}/kT},}
where the index i now specifies a particular state rather than the set of all states with energy




ε

i




{\displaystyle \varepsilon _{i}}
, and



Z
=

∑

i



e

−

ε

i



/

k
T




{\textstyle Z=\sum _{i}e^{-\varepsilon _{i}/kT}}
.

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