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## Maxwell–Boltzmann statistics
In statistical mechanics, The expected number of particles with energy ε where: *N*_{i}is the number of particles in state*i*- ε
_{i}is the energy of the*i*-th state *g*_{i}is the degeneracy of energy level*i*, the number of particle's states (excluding the "free particle" state) with energy ε_{i}- 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 the exponential function
Equivalently, the distribution is sometimes expressed as where the index
Fermi-Dirac and Bose-Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable". The quantum effects appear if the concentration of particles ( Maxwell-Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words the configuration of particle ## Additional recommended knowledge
## A derivation of the Maxwell–Boltzmann distributionIn this particular derivation, the Boltzmann distribution will be derived using the assumption of distinguishable particles, even though the Suppose we have a number of energy levels, labelled by index The number of different ways of performing an ordered selection of one object from where the extended product is over all boxes containing one or more objects. If the For example, suppose we have three particles,
The six ways are calculated from the formula: We wish to find the set of Using Stirling's approximation for the factorials and taking the derivative with respect to It can be shown thermodynamically that β = 1/ Note that the above formula is sometimes written: where Alternatively, we may use the fact that to obtain the population numbers as where ## Another derivationIn the above discussion, the Boltzmann distribution function was obtained via directly analysing the multiplicities of a system. Alternatively, one can make use of the canonical ensemble. In a canonical ensemble, a system is in thermal contact with a reservoir. While energy is free to flow between the system and the reservoir, the reservoir is thought to have infinitely large heat capacity as to maintain constant temperature, In the present context, our system is assumed to be have energy levels ε If our system is in state , then there would be a corresponding number of microstates available to the reservoir. Call this number . By assumption, the combined system (of the system we are interested in and the reservoir) is isolated, so all microstates are equally probable. Therefore, for instance, if , we can conclude that our system is twice as likely to be in state than . In general, if is the probability that our system is in state , Since the entropy of the reservoir , the above becomes Next we recall the thermodynamic identity: - .
In a canonical ensemble, there is no exchange of particles, so the - ,
where and denote the energies of the reservoir and the system at - ,
which implies, for any state - ,
where - ,
where the index where, with obvious modification, , this is the same result as before. ## Comments- Notice that in this formulation, the initial assumption "...
*suppose the system has total N particles*..." is dispensed with. Indeed, the number of particles possessed by the system plays no role in arriving at the distribution. Rather, how many particles would occupy states with energy ε_{i}follows as an easy consequence.
- What has been presented above is essentially a derivation of the canonical partition function. As one can tell by comparing the definitions, the Boltzman sum over states is really no different from the canonical partition function.
- Exactly the same approach can be used to derive Fermi–Dirac and Bose–Einstein statistics. However, there one would replace the canonical ensemble with the grand canonical ensemble, since there is exchange of particles between the system and the reservoir. Also, the system one considers in those cases is a single particle
*state*, not a particle. (In the above discussion, we could have assumed our system to be a single atom.)
## Limits of applicabilityThe Bose–Einstein and Fermi–Dirac distributions may be written: Assuming the minimum value of ε For an ideal gas, we can calculate the chemical potential using the development in the Sackur–Tetrode article to show that: where
## References**^***Z*is sometimes called the**Boltzmann sum over states**.
Carter, Ashley H., "Classical and Statistical Thermodynamics", Prentice-Hall, Inc., 2001, New Jersey. ## See alsoCategories: Statistical mechanics | Particle statistics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Maxwell–Boltzmann_statistics". A list of authors is available in Wikipedia. |