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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.
Contents
- Applications
- Limits of applicability
- Derivations of MaxwellBoltzmann statistics
- Derivation from microcanonical ensemble
- Derivation from canonical ensemble
- References
The expected number of particles with energy
where:
Equivalently, the number of particles is sometimes expressed as
where the index i now specifies a particular state rather than the set of all states with energy
Applications
Maxwell–Boltzmann statistics may be used to derive the Maxwell–Boltzmann distribution (for an ideal gas of classical particles in a three-dimensional box). However, they apply to other situations as well. Maxwell–Boltzmann statistics can be used to extend that distribution to particles with a different energy–momentum relation, such as relativistic particles (Maxwell–Jüttner distribution). In addition, hypothetical situations can be considered, such as particles in a box with different numbers of dimensions (four-dimensional, two-dimensional, etc.).
Limits of applicability
Maxwell–Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words, the configuration of particle A in state 1 and particle B in state 2 is different from the case in which particle B is in state 1 and particle A is in state 2. This assumption leads to the proper (Boltzmann) statistics of particles in the energy states, but yields non-physical results for the entropy, as embodied in the Gibbs paradox.
At the same time, there are no real particles which have the characteristics required by Maxwell–Boltzmann statistics. Indeed, the Gibbs paradox is resolved if we treat all particles of a certain type (e.g., electrons, protons, etc.) as indistinguishable, and this assumption can be justified in the context of quantum mechanics. Once this assumption is made, the particle statistics change. Quantum particles are either bosons (following instead Bose–Einstein statistics) or fermions (subject to the Pauli exclusion principle, following instead Fermi–Dirac statistics). Both of these quantum statistics approach the Maxwell–Boltzmann statistics in the limit of high temperature and low particle density, without the need for any ad hoc assumptions. The Fermi–Dirac and Bose–Einstein statistics give the energy level occupation as:
It can be seen that the condition under which the Maxwell–Boltzmann statistics are valid is when
where
Maxwell–Boltzmann statistics are particularly useful for studying gases that are not very dense. Note, however, that all of these statistics assume that the particles are non-interacting and have static energy states.
Derivations of Maxwell–Boltzmann statistics
Maxwell–Boltzmann statistics can be derived in various statistical mechanical thermodynamic ensembles:
In each case it is necessary to assume that the particles are non-interacting, and that multiple particles can occupy the same state and do so independently.
Derivation from microcanonical ensemble
Suppose we have a container with a huge number of very small particles all with identical physical characteristics (such as mass, charge, etc.). Let's refer to this as the system. Assume that though the particles have identical properties, they are distinguishable. For example, we might identify each particle by continually observing their trajectories, or by placing a marking on each one, e.g., drawing a different number on each one as is done with lottery balls.
The particles are moving inside that container in all directions with great speed. Because the particles are speeding around, they possess some energy. The Maxwell–Boltzmann distribution is a mathematical function that speaks about how many particles in the container have a certain energy.
In general, there may be many particles with the same amount of energy
To begin with, let's ignore the degeneracy problem: assume that there is only one way to put
The number of different ways of performing an ordered selection of one single object from N objects is obviously N. The number of different ways of selecting two objects from N objects, in a particular order, is thus N(N − 1) and that of selecting n objects in a particular order is seen to be N!/(N − n)!. It is divided by the number of permutations, n!, if order does not matter. The binomial coefficient, N!/(n!(N − n)!), is, thus, the number of ways to pick n objects from
and because not even a single object is to be left outside the boxes, implies that the sum made of the terms Na, Nb, Nc, Nd, Ne, ..., Nk must equal N, thus the term (N - Na - Nb - Nc - ... - Nl - Nk)! in the relation above evaluates to 0!. (0!=1) which makes possible to write down that relation as
This is just the multinomial coefficient, the number of ways of arranging N items into k boxes, the i-th box holding Ni items, ignoring the permutation of items in each box.
Now going back to the degeneracy problem which characterizes the reservoir of particles. If the i-th box has a "degeneracy" of
This is the form for W first derived by Boltzmann. Boltzmann's fundamental equation
The Maxwell–Boltzmann distribution follows from this Bose–Einstein distribution for temperatures well above absolute zero, implying that
to write:
Using the fact that
This is essentially a division by N! of Boltzmann's original expression for W, and this correction is referred to as correct Boltzmann counting.
We wish to find the
Finally
In order to maximize the expression above we apply Fermat's theorem (stationary points), according to which local extrema, if exist, must be at critical points (partial derivatives vanish):
By solving the equations above (
Substituting this expression for
or, rearranging:
Boltzmann realized that this is just an expression of the Euler-integrated fundamental equation of thermodynamics. Identifying E as the internal energy, the Euler-integrated fundamental equation states that :
where T is the temperature, P is pressure, V is volume, and μ is the chemical potential. Boltzmann's famous equation
Note that the above formula is sometimes written:
where
Alternatively, we may use the fact that
to obtain the population numbers as
where Z is the partition function defined by:
In an approximation where εi is considered to be a continuous variable, the Thomas-Fermi approximation yields a continuous degeneracy g proportional to
which is just the Maxwell-Boltzmann distribution for the energy.
Derivation from canonical ensemble
In 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, T, for the combined system.
In the present context, our system is assumed to have the energy levels
If our system is in state
Since the entropy of the reservoir
Next we recall the thermodynamic identity (from the first law of thermodynamics):
In a canonical ensemble, there is no exchange of particles, so the
where
which implies, for any state s of the system
where Z is an appropriately chosen "constant" to make total probability 1. (Z is constant provided that the temperature T is invariant.) It is obvious that
where the index s runs through all microstates of the system. Z is sometimes called the Boltzmann sum over states (or "Zustandsumme" in the original German). If we index the summation via the energy eigenvalues instead of all possible states, degeneracy must be taken into account. The probability of our system having energy
where, with obvious modification,
this is the same result as before.
Comments on this derivation: