Maxwell–Boltzmann statistics
Encyclopedia
In statistical mechanics
, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium
, when the temperature is high enough and density is low enough to render quantum effects negligible.
The expected number of particles with energy
for Maxwell–Boltzmann statistics is 
where:
where:
Equivalently, the distribution is sometimes expressed as
where the index i now specifies a particular state rather than the set of all states with energy
, and 
All of those tiny particles are moving inside that container in all directions with great speed. Because the particles are speeding around, they do possess some energy. The Maxwell–Boltzmann distribution is a mathematical function that speaks about how many particles in the container have a certain energy.
It can be so that many particles have the same amount of energy
. The number of particles with the same energy 
is 
. The number of particles possessing another energy 
is 
. In physical speech this statement is lavishly inflated into something complicated which states that those many particles 
with the same energy amount 
, all occupy a so called "energy level" 
. The concept of energy level is used to graphically/mathematically describe and analyse the properties of particles and events experienced by them. Physicists take into consideration the ways particles arrange themself and thus there is more than one way of occupying an energy level and that's the reason why the particles were tagged like lottery ball, to know the intentions of each one of them.
To begin with, let's ignore the degeneracy problem: assume that there is only one single way to put
particles into energy level 
. What follows next is a bit of combinatorial thinking which has little to do in accurately describing the reservoir of particles.
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)! . The number of ways of selecting 2 objects from N objects without regard to order is N(N − 1) divided by the number of ways 2 objects can be ordered, which is 2! . It can be seen that the number of ways of selecting n objects from 
objects without regard to order is the binomial coefficient: N!/(n!(N − n)!). If we now have a set of boxes labelled a, b, c, d, e, ..., k, then the number of ways of selecting Na objects from a total of N objects and placing them in box a, then selecting Nb objects from the remaining N − Na objects and placing them in box b, then selecting Nc objects from the remaining N − Na − Nb objects and placing them in box c, and continuing until no object is left outside is
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! which makes possible to write down that relation as
Now going back to the degeneracy problem which characterize the reservoir of particles. If the i-th box has a "degeneracy" of
, that is, it has 
"sub-boxes", such that any way of filling the i-th box where the number in the sub-boxes is changed is a distinct way of filling the box, then the number of ways of filling the i-th box must be increased by the number of ways of distributing the 
objects in the 
"sub-boxes". The number of ways of placing 
distinguishable objects in 
"sub-boxes" is 
. Thus the number of ways 
that a total of 
particles can be classified into energy levels according to their energies, while each level 
having 
distinct states such that the i-th level accommodates 
particles is:
This is the form for W first derived by Boltzmann
. Boltzmann's fundamental equation
relates the thermodynamic entropy
S to the number of microstates W, where k is the Boltzmann constant. It was pointed out by Gibbs however, that the above expression for W does not yield an extensive entropy, and is therefore faulty. This problem is known as the Gibbs paradox
The problem is that the particles considered by the above equation are not indistinguishable
. In other words, for two particles (A and B) in two energy sublevels the population represented by [A,B] is considered distinct from the population [B,A] while for indistinguishable particles, they are not. If we carry out the argument for indistinguishable particles, we are led to the Bose-Einstein expression for W:
Both the Maxwell-Boltzmann distribution and the Bose-Einstein distribution are only valid for temperatures well above absolute zero, implying that
. The Maxwell-Boltzmann distribution also requires low density, implying that 
. Under these conditions, we may use Stirling's approximation for the factorial:
to write:
Using the fact that
for 
we can again use Stirlings approximation to write:
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
for which the function 
is maximized, while considering the constraint that there is a fixed number of particles 
and a fixed energy 
in the container. The maxima of 
and 
are achieved by the same values of 
and, since it is easier to accomplish mathematically, we will maximize the latter function instead. We constrain our solution using Lagrange multipliers
forming the function:
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 (
) we arrive to an expression for 
:
Substituting this expression for
into the equation for 
and assuming that 
yields:
or, differentiating and rearranging:
Boltzmann realized that this is just an expression of the second law of thermodynamics
. Identifying dE as the internal energy, the second law of thermodynamics states that for variation only in entropy (S) and particle number (N):
where T is the temperature
and μ is the chemical potential
. Boltzmann's famous equation
is the realization that the entropy is proportional to 
with the constant of proportionality being Boltzmann's constant. It follows immediately that 
and 
so that the populations may now be written:
Note that the above formula is sometimes written:
where
is the absolute activity
.
Alternatively, we may use the fact that
to obtain the population numbers as
where Z is the partition function
defined by:
. 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
with degeneracies 
. As before, we would like to calculate the probability that our system has energy 
.
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 (from the first law of thermodynamics
):
In a canonical ensemble, there is no exchange of particles, so the
term is zero. Similarly, 
This gives
where
and 
denote the energies of the reservoir and the system at 
, respectively. For the second equality we have used the conservation of energy. Substituting into the first equation relating 
:
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
is simply the sum of the probabilities of all corresponding microstates:
where, with obvious modification,
this is the same result as before.
and Fermi–Dirac distributions may be written:
Assuming the minimum value of
is small, it can be seen that the condition under which the Maxwell–Boltzmann distribution is valid is when
For an ideal gas
, we can calculate the chemical potential using the development in the Sackur–Tetrode article to show that:
where
is the total internal energy, 
is the entropy
,
is the volume, and 
is the thermal de Broglie wavelength. The condition for the applicability of the Maxwell–Boltzmann distribution for an ideal gas is again shown to be
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium
Thermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...
, when the temperature is high enough and density is low enough to render quantum effects negligible.
The expected number of particles with energy
for Maxwell–Boltzmann statistics is where:where:
is the number of particles in state i
is the energyEnergyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of the i-th state
is the degeneracyDegenerate energy levelIn physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, and in Quantum Mechanics it is represented mathematically by the Hamiltonian for the system having more...
of energy level i, the number of particle's states (excluding the "free particle" state) with energy
- μ is the chemical potentialChemical potentialChemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
- k is Boltzmann's constant
- T is absolute temperatureTemperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
- N is the total number of particles
-
- Z is the partition functionPartition function (statistical mechanics)Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
- Z is the partition function
- e(...) is the exponential functionExponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
- e(...) is the exponential function
Equivalently, the distribution is sometimes expressed as
where the index i now specifies a particular state rather than the set of all states with energy
, and A derivation of the Maxwell–Boltzmann distribution
Suppose we have a container with a huge number of very small identical particles. Although the particles are identical, we still identify them by drawing numbers on them in the way lottery balls are being labelled with numbers and even colors.All of those tiny particles are moving inside that container in all directions with great speed. Because the particles are speeding around, they do possess some energy. The Maxwell–Boltzmann distribution is a mathematical function that speaks about how many particles in the container have a certain energy.
It can be so that many particles have the same amount of energy
. The number of particles with the same energy is . The number of particles possessing another energy is . In physical speech this statement is lavishly inflated into something complicated which states that those many particles with the same energy amount , all occupy a so called "energy level" . The concept of energy level is used to graphically/mathematically describe and analyse the properties of particles and events experienced by them. Physicists take into consideration the ways particles arrange themself and thus there is more than one way of occupying an energy level and that's the reason why the particles were tagged like lottery ball, to know the intentions of each one of them.To begin with, let's ignore the degeneracy problem: assume that there is only one single way to put
particles into energy level . What follows next is a bit of combinatorial thinking which has little to do in accurately describing the reservoir of particles.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)
objects without regard to order is the binomial coefficient: N!/(n!(N − n)!). If we now have a set of boxes labelled a, b, c, d, e, ..., k, then the number of ways of selecting Na objects from a total of N objects and placing them in box a, then selecting Nb objects from the remaining N − Na objects and placing them in box b, then selecting Nc objects from the remaining N − Na − Nb objects and placing them in box c, and continuing until no object is left outside isand 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! which makes possible to write down that relation as
Now going back to the degeneracy problem which characterize the reservoir of particles. If the i-th box has a "degeneracy" of
, that is, it has "sub-boxes", such that any way of filling the i-th box where the number in the sub-boxes is changed is a distinct way of filling the box, then the number of ways of filling the i-th box must be increased by the number of ways of distributing the objects in the "sub-boxes". The number of ways of placing distinguishable objects in "sub-boxes" is . Thus the number of ways that a total of particles can be classified into energy levels according to their energies, while each level having distinct states such that the i-th level accommodates particles is:This is the form for W first derived by Boltzmann
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
. Boltzmann's fundamental equation
relates the thermodynamic entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
S to the number of microstates W, where k is the Boltzmann constant. It was pointed out by Gibbs however, that the above expression for W does not yield an extensive entropy, and is therefore faulty. This problem is known as the Gibbs paradox
Gibbs paradox
In statistical mechanics, a semi-classical derivation of the entropy that doesn't take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive...
The problem is that the particles considered by the above equation are not indistinguishable
Identical particles
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...
. In other words, for two particles (A and B) in two energy sublevels the population represented by [A,B] is considered distinct from the population [B,A] while for indistinguishable particles, they are not. If we carry out the argument for indistinguishable particles, we are led to the Bose-Einstein expression for W:
Both the Maxwell-Boltzmann distribution and the Bose-Einstein distribution are only valid for temperatures well above absolute zero, implying that
. The Maxwell-Boltzmann distribution also requires low density, implying that . Under these conditions, we may use Stirling's approximation for the factorial:to write:
Using the fact that
for we can again use Stirlings approximation to write: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
for which the function is maximized, while considering the constraint that there is a fixed number of particles and a fixed energy in the container. The maxima of and are achieved by the same values of and, since it is easier to accomplish mathematically, we will maximize the latter function instead. We constrain our solution using Lagrange multipliersLagrange multipliers
In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
forming the function:
Finally
In order to maximize the expression above we apply Fermat's theorem (stationary points)
Fermat's theorem (stationary points)
In mathematics, Fermat's theorem is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point...
, according to which local extrema, if exist, must be at critical points (partial derivatives vanish):
By solving the equations above (
) we arrive to an expression for :Substituting this expression for
into the equation for and assuming that yields:or, differentiating and rearranging:
Boltzmann realized that this is just an expression of the second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
. Identifying dE as the internal energy, the second law of thermodynamics states that for variation only in entropy (S) and particle number (N):
where T is the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
and μ is the chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
. Boltzmann's famous equation
is the realization that the entropy is proportional to with the constant of proportionality being Boltzmann's constant. It follows immediately that and so that the populations may now be written:Note that the above formula is sometimes written:
where
is the absolute activityActivity (chemistry)
In chemical thermodynamics, activity is a measure of the “effective concentration” of a species in a mixture, meaning that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution.By convention, activity...
.
Alternatively, we may use the fact that
to obtain the population numbers as
where Z is the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
defined by:
Another derivation (not as fundamental)
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 ensembleCanonical ensemble
The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
. 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
with degeneracies . As before, we would like to calculate the probability that our system has energy .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 becomesNext we recall the thermodynamic identity (from the first law of thermodynamics
First law of thermodynamics
The first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...
):
In a canonical ensemble, there is no exchange of particles, so the
term is zero. Similarly, This giveswhere
and denote the energies of the reservoir and the system at , respectively. For the second equality we have used the conservation of energy. Substituting into the first equation relating :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
is simply the sum of the probabilities of all corresponding microstates: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
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 Boltzmann 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–EinsteinBose–Einstein statisticsIn statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
statistics. However, there one would replace the canonical ensemble with the grand canonical ensembleGrand canonical ensembleIn statistical mechanics, a grand canonical ensemble is a theoretical collection of model systems put together to mirror the calculated probability distribution of microscopic states of a given physical system which is being maintained in a given macroscopic state...
, 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 applicability
The Bose–EinsteinBose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
and Fermi–Dirac distributions may be written:
Assuming the minimum value of
is small, it can be seen that the condition under which the Maxwell–Boltzmann distribution is valid is whenFor an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
, we can calculate the chemical potential using the development in the Sackur–Tetrode article to show that:
where
is the total internal energy, is the entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
,
is the volume, and is the thermal de Broglie wavelength. The condition for the applicability of the Maxwell–Boltzmann distribution for an ideal gas is again shown to be