In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It 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 B T = N Z g i e − ε i / k B T , {\displaystyle \langle N_{i}\rangle ={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}}}={\frac {N}{Z}}\,g_{i}e^{-\varepsilon _{i}/k_{\text{B}}T},}
where:
ε i {\displaystyle \varepsilon _{i}} is the energy of the ith 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, kB is the Boltzmann constant, T is absolute temperature, N is the total number of particles: N = ∑ i N i {\displaystyle \textstyle N=\sum _{i}N_{i}} , Z is the partition function: Z = ∑ i g i e − ε i / k B T {\displaystyle \textstyle Z=\sum _{i}g_{i}e^{-\varepsilon _{i}/k_{\text{B}}T}} , e is Euler's number Equivalently, the number of particles is sometimes expressed as
⟨ N i ⟩ = 1 e ( ε i − μ ) / k B T = N Z e − ε i / k B T , {\displaystyle \langle N_{i}\rangle ={\frac {1}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}}}={\frac {N}{Z}}\,e^{-\varepsilon _{i}/k_{\text{B}}T},}
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 B T {\textstyle Z=\sum _{i}e^{-\varepsilon _{i}/k_{\text{B}}T}} .
History
Maxwell–Boltzmann statistics grew out of the Maxwell–Boltzmann distribution, most likely as a distillation of the underlying technique. The distribution was first derived by Maxwell in 1860 on heuristic grounds. Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution. The distribution can be derived on the ground that it maximizes the entropy of the system.
Relation with Maxwell–Boltzmann Distribution Maxwell–Boltzmann distribution and Maxwell–Boltzmann statistics are closely related. Maxwell–Boltzmann statistics is a more general principle in statistical mechanics that describes the probability of a classical particle being in a particular energy state:
P i = e − E i / k B T Z {\displaystyle P_{i}={\frac {e^{-E_{i}/k_{\text{B}}T}}{Z}}}
where:
Z {\displaystyle Z} is the partition function: Z = ∑ i e − E i / k B T {\displaystyle \textstyle Z=\sum _{i}e^{-E_{i}/k_{\text{B}}T}} ,
E i {\displaystyle E_{i}} is the energy of state i {\displaystyle i} ,
k B {\displaystyle k_{\text{B}}} is the Boltzmann constant,
T {\displaystyle T} is the absolute temperature. Maxwell–Boltzmann distribution is a specific application of Maxwell–Boltzmann statistics to the kinetic energies of gas particles. The distribution of velocities (or speeds) of particles in an ideal gas follows from the statistical assumption that the energy levels of a gas molecule are given by its kinetic energy:
f ( v ) = ( m 2 π k B T ) 3 / 2 4 π v 2 e − m v 2 2 k B T {\displaystyle f(v)=\left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}4\pi v^{2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}}
where:
f ( v ) {\displaystyle f(v)} is the probability density function of particle speeds,
m {\displaystyle m} is the mass of a particle,
k B {\displaystyle k_{\text{B}}} is the Boltzmann constant,
T {\displaystyle T} is the absolute temperature,
v {\displaystyle v} is the speed of the particle.
Derivation We can deduce the Maxwell–Boltzmann distribution from Maxwell–Boltzmann statistics, starting with the Maxwell–Boltzmann probability for energy states and substituting the kinetic energy E = 1 2 m v 2 {\displaystyle E={\tfrac {1}{2}}mv^{2}} to express the probability in terms of velocity:
P ( E ) = 1 Z exp ( − E k B T ) → P ( v ) = 1 Z exp ( − m v 2 2 k B T ) {\displaystyle {\begin{aligned}P(E)&={\frac {1}{Z}}~\exp \left({\frac {-E}{k_{\text{B}}T}}\right)\\\rightarrow P(v)&={\frac {1}{Z}}~\exp \left({\frac {-mv^{2}}{2k_{\text{B}}T}}\right)\end{aligned}}}
In 3D, this is proportional to the surface area of a sphere, 4 π v 2 {\displaystyle 4\pi v^{2}} . Thus, the probability density function (PDF) for speed v {\displaystyle v} becomes:
f ( v ) = C ⋅ 4 π v 2 exp ( − m v 2 2 k B T ) {\displaystyle f(v)=C\cdot 4\pi v^{2}\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right)}
To find the normalization constant C {\displaystyle C} , we require the integral of the probability density function over all possible speeds to be unity:
∫ 0 ∞ f ( v ) d v = 1 → C ∫ 0 ∞ 4 π v 2 exp ( − m v 2 2 k B T ) d v = 1 {\displaystyle {\begin{aligned}\int _{0}^{\infty }f(v)\,dv&=1\\\rightarrow C\int _{0}^{\infty }4\pi v^{2}\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right)dv&=1\end{aligned}}}
Evaluating the integral using the known result ∫ 0 ∞ v 2 e − a v 2 d v = π 4 a 3 / 2 {\displaystyle \int _{0}^{\infty }v^{2}e^{-av^{2}}dv={\frac {\sqrt {\pi }}{4a^{3/2}}}} , with a = m 2 k B T {\displaystyle a={\frac {m}{2k_{\text{B}}T}}} , we obtain:
C ⋅ 4 π ⋅ π 4 ( m 2 k B T ) 3 / 2 = 1 → C = ( m 2 π k B T ) 3 / 2 {\displaystyle {\begin{aligned}C\cdot 4\pi \cdot {\frac {\sqrt {\pi }}{4\left({\frac {m}{2k_{\text{B}}T}}\right)^{3/2}}}=1\quad \rightarrow C=\left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}\end{aligned}}}
Therefore, the Maxwell–Boltzmann speed distribution is:
f ( v ) = ( m 2 π k B T ) 3 / 2 4 π v 2 exp ( − m v 2 2 k B T ) {\displaystyle f(v)=\left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}4\pi v^{2}\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right)}
Applicability
Maxwell–Boltzmann statistics is used to derive the Maxwell–Boltzmann distribution of an ideal gas. However, it can also be used to extend that distribution to particles with a different energy–momentum relation, such as relativistic particles (resulting in Maxwell–Jüttner distribution), and to other than three-dimensional spaces. Maxwell–Boltzmann statistics is 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 that 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 principally indistinguishable. Once this assumption is made, the particle statistics change. The change in entropy in the entropy of mixing example may be viewed as an example of a non-extensive entropy resulting from the distinguishability of the two types of particles being mixed. Quantum particles are either bosons (following 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.
Derivations Maxwell–Boltzmann statistics can be derived in various statistical mechanical thermodynamic ensembles:
The grand canonical ensemble, exactly. The canonical ensemble, exactly. The microcanonical ensemble, but only in the thermodynamic limit. 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 describes about how many particles in the container have a certain energy. More precisely, the Maxwell–Boltzmann distribution gives the non-normalized probability (this means that the probabilities do not add up to 1) that the state corresponding to a particular energy is occupied. In general, there may be many particles with the same amount of energy ε {\displaystyle \varepsilon } . Let the number of particles with the same energy ε 1 {\displaystyle \varepsilon _{1}} be N 1 {\displaystyle N_{1}} , the number of particles possessing another energy ε 2 {\displaystyle \varepsilon _{2}} be N 2 {\displaystyle N_{2}} , and so forth for all the possible energies { ε i ∣ i = 1 , 2 , 3 , … } {\displaystyle \{\varepsilon _{i}\mid i=1,2,3,\ldots \}} . To describe this situation, we say that N i {\displaystyle N_{i}} is the occupation number of the energy level i . {\displaystyle i.} If we know all the occupation numbers { N i ∣ i = 1 , 2 , 3 , … } {\displaystyle \{N_{i}\mid i=1,2,3,\ldots \}} , then we know the total energy of the system. However, because we can distinguish between which particles are occupying each energy level, the set of occupation numbers { N i ∣ i = 1 , 2 , 3 , … } {\displaystyle \{N_{i}\mid i=1,2,3,\ldots \}} does not completely describe the state of the system. To completely describe the state of the system, or the microstate, we must specify exactly which particles are in each energy level. Thus when we count the number of possible states of the system, we must count each and every microstate, and not just the possible sets of occupation numbers. To begin with, assume that there is only one state at each energy level i {\displaystyle i} (there is no degeneracy). What follows next is a bit of combinatorial thinking which has little to do in accurately describing the reservoir of particles. For instance, let's say there is a total of k {\displaystyle k} boxes labelled a , b , … , k {\displaystyle a,b,\ldots ,k} . With the concept of combination, we could calculate how many ways there are to arrange N {\displaystyle N} into the set of boxes, where the order of balls within each box isn’t tracked. First, we select N a {\displaystyle N_{a}} balls from a total of N {\displaystyle N} balls to place into box a {\displaystyle a} , and continue to select for each box from the remaining balls, ensuring that every ball is placed in one of the boxes. The total number of ways that the balls can be arranged is
W = N ! N a ! ( N − N a ) ! × ( N − N a ) ! N b ! ( N − N a − N b ) ! × ( N − N a − N b ) ! N c ! ( N − N a − N b − N c ) ! × ⋯ × ( N − ⋯ − N ℓ ) ! N k ! ( N − ⋯ − N ℓ − N k ) ! = N ! N a ! N b ! N c ! ⋯ N k ! ( N − N a − ⋯ − N ℓ − N k ) ! {\displaystyle {\begin{aligned}W&={\frac {N!}{N_{a}!{\cancel {(N-N_{a})!}}}}\times {\frac {\cancel {(N-N_{a})!}}{N_{b}!{\cancel {(N-N_{a}-N_{b})!}}}}\times {\frac {\cancel {(N-N_{a}-N_{b})!}}{N_{c}!{\cancel {(N-N_{a}-N_{b}-N_{c})!}}}}\times \cdots \times {\frac {\cancel {(N-\cdots -N_{\ell })!}}{N_{k}!(N-\cdots -N_{\ell }-N_{k})!}}\\[8pt]&={\frac {N!}{N_{a}!N_{b}!N_{c}!\cdots N_{k}!(N-N_{a}-\cdots -N_{\ell }-N_{k})!}}\end{aligned}}}
As every ball has been placed into a box, ( N − N a − N b − ⋯ − N k ) ! = 0 ! = 1 {\displaystyle (N-N_{a}-N_{b}-\cdots -N_{k})!=0!=1} , and we simplify the expression as
W = N ! ∏ ℓ = a , b , … k 1 N ℓ ! {\displaystyle W=N!\prod _{\ell =a,b,\ldots }^{k}{\frac {1}{N_{\ell }!}}}
This is just the multinomial coefficient, the number of ways of arranging N items into k boxes, the lth box holding Nl items, ignoring the permutation of items in each box. Now, consider the case where there is more than one way to put N i {\displaystyle N_{i}} particles in the box i {\displaystyle i} (i.e. taking the degeneracy problem into consideration). If the i {\displaystyle i} th box has a "degeneracy" of g i {\displaystyle g_{i}} , that is, it has g i {\displaystyle g_{i}} "sub-boxes" ( g i {\displaystyle g_{i}} boxes with the same energy ε i {\displaystyle \varepsilon _{i}} . These states/boxes with the same energy are called degenerate states.), such that any way of filling the i {\displaystyle 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 ith box must be increased by the number of ways of distributing the N i {\displaystyle N_{i}} objects in the g i {\displaystyle g_{i}} "sub-boxes". The number of ways of placing N i {\displaystyle N_{i}} distinguishable objects in g i {\displaystyle g_{i}} "sub-boxes" is g i N i {\displaystyle g_{i}^{N_{i}}} (the first object can go into any of the g i {\displaystyle g_{i}} boxes, the second object can also go into any of the g i {\displaystyle g_{i}} boxes, and so on). Thus the number of ways W {\displaystyle W} that a total of N {\displaystyle N} particles can be classified into energy levels according to their energies, while each level i {\displaystyle i} having g i {\displaystyle g_{i}} distinct states such that the ith level accommodates N i {\displaystyle N_{i}} particles is:
W = N ! ∏ i g i