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In  classical [[statistical physics]], the '''Boltzmann distribution'''  expresses the  relative probability  that a subsystem  of a physical system has a certain [[energy]]. The subsystem must part of a physical system that is in thermal equilibrium, that is, the system must have a well-defined (absolute) [[temperature]]. For instance, a subsystem can be a single [[molecule]] in,  say, one [[mole]] of an [[ideal gas law|ideal gas]]. Then the Boltzmann distribution applies to the energies of the individual gas molecules, provided the  ideal gas is in thermal equilibrium.     
In  classical [[statistical physics]], the '''Boltzmann distribution'''  expresses the  relative probability  that a subsystem  of a physical system has a certain [[energy]]. The subsystem must part of a physical system that is in thermal equilibrium, that is, the system must have a well-defined (absolute) [[temperature]]. For instance, a subsystem can be a single [[molecule]] in,  say, one [[mole]] of an [[ideal gas law|ideal gas]]. Then the Boltzmann distribution applies to the energies of the individual gas molecules, provided the  ideal gas is in thermal equilibrium.     


The Boltzmann distribution (also known as the '''Maxwell-Boltzmann distribution''') was  proposed in 1859 by the Scotsman [[James Clerk Maxwell]] for the statistical distribution of the [[kinetic energy|kinetic energies]] of ideal gas molecules.  Consider an ideal gas of absolute temperature ''T''. Let ''n''<sub>1</sub> be the number of molecules with kinetic energy ''E''<sub>1</sub> and  ''n''<sub>2</sub> be the number  with kinetic energy ''E''<sub>2</sub>, then according to the Maxwell-Boltzmann distribution law the relative probability is the ratio,
The Boltzmann distribution (also known as the '''Maxwell-Boltzmann distribution''') was  proposed in 1859 by the Scotsman [[James Clerk Maxwell]] for the statistical distribution of the [[kinetic energy|kinetic energies]] of ideal gas molecules.  Consider an ideal gas of absolute temperature ''T'' in a vessel with volume ''V''. Let ''n''<sub>1</sub> be the number of molecules in ''V'' with kinetic energy ''E''<sub>1</sub> and  ''n''<sub>2</sub> be the number in ''V'' with kinetic energy ''E''<sub>2</sub>, then according to the Maxwell-Boltzmann distribution law the relative probability is the ratio,
:<math>
:<math>
\frac{n_1}{n_2} = \frac{e^{-E_1/(kT)}}{e^{-E_2/(kT)}},  \qquad\qquad \qquad\qquad (1)
\frac{n_1}{n_2} = \frac{e^{-E_1/(kT)}}{e^{-E_2/(kT)}},  \qquad\qquad \qquad\qquad (1)
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where ''k'' is the [[Boltzmann constant]]. Most noticeable in this expression are (i) the energy in an exponential, (ii) the inverse temperature in the exponent, and (iii) the appearance of the natural constant ''k''.  Note that an argument of an exponential must be dimensionless and that accordingly ''kT'' has the dimension energy.
where ''k'' is the [[Boltzmann constant]]. Most noticeable in this expression are (i) the energy in an exponential, (ii) the inverse temperature in the exponent, and (iii) the appearance of the natural constant ''k''.  Note that an argument of an exponential must be dimensionless and that accordingly ''kT'' has the dimension energy.


As discovered in 1871 by the Austrian [[Ludwig Boltzmann]], the molecular energies in equation (1) may, in addition to the translational energy considered by Maxwell, contain rotational and vibrational energies of the molecules. Also  interactions with an external field may be included. If a system, as for instance a column of air of constant temperature (zero [[lapse rate]]), is in the [[gravitation|gravitational field]] of the Earth, each molecular energy contains the additional term ''mgh'', where ''m'' is the molecular mass, ''g'' the [[gravitational  acceleration]] and ''h'' the height of the molecule above the surface of the Earth. Thus, the ratio  of the number of molecules at height ''h''<sub>1</sub> and  ''h''<sub>2</sub> is,
As discovered in 1871 by the Austrian [[Ludwig Boltzmann]], the molecular energies in equation (1) may, in addition to the translational energy considered by Maxwell, contain rotational and vibrational energies of the molecules. Also  interactions with an external field may be included. If a system, as for instance a column of air of constant temperature (zero [[atmospheric lapse rate]]), is in the [[gravitation|gravitational field]] of the Earth, each molecular energy contains the additional term ''mgh'', where ''m'' is the molecular mass, ''g'' the [[gravitational  acceleration]] and ''h'' the height of the molecule above the surface of the Earth. Thus, the ratio  of the number density of molecules at height ''h''<sub>1</sub> and  ''h''<sub>2</sub> with velocity ''v''<sub>1</sub> and ''v''<sub>2</sub>, respectively, is given by the following Maxwell-Boltzmann distribution function,
:<math>
:<math>
\frac{n(h_1)}{n(h_2)} = \frac{e^{-(E_1 +mgh_1)/(kT)}}{e^{-(E_2 +mgh_2)/(kT)}}  
\frac{n(h_1, v_1)}{n(h_2, v_1)} = \frac{e^{-(E_1 +mgh_1)/(kT)}}{e^{-(E_2 +mgh_2)/(kT)}} .  
\approx \exp\Big[ - \frac{mg(h_1-h_2)}{kT}\Big] ,
</math>
</math>
where we made the assumption that the difference in molecular  kinetic energies at the two heights is small in comparison  to the difference in potential energies:  
The kinetic energies ''E''<sub>1</sub> and ''E''<sub>2</sub> are quadratic functions of the velocities. When we integrate the left- and right-hand side over all velocities in numerator and denominator we get on the right-hand side the same (finite)  integrals in numerator and denominator that cancel each other. What remains is the ''[[barometer formula]]''<ref>D. ter Haar, ''Elements of Statistical Mechanics'', Holt, Rinehart and Winston, New York (1961), p. 20</ref>
:''E''<sub>2</sub> &minus; ''E''<sub>1</sub> << ''mg''(''h''<sub>2</sub>&minus;''h''<sub>1</sub>),
:<math>
so that the difference in kinetic energies may be omitted.
\frac{n(h_1)}{n(h_2)} = \exp\Big[ - \frac{mg(h_1-h_2)}{kT}\Big],
</math>
where ''n''(''h''<sub>1</sub>) is  the total (integrated over all velocities) number density of molecules at height ''h''<sub>1</sub> and ''n''(''h''<sub>2</sub>) is the same at ''h''<sub>2</sub>.
 
==Generalization==
==Generalization==
A few years after Boltzmann,  the American [[Josiah Willard Gibbs]]  gave a further formalization and generalization (ca. 1877).  Gibbs introduced what he called an ''ensemble'', a "supersystem"  consisting of a statistically large number of identical systems.  For instance, the systems may  be identical vessels containing the same number of the same real gas  molecules at the same temperature and pressure. Further Gibbs assumed that the ensemble, just like a system of gas molecules,  is in thermal equilibrium, i.e., the systems are in thermal contact so that they can exchange [[heat]]. In the example that the systems are gas-filled vessels, this is achieved by requiring that the vessels are in mutual contact through heat-conducting walls.   
A few years after Boltzmann,  the American [[Josiah Willard Gibbs]]  gave a further formalization and generalization (ca. 1877).  Gibbs introduced what he called an ''ensemble'', a "supersystem"  consisting of a statistically large number of identical systems.  For instance, the systems may  be identical vessels containing the same number of the same real gas  molecules at the same temperature and pressure. Further Gibbs assumed that the ensemble, just like a system of gas molecules,  is in thermal equilibrium, i.e., the systems are in thermal contact so that they can exchange [[heat]]. In the example that the systems are gas-filled vessels, this is achieved by requiring that the vessels are in mutual contact through heat-conducting walls.   


Gibbs  assumed<ref>To get the history correct: Gibbs made a few very fundamental assumptions from which he ''derived'' the equivalence of equation (1).</ref>  that the Maxwell-Boltzmann law, equation (1), holds for the energies of the systems in the ensemble.  This generalization, applied to the example of an ensemble consisting of gas-filled vessels, means that the energy of a  gas molecule in equation (1) is replaced by the total energy  of the  molecules in a  vessel;  a "one-molecule" energy is promoted to a "one-vessel" energy.  Because of molecular interactions (that are absent in an ideal gas), a one-vessel energy  cannot be written as a sum of one-molecule energies. This is the main reason for Gibbs' generalization from a single vessel of gas to an ensemble of vessels, or, in more general terms, the generalization from an ideal gas to an ensemble of (rather arbitrary) systems. Just as the molecules in an ideal gas, the systems in the ensemble do not interact other than by exchanging heat.
Gibbs  assumed<ref>In fact, Gibbs made a few assumptions, more fundamental than equation (1), from which he ''derived'' the equivalent of equation (1).</ref>  that the Maxwell-Boltzmann law, equation (1), holds for the energies of the systems in the ensemble.  This generalization, applied to the example of an ensemble consisting of gas-filled vessels, means that the energy of a  gas molecule in equation (1) is replaced by the total energy  of the  molecules in a  vessel;  a "one-molecule" energy is promoted to a "one-vessel" energy.  Because of molecular interactions (that are absent in an ideal gas), a one-vessel energy  cannot be written as a sum of one-molecule energies. This is the main reason for Gibbs' generalization from a single vessel of gas to an ensemble of vessels, or, in more general terms, the generalization from an ideal gas to an ensemble of (rather arbitrary) systems. Just as the molecules in an ideal gas, the systems in the ensemble do not interact other than by exchanging heat.
   
   
The absolute probability for a system to have  total energy ''&epsilon;''<sub>''j''</sub> can be obtained from equation (1) by normalizing. Let us assume, for convenience sake, that the one-system energies are discrete (as they often are in [[quantum mechanics]]) running from &epsilon;<sub>0</sub> to &epsilon;<sub>&infin;</sub> , then
The absolute probability for a system to have  total energy ''&epsilon;''<sub>''j''</sub> can be obtained from equation (1) by normalizing. Let us assume, for convenience sake, that the one-system energies are discrete (as they often are in [[quantum mechanics]]) running from &epsilon;<sub>0</sub> to &epsilon;<sub>&infin;</sub> , then
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</math>
</math>
where ''N''<sub>''j''</sub> is the number of systems that have energy ''&epsilon;''<sub>''j''</sub> and ''N'' is the total number of systems in the ensemble (''N'' must be very large&mdash;go to infinity&mdash;for statistics to apply).   
where ''N''<sub>''j''</sub> is the number of systems that have energy ''&epsilon;''<sub>''j''</sub> and ''N'' is the total number of systems in the ensemble (''N'' must be very large&mdash;go to infinity&mdash;for statistics to apply).   
The ratio is a probability  written as
The ratio is a Boltzmann probability  written as
:<math>
:<math>
\frac{N_j}{N} \equiv \mathcal{P}(\epsilon_j) = \frac{e^{-\epsilon_j/(kT)}}{Q}.  
\frac{N_j}{N} \equiv \mathcal{P}(\epsilon_j) = \frac{e^{-\epsilon_j/(kT)}}{Q}.  
</math>
</math>
The quantity ''Q'' is known as the ''[[Partition function (statistical physics) |partition function]]''  of the system. It is a sum over all energies that the system can have. In the older literature ''Q'' is called ''Zustandssumme'', which is German for "sum over states", and is often denoted by ''Z''.  When we consider again the example of a vessel  with  interacting molecules at certain pressure and temperature, the sum in the partition function is over the possible total  energies &epsilon;<sub>i</sub> of the molecules in the vessel.  
The quantity ''Q'' is known as the ''[[Partition function (statistical physics) |partition function]]''  of the system. It is a sum over all energies that the system can have. In the older literature ''Q'' is called ''Zustandssumme'', which is German for "sum over states", and is often denoted by ''Z''.  When we consider again the example of a vessel  with  interacting molecules at certain pressure and temperature, the sum in the partition function is over the possible total  energies &epsilon;<sub>i</sub> of all the molecules in the vessel.  


In classical statistical physics, where energies are not discrete, the partition function of a system of ''n'' molecules is not a sum over the possible energies of the system, but an integral over the 6<sup>''n''</sup>-dimensional [[phase space]] (space of momenta and positions). In the framework of the "old quantum theory"  it was  discovered in 1912  that the classical partition function of ''n'' molecules  must multiplied by a quantum factor. The classical partition function is,
In classical statistical physics, where energies are not discrete, the partition function of a system of ''n'' molecules is not a sum over the possible energies of the system, but an integral over the 6''n''-dimensional [[phase space]] (space of momenta and positions). In the framework of the "old quantum theory"  it was  discovered in 1912  that the classical partition function of ''n'' molecules  must multiplied by a quantum factor. The classical partition function is,
:<math>
:<math>
Q_\mathrm{class} = \frac{1}{n! h^{3n}} \int e^{-\epsilon(p,q)/(kT)}  
Q_\mathrm{class} = \frac{1}{n! h^{3n}} \int e^{-\epsilon(p,q)/(kT)}  
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where ''h'' is [[Planck's constant]] and ''n''! = 1 &times; 2&times; ... &times; ''n'' (the [[factorial]] of ''n'').
where ''h'' is [[Planck's constant]] and ''n''! = 1 &times; 2&times; ... &times; ''n'' (the [[factorial]] of ''n'').


As a final remark: in quantum statistical thermodynamics the Boltzmann distribution appears as the high-temperature limit of both [[Bose-Einstein statistics]] (valid for [[boson]]s) and [[Fermi-Dirac statistics]] (valid for [[fermions]]).  
As a final remark: in quantum statistical thermodynamics the Boltzmann distribution appears as the high-temperature limit of both [[Bose-Einstein statistics]] (valid for [[boson]]s) and [[Fermi function|Fermi-Dirac statistics]] (valid for [[fermions]]).


==Note==
==Note==
<references />
<references />

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In classical statistical physics, the Boltzmann distribution expresses the relative probability that a subsystem of a physical system has a certain energy. The subsystem must part of a physical system that is in thermal equilibrium, that is, the system must have a well-defined (absolute) temperature. For instance, a subsystem can be a single molecule in, say, one mole of an ideal gas. Then the Boltzmann distribution applies to the energies of the individual gas molecules, provided the ideal gas is in thermal equilibrium.

The Boltzmann distribution (also known as the Maxwell-Boltzmann distribution) was proposed in 1859 by the Scotsman James Clerk Maxwell for the statistical distribution of the kinetic energies of ideal gas molecules. Consider an ideal gas of absolute temperature T in a vessel with volume V. Let n1 be the number of molecules in V with kinetic energy E1 and n2 be the number in V with kinetic energy E2, then according to the Maxwell-Boltzmann distribution law the relative probability is the ratio,

where k is the Boltzmann constant. Most noticeable in this expression are (i) the energy in an exponential, (ii) the inverse temperature in the exponent, and (iii) the appearance of the natural constant k. Note that an argument of an exponential must be dimensionless and that accordingly kT has the dimension energy.

As discovered in 1871 by the Austrian Ludwig Boltzmann, the molecular energies in equation (1) may, in addition to the translational energy considered by Maxwell, contain rotational and vibrational energies of the molecules. Also interactions with an external field may be included. If a system, as for instance a column of air of constant temperature (zero atmospheric lapse rate), is in the gravitational field of the Earth, each molecular energy contains the additional term mgh, where m is the molecular mass, g the gravitational acceleration and h the height of the molecule above the surface of the Earth. Thus, the ratio of the number density of molecules at height h1 and h2 with velocity v1 and v2, respectively, is given by the following Maxwell-Boltzmann distribution function,

The kinetic energies E1 and E2 are quadratic functions of the velocities. When we integrate the left- and right-hand side over all velocities in numerator and denominator we get on the right-hand side the same (finite) integrals in numerator and denominator that cancel each other. What remains is the barometer formula[1]

where n(h1) is the total (integrated over all velocities) number density of molecules at height h1 and n(h2) is the same at h2.

Generalization

A few years after Boltzmann, the American Josiah Willard Gibbs gave a further formalization and generalization (ca. 1877). Gibbs introduced what he called an ensemble, a "supersystem" consisting of a statistically large number of identical systems. For instance, the systems may be identical vessels containing the same number of the same real gas molecules at the same temperature and pressure. Further Gibbs assumed that the ensemble, just like a system of gas molecules, is in thermal equilibrium, i.e., the systems are in thermal contact so that they can exchange heat. In the example that the systems are gas-filled vessels, this is achieved by requiring that the vessels are in mutual contact through heat-conducting walls.

Gibbs assumed[2] that the Maxwell-Boltzmann law, equation (1), holds for the energies of the systems in the ensemble. This generalization, applied to the example of an ensemble consisting of gas-filled vessels, means that the energy of a gas molecule in equation (1) is replaced by the total energy of the molecules in a vessel; a "one-molecule" energy is promoted to a "one-vessel" energy. Because of molecular interactions (that are absent in an ideal gas), a one-vessel energy cannot be written as a sum of one-molecule energies. This is the main reason for Gibbs' generalization from a single vessel of gas to an ensemble of vessels, or, in more general terms, the generalization from an ideal gas to an ensemble of (rather arbitrary) systems. Just as the molecules in an ideal gas, the systems in the ensemble do not interact other than by exchanging heat.

The absolute probability for a system to have total energy εj can be obtained from equation (1) by normalizing. Let us assume, for convenience sake, that the one-system energies are discrete (as they often are in quantum mechanics) running from ε0 to ε , then

which gives

where Nj is the number of systems that have energy εj and N is the total number of systems in the ensemble (N must be very large—go to infinity—for statistics to apply). The ratio is a Boltzmann probability written as

The quantity Q is known as the partition function of the system. It is a sum over all energies that the system can have. In the older literature Q is called Zustandssumme, which is German for "sum over states", and is often denoted by Z. When we consider again the example of a vessel with interacting molecules at certain pressure and temperature, the sum in the partition function is over the possible total energies εi of all the molecules in the vessel.

In classical statistical physics, where energies are not discrete, the partition function of a system of n molecules is not a sum over the possible energies of the system, but an integral over the 6n-dimensional phase space (space of momenta and positions). In the framework of the "old quantum theory" it was discovered in 1912 that the classical partition function of n molecules must multiplied by a quantum factor. The classical partition function is,

where h is Planck's constant and n! = 1 × 2× ... × n (the factorial of n).

As a final remark: in quantum statistical thermodynamics the Boltzmann distribution appears as the high-temperature limit of both Bose-Einstein statistics (valid for bosons) and Fermi-Dirac statistics (valid for fermions).

Note

  1. D. ter Haar, Elements of Statistical Mechanics, Holt, Rinehart and Winston, New York (1961), p. 20
  2. In fact, Gibbs made a few assumptions, more fundamental than equation (1), from which he derived the equivalent of equation (1).