Namespaces
Variants
Actions

Difference between revisions of "Boltzmann statistics"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
Statistics used in a system of non-interacting particles obeying the laws of classical mechanics (a classical ideal gas). The distribution of particles of an ideal gas (since they do not interact) is not considered in the phase space of all particles (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168501.png" />-space) as in Gibbs statistical mechanics (cf. [[Gibbs distribution|Gibbs distribution]]), but in the phase space of coordinates and momenta of one particle (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168502.png" />-space). This is because for an ideal gas the phase volume is preserved in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168503.png" />-space (a special case of one of the [[Liouville theorems|Liouville theorems]]).
+
<!--
 +
b0168501.png
 +
$#A+1 = 39 n = 0
 +
$#C+1 = 39 : ~/encyclopedia/old_files/data/B016/B.0106850 Boltzmann statistics
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
According to the Boltzmann statistics, this phase space is subdivided into a large number of small cells with a phase volume such that each cell contains an even larger number of particles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168504.png" />, and all possible distributions of the particles over these cells are considered. The phase volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168505.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168506.png" />-th cell is its volume in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168507.png" />-space in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168508.png" />-units, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b0168509.png" /> is the Planck constant (a universal constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685010.png" />). The meaning of such a dimensionless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685011.png" /> is the maximum number of possible micro-states in cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685012.png" />, since the smallest value of the product of each pair of coordinates and momenta is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685013.png" /> according to quantum mechanics, and the particle has three degrees of freedom.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
Statistical mechanics is based on the assumption that all microscopic states corresponding to a given total energy and a given number of particles are equally probable. The number of different modes of distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685014.png" /> particles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685015.png" /> cells of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685016.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685017.png" /> particles each is
+
Statistics used in a system of non-interacting particles obeying the laws of classical mechanics (a classical ideal gas). The distribution of particles of an ideal gas (since they do not interact) is not considered in the phase space of all particles (the  $  \Gamma $-
 +
space) as in Gibbs statistical mechanics (cf. [[Gibbs distribution|Gibbs distribution]]), but in the phase space of coordinates and momenta of one particle (the  $  \mu $-
 +
space). This is because for an ideal gas the phase volume is preserved in the  $  \mu $-
 +
space (a special case of one of the [[Liouville theorems|Liouville theorems]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685018.png" /></td> </tr></table>
+
According to the Boltzmann statistics, this phase space is subdivided into a large number of small cells with a phase volume such that each cell contains an even larger number of particles  $  N _ {i} $,
 +
and all possible distributions of the particles over these cells are considered. The phase volume  $  G _ {i} $
 +
of the  $  i $-
 +
th cell is its volume in the  $  \mu $-
 +
space in  $  h  ^ {3} $-
 +
units, where  $  h $
 +
is the Planck constant (a universal constant  $  h = 6.62 \times 10  ^ {-27} erg \times sec $).  
 +
The meaning of such a dimensionless  $  G _ {i} $
 +
is the maximum number of possible micro-states in cell  $  i $,
 +
since the smallest value of the product of each pair of coordinates and momenta is equal to  $  h $
 +
according to quantum mechanics, and the particle has three degrees of freedom.
  
It is assumed that the particles are totally independent, that they are distinguishable and that the state remains unchanged by rearrangements of particles within the same cell. In Boltzmann statistics this magnitude defines the statistical weight, or the thermodynamic probability, of a state (unlike in ordinary probability, it is not normalized to one). In computing the statistical weight it is assumed that a rearrangement of identical particles does not result in a change of state, and for this reason the phase volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685019.png" /> must be divided by a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685020.png" />:
+
Statistical mechanics is based on the assumption that all microscopic states corresponding to a given total energy and a given number of particles are equally probable. The number of different modes of distribution of $  N $
 +
particles over  $  M $
 +
cells of size  $  G _ {i} $
 +
containing  $  N _ {i} $
 +
particles each is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685021.png" /></td> </tr></table>
+
$$
 +
W _ {B} ( \dots N \dots )  = N!
 +
\prod _ {1 \leq  i \leq  M }
 +
 
 +
\frac{G _ {i} ^ {N _ {i} } }{N _ {i} ! }
 +
,\ \
 +
= \sum _ { i } N _ {i} .
 +
$$
 +
 
 +
It is assumed that the particles are totally independent, that they are distinguishable and that the state remains unchanged by rearrangements of particles within the same cell. In Boltzmann statistics this magnitude defines the statistical weight, or the thermodynamic probability, of a state (unlike in ordinary probability, it is not normalized to one). In computing the statistical weight it is assumed that a rearrangement of identical particles does not result in a change of state, and for this reason the phase volume  $  W _ {B} $
 +
must be divided by a factor  $  N ! $:
 +
 
 +
$$
 +
W ( \dots N \dots )  = \
 +
 
 +
\frac{W _ {B} }{N ! }
 +
.
 +
$$
  
 
Phases with volume reduction as above are said to be generic phases (as distinct from the original specific phases).
 
Phases with volume reduction as above are said to be generic phases (as distinct from the original specific phases).
Line 15: Line 57:
 
All microscopic states with different particle distributions over phase cells when the number of particles
 
All microscopic states with different particle distributions over phase cells when the number of particles
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685022.png" /></td> </tr></table>
+
$$
 +
= \sum _ { i=1 } ^ { M }  N _ {i}  $$
  
 
and the total energy
 
and the total energy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685023.png" /></td> </tr></table>
+
$$
 +
= \sum _ { i=1 } ^ { M }
 +
\epsilon _ {i} N _ {i}  $$
 +
 
 +
( $  \epsilon _ {i} $
 +
is the energy of the particles in the  $  i $-
 +
th cell) are given, correspond to the same macroscopic state.
 +
 
 +
It is assumed that the distribution of the particles in a state of statistical equilibrium is the most-probable distribution, i.e. corresponds to the maximum  $  W( \dots N _ {i} \dots ) $
 +
for a given number of particles  $  N $
 +
and energy  $  E $.  
 +
The problem of an arbitrary extremum  $  W ( \dots N _ {i} \dots ) $
 +
for given  $  N $
 +
and  $  E $
 +
yields the following [[Boltzmann distribution|Boltzmann distribution]] for the average number of particles in a cell:
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685024.png" /> is the energy of the particles in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685025.png" />-th cell) are given, correspond to the same macroscopic state.
+
$$
 +
\overline{ {n _ {i} }}\;  = \
  
It is assumed that the distribution of the particles in a state of statistical equilibrium is the most-probable distribution, i.e. corresponds to the maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685026.png" /> for a given number of particles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685027.png" /> and energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685028.png" />. The problem of an arbitrary extremum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685029.png" /> for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685031.png" /> yields the following [[Boltzmann distribution|Boltzmann distribution]] for the average number of particles in a cell:
+
\frac{\overline{ {N _ {i} }}\; }{G _ {i} }
 +
  = \
 +
e ^ {( \mu - \epsilon _ {i} )/kT } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685032.png" /></td> </tr></table>
+
where  $  k $
 +
is the Boltzmann constant (a universal constant  $  k = 1.38 \times 10  ^ {-16} erg/degree $),
 +
$  T $
 +
is the absolute temperature, and  $  \mu $
 +
is the chemical potential, defined by the condition  $  N = \sum _ {i} {N _ {i} } $.
 +
In the particular case of a potential field  $  U( \mathbf r ) $:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685033.png" /> is the Boltzmann constant (a universal constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685034.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685035.png" /> is the absolute temperature, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685036.png" /> is the chemical potential, defined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685037.png" />. In the particular case of a potential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685038.png" />:
+
$$
 +
\epsilon _ {i}  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016850/b01685039.png" /></td> </tr></table>
+
\frac{p _ {i}  ^ {2} }{2m}
 +
+
 +
U( \mathbf r _ {i} ) .
 +
$$
  
 
Boltzmann statistics is a special case of Gibbs statistics — the canonical ensemble for a gas consisting of non-interacting particles. The Boltzmann statistics is the limit case of [[Fermi–Dirac statistics|Fermi–Dirac statistics]] and [[Bose–Einstein statistics|Bose–Einstein statistics]] at sufficiently-high temperatures, when quantum effects can be neglected.
 
Boltzmann statistics is a special case of Gibbs statistics — the canonical ensemble for a gas consisting of non-interacting particles. The Boltzmann statistics is the limit case of [[Fermi–Dirac statistics|Fermi–Dirac statistics]] and [[Bose–Einstein statistics|Bose–Einstein statistics]] at sufficiently-high temperatures, when quantum effects can be neglected.

Latest revision as of 10:59, 29 May 2020


Statistics used in a system of non-interacting particles obeying the laws of classical mechanics (a classical ideal gas). The distribution of particles of an ideal gas (since they do not interact) is not considered in the phase space of all particles (the $ \Gamma $- space) as in Gibbs statistical mechanics (cf. Gibbs distribution), but in the phase space of coordinates and momenta of one particle (the $ \mu $- space). This is because for an ideal gas the phase volume is preserved in the $ \mu $- space (a special case of one of the Liouville theorems).

According to the Boltzmann statistics, this phase space is subdivided into a large number of small cells with a phase volume such that each cell contains an even larger number of particles $ N _ {i} $, and all possible distributions of the particles over these cells are considered. The phase volume $ G _ {i} $ of the $ i $- th cell is its volume in the $ \mu $- space in $ h ^ {3} $- units, where $ h $ is the Planck constant (a universal constant $ h = 6.62 \times 10 ^ {-27} erg \times sec $). The meaning of such a dimensionless $ G _ {i} $ is the maximum number of possible micro-states in cell $ i $, since the smallest value of the product of each pair of coordinates and momenta is equal to $ h $ according to quantum mechanics, and the particle has three degrees of freedom.

Statistical mechanics is based on the assumption that all microscopic states corresponding to a given total energy and a given number of particles are equally probable. The number of different modes of distribution of $ N $ particles over $ M $ cells of size $ G _ {i} $ containing $ N _ {i} $ particles each is

$$ W _ {B} ( \dots N \dots ) = N! \prod _ {1 \leq i \leq M } \frac{G _ {i} ^ {N _ {i} } }{N _ {i} ! } ,\ \ N = \sum _ { i } N _ {i} . $$

It is assumed that the particles are totally independent, that they are distinguishable and that the state remains unchanged by rearrangements of particles within the same cell. In Boltzmann statistics this magnitude defines the statistical weight, or the thermodynamic probability, of a state (unlike in ordinary probability, it is not normalized to one). In computing the statistical weight it is assumed that a rearrangement of identical particles does not result in a change of state, and for this reason the phase volume $ W _ {B} $ must be divided by a factor $ N ! $:

$$ W ( \dots N \dots ) = \ \frac{W _ {B} }{N ! } . $$

Phases with volume reduction as above are said to be generic phases (as distinct from the original specific phases).

All microscopic states with different particle distributions over phase cells when the number of particles

$$ N = \sum _ { i=1 } ^ { M } N _ {i} $$

and the total energy

$$ E = \sum _ { i=1 } ^ { M } \epsilon _ {i} N _ {i} $$

( $ \epsilon _ {i} $ is the energy of the particles in the $ i $- th cell) are given, correspond to the same macroscopic state.

It is assumed that the distribution of the particles in a state of statistical equilibrium is the most-probable distribution, i.e. corresponds to the maximum $ W( \dots N _ {i} \dots ) $ for a given number of particles $ N $ and energy $ E $. The problem of an arbitrary extremum $ W ( \dots N _ {i} \dots ) $ for given $ N $ and $ E $ yields the following Boltzmann distribution for the average number of particles in a cell:

$$ \overline{ {n _ {i} }}\; = \ \frac{\overline{ {N _ {i} }}\; }{G _ {i} } = \ e ^ {( \mu - \epsilon _ {i} )/kT } , $$

where $ k $ is the Boltzmann constant (a universal constant $ k = 1.38 \times 10 ^ {-16} erg/degree $), $ T $ is the absolute temperature, and $ \mu $ is the chemical potential, defined by the condition $ N = \sum _ {i} {N _ {i} } $. In the particular case of a potential field $ U( \mathbf r ) $:

$$ \epsilon _ {i} = \ \frac{p _ {i} ^ {2} }{2m} + U( \mathbf r _ {i} ) . $$

Boltzmann statistics is a special case of Gibbs statistics — the canonical ensemble for a gas consisting of non-interacting particles. The Boltzmann statistics is the limit case of Fermi–Dirac statistics and Bose–Einstein statistics at sufficiently-high temperatures, when quantum effects can be neglected.

The Boltzmann statistics was proposed by L. Boltzmann in 1868–1871.

References

[1] J.E. Mayer, M. Goeppert-Mayer, "Statistical mechanics" , Wiley (1940)
[2] A. Sommerfeld, "Thermodynamics and statistical mechanics" , Acad. Press (1956) (Translated from German)
[3] E. Schrödinger, "Statistical thermodynamics" , Cambridge Univ. Press (1948)
[4] R. Fowler, E. Guggenheim, "Statistical thermodynamics" , Cambridge Univ. Press (1960)
How to Cite This Entry:
Boltzmann statistics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boltzmann_statistics&oldid=11557
This article was adapted from an original article by D.N. Zubarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article