Boltzmann distribution

The statistical equilibrium distribution function $ f ( \mathbf p , \mathbf r ) $ of the momenta $ \mathbf p $ and coordinates $ \mathbf r $ for the particles of an ideal gas, the molecules of which obey the laws of classical mechanics, in an external potential field:

$$ \tag{1 } f ( \mathbf p , \mathbf r ) = A \mathop{\rm exp} \ \left \{ - \frac{ \frac{\mathbf p ^ {2} }{2m} +U( \mathbf r ) }{kT} \right \} . $$

Here $ k $ is the Boltzmann constant (a universal constant: $ k = 1.38 $ times $ 10 ^ {-16} erg/degree $), $ T $ is the absolute temperature, $ \mathbf p ^ {2} /2m $ is the kinetic energy of the particle, $ U ( \mathbf r ) $ is the potential energy of the particle in the field, and the constant $ A $ is defined by normalization over a dimensionless phase volume:

$$ \int\limits \int\limits f ( \mathbf p , \mathbf r ) \frac{d ^ {3} \mathbf p d ^ {3} \mathbf r }{h ^ {3} } = N. $$

Here $ N $ is the total number of particles, $ h $ is the Planck constant (a universal constant $ h = 6.62 \times 10 ^ {-27} erg \times sec $),

$$ d ^ {3} \mathbf p = dp _ {x} dp _ {y} dp _ {z} ,\ \ d ^ {3} \mathbf r = dx dy dz . $$

$ A $ can also be defined by the condition of normalization in the space of velocities and coordinates, which is more usual in the kinetic theory of gases:

$$ \int\limits \int\limits f( \mathbf p , \mathbf r ) d ^ {3} \mathbf v d ^ {3} \mathbf r = N, $$

$$ \mathbf v = \frac{\mathbf p }{m} ,\ d ^ {3} \mathbf v = dv _ {x} dv _ {y} dv _ {z} . $$

The Boltzmann distribution is a consequence of the Boltzmann statistics for an ideal gas, and is a particular case of the Gibbs distribution

$$ \rho ( \mathbf p _ {1} , \mathbf r _ {1} \dots \mathbf p _ {N} , \mathbf r _ {N} ) = \ { \frac{1}{Z} } e ^ {- \frac{H( \mathbf p _ {1} , \mathbf r _ {1} \dots \mathbf p _ {N} , \mathbf r _ {N} ) }{kT} } $$

for an ideal gas, when

$$ H = \sum _ { i } \frac{\mathbf p _ {i} ^ {2} }{2m} + \sum _ { i } U ( \mathbf r _ {i} ) , $$

and the canonical Gibbs distribution becomes the product of the Boltzmann distributions for individual particles. The Boltzmann distribution is the limiting case of quantum statistics for an ideal gas at sufficiently high temperatures, when quantum effects can be neglected. The average occupation number of the $ i $- th quantum state of a particle is

$$ \tag{2 } \overline{ {n _ {i} }}\; = \ e ^ {( \mu - \epsilon _ {i} ) / kT } , $$

where $ \epsilon _ {i} $ is the energy corresponding to the $ i $- th quantum state of the particle and $ \mu $ is the chemical potential defined by the condition $ \sum \overline{ {n _ {i} }}\; = N $. Formula (2) is valid for temperatures and densities at which the average distance between the particles is larger than the ratio between the Planck constant $ h $ and the modulus of the average thermal velocity

$$ \left ( \frac{V}{N} \right ) ^ {1/3} \gg \ { \frac{h}{\sqrt {m kT }} } . $$

The Maxwell distribution is a special case of the Boltzmann distribution (1) for $ U = 0 $:

$$ \tag{3 } f( \mathbf p ) = \frac{N}{V} \left ( { \frac{m}{2 \pi kT } } \right ) ^ {3/2} e ^ {- \mathbf p ^ {2} / 2mkT } . $$

The distribution function (1) is sometimes referred to as the Maxwell–Boltzmann distribution, the term Boltzmann distribution being reserved for the distribution function (1) integrated over all momenta of particles representing the density of the number of particles at the point $ \mathbf r $:

$$ \tag{4 } n( \mathbf r ) = n _ {0} e ^ { - {U ( \mathbf r ) } / kT } , $$

where $ n _ {0} $ is the density of the number of particles corresponding to the point at which $ U = 0 $. The relative densities of the number of particles at different points depend on the differences between the potential energies at these points:

$$ \frac{n _ {1} }{n _ {2} } = \ e ^ {- {\Delta U } / kT } , $$

where $ \Delta U = U(r _ {1} ) - U(r _ {2} ) $. A particular case of (4) yields the barometric formula, which defines the particle densities in the gravity field above the surface of the Earth:

$$ \tag{5 } n(z) = n _ {0} e ^ {- mgz / kT } , $$

where $ g $ is the acceleration of gravity, $ m $ is the mass of the particle, $ z $ is the altitude above the Earth's surface, and $ n _ {0} $ is the density at $ z = 0 $.

The Boltzmann distribution of a mixture of several gases with different masses shows that the partial density distributions of the particles for each individual component is independent of that of other components. For a gas in a rotating vessel, $ U( \mathbf r ) $ is the field of the centrifugal forces:

$$ U ( \mathbf r ) = - \frac{m \omega ^ {2} \mathbf r ^ {2} }{2} , $$

where $ \omega $ is the angular velocity of rotation.

For references, see Boltzmann statistics.