Fermi-Dirac statistics

Fermi statistics

Quantum statistics applied to systems of identical particles with half-integral spin ( $ 1/2, 3/2, 5/2 ,\dots $ in units $ \hbar = 1.05 \times 10 ^ {-} 27 \mathop{\rm erg} \cdot \mathop{\rm sec} $). It was proposed by E. Fermi (1926), and its quantum-mechanical meaning was elucidated by P. Dirac (1926). According to Fermi–Dirac statistics there can be at most one particle in each quantum state (the Pauli principle). For a system of particles subject to Fermi–Dirac statistics, the states are described by wave functions that are anti-symmetric relative to permutations of particles (that is, of their coordinates and spins), while for Bose–Einstein statistics they are symmetric.

The quantum state of an ideal gas is defined by giving the totality of occupation numbers of the levels of the system in the space of momenta $ p $ and spins $ \sigma $, $ \\{ n _ {p \sigma } \\} $, where each $ n _ {p \sigma } $ indicates the number of particles with momentum $ p $ and spin $ \sigma $. In the case of Fermi–Dirac statistics $ n _ {p \sigma } $ can be equal to zero or one.

A gas is a system of a very large number of particles, and therefore its quantum levels are very densely distributed and tend to a continuous spectrum as the volume tends to infinity. It is convenient to group the energy levels in small cells containing $ G _ {i} $ levels in a cell. To each cell corresponds an average energy $ \epsilon _ {i} $, and the number $ G _ {i} $ is assumed to be very large. The quantum-mechanical state of the system is defined by the collection $ \\{ N _ {i} \\} $, where $ N $ is the number of particles in a cell, that is, the sum of the $ n _ {p \sigma } $ for the levels in the cell. The number of different distributions over the cells (that is, the statistical weight of the state of the ideal Fermi–Dirac gas) is equal to

$$ \tag{1 } W \\{ N _ {i} \\} = \ \prod _ { i } \frac{G _ {i} ! }{N _ {i} ! ( G _ {i} - N _ {i} )! } $$

and determines the probability of that distribution of the particles over the cells characterized by the occupation numbers $ N _ {1} , N _ {2} ,\dots $. The statistical weight is calculated by means of combinatorial analysis, taking account of the indistinguishability of the particles and the fact that there can be at most one particle in each state.

The most-probable distribution of the particles over the quantum states, corresponding to a given energy $ E $ and number of particles $ N $,

$$ \tag{2 } E = \sum _ { i } \epsilon _ {i} N _ {i} ,\ \ N = \sum _ { i } N _ {i} , $$

is found from the extremum of the statistical weight (1) under the additional conditions (2). The corresponding average occupation numbers are equal to

$$ \tag{3 } \overline{n}\; _ {i} = \ \frac{\overline{N}\; _ {i} }{G _ {i} } = \ { \frac{1}{e ^ {\beta ( \epsilon _ {i} - \mu ) } + 1 } } , $$

where $ \mu $ is the chemical potential, $ \beta = 1/kT $, $ k $ is the Boltzmann constant (a universal constant, $ k = 1.38 \times 10 ^ {-} 16 \mathop{\rm erg} / \mathop{\rm deg} $), and $ T $ is the absolute temperature. The magnitudes $ \beta $ and $ \mu $ are found from the conditions (2).

The entropy of an ideal Fermi gas is defined as the logarithm of the statistical weight (1) for the most-probable distribution (3)

$$ S = k \mathop{\rm ln} W \\{ \overline{n}\; _ {i} \\} = \ \- k \sum _ { i } G _ {i} \\{ \overline{n}\; _ {i} \mathop{\rm ln} \overline{n}\; _ {i} + ( 1 - \overline{n}\; _ {i} ) \mathop{\rm ln} ( 1 - \overline{n}\; _ {i} ) \\} , $$

where the summation is over all cells. The entropy can be used to calculate the free energy and other thermodynamical functions.

In the case of a non-ideal Fermi gas, the calculation of the thermodynamical functions is a tricky problem, and cannot be reduced to a simple problem in combinatorial analysis. Their calculation is based on the Gibbs method, taking account of Fermi–Dirac statistics. If the Hamilton operator $ H $ of the system is known, then the free energy is equal to

$$ F = - kT \mathop{\rm ln} \ \mathop{\rm Tr} \mathop{\rm exp} \left \\{ - \frac{H }{kT } \right \\} , $$

where the trace operator is taken over states satisfying the requirements of Fermi–Dirac statistics, that is, over anti-symmetric wave functions. This can be achieved if one uses a representation for $ H $ in which its action is defined on the space of wave functions and occupation numbers, that is, if one passes to the second quantization representation.

For references see Bose–Einstein statistics.