Degeneracy pressure is the pressure exerted by a dense system of fermions because the Pauli exclusion principle prevents identical particles from occupying the same single-particle quantum state. Unlike the thermal pressure of a classical gas, it remains nonzero at absolute zero. It is central to the structure of white dwarfs and contributes, together with nuclear interactions, to the support of neutron stars against gravitational collapse. (damtp.cam.ac.uk)
Physical origin
Fermions, including electrons and neutrons, obey Fermi–Dirac statistics. At zero temperature, noninteracting fermions fill the lowest available energy states successively rather than all settling into a single state. The highest occupied energy is the Fermi energy. For free particles, filling higher-energy states means occupying states with greater momentum. Consequently, a filled system has substantial particle motion even when thermal excitation is absent. (damtp.cam.ac.uk)
Compression increases this energy. In a box, the allowed single-particle energies rise as the box shrinks; for nonrelativistic particles, they scale inversely with the square of its linear dimension. The work required to compress the occupied states appears macroscopically as pressure. If (U_0) is the ground-state energy of (N) particles in volume (V), then
[ P_0=-\left(\frac{\partial U_0}{\partial V}\right)_N. ]
Thus degeneracy pressure exists even in a model without particle–particle interactions: it is not an additional fundamental repulsive force, but a consequence of the allowed quantum states and their dependence on volume. (paradigms.oregonstate.edu)
When matter becomes degenerate
For a nonrelativistic gas, the characteristic temperature is the Fermi temperature,
[ T_F=\frac{E_F}{k_B}, ]
where (k_B) is the Boltzmann constant. A gas is strongly degenerate when its temperature satisfies (T\ll T_F). “Cold” therefore means cold relative to the Fermi energy, not necessarily cold on an everyday temperature scale. Dense stellar matter can be strongly degenerate at temperatures of millions of kelvin. (damtp.cam.ac.uk)
At finite temperature, the mean occupation of a single-particle state of energy (\epsilon) is
[ f(\epsilon)= \frac{1}{\exp[(\epsilon-\mu)/(k_BT)]+1}, ]
where (\mu) is the chemical potential. At zero temperature this distribution becomes a step: states below (E_F) are occupied and those above it are empty. Warming the gas smooths the boundary, with the main occupation changes occurring near the Fermi energy. (damtp.cam.ac.uk)
There is no sharp phase boundary between a degenerate ideal Fermi gas and a classical gas. Their behavior crosses over continuously as temperature increases relative to (T_F). (arxiv.org)
Equation of state
The simplest quantitative treatment assumes a uniform, three-dimensional gas of noninteracting spin-(\tfrac12) particles, with both spin states equally populated. Write (n=N/V) for particle number density and (\hbar=h/(2\pi)), where (h) is the Planck constant. Counting the occupied states gives the Fermi momentum
[ p_F=\hbar(3\pi^2n)^{1/3}. ]
The resulting equation of state depends on whether particle motion is nonrelativistic or relativistic. (damtp.cam.ac.uk)
Nonrelativistic limit
When (p_F\ll mc), where (m) is the particle mass and (c) the speed of light,
[ E_F=\frac{p_F^2}{2m}. ]
The zero-temperature kinetic energy and pressure are
[ U_0=\frac35NE_F, \qquad P_0=\frac25nE_F =\frac{\hbar^2}{5m}(3\pi^2)^{2/3}n^{5/3}. ]
Hence (P_0\propto n^{5/3}). At equal number density, lighter particles exert greater nonrelativistic degeneracy pressure, because the pressure is inversely proportional to their mass. This explains why electron pressure dominates over the ideal-gas degeneracy contribution of much heavier nuclei in white-dwarf models. (damtp.cam.ac.uk)
Ultrarelativistic limit
When (p_F\gg mc), particle energy is approximately (pc). The pressure becomes
[ P_0=\frac14np_Fc =\frac{\hbar c}{4}(3\pi^2)^{1/3}n^{4/3}. ]
It therefore grows as (n^{4/3}), more slowly with compression than in the nonrelativistic regime. This change is essential to the existence of a limiting mass for white dwarfs. Between the two limits, the equation of state must retain the full relativistic energy–momentum relation. (maths.cam.ac.uk)
Temperature dependence
Degeneracy pressure is often called temperature-independent, but this is exact only for its zero-temperature value. For the three-dimensional nonrelativistic ideal Fermi gas at fixed number density, the low-temperature expansion gives
[ P(n,T)=P_0(n) \left[ 1+\frac{5\pi^2}{12} \left(\frac{T}{T_F}\right)^2 +O!\left(\frac{T}{T_F}\right)^4 \right]. ]
Thermal corrections are consequently small when (T\ll T_F). At sufficiently high temperature and low quantum-state occupancy, the pressure approaches the classical ideal-gas result (P=nk_BT). (arxiv.org)
The finite-temperature pressure is not generally obtained by simply adding (nk_BT) to the zero-temperature degeneracy pressure. Both regimes arise from the same Fermi–Dirac distribution, and the crossover must be calculated consistently. (arxiv.org)
Role in compact stars
White dwarfs
A white dwarf contains a dense population of electrons amid atomic nuclei. The nuclei supply most of its mass, while degenerate electrons supply most of its supporting pressure. This allows the remnant to remain in equilibrium against gravity while cooling, without requiring ongoing nuclear fusion to maintain that support. More massive white dwarfs are generally smaller and denser. (damtp.cam.ac.uk)
As density increases, the electrons become relativistic and the pressure law approaches the softer (n^{4/3}) scaling. For a cold, nonrotating white dwarf with about two nucleons per electron, the idealized Chandrasekhar limit is approximately (1.4) solar masses. It is a limit on the mass of the white-dwarf configuration, not a universal threshold for the initial mass of its progenitor star. (nobelprize.org)
Neutron stars
In a neutron star, neutron degeneracy contributes to the pressure, but treating the star as an ideal neutron gas is inadequate. Interactions among nuclear particles, including those associated with the strong interaction, are important to its equation of state. Its equilibrium structure must also be treated within general relativity. (astro.princeton.edu)
The maximum supported neutron-star mass consequently depends on the dense-matter equation of state rather than on neutron degeneracy pressure alone. Pressure near nuclear density also has a major influence on the stellar radius. (arxiv.org)
Historical development
In 1926, Ralph H. Fowler applied the newly developed quantum statistics of fermions to dense stellar matter, showing how electron degeneracy could explain the support of white dwarfs after substantial energy loss. This replaced an explanation based solely on classical thermal motion. (nobelprize.org)
During the early 1930s, Subrahmanyan Chandrasekhar developed the relativistic theory of white-dwarf structure and its limiting mass. The decisive feature was the transition from the nonrelativistic (5/3) pressure exponent to the ultrarelativistic (4/3) exponent. His work connected microscopic quantum statistics to the possible equilibrium configurations of an entire star. (nobelprize.org)
Scope and limitations
The ideal Fermi-gas formulas provide a baseline, not a complete description of every degenerate material. They assume a specified dimensionality, particle dispersion relation, spin population, and absence of interactions. Electron systems in solids can require a different description of their available states, while neutron-star matter requires nuclear interactions and relativistic stellar structure. (arxiv.org)
Degeneracy also does not make matter incompressible. Pressure remains finite at every finite density in the ideal model and increases continuously under compression. Nor does gravitational instability mean that the Pauli exclusion principle has stopped operating: a configuration can lack sufficient pressure support even while its particles continue to obey fermionic quantum statistics. (paradigms.oregonstate.edu)
References
- Pressure and entropy of a degenerate Fermi gas — Handoutparadigms.oregonstate.edu
- Thermodynamics of the nonrelativistic free-electron Fermi gas in one, two, and three dimensions from the degenerate to the nondegenerate temperature regimearxiv.org
- University of Cambridge: Mathematical Tripos Part II, 2004maths.cam.ac.uk
- Subrahmanyan Chandrasekhar — Nobel Lecturenobelprize.org
- Subrahmanyan Chandrasekhar — Factsnobelprize.org
- Stars in an Exoplanet Worldscience.nasa.gov
- Modelling and measuringasd.gsfc.nasa.gov
- Neutron Starsastro.princeton.edu
- Neutron star equilibrium configurations within a fully relativistic theory with strong, weak, electromagnetic, and gravitational interactionsarxiv.org
- Neutron Star Structure and the Equation of Statearxiv.org