In physics, a photon gas is a gas-like collection of

photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...

s, which has many of the same properties of a conventional gas like

hydrogen Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...

neon Neon is a chemical element; it has symbol Ne and atomic number 10. It is the second noble gas in the periodic table. Neon is a colorless, odorless, inert monatomic gas under standard conditions, with approximately two-thirds the density of ...

– including pressure, temperature, and entropy. The most common example of a photon gas in equilibrium is the

black-body radiation Black-body radiation is the thermal radiation, thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific ...

. Photons are part of a family of particles known as

boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...

s, particles that follow Bose–Einstein statistics and with integer spin. A gas of bosons with only one type of particle is uniquely described by three state functions such as the

temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

, and the

number of particles In thermodynamics, the particle number (symbol ) of a thermodynamic system is the number of constituent particles in that system. The particle number is a fundamental thermodynamic property which is conjugate to the chemical potential. Unlike m ...

. However, for a black body, the

energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...

distribution is established by the interaction of the photons with matter, usually the walls of the container, and the number of photons is not conserved. As a result, the

chemical potential In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...

of the black-body photon gas is zero at thermodynamic equilibrium. The number of state variables needed to describe a black-body state is thus reduced from three to two (e.g. temperature and volume).

Thermodynamics of a black body photon gas

In a classical

ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...

with massive particles, the energy of the particles is distributed according to a

Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...

. This distribution is established as the particles collide with each other, exchanging energy (and momentum) in the process. In a photon gas, there will also be an equilibrium distribution, but photons do not collide with each other (except under very extreme conditions, see two-photon physics), so the equilibrium distribution must be established by other means. The most common way that an equilibrium distribution is established is by the interaction of the photons with matter. Translated in See als

* If the photons are absorbed and emitted by the walls of the system containing the photon gas, and the walls are at a particular temperature, then the equilibrium distribution for the photons will be a black-body distribution at that temperature. A very important difference between a generic Bose gas (gas of massive bosons) and a photon gas with a black-body distribution is that the number of photons in the photon gas is not conserved. A photon can be created upon thermal excitation of an atom in the wall into an upper electronic state, followed by the emission of a photon when the atom falls back to a lower energetic state. This type of photon generation is called thermal emission. The reverse process can also take place, resulting in a photon being destroyed and removed from the gas. It can be shown that, as a result of such processes there is no constraint on the number of photons in the system, and the

of the photons must be zero for black-body radiation. The thermodynamics of a black-body photon gas may be derived using quantum statistical mechanical arguments, with the radiation field being in equilibrium with the atoms in the wall. The derivation yields the spectral energy density , which is the energy of the radiation field per unit volume per unit frequency interval, given by: Planck's law Translated in

u(\nu, T) = \frac ~ \frac,

where is the

Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...

, is the speed of light, is the frequency, is the Boltzmann constant, and is temperature. Integrating over frequency and multiplying by the volume, , gives the

internal energy The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...

of a black-body photon gas:

U = \left(\frac\right) V T^4.

The derivation also yields the (expected) number of photons :

N = \left(\frac\right)VT^3,

where

\zeta(n)

is the Riemann zeta function. Note that for a particular temperature, the

particle number In thermodynamics, the particle number (symbol ) of a thermodynamic system is the number of constituent particles in that system. The particle number is a fundamental thermodynamic property which is conjugate to the chemical potential. Unlike m ...

varies with the volume in a fixed manner, adjusting itself to have a constant density of photons. If we note that the equation of state for an ultra-relativistic quantum gas (which inherently describes photons) is given by

U = 3PV,

then we can combine the above formulas to produce an equation of state that looks much like that of an ideal gas:

PV = \frac NkT \approx 0.9\, NkT.

The following table summarizes the thermodynamic state functions for a black-body photon gas. Notice that the pressure can be written in the form

P=b T^4

, which is independent of volume (''b'' is a constant). Within the table,

\hbar

refers to the

reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...

, i.e.,

Isothermal transformations

As an example of a thermodynamic process involving a photon gas, consider a cylinder with a movable piston. The interior walls of the cylinder are "black" in order that the temperature of the photons can be maintained at a particular temperature. This means that the space inside the cylinder will contain a blackbody-distributed photon gas. Unlike a massive gas, this gas will exist without the photons being introduced from the outside – the walls will provide the photons for the gas. Suppose the piston is pushed all the way into the cylinder so that there is an extremely small volume. The photon gas inside the volume will press against the piston, moving it outward, and in order for the transformation to be isothermic, a counter force of almost the same value will have to be applied to the piston so that the motion of the piston is very slow. This force will be equal to the pressure times the cross sectional area () of the piston. This process can be continued at a constant temperature until the photon gas is at a volume . Integrating the force over the distance () traveled yields the total work done to create this photon gas at this volume

W = -\int_0^ P (A \mathrmx) ,

where the relationship has been used. Defining

b = \frac.

The pressure is

P(x) = \frac\,.

Integrating, the work done is just

W = -\frac = -\frac.

The amount of heat that must be added in order to create the gas is

Q = U - W = H_0\,.

where ''H₀'' is the enthalpy at the end of the transformation. It is seen that the enthalpy is the amount of energy needed to create the photon gas.

Photon gases with tunable chemical potential

In low-dimensional systems, for example in dye-solution filled optical microcavities with a distance between the resonator mirrors in the wavelength range where the situation becomes two-dimensional, also photon gases with tunable chemical potential can be realized. Such a photon gas in many respects behaves like a gas of material particles. One consequence of the tunable chemical potential is that at high phase space densities then Bose-Einstein condensation of photons is observed.

Thermodynamics of a black body photon gas

Isothermal transformations

Photon gases with tunable chemical potential

See also

References

Further reading