Bose gas

Bosons are quantum mechanical particles that follow Bose–Einstein statistics, or equivalently, that possess integer spin.

Additionally, some quasiparticles in more complex systems can also be considered bosons like the plasmons (quanta of charge density waves).

The phonon gas, also known as Debye model, is an example where the normal modes of vibration of the crystal lattice of a metal, can be treated as effective massless bosons.

Peter Debye used the phonon gas model to explain the behaviour of heat capacity of metals at low temperature.

The equivalent model for half-integer particles (like electrons or helium-3 atoms), that follow Fermi–Dirac statistics, is called the Fermi gas (an ensemble of non-interacting fermions).

The grand potential for a Bose gas is given by: where each term in the sum corresponds to a particular single-particle energy level εi; gi is the number of states with energy εi; z is the absolute activity (or "fugacity"), which may also be expressed in terms of the chemical potential μ by defining: and β defined as: where kB is the Boltzmann constant and T is the temperature.

: The degeneracy dg may be expressed for many different situations by the general formula: where α is a constant, Ec is a critical energy, and Γ is the gamma function.

For example, for a massive Bose gas in a box, α = 3/2 and the critical energy is given by: where Λ is the thermal wavelength,[clarification needed] and f is a degeneracy factor (f = 1 for simple spinless bosons).

The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy.

The above problem raises the question for α > 1: if a Bose gas with a fixed number of particles is lowered down below the critical temperature, what happens?

[2] Practically however, the aforementioned theoretical flaw is a minor issue, as the most unrealistic assumption is that of non-interaction between bosons.

The interactions significantly change the physics of how a condensate of bosons behaves: the ground state spreads out, the chemical potential saturates to a positive value even at zero temperature, and the fluctuation problem disappears (the compressibility becomes finite).

For smaller, mesoscopic, systems (for example, with only thousands of particles), the ground state term can be more explicitly approximated by adding in an actual discrete level at energy ε=0 in the grand potential: which gives instead N0 = ⁠g0z/1 − z⁠.

Figure 1 shows the results of the solution to this equation for α = 3/2, with k = εc = 1, which corresponds to a gas of bosons in a box.

As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature.

This high variance is due to the choice of using the grand canonical ensemble for the entire system, including the condensate state.

For example, the relationship between internal energy and the product of pressure and volume is the same as that for a classical ideal gas over all temperatures: A similar situation holds for the specific heat at constant volume The entropy is given by: Note that in the limit of high temperature, we have which, for α = 3/2 is simply a restatement of the Sackur–Tetrode equation.

In one dimension bosons with delta interaction behave as fermions, they obey Pauli exclusion principle.

Pressure vs temperature curves of classical and quantum ideal gases ( Fermi gas , Bose gas) in three dimensions. The Bose gas pressure is lower than an equivalent classical gas, especially below the critical temperature (marked with ★) where particles begin moving en masse into the zero-pressure condensed phase.
Figure 1: Various Bose gas parameters as a function of normalized temperature τ . The value of α is 3/2. Solid lines are for N = 10 000 , dotted lines are for N = 1000 . Black lines are the fraction of excited particles, blue are the fraction of condensed particles. The negative of the chemical potential μ is shown in red, and green lines are the values of z . It has been assumed that k = ε c = 1 .