The quantum principle of wave-particle duality associates a wave with every particle [Dualité Onde CORPUSCULE]. The wavelength of a particle – i.e. the spatial scale on which the wave oscillations take place – increases when the temperature decreases. So in order to see the undulatory behaviour of a particle, its wavelength must be higher than its size, which is only true at a low enough temperature.
The transition from a classical gas to a Bose-Einstein condensate happens when the wavelength associated to a particle becomes equal to the distance between two particles. In that situation, the quantum effects are stronger and the waves of each particle cover one another, with the consequence of making it impossible to differentiate one particle from another. This is when it becomes possible to accumulate a large number of bosons in one single quantum state.
Transition towards a Bose-Einstein condensate. At high temperature, atoms behave as billiard balls (up). When the temperature is slow enough and the wavelength (de Broglie wavelength) bigger than the size of the atom: the quantum effects become stronger. When the length becomes equal to the distance between atoms, the unique wavelength of the condensate appears.
