Bose condensation of finite-lifetime particles with excitons as an example. S.G. Tikhodeev in Zh. Eksp. Teor. Fiz 97:681 (1990)  What the paper says!?

This works revisits the Fröhlich mechanism of condensation for 3D excitons, describing the Bose-Einstein condensation of finite-lifetime, non-interacting particles under steady incoherent pumping, which it extends with a transferred-momentum dependent form of the exciton-phonon interaction. It describes analytically a quantum (T=0) phase transition.

The goal of the present study is to analyze the possibil- ity of forming a Bose condensate in a system of nonequilibri-

um particles with a finite lifetime.

There are many approximations but some basic features can be identified:

many of the rules for Bose condensation in a sys- tem of particles with a finite lifetime have become clear from

this simplest [Fröhlich] model.

Unlike Fröhlich, only T=0 is assumed, but it takes «into account the dependence of the matrix element on the transmitted momentum.» Analytical results are obtained for constant matrix elements (apparently an even greater approximation than Fröhlich, who also include finite lifetime, whose form furthermore gives the Bose-Einstein distribution):

As regards the form of the function W, in actuality the matrix element for phonon interaction usually decreases as the phonon momentum decreases. Therefore W(p,q) does not have the form of Eq. (1.3), and the steady-state distribu-

tion is not Bose-Einstein.

The Author speaks of a weakly nonideal gas, but in the sense of having interactions mediated by phonons: there are no interactions between excitons. This terminology of non-ideal seems wrong to me.

The model reads:

and the Fröhlich/Duffield versions is for a special type of $W$, namely:

The equation are homogenous and therefore if $n_0=0$ at $t=0$, then it remains zero at all times (problem of the seed):

the closer a level is to zero (at which condensation takes place), the longer it takes to establish a stationary popula- tion. If at the initial time the system has no Bose condensate, then the steady-state value of its density is established, for- mally, at infinite time (that is, it diverges in the statistical

limit).

The Author finds a $1/p^2$ divergence (after his Eq. (1.9)) for momentum-dependent interactions and concludes that in this physical case,

Since the function should be integrable for $p\to0$ [...] in one-dimensional and two-dimensional systems Bose con- densation in a system of particles with a finite lifetime is

impossible also for T = 0.

However for $\nu=0$ (W is constant), then condensation is possible in all dimensions (but get destroyed by $T\neq0$).

for pulsed excitation, Bose condensation of particles with a finite life-time formally does not, in general, take place, no matter how high the excitation intensity. In a real experimental situa- tion, it is necessary to choose the length and amplitude of the excitation pulse according to the spectral resolution attaina- ble in studying the distribution function in order to observe

Bose condensation.

This appears to be the exact same text as Ref. .