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A condensate of pairs in an isolated, homogeneous, unpolarised, finite-sizespin 1/2 Fermi gas at low nonzero temperature T, undergoes with time a phasechange with a random component, due to coupling to the gas thermal phonons.With the quantum second Josephson relation relating the derivative of thecondensate phase operator to the phonon occupation numbers, and linearisedkinetic equations giving the evolution of occupation number fluctuations, weaccess the behaviour of the phase change variance at times much longer than thephonon collision time. The case of a convex phonon branch is similar to theBose gas case: the leading collisional processes are the Beliaev-Landau3-phonons processes, and the variance is the sum of a ballistic term and of adelayed diffusive term, whose analytical expressions are given in thethermodynamic limit. The concave case is more exotic. It is analysed at timescales $T^{-9}$, allowing one to restrict to 2 phonons $\rightarrow$ 2 phononssmall-angle Landau-Khalatnikov processes. The total number of phonons isconserved and the phonon mean occupation numbers at equilibrium can exhibit anegative chemical potential, assumed isotropic. The phase change variance isthen the sum of a ballistic term, of a diffusive term, of exotic subsubleadingterms and of a constant term. The analytic expression of some of thecorresponding coefficients is obtained, as well as the diverging leadingbehavior of the other ones when the phonon chemical potential tends to 0. Whenthis chemical potential is 0, the variance sub-ballistic part becomessuperdiffusive, with an exponent 5/3 and an exactly-known coefficient. For anonzero infinitesimal phonon chemical potential, a law is found, interpolatingbetween superdiffusive and diffusive phase spreading. Also new results areobtained on the phonon Landau-Khalatnikov damping rate, in particular atnegative phonon chemical potential.

Phase spreading and temporal coherence of a pair-condensed Fermi gas at low temperature