Landau superfluids as nonequilibrium stationary states

Abstract

We define a superfluid state to be a nonequilibrium stationary state (NESS), which, at zero temperature, satisfies certain metastability conditions, which physically express that there should be a sufficiently small energy-momentum transfer between the particles of the fluid and the surroundings (e.g., pipe). It is shown that two models, the Girardeau model and the Huang-Yang-Luttinger (HYL) model describe superfluids in this sense and, moreover, that, in the case of the HYL model, the metastability condition is directly related to Nozières' conjecture that, due to the repulsive interaction, the condensate does not suffer fragmentation into two (or more) parts, thereby assuring its quantum coherence. The models are rigorous examples of NESS in which the system is not finite, but rather a many-body system.

Equilibrium and Non-equilibrium Properties of Superfluids and Superconductors

“…We showed that current-carrying states cannot be thermal equilibrium states (theorem 1). In the case of superfluidity, it was demonstrated, together with results of [Wre15a], that ground states cannot be equilibrium states if a certain condition on the limit points (possibly along subsequences) of the energy-momentum spectrum of finite systems is met (corollary 1), and that in the latter case the systems describe non-equlibrium stationary states (NESS). In the Girardeau-Lieb-Liniger model it was shown that the conditions of corollary 1 are indeed met, and that the energy spectrum of the infinite system is unbounded from below (proposition 1).…”

“…Since, for v satisfying (1.42.1), ǫ 1 ( k i ) + v · k i − k 2 i /2 ≥ 0 for all i = 1, · · · , r, and similarly for the other excitations (excluding the umklapp excitations, which we have shown to be absent), it follows that the restriction of H N,L − E 0 N,L + v · P N,L to R c,d N,L is positive, which is (1.36.2). The fact that the subspace R c,d N,L does not shrink to the empty set in the thermodynamic limit, i.e., (1.36.3), is easily seen to be true for any r ≥ 1, see [Wre15a].…”

“…Moreover, the general basic assumption that the time and space translations define automorphisms of the (weakly closed) quasilocal algebra, and even the existence of the physical Hamiltonian H ω and physical momentum P ω , very natural for any system, have been proved only for the free Bose gas and the BCS model [DS70]. It is therefore better [Wre15a] to base one's discussion on the Green's function for the finite system [BR97], defined as…”

“…In [Wre15a] we defined the following concept (see, e.g., [BB03], for the concept of support of a distribution): emsp ∞ ( v) ≡ The support of the Fourier transform of G v (A * , A; t, x) as a tempered distribution…”

“…Much less understood than the coherence properties, and still a conceptual challenge in several respects is the issue of whether the properties of superfluids and superconductors mentioned above, are equilibrium or nonequilibrium properties [Wre15a]. This issue is also relevant to a third class of phenomena, that of magnetism: all three have in common the fact that they are macroscopic (i.e, due to infinite number of degrees of freedom) manifestations of quantum mechanics.…”