Physical systems typically decay as power laws (τ⁻ᵃ) or exponentials (e⁻ᵇᵗ). These are the two attractors of late-time behavior — algebraic relaxation for systems near criticality, exponential for systems with a gap. The space between them is usually empty.
Superfluid Bjorken flow fills it.
In an expanding superfluid described by Mueller-Israel-Stewart theory with broken U(1) symmetry, the late-time behavior contains factors of the form τ⁻ᵃ ˡⁿ τ — decay that is slower than any exponential but faster than any fixed power law. The exponent itself grows logarithmically with time, producing a mathematical object that lives in the gap between the two standard categories.
The form emerges from a transseries expansion — a generalization of asymptotic series that includes exponentially small corrections organized by their decay rates. The condensate relaxation rate determines whether the approach to equilibrium is monotonic or oscillatory. When the relaxation is slow, the system decays smoothly. When fast, it oscillates while decaying — the condensate overshoots and undershoots on its way to equilibrium, each oscillation damped by the in-between factor.
These oscillations may be observable in heavy-ion collision experiments, where quark-gluon plasma undergoes rapid longitudinal expansion. The mathematical novelty has a potential experimental signature.
The deeper point is structural. The gap between power-law and exponential decay is not empty — it is populated by forms that arise specifically from spontaneous symmetry breaking under expansion. A system that both expands and has a broken symmetry cannot decay in either standard way. The expansion pulls toward power-law; the broken symmetry pushes toward exponential. Their competition produces a hybrid that belongs to neither category.
Some forms exist only in the space between the known ones. The classification of decay has an occupied middle.