Conditional ensemble averages and ground-state fluctuations in few-body ideal Bose gases

Abstract

Abstract The grand canonical catastrophe—characterized by order-unity relative fluctuations of the ground-state occupation in the grand canonical ensemble (GCE) at low temperatures—is a hallmark of ensemble inequivalence in ideal Bose gases and has been observed in photon Bose–Einstein condensates. In this work, we investigate an alternative class of observables defined via conditional ensemble averages , namely ⟨ n 0 / N ⟩ and ⟨ δ n 0 2 / N 2 ⟩ , where the average excludes the empty-system sector ( N = 0 ). Using the exactly solvable model of an ideal Bose gas in a one-dimensional harmonic trap, we show that these conditional averages reproduce microcanonical predictions even for modest particle numbers ( N ∼ 10 ) and exhibit vanishing relative fluctuations as temperature approaches zero. We stress that this does not contradict the standard GCE result ⟨ δ n 0 2 ⟩ → ⟨ N ⟩ 2 , which remains thermodynamically valid for open systems. Instead, our framework provides a natural description for experimental scenarios involving post-selection or implicit conditioning on non-empty realizations—common in few-body quantum gas experiments. Thus, the absence of large fluctuations in these conditional averages reflects a change in the observable, not a failure of ensemble theory.