No Anomalous Fluctuations Exist in Stable Equilibrium Systems

An equilibrium statistical system is known to be stable if the fluctuations of global observables are normal, when their dispersions are proportional to the number of particles, or to the system volume. A general theorem is rigorously proved for the case, when an observable is a sum of linearly independent terms: The dispersion of a global observable is normal if and only if all partial dispersions of its terms are normal, and it is anomalous if and only if at least one of the partial dispersions is anomalous. This theorem, in particular, rules out the possibility that in a stable system with Bose-Einstein condensate some fluctuations of either condensed or noncondensed particles could be anomalous. The conclusion is valid for arbitrary systems, whether uniform or nonuniform, interacting weakly or strongly. The origin of fictitious fluctuation anomalies, arising in some calculations, is elucidated. 05.70.-a, 05.30.Jp, 03.75.-b, 67.40.-w Typeset using REVTEX 1 The problem of fluctuations of observable quantities is among the most important questions in statistical mechanics, being related to the fundamentals of the latter. A great revival, in recent years, of interest to this problem is caused by intensive experimental and theoretical studies of Bose-Einstein condensation in dilute atomic gases (see e.g. reviews [1– 4]). The fact that the ideal uniform Bose gas possesses anomalously large number-of-particle fluctuations has been known long ago [5,6], which has not been of much surprise, since such an ideal gas is an unrealistic and unstable system. However, as has been recently suggested in many papers, similar anomalous fluctuations could appear in real interacting Bose systems. A number of recent publications has addressed the problem of fluctuations in Bose gas, proclaiming controversial statements of either the existence or absence of anomalous fluctuations (see discussion in review [7]). So that the issue has not been finally resolved. In the present paper, the problem of fluctuations is considered from the general point of view, independent of particular models or calculational methods. A general theorem is rigorously proved, from which it follows that there are no anomalous fluctuations in any stable equilibrium systems. It is worth stressing that no phase transitions are considered in this paper. As can be easily inferred from any textbook on thermodynamics or statistical mechanics, the points of phase transitions are, by definition, the points of instability. A phase transition occurs exactly because one phase becomes unstable and has to change to another stable phase. It is well known that at the points of second-order phase transitions fluctuations do become anomalous, yielding divergent susceptibilities, as it should be at the points of instability. After a phase transition has occurred, the system, as is also well known, becomes stable and susceptibilities go finite. However, in many papers on Bose systems, the claims are made that fluctuations remain anomalous far below the condensation point, in the whole region of the Bose-condensed system. As is shown below, these claims are incorrect, since such a system with anomalous fluctuations possesses a divergent compressibility, thus, being unstable. Observable quantities are represented by Hermitian operators from the algebra of ob2 servables. Let  be an operator from this algebra. Fluctuations of the related observable quantity are quantified by the dispersion ∆(Â) ≡ <  > − <  > , (1) where < . . . > implies equilibrium statistical averaging. The dispersion itself can be treated as an observable quantity, which is the average of an operator (Â− <  >), since each dispersion is directly linked to a measurable quantity. For instance, the dispersion for the number-of-particle operator N̂ defines the isothermal compressibility κT ≡ − 1 V (