Small Thermal Fluctuations on a Large Domain

Weak first-order phase transitions proceed with percolation of new phase. The kinematics of this process is clarified from the point of view of subcritical bubbles. We examine the effect of small subcritical bubbles around a large domain of asymmetric phase by introducing an effective geometry. The percolation process can be understood as a perpetual growth of the large domain aided by the small subcritical bubbles. Recently crucial effects of small rapid fluctuations -subcritical bubbles[1] are actively investigated for the electroweak baryogenesis[2] and for the inflationary cosmology. We could previously show, by studying the effect of subcritical bubbles, that the phase mixing is achieved in weakly first order phase transitions. However the global achievement of the phase transition -percolationhas not yet been explained from the point of view of subcritical bubbles. The difficulty lies on the previous treatment of subcritical bubbles; one always assumed spherically symmetry for the configuration of subcritical bubbles and interactions among them were neglected. However, it is possible to consider these effects easily within the point of view of subcritical bubbles. After the phase mixing is attained, there appear large and stable domains of asymmetric phase. If this domain were isolated in thermal fluctuations, then this domain would eventually collapse due to the surface tension of itself. However if we properly consider the effect of small fluctuations of subcritical bubbles around the large domain, this domain is stabilized against the collapse and eventually grows. This is the mechanism which we would like to clarify in this paper. This growth can be seen as the process of percolation of the system. Our study has been inspired by the interesting work by Gleiser et al.[3]. However in our case, both the collapse and growth of the domain are automatically taken into account and we worry about the introducing an extra term which guarantees the decay as claimed in [3]. In this letter, we first show that a large asymmetric domain is stabilized by subcritical bubbles around the wall. For simplicity, we assume that the shape of the large domain is spherical with radius R(t) and volume V+(t). Hereafter we simply call the subcritical bubbles as bubbles. The time variation of the volume V+(t) is composed from the kinematical and subcritical bubble contributions. ; dV+(t) dt = 4πR dR dt + (Γ+∆V ) 4π 3 〈R〉+ − (Γ0∆V ) 4π 3 〈R〉0, (2) where only the contribution from bubbles near wall of the large domain is included in the second and third terms in the right-hand-side of this equation. Γ+ and Γ0 are the creation rates per unit volume of a bubble and an anti-bubble, respectively: Γ+ ≃ m+(T )exp [−βF+(〈R〉+)] (3) and Γ0 ≃ m0(T )exp [−βF0(〈R〉0)] , (4) where F0,+ and 〈R〉0,+ are the free energy and the averaged size of a bubble and an antibubble, respectively. As F0 ≃ F+, m0(T ) ≃ m+(T ) and 〈R〉+ ≃ 〈R〉0 := 〈R〉 near T = Tc at which two vacua degenerate, approximately Γ+ = Γ0 =: Γ holds and therefore dV+(t) dt = 4πR dR dt + Γ (∆V −∆V ) 4π 3 〈R〉. (5) Furthermore, as ∆V −∆V ′ ≃ 32πR〈R〉2, this equation reduces to dV+(t) dt = 4πR dR dt + 128π 3 ΓR〈R〉. (6) This is the relation between the spherical symmetric volume and its radius modified by the bubbles around the large domain. A cute and elegant way to express this important relation is to introduce a fictitious geometry of non-Euclidean space whose metric is given by dl = dr + a(t)rdΩ2. (7) Deviation of the “scale factor” a from 1 represents the non-trivial effect from the last term in eq.(6). The time variation of the volume in this geometry (7) becomes dV+ dt = d dt (