Condensation of fluctuations in and out of equilibrium

Condensation of fluctuations is an interesting phenomenon conceptually distinct from condensation on average. One striking feature is that, contrary to what happens on average, condensation of fluctuations may occur even in the absence of interaction. The explanation emerges from the duality between large deviation events in the given system and typical events in a new and appropriately biased system. This phenomenon is investigated in the context of the Gaussian model, chosen as a paradigmatical noninteracting system, before and after an instantaneous temperature quench. It is shown that the bias induces a mean-field-like effective interaction responsible for the condensation on average. Phase diagrams, covering both the equilibrium and the off-equilibrium regimes, are derived for observables representative of generic behaviors.

Figure 1

Spectra of inverse effective temperatures (top) and of ${\rho }_k$ for the order parameter sample variance (bottom) with $r=1,T_I=1$, and $T_F=0.2$.

Figure 2

Typical behavior of $F_{\mathcal{M}}(z,t)$, obtained with ${\mu }_k=2,t=2,r=1,T_I=1$, and $T_F=0.2$. Top panel: graphical solution of Eq. (31) with $d=1$. Bottom panel: graphical solution of Eq. (36) with $d=3$ and for $m<m_C,m=m_C$, and $m>m_C$.

Figure 3

Rate function $I_{\mathcal{S}}(s,t)$ for the sample variance discussed in Sec. 5, $s_C$ denotes the critical threshold. Parameters ${\mu }_k=2,t=2,r=1,T_I=1,T_F=0.2$, and $d=3$.

Figure 4

Top panel: Phase diagram of order parameter sample variance. The upper horizontal dashed line corresponds to $s_C\left(T_F\right)$. The green line is the plot of $\langle s\left(t\right)\rangle$. The lower dashed line is the difference $[s_C\left(t\right)-\langle s\left(t\right)\rangle ]$. Bottom panel: Energy phase diagram. In both cases: $r=1,T_I=1,T_F=0.2$, and $d=3$.