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We analyze multi-point correlation functions of a tracer in an incompressibleflow at scales far exceeding the scale $L$ at which fluctuations are generated(quasi-equilibrium domain) and compare them with the correlation functions atscales smaller than $L$ (turbulence domain). We demonstrate that the scaleinvariance can be broken in the equilibrium domain and trace this breakdown tothe statistical integrals of motion (zero modes) as has been done before forturbulence. Employing Kraichnan model of short-correlated velocity we identifythe new type of zero modes, which break scale invariance and determine ananomalously slow decay of correlations at large scales

Anomalous Scaling of a Passive Scalar in Turbulence and in Equilibrium