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Fluctuations in the critical region

A qualitative theory of fluctuations in the critical region, including the critical point, is developed using the example of a classical system describable by the Gizburg-Landau equation with a random source. The starting points are the conditions for consistency of two limits: passage to the critical point (T-T,) and the thermodynamic limit. Depending on the value of 7 = (T Tc)/Tc four subregions can be distinguished: 1) The critical point and its immediate neighborhood, in which the parameter e a l/N is not small. The critical indices fl, y , and v characterize the power dependences on N. 2) The region of scale invariance. The passage to this from subregion 1 occurs when the thermodynamic l i t is taken. The same critical indices determine the power dependences on M. In the regions 1 and 2 the renonnalized Gizburg parameter is not a small parameter. 3) The crossover region between region 2 and the region of applicability of the Landau theory. 4) The region of applicability of the Landau theory. The relationship between the critical indices in the regions 2 and 4 is determined by the " W i n parameter" ew, the values of which lie in the interval 1 z e w 20. The value 1 corresponds to the region of scale invariance and the value 0 to the region in which the Landau theory is valid. In region 4 the Ginzburg parameter is small. The respective expressions for the correlation times of fluctuations of the order parameter and the volume-averaged polarization are given. They are valid for all the regions indicated. For T < T, the dependences of these times on N and T are substantially different. The fonner decreases in accordance with the Curie law with increase of M, and the latter tends to infinity as N m .