Order, Pattern Selection and Noise in Low Dimensional Systems

Abstract

In this paper I will address some aspects of the role of noise in pattern formation problems. Pattern formation is associated with spatial symmetry breaking, but broken symmetries might be restored by noise. Equilibrium phase transitions is a well known class of problems associated with symmetry breaking. A phase transition takes place when, in the thermodynamic limit, thermal fluctuations are not able to mix states with different symmetries. This only occurs for a large enough spatial dimensionality, which depends on the type of broken symmetry. An example of the opposite situation of symmetry restoring by noise is given by the ordinary laser instability. The proper description of this system is zero dimensional and spontaneous emission noise restores the phase symmetry of the lasing field after a time of the order of the coherence time. Pattern formation usually occurs in open systems of low dimensionality which correspond to situations somehow intermediate between the two discussed above.