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The Kuramoto model has been introduced in order to describe synchronizationphenomena observed in groups of cells, individuals, circuits, etc... We look atthe Kuramoto model with white noise forces: in mathematical terms it is a setof N oscillators, each driven by an independent Brownian motion with a constantdrift, that is each oscillator has its own frequency, which, in general,changes from one oscillator to another (these frequencies are usually taken tobe random and they may be viewed as a quenched disorder). The interactionsbetween oscillators are of long range type (mean field). We review some resultson the Kuramoto model from a statistical mechanics standpoint: we give inparticular necessary and sufficient conditions for reversibility and we pointout a formal analogy, in the N to infinity limit, with local mean field modelswith conservative dynamics (an analogy that is exploited to identify inparticular a Lyapunov functional in the reversible set-up). We then focus onthe reversible Kuramoto model with sinusoidal interactions in the N to infinitylimit and analyze the stability of the non-trivial stationary profiles arisingwhen the interaction parameter K is larger than its critical value K_c. Weprovide an analysis of the linear operator describing the time evolution in aneighborhood of the synchronized profile: we exhibit a Hilbert space in whichthis operator has a self-adjoint extension and we establish, as our mainresult, a spectral gap inequality for every K>K_c.

Dynamical aspects of mean field plane rotators and the Kuramoto model