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We consider the problem of building a continuous stochastic model, i.e. aLangevin or Fokker-Planck equation, through a well-controlled coarse-grainingprocedure. Such a method usually involves the elimination of the fast degreesof freedom of the "bath" to which the particle is coupled. Specifically, welook into the general case where the bath may be at negative temperatures, asfound - for instance - in models and experiments with bounded effective kineticenergy. Here, we generalise previous studies by considering the case in whichthe coarse-graining leads to (i) a renormalisation of the potential felt by theparticle, and (ii) spatially dependent viscosity and diffusivity. In addition,a particular relevant example is provided, where the bath is a spin system anda sort of phase transition takes place when going from positive to negativetemperatures. A Chapman-Enskog-like expansion allows us to rigorously derivethe Fokker-Planck equation from the microscopic dynamics. Our theoreticalpredictions show an excellent agrement with numerical simulations.

Derivation of a Langevin equation in a system with multiple scales: the case of negative temperatures