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

We consider the problem of building a continuous stochastic model, i.e., a Langevin or Fokker-Planck equation, through a well-controlled coarse-graining procedure. Such a method usually involves the elimination of the fast degrees of freedom of the “bath” to which the particle is coupled. Specifically, we look into the general case where the bath may be at negative temperatures, as found, for instance, in models and experiments with bounded effective kinetic energy. Here, we generalize previous studies by considering the case in which the coarse graining leads to (i) a renormalization of the potential felt by the particle, and (ii) spatially dependent viscosity and diffusivity. In addition, a particular relevant example is provided, where the bath is a spin system and a sort of phase transition takes place when going from positive to negative temperatures. A Chapman-Enskog-like expansion allows us to rigorously derive the Fokker-Planck equation from the microscopic dynamics. Our theoretical predictions show excellent agreement with numerical simulations.

Figure 1

Equilibrium PDF of the particle variables $(x,p)$. The top (bottom) panels correspond to $\beta =2$ ($\beta =-2$). Histograms are computed from numerical simulations of the microscopic dynamics (14); red solid lines are the best fits to the Boltzmann distribution (2a). Parameters: $N={10}^4$, $\alpha =10$, $\mu ={10}^{-2}$, $dt={\left(\alpha N\right)}^{-1}$.

Figure 2

Equilibrium autocorrelation functions. Specifically, we consider two observables, $sinx$ above and $sinp$ below, for $\beta =2$ (left panels) and $\beta =-2$ (right panels). Red circles represent the simulations of the original dynamics (14), whereas blue diamonds are the numerical integration of the Fokker-Plank equation (23), with a time step $h={10}^{-4}$. Other parameters as in Fig. 1.

Figure 3

Relaxation to equilibrium. We plot the time evolution of the averages of $cosx$ (top) and $cosp$ (bottom), for $\beta =2$ (left panels) and $\beta =-2$ (right panels). In all cases, the particle starts from the initial condition $\mathrm{\Gamma }\left(0\right)=\left(x\right(0),p(0\left)\right)=(1,1)$. As in Fig. 2, the original dynamics (red circles) is compared with the Fokker-Planck equation (blue diamonds), with a time step $h={10}^{-4}$. Other parameters as in Fig. 1. Green dashed lines are $\langle cosx\rangle$ and $\langle cosp\rangle$ analytically computed averaging over the theoretical equilibrium distributions.