General dynamic equation regime

If Xe is somewhat larger, then there may arise an effective time scale Xr > Xe, with 5, < Xr sueh that the environment has some memory of the particle s previous history and therefore responds accordingly. This is the regime of the generalized Langevin equation (GLE) with colored friction. - In all these cases, the environment is sufficiently large that the particle is unable to affect the environment s equilibrium properties. Likewise, the environment is noninteracting with the rest of the universe such that its properties are independent of the absolute time. All of these systems, therefore, describe the dynamics of a stochastic particle in a stationary —albeit possibly colored— environment. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

Generally, synchronization is observed for high-order resonances nco mtoo as well. In this case the dynamics of the generalized phase difference ip = m(p — mot, where n, m are integers, is described by the equation similar to Eq. (13.3), namely by d ip)/dt = — mo — rmoo) + sq ip). Synchronous regime then means perfect entrainment of the oscillator frequency at the rational multiple of the forcing frequency, fl = lo, as well as phase locking... 

In general, the presence of a destabilizing factor is insufficient to ensure instability and there will be stable and unstable regimes that depend on the particular values of the coefficients in the kinetic equations. A set of parameters separating two structurally stable regimes with different qualitative dynamics is called a bifurcation point. At bifurcation points, differential equations are not structurally stable. There is an extensive literature on bifurcation theory, but it will not be reviewed here. 

It seems that all the successfiilly elucidated dynamical imiversality classes adopt as their minimal models the ones described in terms of nonlinear Langevin equations (33). This indicates the general correctness of Onsager s principle as a fimdamental principle (beyond the linear regime). Just as equilibrium statistical mechanics is a statistical framework based on the principle (of equal... 

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