Several methods for nonequilibrium computer simulation of plane Couette flow are analyzed by kinetic theory. The boundary-value problem for the nonlinear Boltzmann equation is related to the stochastic, Lees-Edwards, and "non-Newtonian" dynamics methods. It is found that the kinetictheory and computer simulation methods can be put into close correspondence, except for one form of the non-Newtonian equations of motion. The effects of homogeneous, nonconservative forces used to maintain constant temperature are also studied. For a special interatomic force law exact scaling relations are obtained to relate isothermal and nonisothermal solutions to the Boltzmann equation. For other force laws this scaling relationship is only approximate.

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We analyse the statistical physics of self-propelled particles from a general theoretical framework that properly describes the most salient characteristics of active motion in arbitrary spatial dimensions. Such a framework is devised in terms of a Smoluchowski-like equation for the probability density of finding a particle at a given position, that carries the Brownian component of the motion due to thermal fluctuations, and the active component due to the intrinsic persistent motion of the particle. The active probability current not only considers the gradient of the probability density at the current time, as in the standard Fick's law, but also the gradient of the probability density at all previous times weighted by a memory function that entails the main features of active motion. We focus in the consequences when the memory function depends only on time and decays as a power law in the short-time regime, and exponentially in the long-time one. In addition, we found analytical expressions for the Intermediate Scattering Function and the time dependence of the mean-squared displacement and the kurtosis. * fjsevilla@fisica.unam.mx † pcastrov@unach.mx

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We generalize fluctuating hydrodynamics to study the effect of fractional time derivatives on the light-scattering spectrum of a suspension in a viscoelastic solvent under an external density gradient. Viscoelasticity introduces additional memory effects into the fluctuating hydrodynamic equations, causing the time scales associated with the mesoscopic variables and those of the microscopic events to be no longer well separated. This situation is taken into account by introducing Caputo's fractional time derivative into the description. The structure factor of the suspension is calculated, and we find that its nonequilibrium correction is an odd function of the frequency. It exhibits a shift towards negative frequencies proportional to the magnitude of the imposed gradient. We consider solvents that are described by Maxwell's or power-law rheological equations of state. The fractional structure factor is compared with the nonfractional one, and it is found that the ratio of the former to the latter may be positive and up to two orders of magnitude for both types of viscoelasticity. This prediction of our model calculation suggests that this relative change might be measurable.