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Register a new userTemperature Expansions in the Square-Shoulder Fluid I: the Wiener-Hopf Function
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We investigate the spatial structure of dense square-shoulder fluids. To this end we derive analytical perturbative solutions of the Ornstein-Zernike equation in the low- and high-temperature limits as expansions around the known hard sphere solutions. We then discuss the suitability of perturbative approaches in relation to the Ornstein-Zernike equation.
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In a companion paper, we derived analytical expressions for the structure factor of the square-shoulder potential in a perturbative way around the high- and low-temperature regimes. Here, various physical properties of these solutions are derived. In particular, we investigate the large wave number sector, and relate it to the contact values of the pair-correlation function. Then, thermoelastic properties of the square-shoulder fluids are discussed.
To understand the non-exponential relaxation associated with solvation dynamics in the ionic liquid 1-ethyl-3-methylimidazolium hexafluorophosphate, we study power spectra of the fluctuating Franck-Condon energy gap of a diatomic probe solute via molecular dynamics simulations. Results show 1/f dependence in a wide frequency range over 2 to 3 decades, indicating distributed relaxation times. We analyze the memory function and solvation time in the framework of the generalized Langevin equation using a simple model description for the power spectrum. It is found that the crossover frequency toward the white noise plateau is directly related to the time scale for the memory function and thus the solvation time. Specifically, the low crossover frequency observed in the ionic liquid leads to a slowly-decaying tail in its memory function and long solvation time. By contrast, acetonitrile characterized by a high crossover frequency and (near) absence of 1/f behavior in its power spectra shows fast relaxation of the memory function and single-exponential decay of solvation dynamics in the long-time regime.
In V-T theory the atomic motion is harmonic vibrations in a liquid-specific potential energy valley, plus transits, which move the system rapidly among the multitude of such valleys. In its first application to the self intermediate scattering function (SISF), V-T theory produced an accurate account of molecular dynamics (MD) data at all wave numbers q and time t. Recently, analysis of the mean square displacement (MSD) resolved a crossover behavior that was not observed in the SISF study. Our purpose here is to apply the more accurate MSD calibration to the SISF, and assess the results. We derive and discuss the theoretical equations for vibrational and transit contributions to the SISF. The time evolution is divided into three successive intervals: the vibrational interval when the vibrational contribution alone accurately accounts for the MD data; the crossover when the vibrational contribution saturates and the transit contribution becomes resolved; and the diffusive interval when the transit contribution alone accurately accounts for the MD data. The resulting theoretical error is extremely small at all q and t. Comparison of V-T and mode-coupling theories for the MSD and SISF reveals that, while their formulations differ substantially, their underlying atomic motions are in logical correspondence.
The response of an isolated granular fluid to small perturbations of the hydrodynamic fields is considered. The corresponding linear response functions are identified in terms of a formal solution to the Liouville equation including the effects of the cooling reference state. These functions are evaluated exactly in the asymptotic long wavelength limit and shown to represent hydrodynamic modes. More generally, the linear granular Navier-Stokes equations for the response functions and related Langevin equations are obtained from an extension of Moris identity. The resulting Green-Kubo expressions for transport coefficients are compared and contrasted with those for a molecular fluid. Next the response functions are described in terms of an effective dynamics in the single particle phase space. A closed linear kinetic equation is obtained formally in terms of a linear two particle functional. This closure is evaluated for two examples: a short time Markovian approximation, and a low density expansion on length and time scales of the mean free time and mean free path. The former is a generalization of the revised Enskog kinetic theory to include velocity correlations. The latter is an extension of the Boltzmann equation to include the effects of recollisions (rings) among the particles.
Newton viscosity law for the momentum flux and Fouriers law for the heat flux define Navier-Stokes hydrodynamics for a simple, one component fluid. There is ample evidence that a hydrodynamic description applies as well to a mesoscopic granular fluid with the same form for Newtons viscosity law. However, theory predicts a qualitative difference for Fouriers law with an additional contribution from density gradients even at uniform temperature. The reasons for the absence of such terms for normal fluids are indicated, and a related microscopic explanation for their existence in granular fluids is presented.
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