Hydrodynamic mode

Out of the five hydrodynamic modes, the polarized inelastic light scattering experiment can probe only the tliree modes represented by equation (A3.3.18), equation (A3.3.19) and equation (A3.3.20). The other two modes, which are in equation (A3.3.17), decouple from the density fluctuations diese are due to transverse... 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

Fig. 4.4.2 The discrete data points represent Taylor-Couette-Poiseuille flow regimes observed with MRI for r = 0.5 [41]. The curved boundaries were obtained for r = 0.77 with optical techniques [38]. The two inserts show MRI spin-tagging FLASH images of the SHV and PTV hydrodynamic modes.

This section focuses on steady and unsteady hydrodynamic modes that emerge as the rotational speed of the inner cylinder (expressed by Ta) and pressure-driven axial flow rate (scaled by Re) are varied, while the outer cylinder is kept fixed. These modes constitute primary, secondary and higher order bifurcations, which break the symmetry of the base helical Couette-Poiseuille (CP) flow and represent drastic changes in flow structure. Figure 4.4.2 presents a map of observed hydrodynamic modes in the (Ta, Re) space, and marks the domain where all of the hydrodynamic modes that interest us appear. We will return to this figure shortly. 

Fig. 3.151. Electropherogram of a dye mixture obtained under optimized conditions voltage 15 kV and detection 460 nm. Injection was made in hydrodynamic mode by 4 s. The dye concentrations in the standard were 15 jUg/ml except methyl orange (30, ug/inl) in the spiked river water they were 20 jUg/ml except methyl orange (40 pg/ml). Reprinted with permission from M. Perez-Urquiza et al. [193],...

Step 6 The sample is preferentially injected by hydrodynamic mode since in this way a representative amount of sample solution is introduced in the capillary inlet. The outlet of the capillary is maintained in the dip-vial. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

These spatially periodic solutions are known as the hydrodynamic modes. The decay rates s of these solutions are obtained by solving an eigenvalue problem for the five linearized hydrodynamic equations. These decay rates define the so-called dispersion relations of the five hydrodynamic modes. The dispersion relations can be expanded in powers of the wavenumber k as shown in Table 1. The sound modes are propagative because their dispersion relations include the imaginary term iUsk with the sound velocity... 

The shear and thermal modes are not propagative. All the hydrodynamic modes are damped with a relaxation rate proportional to the square of the wavenumber so that their damping vanishes as the wavelength of the mode tends to infinity, which has its origin in the fact that these five modes are associated with the five... 

Figure 1. Schematic dispersion relations of the five hydrodynamic modes of a fluid with one component.

The dispersion relations of the five hydrodynamic modes are depicted in Fig. 1. Beyond the hydrodynamic modes, there may exist kinetic modes that are not associated with conservation laws so that their decay rate does not vanish with the wavenumber. These kinetic modes are not described by the hydrodynamic equations but by the Boltzmann equation in dilute fluids. The decay rates of the kinetic modes are of the order of magnitude of the inverse of the intercollisional time. 

A major preoccupation of nonequilibrium statistical mechanics is to justify the existence of the hydrodynamic modes from the microscopic Hamiltonian dynamics. Boltzmann equation is based on approximations valid for dilute fluids such as the Stosszahlansatz. In the context of Boltzmann s theory, the concept of hydrodynamic modes has a limited validity because of this approximation. We may wonder if they can be justified directly from the microscopic dynamics without any approximation. If this were the case, this would be great progress... 

The dispersion relations of the two modes are depicted in Fig. 2. The reactive mode is one of the kinetic modes existing beside the hydrodynamic modes such as the diffusive mode. Here also, we may wonder if these modes can be justified from the microscopic dynamics. 

The generalized eigenvalue Sk is a Pollicott-Ruelle resonance associated with the eigenstate k- The hydrodynamic modes can be identified as the eigenstates associated with eigenvalues Sk vanishing with the wavenumber k. 

At the microscopic level of description, the hydrodynamic mode of diffusion is dehned as the Liouvillian eigenstate ... 

The zeta function methods have proved to be extremely powerful to obtain the resonances of classical scattering systems, which give the quasiclassical reaction rates [61]. In transport processes, the classical resonances give the dispersion relations that characterize the relaxation of hydrodynamic modes [64], These results bring about a new understanding of the problem of irreversibility at the classical level, as discussed elsewhere [64],... 

A third kind of slowness, that due to hydrodynamic modes, has been discussed already. It is difficult to do anything about these slow collective modes, but fortunately they cannot cost very many orders of magnitude in a system of a few thousand atoms or less. 

II. Hydrodynamic Modes The Basic Dynamical Variables in Liquids HI. Slow Dynamics at Large Wavenumbers de Gennes Narrowing... 

MCT can be best viewed as a synthesis of two formidable theoretical approaches, namely the renormalized kinetic theory [5-9] and the extended hydrodynamic theory [10]. While the former provides the method to treat both the very short and the very long time responses, it often becomes intractable in the intermediate times. This is best seen in the calculation of the velocity time correlation function of a tagged atom or a molecule. The extended hydrodynamic theory provides the simplicity in terms of the wavenumber-dependent hydrodynamic modes. The decay of these modes are expressed in terms of the wavenumber- and frequency-dependent transport coefficients. This hydrodynamic description is often valid from intermediate to long times, although it breaks down both at very short and at very long times, for different reasons. None of these two approaches provides a self-consistent description. The self-consistency enters in the determination of the time correlation functions of the hydrodynamic modes in terms of the... 

Even with all of its sophistication, the mode coupling theory is still a perturbation theory where dynamics is described in terms of a subset of dynamical variables chosen from the products of hydrodynamic modes. It fails, for example, to describe rare events, such as the activated processes or stringlike cooperative motions often found to dictate dynamics in glassy liquids [13-15]. 

H. HYDRODYNAMIC MODES THE BASIC DYNAMICAL VARIABLES IN LIQUIDS... 

Physically, the Brillouin spectrum arises from the inelastic interaction between a photon and the hydrodynamics modes of the fluid. The doublets can be regarded as the Stokes and anti-Stokes translational Raman spectrum of the liquid. These lines arise due to the inelastic collision between the photon and the fluid, in which the photon gains or loses energy to the phonons (the propagating sound modes in the fluid) and thus suffer a frequency shift. The width of the band gives the lifetime ( 2r)-1 of a classical phonon of wavenumber q. The Rayleigh band, on the other hand, represents the... 

There exists another prescription to extend the hydrodynamical modes to intermediate wavenumbers which provides similar results for dense fluids. This was done by Kirkpatrick [10], who replaced the transport coefficients appearing in the generalized hydrodynamics by their wavenumber and frequency-dependent analogs. He used the standard projection operator technique to derive generalized hydrodynamic equations for the equilibrium time correlation functions in a hard-sphere fluid. In the short-time approximation the frequency dependence of the memory kernel vanishes. The final result is a... 

Finally, note that the relaxation equation [Eq. (76)] is usually written in terms of the hydrodynamic modes. In many problems of chemical interest, nonhydrodynamic modes such as intramolecular vibration, play an important role [50]. Presence of such coupling creates an extra channel for dissipation. Thus, the memory kernel, T, gets renormalized and acquires an additional frequency-dependent term [16, 43]. 

As discussed before, the viscoelastic model is known to provide a correct description of F(qz) in the intermediate density regime. Even in a supercooled liquid, it can provide correct short-time description, but fails in the long time, where the contribution from the hydrodynamic modes become important. 

In the supercooled liquid, the important part of the memory kernel is its long-time part, r (q, t). The recollision term contains the contribution from the hydrodynamic modes. As discussed by many authors [3, 30, 34], among all the hydrodynamic modes the density fluctuation is found to yield the main contribution to the memory kernel in the supercooled fluid regime. 

Let us first rewrite the expression for the friction in terms of the binary friction and die contributions from different hydrodynamic modes ... 

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