Kinetic theory, renormalized

Andersen, H. C. Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. III. Cluster Analysis of the Renormalized Interactions and a Second Diagrammatic Representation of the Correlation Functions. J. Phys. Chem. B 2003, 107, 10234-10242. 

VI. Renormalized Kinetic Theory Microscopic Approach to the Mode Coupling Theory... 

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... 

VI. RENORMALIZED KINETIC THEORY MICROSCOPIC APPROACH TO THE MODE COUPLING THEORY... 

In Sections V and VI, a brief history of the developments of the MCT from the hydrodynamic approach (Critical Phenomena) and the renormalized kinetic theory approach has been presented. The basic concept of MCT is to use the product of the slow (hydrodynamic) variables to span the orthogonal subspace of the fast variables. 

In the formulations developed from the renormalized kinetic theory approach, these self-consistencies were avoided either by using values obtained from computer simulation and experiments or by using some exactly known limiting values for the transport coefficient. For example, in the treatment of Mazenko [5-7], and of Mehaffey and Cukier s [8] the transport coefficients are replaced by their Enskog values. In the theory developed by Sjogren and Sjolander [9], the velocity autocorrelation function is required as an input that was obtained from the computer simulated values. This limits the validity of the theories only to certain regimes and for certain systems where the experimental or computer-simulated results are available. 

It should be noted that the integrodifferential equations equivalent to Eqs. (3.168) were first obtained with the fully renormalized kinetic theory of Yang, Lee, and Shin (YLS) [132]. However, they had not confined themselves to the original integral equations (3.168) but reduced them to the differential rate equations by the incorrect substitution used instead of (3.164) ... 

The bases for such a generalized kinetic theory were set during the 1970s and the early 1980s by several independent workers. The final form of the theory is largely owed to the works of Sjogren et al. [30, 31], which are based on the results of the so-called renormalized kinetic theories by Mazenko and co-workers [32]. 

Chapter 4. Renormalized Kinetic Theory of Dense Fluids Gene F. Mazenko and Sidney Yip... 

The main objective of our theory is to develop useful methods for approximating the memory function. Two basic motivations characterize our analysis in Section 3, the systematic inclusion of the exact correlation functions and the expression of the memory function in terms of an effective two-body problem, which constitutes a second renormalization. The second motivation is particularly important in setting up approximation schemes since we are conditioned to think in terms of two-body collision processes. In Section 4 we use these ideas to arrive at an approximate expression for the memory function. The memory function is the sum of a Boltzmann-Enskog term and a recollision term that describes the correlated dynamical processes mentioned above. We will call this theory the fully renormalized kinetic theory because the medium effects involving particles other than the interacting binary cluster are taken into account and also because bare interactions are replaced by effective interactions, which depend on the equilibrium correlations in the fluid. 

In the formalism of fully renormalized kinetic theory (FRKT) one directs one s attention not to the phase space correlation function 0(12) but to the associated memory function < (12), which is defined by " ... 

We have studied a kinetic theory approach to the calculation of dynamical correlations in a dense fluid. By considering density correlations in phase space one can discuss the dynamics in terms of renormalized molecular interactions that take into account the existence of structural correlations in the fluid. The advantage of this approach is that through the properties of a single function, the phase space memory function, one can treat in a unified manner a variety of correlation functions without restriction to either wavelength-frequency domain or fluid density. 

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