Subsystems separate scalings

The first step is to identify the system and initial and boundary conditions for the problem. This is followed by the selection of the major subsystems and identification of the interfacial conditions between them. Then the scaling study is carried out for each subsystem separately. 

This will lead to two separate energy scales at the expense of losing the physical time information of the quantum subsystem dynamics. The according Lagrangian (Eq. 1) reads ... 

In the previous section, we have shown that switching the picture from the nearly integrable Hamiltonian to the Hamiltonian with internal structures may make it possible to solve several controversial issues listed in Section IV. In this section we shall examine the validity of an alternative scenario by reconsidering the analyses done in MD simulations of liquid water. As mentioned in Section III, since a water molecule is modeled by a rigid rotor, and has both translational and rotational degrees of freedom. So, the equation of motion involves the Euler equation for the rigid body, coupled with ordinary Hamiltonian equations describing the translational motions. The precise Hamiltonian is therefore different from that of the Hamiltonian in Eq. (1), but they are common in that the systems have internal structures, and the separation of the time scale between subsystems appears if system parameters are appropriately set. 

The experimental assessments so far have yielded favorable results as far as achieving the partitioning objectives. Of course, these tests are not conclusive, since they have been performed on only a very small scale and only separate subsystems have been examined. However, there does not appear to be any fundamental reason that precludes achieving the stated partitioning goals and, in fact, it may be possible to exceed them. 

This vast spectral bandwidth illustrates the necessity of a reliable scale and time resolved decomposition of available observations to separate and describe single processes as individual parts of the whole system. Often, the comlex interplay between climate subsystems plays an essential role and the understanding of coupling mechanisms is of crucial importance for the study and prediction of at first sight independent phenomena. Continuous wavelet transformation (CWT) is the prototypic instrument to address these tasks As an important application, it transforms time series to the time/scale domain for estimating the linear non-stationary spectral properties of the underlying process. 

For strongly asymmetric mixtures (e.g., mixtures where the A-chains are stiff while the B-chains are flexible) the semi-grandcanonical approach is clearly not feasible, and one must work in a canonical ensemble where both the number of A-chains nA and the number of B-chains nB are fixed. However, the finite size scaling ideas for PL(M) as exposed above still can be exploited if one considers the order parameter M in L x L subsystems of a much larger system [267]. The usefulness of this concept was demonstrated earlier for Ising models and Len-nard-Jones fluids [268-271]. Gauger and Pakula [267] find an entropy-driven phase separation without any intermolecular interactions. 

To interpret Eq. (5.186), we first note that this expression describes a scaling of the pressure. The system of JV particles is (virtually) separated into NjK subsystems, every one of which contains K interacting particles. Each of these subsystems occupies the entire volume V and contributes a partial pressure p K g,k T/V). If it were permissible to neglect interactions among particles assigned to different subsystems, the total pressure would equal N/K)p K g,k T/V). However, particles belonging to different subsystems do indeed interact with one another, so that it is necessary to introduce a corrective factor, namely, the renormalized compressibility factor functional equation, expressing the requirement that the value of the pressure be unaltered by the virtual subdivision of the system. 

For large-scale systems, the description must be of a nature which, via the definition of suitable interfaces, permits a breakdown into subsystems that may be subjected to separate analyses. 

Abstract In this chapter we generalize the birth-death process analyzed in the previous chapter to account for enzymatic molecule synthesis, rather than simple Poissonian production. To facilitate the analysis we assume a time-scale separation in the enzymatic reactions, and use it to reduce the complexity of the complete system. With this simplification the generalized birth-death process can be separated into two different subsystems that can be studied separately, and correspond to the systems studied in Chaps. 3 and 4. The simplification procedure, introduced in Sect. 5.1, is a very useful mathematical tool way beyond the scope of the present chapter. 

The relaxation time of the fast subsystem is proportional to k + kj), while the relaxation time of the slow subsystem is proportional to Ym - Thus, the inequality ym k, kp guaranties a clear separation of scales in the relaxation... 

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