Reduced descriptions

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... 

Therefore, modeling a protein molecule amounts to deciding on the atoms considered to be essential and to specifying the contribution of the various interactions to the potential. Since the work to find the global minimizer increases drastically (and possibly exponentially) with the dimension of x, it is customary to use for larger proteins a reduced description that treats only very few atoms in each amino acid as essential. 

We consider a biological macromolecule in solution. Let X and Y represent the degrees of freedom of the solute (biomolecule) and solvent, respectively, and let U(X, Y) be the potential energy function. The thermal properties of the system are averages over a Boltzmann distribution P(X, Y) that depends on both X and Y. To obtain a reduced description in terms of the solute only, the solvent degrees of freedom must be integrated out. The reduced probability distribution P is... 

Similar to quantum mechanics, which can be formulated in terms of different quantities in addition to the traditional wave function formulation, in quantum chemistry a number of alternative tools are developed for this purpose, which may be useful in the context of the present book. We have already described different approximate models of representing the electronic structure using (many-electronic) wave functions. The coordinate and second quantization representations were employed to get this. However, the entire amount of information contained in the many-electron wave function taken in whatever representation is enormously large. In fact it is mostly excessive for the purpose of describing the properties of any molecular system due to the specific structure of the operators to be averaged to obtain physically relevant information and for the symmetry properties of the wave functions the expectation values have to be calculated over. Thus some reduced descriptions are possible, which will be presented here for reference. 

Experimental observations of the time evolution of externally unforced macroscopic systems on the level meSo l show that the level eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if Cmeso is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here Ath) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for Ath (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description (i) Mslow/ (ii) Aneso 2 -> Ath, (ui) Aneso l -> Aneso 2, and (iv) Aneso i —> Aneso 2 —> Ath- The first two are the same as (111). We now turn our attention to the third one, that is,... 

While there is, of course, no interest in the slow time evolution in the discussion of meso 1 —> Ath, since the slow time evolution is in this case no time evolution, this point is the main focus of the investigation of reduced descriptions (see, e.g., Gorban and Karlin, 2005 Yablonskii et al., 1991). The lack of interest in the point (III) (see (114)) in the context of ineSo i —> Ath then also means the lack of interest in the manifold Meth, i.e., in the point (II) (see (113)). 

We use a reduced description of the system (or process) of interest. In many cases, we seek simplified descriptions of physical processes by focusing on a small subsystem or on a few observables that characterize the process of interest. These observables can be macroscopic, for example, the energy, pressure, temperature, etc., or microscopic, for example, the center of mass position, a particular bond length, or the internal energy of a single molecule. In the reduced space of these important observables, the microscopic influence of the other 10 degrees of... 

Joint probabilities, conditional probabilities, and reduced descriptions... 

Reduced descriptions are not necessarily obtained in terms of the original random variables. For example, given the probability density P x,y we may want the probability of the random variable... 

This evolution equation demonstrates the way in which a reduced description (see Section 1.5.1) yields dynamics that is qualitatively different than the fundamental one A complete description of the assumed classical system involves the solution of a huge number of coupled Newton equations for all particles. Focusing on the position of one particle and realizing that the ensuing description has to be probabilistic, we find (in the present case experimentally) that the evolution is fundamentally different. For example, in the absence of external forces the particle position changes linearly with time, x = vZ, while (see below) Eq. (1,200) implies that the mean square displacement x changes linearly with time. Clearly the reason for this is... 

In the previous chapter we have seen how spatial correlation functions express useful structural infonnation about our system. This chapter focuses on time correlation functions (see also Section 1.5.4) tlrat, as will be seen, convey important dynamical information. Time correlation functions will repeatedly appear in our future discussions of reduced descriptions of physical systems. A typical task is to derive dynamical equations for the time evolution of an interesting subsystem, in which only relevant information about the surrounding thermal environment (bath) is included. We will see that dynamic aspects of this relevant information usually enter via time correlation functions involving bath variables. Another type of reduction aims to derive equations for the evolution of macroscopic variables by averaging out microscopic information. This leads to kinetic equations that involve rates and transport coefficients, which are also expressed as time correlation functions of microscopic variables. Such functions are therefore instrumental in all discussions that relate macroscopic dynamics to microscopic equations of motion. 

How do the definitions (6.3) and (6.6) relate to each other While a formal connection can be made, it is more important at this stage to understand their range of applicability. The definition (6.6) involves the detailed time evolution of all particles in the system. Equation (6.3) becomes useful in reduced descriptions of the system of interest. In the present case, if we are interested only in the mutual dynamics of the observables A and B we may seek a description in the subspace of these variables and include the effect of the huge number of all other microscopic variables only to the extent that it affects the dynamics of interest. This leads to a reduced space dynamics that is probabilistic in nature, where the functions P(B,t2, A,t].) and P(B, Z2 A, Zi) emerge. We will dwell more on these issues in Chapter 1. Common procedures for evaluating time correlation functions are discussed in Section 7.4.1. 

Why has the Markovian time evolution (7.50) of a system with two degrees of freedom become a non-Markovian description in the subspace of one of them Equation (7.51) shows that this results from the fact that Z2(t) responds to the historical time evolution of zi, and therefore depends on past values of zi, not only on its value at time t. More generally, consider a system A B made of a part (subsystem) A that is relevant to us as observers, and another part, B, that affects the relevant subsystem through mutual interaction but is otherwise uninteresting. The non-Markovian behavior of the reduced description of the physical subsystem A reflects the fact that at any time t subsystem A interacts with the rest of the total system, that is, with B, whose state is affected by its past interaction with A. In effect, the present state of B carries the memory of past states of the relevant subsystem A. 

It is important to point out that this does not imply that Markovian stochastic equations cannot be used in descriptions of condensed phase molecular processes. On the contrary, such equations are often applied successfully. The recipe for a successful application is to be aware of what can and what cannot be described with such approach. Recall that stochastic dynamics emerge when seeking coarsegrained or reduced descriptions of physical processes. The message from the timescales comparison made above is that Markovian descriptions are valid for molecular processes that are slow relative to environmental relaxation rates. Thus, with Markovian equations of motion we cannot describe molecular nuclear motions in detail, because vibrational periods (10 " s) are short relative to environmental relaxation rates, but we should be able to describe vibrational relaxation processes that are often much slower, as is shown in Section 8.3.3. 

The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (r (f), p (f)) for a given initial condition (r (0), p (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1,2,2)) that describes the time evolution of the phase space probability density p f). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1-5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. 

In Chapter 7 (see in particular Section 7,2) we have motivated the use of reduced descriptions of dynamical processes, where we focus on the dynamics of the subsystem of interest under the influence of its environment. This leads to reduced descriptions of dynamical processes whose stochastic nature stems from the incomplete knowledge of the state of the bath. The essence of a reduction process is exemplified by the relationship... 

A reduced description of the subsystem S alone will provide a density operator... 

When V 0 transitions between L and R can take place, and their populations evolve in time. Defining the total L and R populations by our goal is to characterize the kinetics of the L R process. This is a reduced description because we are not interested in the dynamics of individual level /) and r), only in the overall dynamics associated with transitions between the L and R species. Note that reduction can be done on different levels, and the present focus is on Pl and Pr and the transitions between them. This reduction is not done by limiting attention to a small physical subsystem, but by focusing on a subset of density-matrix elements or, rather, their combinations. 

---