Microscopic dynamics

Chapter 6 provided an overview of the characteristics of the mechanical and dielectric behavior of polymer systems. We discussed the material properties as they are described by the various response functions and were in particular concerned with the pronounced effects of temperature. These macroscopic properties have a microscopic basis. So far, we have addressed this basis only in qualitative terms. This chapter deals with the microscopic dynamics and hereby, in particular, with some models yielding a quantitative description for observed macroscopic properties. 

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Progress in the theoretical description of reaction rates in solution of course correlates strongly with that in other theoretical disciplines, in particular those which have profited most from the enonnous advances in computing power such as quantum chemistry and equilibrium as well as non-equilibrium statistical mechanics of liquid solutions where Monte Carlo and molecular dynamics simulations in many cases have taken on the traditional role of experunents, as they allow the detailed investigation of the influence of intra- and intemiolecular potential parameters on the microscopic dynamics not accessible to measurements in the laboratory. No attempt, however, will be made here to address these areas in more than a cursory way, and the interested reader is referred to the corresponding chapters of the encyclopedia. 

Of great interest to physical chemists and chemical physicists are the broadening mechanisms of Raman lines in the condensed phase. Characterization of tliese mechanisms provides infomiation about the microscopic dynamical behaviour of material. The line broadening is due to the interaction between the Raman active chromophore and its environment. 

Fernandez A and Colubri A 1998 Microscopic dynamics from a coarsely defined solution to the protein folding problem J. Math. Phys. 39 3167-87... 

The small statistical sample leaves strong fluctuations on the timescale of the nuclear vibrations, which is a behavior typical of any detailed microscopic dynamics used as data for a statistical treatment to obtain macroscopic quantities. 

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

Lattice gases are micro-level rule-based simulations of macro-level fluid behavior. Lattice-gas models provide a powerful new tool in modeling real fluid behavior ([doolenQO], [doolenQl]). The idea is to reproduce the desired macroscopic behavior of a fluid by modeling the underlying microscopic dynamics. 

From an analytic point of view, the techniques of conventional thermodynamics, which describes systems whose microscopic dynamics is reversible, may be formally applied to reversible CA as well. We shall, in fact, follow this course in chapter 4. 

One of the principal themes of this book is the use of CA in modeling real physical systems and dynamics. To this end, it is important to address the question of what fundamental properties of physical systems can be appropriately abstracted from nature and embodied within abstract CA constructs. As observed earlier, some properties such as homogeneity and locality are automatically built in on the generic level. Another such property, namely the reversibility of microscopic dynamics, which plays such a fundamental role in physics, has its analogue in CA dynamics as well. 

A schematic representation of emergence is given in figure 12.6, which depicts the first three levels of a dynamical hierarchy and the rules or laws describing their behavior. The first, or lowest, level might be thought of as the level on which a CA system is usually defined. It consists of the lattice sites and values that define the microscopic dynamics. 

Though statistical models are important, they may not provide a complete picture of the microscopic reaction dynamics. There are several basic questions associated with the microscopic dynamics of gas-phase SN2 nucleophilic substitution that are important to the development of accurate theoretical models for bimolecular and unimolecular reactions.1 Collisional association of X" with RY to form the X-—RY... 

Classical trajectory calculations, performed on the PES1 and PESl(Br) potential energy surfaces described above, have provided a detailed picture of the microscopic dynamics of the Cl- + CH3Clb and Cl" + CH3Br SN2 nucleophilic substitution reactions.6,8,35-38 In the sections below, different aspects of these trajectory studies and their relation to experimental results and statistical theories are reviewed. 

Additional experimental, theoretical, and computational work is needed to acquire a complete understanding of the microscopic dynamics of gas-phase SN2 nucleophilic substitution reactions. Experimental measurements of the SN2 reaction rate versus excitation of specific vibrational modes of RY (equation 1) are needed, as are experimental studies of the dissociation and isomerization rates of the X--RY complex versus specific excitations of the complex s intermolecular and intramolecular modes. Experimental studies that probe the molecular dynamics of the [X-. r - Y]- central barrier region would also be extremely useful. 

The principle of a lattice gas is to reproduce macroscopic behavior by modeling the underlying microscopic dynamics. In order to successfully predict the macro-level behavior of a fluid from micro-level rules, three requirements must be satisfied. First, the number of particles must be conserved and, in most cases, so is the particle momentum. States of all the cells in the neighborhood depend on the states of all the others, but neighborhoods do not overlap. This makes application of conservation laws simple because if they apply to one neighborhood they apply to the whole lattice. 

Linear response theory10 provides a link between the phenomenological description of the kinetics in term of reaction rate constants and the microscopic dynamics of the system [33]. All information needed to calculate the reaction rate constants is contained in the time correlation function... 

It can be shown that the assumption of a weak perturbation central to linear response theory can be relaxed in this case [9]. The equations presented in this section relating the kinetic coefficients with the microscopic dynamics of the system remain valid for arbitrarily strong perturbations. 

In this paper we give a brief review of the relation between microscopic dynamical properties and the Fourier law of heat conduction as well as the connection between anomalous conduction and anomalous diffusion. We then discuss the possibility to control the heat flow. 

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. 

We notice the similarity of the structures of Eq. (101) with the chaos-transport relationship (95). Indeed, both formulas are large-deviation dynamical relationships giving an irreversible property from the difference between two quantities characterizing the dynamical randomness or instability of the microscopic dynamics (see Fig. 16). 

Furthermore, the Onsager symmetry resulting from the time reversibility of the microscopic dynamics stipulates... 

Above Tc the first component/Q< (t) relates to the structural relaxation while below Tc it measures the amount of structural arrest. The second part describes fast motional processes (that would take place in the picosecond range, not accessible by NSE) not related to transport phenomena, is the characteristic time of such fast microscopic dynamics. Concerning the structural relaxation, the following predictions are made ... 

The system is assumed to be in contact with a thermal bath at temperature T. We also assume that the microscopic dynamics of the system is of the Markovian type the probability that the system has a given configuration at a given time only depends on its previous configuration. We then introduce the transition probability Wt(C C ). This denotes the probability for the system to change from C to C at time step k. According to the Bayes formula,... 

The transition probabilities W% C C) cannot be arbitrary but must guarantee that the equilibrium state P C) is a stationary solution of the master equation (5). The simplest way to impose such a condition is to model the microscopic dynamics as ergodic and reversible for a fixed value of X ... 

This relation provides a simple closure to the iGLE in which the microscopic dynamics is connected to the macroscopic behavior. Because of this closure, the microscopic dynamics are said to depend self-consistently on the macroscopic (averaged) trajectory. Formally, this construction is well-defined in the sense that if the true (R(t)) is known a priori, then the system of equations return to that of the iGLE with a known g(t). In practice, the simulations are performed either by iteration of (R(t)) in which a new trajectory is calculated at each step and (R(t)) is revised for the next step, propagation of a large number of trajectories with (R(t)) calculated on-the-fly, or some combination thereof. 

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