Statistic operator dynamic

Equation (30) is quite a beguiling expression. For example, in the classical expression for N(E), equation (12), there is a statistical factor 5 E — H), the flux factor F, and a dynamical factor X- A similar structure exist in the quantum expres.sion, equation (26), where the dynamical factor is the projection operator Xf The manipulations following equation (27), however, lead to the result, equation (30), which appears to have no dynamical information, i.e., only the statistical operator S(E — H) and flux operator F are involved in equation (30). This is an example of the fact that dynamics and statistics are inseparably intertwined in quantum mechanics, e.g., a wave-function describes the dynamical motion of the particles and also their statistics. 

Some authors have described the time evolution of the system by more general methods than time-dependent perturbation theory. For example, War-shel and co-workers have attempted to calculate the evolution of the function /(r, Q, t) defined by Eq. (3) by a semi-classical method [44, 96] the probability for the system to occupy state v]/, is obtained by considering the fluctuations of the energy gap between and 11, which are induced by the trajectories of all the atoms of the system. These trajectories are generated through molecular dynamics models based on classical equations of motion. This method was in particular applied to simulate the kinetics of the primary electron transfer process in the bacterial reaction center [97]. Mikkelsen and Ratner have recently proposed a very different approach to the electron transfer problem, in which the time evolution of the system is described by a time-dependent statistical density operator [98, 99]. 

I now consider statement 3 How should an extension of dynamics be understood In the MPC theory the problem does not exist For the intrinsically stochastic systems there is no need for modifying the laws of dynamics. As for the LPS theory, one notes the presence of two essentially new concepts. The introduction of non-Hilbert functional spaces only concerns the definition of the states of the dynamical system, and not at all the law governing their evolution. It is an important precision introduced in statistical mechanics. The extension of dynamics thus only appears in the operation of regularization of the resonances. This step is also the one that is most difficult to justify rigorously it is related to the (practical) necessity to use perturbation calculus (see Appendix). 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

The goal of most simulation scenarios is to evaluate some aspect of a system s behavior or performance and to quantify, typically in terms of statistics, the results and then use the results for decisions. This requires a simulation environment with software tools that provide insight into a model s dynamic operation. 

We will delay a more detailed discussion of ensemble thermodynamics until Chapter 10 indeed, in this chapter we will make use of ensembles designed to render the operative equations as transparent as possible without much discussion of extensions to other ensembles. The point to be re-emphasized here is that the vast majority of experimental techniques measure molecular properties as averages - either time averages or ensemble averages or, most typically, both. Thus, we seek computational techniques capable of accurately reproducing these aspects of molecular behavior. In this chapter, we will consider Monte Carlo (MC) and molecular dynamics (MD) techniques for the simulation of real systems. Prior to discussing the details of computational algorithms, however, we need to briefly review some basic concepts from statistical mechanics. 

Equation (139) was already discussed elsewhere [22,23,31] as a phenomenological representation of the dynamic equation for the CC law. Thus, Eq. (139) shows that since the fractional differentiation and integration operators have a convolution form, it can be regarded as a consequence of the memory effect. A comprehensive discussion of the memory function (137) properties is presented in Refs. 22 and 23. Accordingly, Eq. (139) holds for some cooperative domain and describes the relaxation of an ensemble of microscopic units. Each unit has its own microscopic memory function m (t), which describes the interaction between this unit and the surroundings (interaction with the statistical reservoir). The main idea of such an interaction was introduced in Refs. 22 and 23 and suggests that mg(f) JT 8(f,- — t) (see Fig. 50). 

In addition to the above mentioned dynamic problems of copolymerization theory this review naturally dwells on more traditional statistical problems of calculation of instantaneous composition, parameters of copolymer molecular structure and composition distribution. The manner of presentation of the material based on the formalism of Markov s chains theory allows one to calculate in the uniform way all the above mentioned copolymer characteristics for the different kinetic models by means of elementary arithmetical operations. In Sect. 3 which is devoted to these problems, one can also find a number of original results concerning the statistical description of the copolymers produced through the complex radical mechanism. 

Examples of calculations of such characteristics for the concrete terpolymer are presented, for instance, in the monograph listed as Ref. [6]. Similar computer calculations of the trajectories i(p) and X(p) which determine the values of the statistical characteristics (5.3) and (5.7) can be in principle carried out for other processes of multicomponent copolymerization. However, the original approach [6, 201, 13-15, 18, 202, 203] allows one to use traditional methods of dynamic systems theory [204] to reveal the main qualitative peculiarities of the behavior of the trajectories X(p) and X(p) only via the elementary arithmetical operations by means of a pocket microcalculator. 

This result is purely statistical. Replacing the distribution function by particular expressions, depending on the temperature, is the last operation When a dynamical process occurs the equilibrium distribution function (maxwellian) should be modified, and the greater the reaction rate compared to the relaxation rates of both the velocities and the intramolecular states, the greater the modification Thus it is only for low reaction rates that equilibrium distribution functions can be inserted in the formulas above, and that the reaction rate depends on the temperature, but neither on the time nor on the concentrations. 

Stochastic modelling has been developing exponentially in all the domains of scientific research since 1950, when the initial efforts for the particularization of the stochastic theory in some practical domains were carried out. In 1960, James R. Newman, who was one of the first scientists in modern statistical theory, wrote the following about the stochastic theory particularization Currently in the period of dynamic indetermination in science, there is a serious piece of research that, if treated realistically, does not involve operations on stochastic processes [4.8]. 

Any discretization of the transfer operator will suffer from the curse of dimension whenever it is based on a uniform partition of all of the hundreds or thousands of degrees of freedom in a typical biomolecular system. Fortunately, chemical observations reveal that -even for larger biomolecules- the curse of dimensionality can be circumvented by exploiting the hierarchical structure of the dynamical and statistical properties of biomolecular systems ... 

The best physical model is the simplest one that can explain all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. 

Role of quantum statistics. When considering complex systems by methods of statistical physics, one operates with their time-dependent distributions. In fermionic systems (see Yu. Ozhigov), statistical requirements imply that we must replace the independent-particle description by a quasiparticle formalism for quantum information processing. Effects of statistical fluctuations on coherent scattering processes (see M. Blaauboer et al.) suggest the need for furher exploration of the role of statistics on the dynamics of entangled systems. 

While considering ensembles of identical particles, one has to take into account their quantum statistical properties. This imposes additional symmetry restrictions on the ensemble dynamics and requires the inclusion of additional symmetry operations into the group of operations. This situation is discussed in the second paper of the chapter for Fermi systems, where the additional operation is antisymmetrization over the wave functions of individual fermionic qubits. 

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