Time hierarchy

Hierarchical Structure. In order to be better able to simulate the hierarchical nature of many real-world complex systems, in which agent behavior can itself be best described as being the result of the collective behavior of some swarm of constituent agents. Swarm is designed so that agents themselves can be swarms of other agents. Moreover, Swarm is designed around a time hierarchy, Thus, Swarm is both a nested hierarchy of swarms and a nested hierarchy of schedules. 

Thermodynamics of nonequilibrium (irreversible) processes is an extension of classical thermodynamics, mainly to open systems. Unfortunately, the Second Law of classical thermodynamics cannot be applied directly to systems where nonequilibrium (i.e., thermodynamically irreversible) pro cesses occur. For this reason, thermodynamics of nonequilibrium processes has used several principal concepts that are supplementary to the classical thermodynamics postulates. In contrast to the postulates, many of the con cepts in thermodynamics of nonequilibrium processes can be mathe maticaUy substantiated by considering, for example, the time hierarchy of the processes involved. 

The putative time hierarchy means that characteristic time X of a change of relevant thermodynamic parameters of the physically small parts under discussion is much longer than time Xx, of the thermal relaxation in these parts but much shorter than time Xl needed to estabhsh the equibbrium in the entire system ... 

Timing Hierarchy of Processes in Complex Dynamic Systems 299... 

TIMING HIERARCHY OF PROCESSES IN COMPLEX DYNAMIC SYSTEMS... 

In the systems that are far from equilibrium, the stratification into sub systems with fast and slow is also possible, with the subsystem with fast internal variables being characterized by the minimum of the relevant Lyapunov function (provided that such a function exists for the particular process scheme). The ways to describe systems that can be stratified in accordance with the timing hierarchy of the processes involved are under intensive study in modem chemical engineering and biophysics. The methods are based on models that take into account mechanistic (deterministic) and statistical degrees of freedom and their contribution to processes of energy transfer and chemical conversions in the systems with a very complicated process hierarchy (for example, catalytic and biological processes). 

The centre manifold theorem may be regarded as a generalization of the Tikhonov theorem to a case when the time hierarchy does not explicitly occur in the examined system, i.e. when the equations cannot be divided into... 

We shall now show an application of the centre manifold theorem to the system without marked time hierarchy (this is the Duffing equation) ... 

Note that it follows from inequalities (5.129b), (5.129d) that a11, a22 have opposite signs, e.g. an >0. This implies that the x variable is of an autocatalytic nature. Furthermore, since an, a22 are proportional to the inverse time hierarchies associated with the x, y variables (when al2 = 0, a21 = 0, Dx = 0, Dy = 0 x exp(a11t), y exp(a22t)) it follows from inequality (5.129b) that the component X should diffuse more rapidly than the component Y [X] (x = [X], y = [Y]). 

Reaction mode analysis was also used for the investigation of time hierarchies of a biochemical kinetic mechanism that describes the carbohydrate uptake and metabolism of bacterium Escherichia coli (Kremling et al. 2004). The Jacobian was calculated at the steady-state point of the system, and the analysis revealed which reaction steps contribute mainly to the reaction modes having different timescales. 

Kremling, A., Fischer, S., Sauter, T., Bettenbrock, K., Gilles, E.D. Time hierarchies in the Escherichia coli carbohydrate uptake and metabolism. Biosystems 73, 57-71 (2004)... 

Election nuclear dynamics theory is a direct nonadiababc dynamics approach to molecular processes and uses an electi onic basis of atomic orbitals attached to dynamical centers, whose positions and momenta are dynamical variables. Although computationally intensive, this approach is general and has a systematic hierarchy of approximations when applied in an ab initio fashion. It can also be applied with semiempirical treatment of electronic degrees of freedom [4]. It is important to recognize that the reactants in this approach are not forced to follow a certain reaction path but for a given set of initial conditions the entire system evolves in time in a completely dynamical manner dictated by the inteiparbcle interactions. 

In fact, there is a hierarchy in calculating molecular properties by additivity of atomic, bond, or group properties, as was pointed out some time ago by Benson [1, 2]. The larger the substructures that have to be considered, the larger the number of inaements that can be derived and the higher the accuracy in the values obtained for a molecular property. 

Figure 12.4. Hierarchy of length scales of structure and time scales of motion in polymers. Tg denotes the glass transition temperature. After Uhlherr and Theodorou (1998) (courtesy Elsevier...

Each one of these four models can be thought of as defining the response of a black-box to a tape of symbols that is fed to it, cyclically, one cell at a time. During each cycle, the black-box reads the symbol at the appropriate cell, responds to that symbol according to a set of model-specific rules, and then moves the tape forward by one cell. What type of computational model the black-box represents depends on its general response. Although a more detailed discussion of each of these types will be given in chapter 6, we mention here that the four models make up what is conventionally called the Chomsky hierarchy Each type of computational... 

Notice that in this general case, correlation functions cannot be solved for directly instead, there is an entire hierarchy of lower-order correlations expressed as functions of higher-order correlations. For example if we take an average of equation 7.79 over all space-time histories, and assume that we have a steady-state so... 

Hierarchical Structures Huberman and Kerzberg [huber85c] show that 1// noise can result from certain hierarchical structures, the basic idea being that diffusion between different levels of the hierarchy yields a hierarchy of time scales. Since the hierarchical dynamics approach appears to be (on the surface, least) very different from the sandpile CA model, it is an intriguing challenge to see if the two approaches are related on a more fundamental level. 

Hierarchical Control, in which a three-level command hierarchy is established each lower-level echelon element keys on those in the next higher echelon on each time step of the evolution. 

Circadian Rhythms. Figure 1 Mammalian circadian system is organized as a hierarchy of multiple clocks thatfunction to synchronize timing of internal processes between each other and with the environment. See details in the text. 

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