Dynamical dimension

The growing computahonal power available to researchers proves an invaluable tool to investigate the dynamic profile of molecules. Molecular dynamics (MD) and Monte Carlo (MC) simulahons have thus become pivotal techniques to explore the dynamic dimension of physicochemical properhes [1]. Furthermore, the powerful computational methods based in parhcular on MIFs [7-10] allow some physicochemical properhes to be computed for each conformer (e.g. virtual log P), suggesting that to the conformahonal space there must correspond a property space covering the ensemble of all possible conformer-dependent property values. 

Sammon, M.P. and Bruce, E.N. 1991. Vagal afferent activity increases dynamical dimension of respiration in rats. J. Appl. Physiol. 70 1748. 

The isotherm of n-butane and 2-butyne sorbed in Silicalite-1 are shown in Figure 11. The amount of 2-butyne sorbed exceeds that of n-butane by -20% at a given pressure and temperature which may be due to bofii the smaller dynamic dimensions of the 2-butyne molecules and the specific interaction of the triple bond wifii the silanol groups which are present in the Silicalite-1 channels and intersections. [7,17] This latter interaction gives an additional lOkJ/mol to the isosteric heat of sorption of 2-butyne over that of n-butane at low loadings. Above loadings of 2 molecules perunit cell (m/uc) die isosteric heats of both sorbates are very similar increasing from 52 to 57 kJmol" because of sorbate-sorbate interactions. 

The corrected diffusion coefficients, Db, obtained using the Darken equation are given in Table 5. The diffiisivities of 2-butyneare shown in this table to be 3-5 times faster in the straight chaimels than those of n-butane and 2-3 times faster in the sinusoidal channels. Both n-butane and 2-butyne diffuse some 6-8 times faster in the straight channels than in the sinusoidal channels. The faster diffusivity of 2-butyne over n-butane is most probably due to a) its smaller dynamic dimensions and b) its inflexibility. 

The present paper steps into this gap. In order to emphasize ideas rather than technicalities, the more complicated PDE situation is replaced here, for the time being, by the much simpler ODE situation. In Section 2 below, the splitting technique of Maas and Pope is revisited in mathematical terms of ODEs and associated DAEs. As implementation the linearly-implicit Euler discretization [4] is exemplified. In Section 3, a cheap estimation technique for the introduced QSSA error is analytically derived and its implementation discussed. This estimation technique permits the desired adaptive control of the QSSA error also dynamically. Finally, in Section 4, the thus developed dynamic dimension reduction method for ODE models is illustrated by three moderate size, but nevertheless quite challenging examples from chemical reaction kinetics. The positive effect of the new dimension monitor on the robustness and efficiency of the numerical simulation is well documented. The transfer of the herein presented techniques to the PDE situation will be published in a forthcoming paper. 

The dynamic dimension reduction algorithm as derived above has been implemented within the linearly implicit Euler discretization with two extrapolation steps only, thus restricting the discretization order to only p = 3. Of course, any other DAE integrator could have been used to demonstrate the performance of the new adaptive technique. For the sake of clarity, we briefly arrange the main steps of the algorithm ... 

Example 1 Hydrogen-Oxygen Combustion [14]. This problem consists of n = 8 chemical species (and ODEs) and 37 elementary chemical reactions. The reaction mechanism is part of a system investigated by U.Maas in [18] and [17]. The problem has been given in [14] as an example, where the traditional QSSA (as mentioned in the Introduction) fails to be applicable. We were therefore interested to see the performance of our dynamic dimension reduction. 

Figure 4.6. Example 3 - Dynamic dimension reduction over two time intervals.

Here, we want to discuss diffusion NMR experiments from a pragmatic point of view in order to show what information can be obtained and how reliable it is, focusing attention on supramolecular objects of intermediate dimensions. In particular, after recalling the principles underlying diffusion NMR spectroscopy and the measurement of the translational self-diffusion coefficient (A) (Section 2), we show how accurate hydrodynamic dimensions can be derived from A once the shape and size of the diffusing particles have been correctly taken into account (Section 3). Later on, the application of diffusion NMR to the study of supramolecular systems is described (Section 4) in terms of determination of the average hydro-dynamic dimensions and thermodynamic parameters of the self-assembly processes. 

While studying dynamic models with three variables we realized, however, that differential flow induced instability may occur in systems of > 3 dynamical dimensions even in the absence of an activator (or unstable subsystem). Such systems cannot exhibit the TI and the origin of the destabilization... 

The calculated is not the actual dimension of the physical or chemical system, but is rather the minimum number of variables that can be used to model the system with acceptable accuracy. For example, a model that is described by an 8-variable ODE, but has dynamical dimension of 2, can be replaced by coupled system of differential and algebraic equations, where the change in values of 2 variables are calculated by ODEs, whilst the values of the other 6 variables can be calculated from these 2 variables using algebraic equations. The actual form of the ODEs or other equivalent time-dependent models can be developed in different ways as will be further discussed in Chap. 7. 

An analysis of the data (e.g. using methods outlined in Sect. 6.5) leads to the determination of highest dynamical dimension... 

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