Slow relaxation dynamics

Fortunately, however, the states that are sampled are the most probable states of the system which make the maximum contribution to the thermodynamic quantities. Thus, we obtain the average values (entropy, specific heat) of the thermodynamic quantities with virtually no error. However, to achieve that kind of accuracy, our system still needs to sample a large number of representative states. The sampling of such a large number of states takes a longer time (due to slow relaxation dynamics) at low temperatures, and the utmost care is needed. 

Li TP, Hassanali AA, Singer SJ (2008) Origin of slow relaxation following photoexcitation of W7 in myoglobin and the dynamics of its hydration layer. J Phys Chem B 112(50) 16121-16134... 

Some gadolinium-based MOF have also revealed the peculiar dynamics of the magnetization with the appearance of the out-of-phase component of the dynamic magnetization in presence of an applied static magnetic field. This behaviour, which is reminiscent of the field-induced slow relaxation that characterize many lanthanide-based SMMs, is however not related to the magnetic anisotropy, which is vanishingly small in most Gd3+ systems. 

Mechanistic Ideas. The ordinary-extraordinary transition has also been observed in solutions of dinucleosomal DNA fragments (350 bp) by Schmitz and Lu (12.). Fast and slow relaxation times have been observed as functions of polymer concentration in solutions of single-stranded poly(adenylic acid) (13 14), but these experiments were conducted at relatively high salt and are interpreted as a transition between dilute and semidilute regimes. The ordinary-extraordinary transition has also been observed in low-salt solutions of poly(L-lysine) (15). and poly(styrene sulfonate) (16,17). In poly(L-lysine), which is the best-studied case, the transition is detected only by QLS, which measures the mutual diffusion coefficient. The tracer diffusion coefficient (12), electrical conductivity (12.) / electrophoretic mobility (18.20.21) and intrinsic viscosity (22) do not show the same profound change. It appears that the transition is a manifestation of collective particle dynamics mediated by long-range forces but the mechanistic details of the phenomenon are quite obscure. 

A. V. Einkelstein, Proteins structural, thermodynamic and kinetic aspects, in Slow Relaxations arul Nonequilibrium Dynamics (J. L. Barrat and J. Kurchan, eds.) Springer-Verlag, Berlin, 2004, pp. 650-703. 

Garrahan and Chandler [230] have recently attempted to rationalize the string-like motion in supercooled liquids based on a completely different concept of dynamic facilitation, derived from the study of magnetic spin models originally developed by Fredrickson and Anderson [231]. Although these spin models seem to exhibit dynamic heterogeneity of some kind and slow relaxation processes, the slowing down of the dynamics in these models is entirely decoupled from the spin model s thermodynamics [116, 230]. In view... 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

A detailed study of model (16) for CO oxidation on polycrystalline platinum was carried out by Makhotkin et al. [139]. Numerical experiments revealed that the bulk diffusion effect on the character of reaction dynamics is rather different and controlled by the following factors (1) the initial composition of catalyst surface and bulk, (2) the steady state of its surface and bulk, and (3) the position of the region for slow relaxations of kinetic origin (see ref. 139). As a rule, diffusion retards the establishment of steady states, but the case in which the attainment of this state is accelerated by diffusion is possible. 

The results of the numerical experiment for system (20) necessitated a general mathematical investigation of slow relaxations in chemical kinetic equations. This study was performed by Gorban et al. [226-228] who obtained several theorems permitting them to associate the existence of slow relaxations in a system of chemical kinetic equations (and, in general, in dynamic systems) with the qualitative changes in the phase portrait with its parameters (see Chap. 7). 

To interpret the problem under discussion concerning slow relaxations in chemistry, it was necessary to clarify what must be regarded as slow relaxations of dynamic systems (i.e. to introduce some reasonable definition). In addition, it was necessary to find connections of slow relaxations with bifurcations and other dynamic peculiarities. This has been done by Gorban et al. [13-19]. 

In this chapter we will suggest a theory of transition processes and slow relaxations in dynamic systems. The inclusion of such mathematical sections to a book on chemical kinetics is dictated by the necessity to understand the details of slow transition processes in the absence of a comprehensive and clear representation of the theory of slow relaxations. 

The majority of the above examples are non-rough (structurally unstable) systems. The rough dynamic systems on the plane cannot demonstrate the properties shown by the above examples. If Tt is specified by a rough individual (without parameters) system on the plane, there cannot exist th, rj2 slow relaxations and rh 2,3 and tj3 slow relaxations can take place only simultaneously. This can be confirmed by the results given below and the data of some classical studies concerning smooth rough two-dimensional systems [20, 21],... 

There are examples when t2, t 2 slow relaxations take place without bifurcations. So far, complete characteristics of these slow relaxations in terms of the limit behaviour for a dynamic system (that is dependent on the parameters) has not been obtained. Only some of the additional sufficient conditions have been defined. 

For t3, tj3 slow relaxations, the necessary and sufficient conditions have been obtained in terms of the limit behaviour of dynamic systems. Note that the (x, -motion is called positively Poisson-stable (P+-stable) if xeco(x, k). 

In the two-dimensional case (two variables) "almost any C1-smooth dynamic system is rough (i.e. at small bifurcations its phase pattern deforms only slightly without qualitative variations). For rough two-dimensional systems, the co-limit set of every motion is either a fixed point or a limit cycle. The stability of these points and cycles can be checked even by a linear approximation. Mutual relationships between six different types of slow relaxations for rough two-dimensional systems are sharply simplified. 

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