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Separation of time scales

Most biochemical reactions are catalyzed by enzymes. Therefore, although quite instructive, the model for the birth-death process studied in the previous chapter is not good enough an approximation in many instances. Typically, an enzymatic process consists of a series of chemical reactions that occur at different rates, and in some occasions it is possible to identify two well-separated time scales. When this occurs, the time-scale separation can be exploited to simplify the analysis of the whole system. Below we introduce a methodology to perform such simplification. [Pg.51]

Consider a system in which molecules of N different chemical species are involved in M chemical reactions. The state of such a system is determined by the set of all the chemical-species molecule counts ni, n2... njv. Regardless of the value of N and the maximum values of ni, n2... njv, the set of all the possible system states is discrete, and in consequence the states can be enumerated. Thus, let us assume that x = 1,2... labels of all the available states. [Pg.51]

When an individual chemical event takes place, the molecule counts of some of the chemical species change, and so does the system state. Taking this into account. [Pg.51]

Santillan, Chemical Kinetics, Stochastic Processes, and Irreversible Thermodynamics, Lecture Notes on Mathematical Modelling in the Life Sciences, [Pg.51]

In the equation above, the probabilities, P, and the propensities, k, have equivalent meanings as those in Eq. (5.1). [Pg.52]


For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... [Pg.851]

Each of the approaches is based on the premise that it makes sense to focus on the Born Oppenheimer potential for the OH stretch for fixed bath variables. Such a potential has vibrational eigenvalues, and for example h times the transition frequency of the fundamental is simply the difference between the first excited and ground state eigenvalues. Thus in essence this is an adiabatic approximation the assumption is that the vibrational chromophore is sufficiently fast compared to the bath coordinates. To the extent that the h times frequency of the chromophore is large compared to kT, and those of the bath are small compared to kT, this separation of time scales exists and so this should be a reasonable approximation. For water, as discussed earlier, some of the bath variables (librations) have frequencies somewhat larger than kT/h, and... [Pg.70]

The propagation in phase space can also be probed in the frequency domain via the vibrational overtone spectra of polyatomic molecules. [30, 34] The separation of time scales can be demonstrated in this fashion and is useful in representing such spectra by the maximal entropy method. [30] Further details can be found in two recent tutorial reviews in Refs. [23] and [24]. [Pg.216]

The fact that there are now two equations, viz. (8.12) and (8.9), implies that no longer is y(t) determined by /(0) alone, but by the initial vaue of a2 as well. One might hope that, after a short initial transient time, o2 adjusts itself by rapidly approaching an asymptotic value depending on the instantaneous y(t) alone, so that a renormalized equation for / holds after the initial transient. However, this is not the case the time scale on which a2 approaches (8.10) is determined by the coefficient a in (8.9) and is therefore comparable to the rate at which / itself varies, see (8.7). There is no separation of time scales and therefore no single equation for / by itself. [Pg.126]

SEPARATION OF TIME SCALES IN THE DYNAMICS OF HIGH MOLECULAR RYDBERG STATES... [Pg.625]

To discuss the separation of time scales, we begin with the argument that a system that reaches the continuum via a narrow bottleneck can exhibit more than one time scale [45a,b,f, 51]. Particular attention will be given to the question of when this will be the case. The argument begins by considering the time evolution in the bound subspace. As is well known [52,53, 54], one can confine attention to the bound levels by the introduction of an effective Hamiltonian H in which the coupling to the continuum is accounted foT by a rate operator T ... [Pg.636]

The classical trajectory simulations of Rydberg molecular states carried out by Levine ( Separation of Time Scales in the Dynamics of High Molecular Rydberg States, this volume) remind me of the related question asked yesterday by Prof. Woste (see Berry et a]., Size-Dependent Ultrafast Relaxation Phenomena in Metal Clusters, this volume). Here I wish to add that similar classical trajectory studies of ionic model clusters of the type A B have been carried out by... [Pg.657]

Prof. Levine ( Separation of Time Scales in the Dynamics of... [Pg.657]

At the heart of the mode coupling theory of liquids is the assumption that a separation of time scale exists between different dynamical events. While the time scale separation between the fast collisional events and the slower collective relaxation is explicitly exploited in the formulation of the theory, there is also an underlying assumption of the separation of length scales between different relaxation modes. Much of the success of MCT depends on the validity of this separation of length and time scales. [Pg.71]

Note that in deriving the contribution from the density fluctuation to the total viscosity, terms of order t2 has not been taken out. In the initial argument of separation of time scale, it was stated that contributions from terms up to order t2 should be included only in the binary term (>yf), and the collective contribution term was expected to start as f4. Thus to take out all the contributions of order t2 from Eq. (196), the short-time dynamics has to be taken out from the propagator as has been done in case of friction. This is achieved by taking out the short-time dynamics from (F(q,t)/S(q))2. Thus the corrected expression for rjspp can now be written as... [Pg.124]

In their important work, Schweizer and Chandler [123] assumed a separation of time scales between the attractive and the repulsive forces. Based on this argument, they could analyze the respective roles of the attractive and the repulsive contributions and were the first to point out that the vibrational dephasing is largely controlled by the attractive forces. MCT theory, on the... [Pg.173]

An issue that has been explored is how the relative distribution of charge and mass affect the viscosity of an ionic liquid. Kobrak and Sandalow [183] pointed out that ionic dynamics are sensitive to the distance between the centers of charge and mass. Where these centers are separated, ionic rotation is coupled to Coulomb interactions with neighboring ions where the centers of charge and mass are the same, rotational motion is, in the lowest order description, decoupled from an applied electric field. This is significant, because the Kerr effect experiments and simulation studies noted in Section III. A imply a separation of time scales for ionic libration (fast) and translation (slow) in ILs. Ions in which charge and mass centers are displaced can respond rapidly to an applied electric field via libration. Time-dependent electric fields are generated by the motion of ions in the liquid... [Pg.104]

If a clear separation of time scales exists for Eqs. 4.52a and 4.52b, as compared to Eqs. 4.52c-4.52e, then the kinetics of surface oxidation-reduction can be decoupled from those of surface species detachment. For example, if Mn(II) oxidation is negligibly slow, Kb - 0 and Eqs. 4.52a and 4.52b can be solved approximately, as is described in either Section 3.4 in connection with Eqs. 3.48 and 3.49, or Section 4.3 in connection with Eq. 4.30. The first approach applies to constant [HSeO ], whereas the second one requires small deviations of the concentrations of the four chemical species in Eqs. 4.52a and 4.52b from their equilibrium values. [Pg.161]

As expected, we find that the total response function Xa=i Ri = Xa=4 Ri = J2i=1 Ri = 0 (i.e., for each possible time ordering) vanishes exactly in the harmonic case, defined by A = 0 and /x2i 2 = 2 /r10 2. Furthermore, it can be easily seen that in the case of a strict separation of time scales of homogeneous and inhomogeneous broadening, the line shape function becomes g(t) = t/T2 + a212/2 and the total response function reduces exactly to the result obtained within a Bloch picture (see, for example Refs. 52 and 75), e.g.,... [Pg.299]


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See also in sourсe #XX -- [ Pg.85 ]




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