Equilibrium correlation

L 1967. Computer Experiments on Classical Fluids. II. Equilibrium Correlation Functions. tysical Review 165 201-204. 

For the identity reactions, the intrinsic barriers are their free energies of activation, which can be determined by tracer studies or less directly by rate-equilibrium correlations. ... 

This contrasts with relation (5.16), which led to a non-physical conservation law for J. Eqs. (5.28) and Eq. (5.30) make it possible to calculate in the high-temperature limit the relaxation of both rotational energy and momentum, avoiding any difficulties peculiar to EFA. In the next section we will find their equilibrium correlation functions and determine corresponding correlation times. 

Thus the nth vibrational spectral moment is equal to an equilibrium correlation function, the nth derivative of the dipole moment autocorrelation function evaluated at t=0. By using the repeated application of the Heisenberg equation of motion ... 

In the manufacture of methyl ethyl ketone from butanol, the product is separated from unreacted butanol by distillation. The feed to the column consists of a mixture of methyl ethyl ketone, 2-butanol and trichloroethane. What would be a suitable phase equilibrium correlation to use in modelling this process ... 

Table 2 gives rate and equilibrium constants for the deprotonation of and nucleophilic addition of water to X-[6+]. These data are plotted as logarithmic rate-equilibrium correlations in Fig. 5, which shows (a) correlations of log ftp for deprotonation of X-[6+] and log Hoh for addition of water to X-[6+] with logXaik and log KR, respectively (b) correlations of log(/cH)soiv for specific-acid-catalyzed cleavage of X-[6]-OH (the microscopic reverse of nucleophilic addition of water to X-[6+]) and log( H)aik for protonation of X-[7] (the microscopic reverse of deprotonation of X-[6+]) with log Xafc and log XR, respectively. 

This time-independent expression obviously has to be identified with the equilibrium distribution. We thus obtain the following functional relation between the equilibrium correlations (k 0) and the velocity distribution ... 

This dynamical formulation of the equilibrium correlations in an interacting system will be the starting point of our analysis of equilibrium electrolytes. Of course, this method gives results analogous to the more usual methods based on the canonical distribution 40... 

The essential characteristic of the equilibrium correlations is that they originate in a system starting from non-correlated states. We recall also that the correct form of the equilibrium correlations can be obtained if one admits that for long times the velocity distribution function takes a Maxwellian form. 

Finally, if Eq. (50) is admitted, one can show that (48) gives the correct form for the equilibrium correlations 1 12.24 the dynamical approach (48) is then equivalent to the expansion in equilibrium clusters (see, for example, ref. 13). 

Figure 1.3. Rate-equilibrium correlation for hydration of carbocations triarylmethyl and 9-aryl-9-fluorenyl ( and , respectively, slope =—0.60) diarylmethyl (A, —0.54) aryltropylium (O, —0.68) 9-xanthylium and cyclic phenyldialkoxycarbocations ( and A, respectively, slope = —0.63).

The adsorption of diatomic or dimeric molecules on a suitable cold crystalline surface can be quite realistically considered in terms of the dimer model in which dimers are represented by rigid rods which occupy the bonds (and associated terminal sites) of a plane lattice to the exclusion of other dimers. The partition function of a planar lattice of AT sites filled with jV dimers can be calculated exactly.7 Now if a single dimer is removed from the lattice, one is left with two monomers or holes which may separate. The equilibrium correlation between the two monomers, however, is appreciable. As in the case of Ising models, the correlation functions for particular directions of monomer-monomer separation can be expressed exactly in terms of a Toeplitz determinant.8 Although the structure of the basic generating functions is more complex than Eq. (12), the corresponding determinant for one direction has been reduced to an equally simple form.9 One discovers that the correlations decay asymptotically only as 1 /r1/2. 

The major advantage of the reactive flux method is that it enables one to initiate trajectories at the barrier top. instead of at reactants or products. Computer time is not wasted by waiting for the particle to escape from the well to the barrier. The method is based on the validity of Onsager s regression hypothesis,97 98 which assures that fluctuations about the equilibrium state decay on the average with the same rate as macroscopic deviations from equilibrium. It is sufficient to know the decay rate of equilibrium correlation functions. There isn t any need to determine the decay rate of the macroscopic population as in the previous subsection. 

Another way of obtaining the characteristic time scale and dynamical range of conformational dynamics is from the equilibrium correlation functions of the FRET efficiency ... 

An attempt to solve the difficulties and inconsistencies arising from an approximated derivation of quantum-classical equations of motion was made some time ago [15] to restore the properties that are expected to hold within a consistent formulation of dynamics and statistical mechanics, and are instead missed by the existing approximate methods. We refer not only to the properties that the Lie brackets, which generate the dynamics, satisfy in a full quantum and full classical formulation, e.g., the bi-linearity and anti-symmetry properties, the Jacobi identity and the Leibniz rule12, but also to statistical mechanical properties, like the time translational invariance of equilibrium correlation functions [see eq.(8)]. 

H. Yamataka, S. Nagase, J. Org. Chem. 53, 3232 (1988). Ah Initio Calculations of Hydrogen Transfers. A Computational Test of Variations in the Transition-State Structure and the Coefficient of Rate-Equilibrium Correlation. 

Let us note that formula (4.33) is a generalisation of the equilibrium correlation function of the normal co-ordinates of the macromolecule in a viscous liquid... 

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

These are exactly the known results (Doi and Edwards 1986, p. 196). The time behaviour of the equilibrium correlation function is described by a formula which is identical to formula for a chain in viscous liquid (equation (4.34)), while the Rouse relaxation times are replaced by the reptation relaxation times. In fact, the chain in the Doi-Edwards theory is considered as a flexible rod, so that the distribution of relaxation times naturally can differ from that given by equation (4.36) the relaxation times can be close to the only disentanglement relaxation time r[ep. 

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