Average reaction force

The mean force therefore includes a contribution from the average mechanical force and another contribution arising from the variations of the volume element associated with the reaction coordinate The free energy change between two states and 2 can be obtained by integrating (23) according to... 

The latter is a well-known quantity in the reaction-diffusion analysis in catalytic media (see Section 8.2.3) and can be written as the ratio between the average reaction rate over the washcoat cross-sectional area at a given axial position and its value at the surface. The former compares the driving force for mass transfer toward the coating, with the total potential for concentration decay (due to mass transfer and surface reaction). For a first-order reaction, 0 reduces to the Carberry number (see Chapter 3), and >/ is a concentration ratio between the averaged value inside the washcoat and the one at the surface. 

If the pressure is on the average AP larger on the left side than on the right side, the horizontal reaction force over length L is ... 

We have derived the ABC formulation of quantum reactive scattering theory, and applied it to the calculation of the initial state selected reaction probability. By exploiting the highly localized nature of forces in reactive scattering, the ABC formulation facilitates the direct calculation of detailed or averaged reaction probabilities while sampling only a finite region of space. 

Extracted parameters of kinetics and EMG data are presented in Table 1. Ground reaction forces were measured in Newton whereas contact time was measured in second. EMG-based average rectified voltage was measured in mV. 

Table 1 Extracted parameters of kinetics and EMG data example of spastic diplegia (Ft and Fd are the ground reaction forces in heel strike and toe off phases respectively Tt and Td are the times in heel strike and toe off phases respectively L and R are left and right sides EMGafv is average rectified voltage)...

The mass distributions of all subjects based on the difference of ground reactions forces of heel strike and toe off phases (Table 2) and EMG-based average rectified voltage were calculated. The mass distribution of the combination of the difference of ground reactions forees of heel strike and toe off phases and EMG-based average rectified voltage was carried out as input data in order to develop the belief decision tree of the cerebral palsy. 

The belief decision tree of cerebral palsy showed an impact of the ground reactions forces and EMG-based average rectified voltage for classifying two types of cerebral palsy such as spastic diplegia and hemiplegia. The combination of belief functions and standard decision tree permitted to develop a universal formalism in case of uncertain classification. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

Where A F(z) is the free energy at z relative to that at the reactant state minimum zr, and the ensemble average < > is obtained by a quantum mechanical effective potential [15]. Note that the inherent nature of quantum mechanics is at odds with a potential of mean force as a function of a finite reaction coordinate. Nevertheless, the reaction coordinate function z[r] can be evaluated from the path centroids r, first recognized by Feynman and Flibbs as the most classical-like variable in quantum statistical mechanics and later explored by many researchers [14, 15]. 

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