Entropy deficiency

The quantitative measure of relaxation towards a thermal distribution is the entropy deficiency which in our case is given by... 

Here v is the (scalar) velocity, f v) is the normalized three-dimensional velocity distribution as determined by the molecular dynamics simulation at a given point in time. f u) is the three dimensional Maxwell-Boltzmann velocity distribution with a temperature determined by the condition that / (i/) has the same mean energy as the velocity distribution obtained in the simulation after a long propagation time. At thermal equilibrium DS = 0 and otherwise it is positive. The larger is DS, the more extreme is the deviation from equilibrium. The results for the entropy deficiency are shown in... 

Fig. 24. The entropy deficiency, Eq. (9) vs. time for clusters of different sizes. For thermalization DS —r 0 and the value of zero is achieved if and only if the system has equilibrated. When most of the atoms are at or near the surface the relaxation is fairly extensive but is not completed before the cluster shatters.

The necessity of the prior distribution pj in Eq. (72) has been rationalized by minimizing the so-called entropy deficiency (or cross entropy [45])... 

The right-hand side of (3.34) is, by definition, the entropy deficiency, DS. It is nonnegative and vanishes if and only if pf = p°. The fraction of the available final states that are actually accessed is... 

Minimum entropy deficiency partitioning of many-electron densities 155... 

It has been shown [20], that this partitioning minimizes the information distance (missing information, entropy deficiency, directed divergence) [107],... 

MINIMUM ENTROPY DEFICIENCY PARTITIONING OF MANY-ELECTRON DENSITIES... 

Postulate II. There exists a functional AW[plp°], called the entropy deficiency (directed divergence or divergence), of the extensive subsystem parameters pa of a composite molecular system, defined for all equilibrium (Hirshfeld) divisions of p and having the following property. The values assumed by pa in the absence of the internal constraints are those that minimize A77 plp° over the manifold of constrained equilibrium states, the latter being defined by the effective subsystem external potentials (v ff(r) = v[fr p r. ... 

Postulate III. The information-entropy deficiency of a composite system and its density are additive over the constituent components. 

An important property of the stockholder molecular fragments is manifested by their equalization of the local values of a given measure of the entropy deficiency density at the corresponding global value, for the system as a whole [30,33]. An example of such locally equalized information quantities are the subsystem-independent local enhancement factors of the Hirshfeld subsystem densities (equation (91a)) ... 

An alternative measure of the subsystem local information distance relative to the corresponding reference density is defined by the entropy deficiency intensive conjugate of the fragment electron density ( extensive state-function). The Kullback-Lei bier functional of equations (92) and (93) gives ... 

We have introduced in the Euler equations (134) the entropy deficiency conjugate of the subsystem overall number of electrons,... 

A similar relation can be derived for the partitioning of the promolecular FF, by reversing the roles of the Hirshfeld and the free reactant densities in the minimum entropy deficiency principle, so that now pH(r) play the role of the reference densities, while p°(r) are the optimum densities satisfying the constraint XaP°(r) = P°(r) ... 

The same proportionality relations follow from the entropy deficiency mles, in which the Kullback-Leibler functional is formulated directly in terms of the... 

Several important properties of these stockholder atoms, the entropy deficiency equilibrium and stable pieces of the molecular electron density, make them... 

The different efficiencies of chemical lasers governed by different kinetic coupling schemes can be derived from a general statistical-thermodynamic approach to work processes in nonequilibrium molecular systems " . The two major components of this approach are the maximum entropy principle and the entropy deficiency function. The entropy deficiency is a generalized thermodynamic potential (free energy). That is, it decreases monotonically in time in spontaneous relaxation processes and provides an upper bound to the thermodynamic work performed by the system in a controlled process. For systems of weakly interacting molecules the entropy deficiency DS[X X ] is given by... 

The surprisal is a measure of the deviation of a single population from its expected value. The two parameters A , and exp(Aa), or similar parameters for other distributions, characterize the deviation of the whole distribution from prior expectation when this can be described by equation (30). Finally, the average vaiue of this deviation can be calculated and it is known as the entropy of the distribution. The entropy deficiency 2,43,93 vibrational distribution is then d ned by... 

The greater the specificity of energy disposal in an exothermic reaction, then the larger is its entropy deficiency. Values of A., and KS,. for some well-studied atom-transfer reactions are listed in Table 1 is the av age fraction of available energy that is released as product vibration, i.e. 

An important generalization of Shannon entropy, called relative (cross) entropy (also known as entropy deficiency, missing information, or directed divergence), has been proposed by Kullback and Leibler [5] and Kullback [6]. It measures the information distance between two (normalized) probability distributions for the same set of events ... 

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