Entropy representation

The two equivalent, but different choices of independent variable, U or S, are distinguished as energy and entropy representations, respectively. 

Hence, when Tk is zero the system is in equilibrium, but for non-zero Tk an irreversible process that takes the system towards equilibrium, occurs. The quantity Tkl which is the difference between intensive parameters in entropy representation, acts as the driving force, or affinity of the non-equilibrium process. 

The affinities that define the rate of entropy production in continuous systems are therefore gradients of intensive parameters (in entropy representation) rather than discrete differences. For instance, the affinities associated with the z-components of energy and matter flow for constituent k, in this notation would be... 

As in energy representation the fundamental thermodynamic equation in entropy representation (3) may also be subjected to Legendre transformation to generate a series of characteristic functions designated as Massieu-Planck (MP) functions, m. The index m denotes the number of intensive parameters introduced as independent variables, i.e. 

In essence, we wish to follow the Gibbs entropy maximization procedure of Section 5.2 backward in time. Specifically, we seek to characterize the final stage of equilibration when effective local equilibrium has been achieved in each small cell n, but cell intensities are not yet equalized throughout the system. To make direct contact with Section 5.2, we shall temporarily revert to the entropy representation (Section 5.3), which generates a scalar product and metric geometry that is conformally equivalent to the (/-based metric... 

Mayer et al. begin with the fundamental equation in the Boltzmann-reduced entropy representation... 

The next steps are the following Step 1 Passage to the entropy representation and specification of the dissipative thermodynamic forces and the dissipative potential E. Step 2 Specification of the thermodynamic potential o. Step 3 Recasting of the equation governing the time evolution of the np-particle distribution function/ p into a Liouville equation corresponding to the time evolution of np particles (or p quasi-particles, Up > iip —see the point 4 below) that then represent the governing equations of direct molecular simulations. 

Equations of state relate intensive properties to extensive properties, and are obtained from the Euler equation as partial derivatives. In the entropy representation, we have the following equations of state ... 

The ultimate goal of a thermodynamic description of molecular systems, however, is to determine the horizontal displacements of the electronic structure (see Section 7), i.e., transitions from one v-representable molecular density to another. In order to relate the information entropy H[p], possibly involving also the reference densities (equation (92)), to the system energetic parameters one uses the generalized variational principle in the entropy representation [108] ... 

As we have demonstrated in this survey, there is a wide range of problems in the theory of electronic structure and chemical reactivity, which can be tackled using concepts and techniques of the density functional and information theories. These two descriptions are complementary in character, providing the energy and entropy representations of molecular systems, respectively. Together they constitute the... 

Figure 19 The Raman spectrum and time cross correlation function when the motion on the excited electronic state potential is anharmonic, compare to Figs. 17 and 18, which are for a harmonic approximation. (Top, a) Computed time correlation function using a wide window function (b) The maximal entropy representation of this function, determined from the spectrum. Note the clear separation of time scales due to the anharmonicity (cf. Fig. 20). (Bottom) The Raman excitation spectrum obtained from the computed time correlation function (a). The arrows are the sequence of computations (a) is determined from the dynamics. The spectrum is determined from (a). The maximum entropy cross-correlation function (b) uses only the spectrum as input.

Illustrative Problem. Begin with the entropy representation of the fundamental equation of thermodynamics for a multicomponent system, S(U, V, all tV,), and derive the Gibbs-Duhem equation. Does this form differ from equation (29-45) in step 4 above ... 

Therefore, the entropy representation of the reaction mechanism reveals the whole complexity of this transformation, while the associated Minimum Energy Path (MEP) profile only localizes the transition state on PES, missing the crucial transitory localization/delocalization and relaxational phenomena involved in this two-step process. 

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