Entropy micro concept

The vectors of Bf express f in the original basis h, see Eqs. (28-29). To prepare for a discussion of proton tunnelling in DNA we will briefly deliberate on the concept of the micro entropy. A direct generalisation of Gibbs entropy follows from the definition below (Coleman s normalisation [27])... 

Generally speaking, the degrees of freedom in many-body systems, such as Ar7, are too many to analyze the phase-space dynamics, and only limited methods originally developed to investigate chaotic systems with a few degrees of freedom can be applicable for the analysis. Seko et al. calculated the phase volume—that is, the configuration entropy—of Ar7 and proposed a new concept of the temperature in micro-clusters based on this phase volume [17], A phase-space analysis seems to be prospective even for many-body systems, such as Ar7. However, most of the currently available methods concern statistical properties. The methods and quantities that are directly related to the dynamics are expected for a detailed analysis. 

The HLD concept has been recently related to the so-called net-average curvature which indicates the size of the oil and water domains in the micro emulsion. For marginal microemulsions, i.e. of the WI or WII type at some distance from optimum, the inverse of the swollen micelle Sauter diameter is proportional to HLD. The zero net curvature at optimum does not result from infinite radius but rather from the coexistence of finite curvatures of opposite signs. For bicontinuous micro emulsions, it is the inverse of the characteristic length which is maximised at HLD = 0. As discussed elsewhere [38], its value at optimum formulation is the maximum distance that a molecule of oil or water can be separated from the surfactant layer and still interacts with it. In other words, it is the length at which the molecular interaction becomes equal to the molecular entropy. 

The constant growth of entropy postulated in the second law is closely related to the following principle By the irreversible process the system is spontaneously directed towards the most frequently occurring set of micro states which corresponds to the same macro state of the system. The macroscopic state of equilibrium - where entropy is at its maximum - is the state which is linked to the largest number of different micro states. This basic concept is the basis of the Boltzmann relation that defines the entropy function S in statistical thermodynamics. 

Through the Boltzmann relation, the macroscopic entropy concept of classical thermodynamics is connected with the molecular statistic quantity 1 that specifies the number of micro states corresponding to the same macro state. In many cases, this relationship can be used for simple, qualitative estimates of the properties and behaviour of the substances. 

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