Entropy, defined

SbS = Smea.s B) (compare to the Kolmogorov-Sinai entropy, defined in equation 4.101). 

An alternative method, which uses the concept of maximum entropy (MaxEnt), appeared to be a formidable improvement in the treatment of diffraction data. This method is based on a Bayesian approach among all the maps compatible with the experimental data, it selects that one which has the highest prior (intrinsic) probability. Considering that all the points of the map are equally probable, this probability (flat prior) is expressed via the Boltzman entropy of the distribution, with the entropy defined as... 

The expressions in the two parentheses can be identified as the surface excess moles and surface excess entropy defined by eqs. (6.2) and (6.5). Equation (6.12) thus reduces to... 

Here, tr is a dominant structural relaxation time of the liquid (e.g., viscosity p cx tr), T is temperature, Sc is the molar configurational entropy (defined... 

Chemical considerations favour eight coordination for Sr " ", but which of the graphs. Figs 11.3(b), (c), or (d), is the most symmetric Rao and Brown (1998) proposed that the entropy, defined by eqn (11.2), can be used as a measure of the degree of symmetry in these cases ... 

These two additional properties of the H in (5.6), together with its mono-tonic decrease, has led to its identification with the entropy defined by the second law of thermodynamics. It must be realized, however, that H is a functional of a non-equilibrium probability distribution, whereas the thermodynamic entropy is a quantity defined for thermodynamic equilibrium states. The present entropy is therefore a generalization of the thermodynamic entropy the generalized entropy is... 

The information entropy defined by Eq. (1.10) is as important as that defined by Eq. (1.3) in chemical engineering because there are a number of phenomena that are controlled by time as a variable in other fields of engineering as well. 

AAhydntion hydration entropy defined as change in entropy from hypothetical corn-... 

Figure 15. Plot of the excess entropy versus the reduced density for three different temperatures. The solid line represents the results from the BB approximation conjugated with the direct formula of Lee, while the open circles correspond to the Johnson et al. [34] data. The dash line stands for the pair contribution S to the excess entropy defined by Eq. (85). Taken from Refs. [69,70],...

The reader is referred to Ashcroft and Mermin (1981) for the partition function expression, as well as the vibrational and entropy defined in terms of Z. 

Before we discuss entropy, define reversible process and reversible cycle. A reversible process is a process in which the original state or condition of a system can be recovered back if the process is done in the opposite direction from that in which it is currently being done. To perform a reversible process, the steps must be conducted very, very slowly, in an inhnitesimal manner, and without friction. From the definition of a reversible process, the dehnition of a reversible cycle follows. A reversible cycle is a cycle in which the reversible process is applied in every step around the cycle. 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

Theorem - Criterion. Given an ensemble of identical systems having a Hamiltonian operator H and a density operator P, and consisting of two or more subensembles each of which is prepared by means of an unambiguous preparation, the entropy defined in terms of availability is either equal to -k Tr(P In 0) if the preparations of the subensembles are identical, or smaller than -k Tr( In P) if the preparations of the subensembles are different. 

Nine years after Shannon s paper, Edwin T. Jaynes published a synthesis of the work of Cox and Shannon (11). In this paper Jaynes presented the "Maximum Entropy Principle" as a principle in general statistical inference, applicable in a wide variety of fields. The principle is simple. If you know something but don t know everything, encode what you know using probabilities as defined by Cox. Assign the probabilities to maximize the entropy, defined by Shannon, consistent with what you know. This is the principle of "minimum prejudice." Jaynes applied the principle in communication theory and statistical physics. It was easy to extend the theory to include classical thermodynamics and supply the equations complementary to the Rothstein paper(12). 

There is a common misconception that the Lyapunov exponents, or the K-entropy defined below, relates directly to the rate of exponential relaxation of observables in chaotic systems. The actual situation is far more complex, as discussed in Section IIC. 

More recently Giaquinta and Giunta [159] proposed another rule to portend the onset of a phase transition. The key quantity is a type of multiparticle-excess entropy, defined... 

This variable is the temperature-dependent component of specific entropy, defined to be zero at a temperature of absolute zero. We will call it simply phi , J/(kgK). Substituting back into equation (16.47) gives ... 

Equation (5.25) suggests that the ratio between the heat (received or rejected) and the temperature might be a system property that might characterise the reversibility of the heat exchange in a cyclic process. By more rigorous reasoning, this observation leads to a new state function, entropy, defined as ... 

Free-energy Term. Entropy, defined in this manner and when multiplied by the absolute temperature, gives a quantity TS which is a measure of that portion of the energy which is imavailable. The term TS subtracted from the enthalpy leaves a quantity which is that portion of the enthalpy in a flow process, or constant-pressure batch process, which is available for useful work. The latter term is called the free energy and is defined as G H--TS. 

The ratios of the two transition entropies defined in Eq. (7.13) are estimated according to the following expression ... 

Thus, we obtain entropy inequality (1.21) for entropies defined relative to the same... 

Such problems, giving more or less only partial interpretation of entropy defined in this chapter in terms of entropies introduced in the remaining chapters, are similar, apparently not incidentally, to the interpretation of statistically defined entropy, cf, e.g., [12, Sect. 11.14]. 

The conformational entropy defined by Eq. 1 is directly related to the flexibility or rigidity of given polymer chains. Thermodynamic properties of the bulk state are known to be largely affected by the flexibility of the polymer chain. The RIS information is fundamentally important in predicting bulk properties of polymers from a given first-order structure. A great deal of effort has been made to elucidate the role of conformational entropy in determining the phase transition behavior of chain molecules. 

As the number of probable states available to a system increases, the uncertainty as to which state the system occupies increases and the entropy defined in terms of probability increases. A statistical interpretation of entropy is related to the uncertainty of knowledge about the state of the system. 

Here we have the entropy defined in terms of something measurable, the heat capacity. Integrating (4.25), we have... 

Equation 8.1 is a statistical definition of entropy. Defining entropy in terms of probability provides a molecular interpretation of entropy changes as well as allowing for the cglpw iqj enfpja plangi fip qip H such as that of an ideal gas. In... 

Finally, we turn to the equation of entropy conservation and look at the specific expressions for the entropy flux and entropy generation terms. With entropy defined in the s = 1 space, the entropy flux from Eq. (5.86) in dimensionless terms is ... 

The bridge which connects the thermodynamic equation (1.2.2) to the statistical definition (1.2.3) is precisely the statistical entropy, defined as ... 

---