Entropy statistical definition

By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

Planck Bed. Ber., 1908, 633) has, from the fundamental statistical definition of entropy, deduced the equation ... 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

What is transported One of the peculiarities of the thermal energy variety is to think of the thermal conduction in terms of energy transported, when other domains consider entities as the transported quantity (charges, momenta, etc.). According to their definition, as entities bear energy, there is no physical consequence due to this disparity. There are naturally historical reasons for this, but also a conceptual difficulty in our modern minds to view the entropy as a quantity able to be transported. This is certainly due to the influence of the statistical definition of entropy as a measure of order/disorder in the system, considered as a whole with implicitly a uniform entropy distribution. 

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... 

In general, the thermodynamic definition of entropy (Equations 8.6 to 8.8) yields the same value for the entropy change of a process as Boltzmann s statistical definition (Equation 8.3) for the same process. Consider, for example, the entropy change in the reversible and isothermal (constant teinperature) mmqn n i n i les of an ideal gas from an initial volume Vi to a inB39B9tft0-6K ile heat... 

This is exactly the same value that we obtained for this process in Example 8.1 using the statistical definition of entropy. 

The fact that this is an abaoltUe value arises from the statistical definition of entropy m equation (11 14). According to this equation the entropy would be zero if the system were known to be in a single quantum state. This point will be discussed in more detail in the next chapter in connexion with the third law. For the moment it may be noted that equation (12 64) leads to an apparent paradox—as T approaches zero it appears that 8 approaches an infinitely negative value, whereas the least value of 8 should be just zero, as occurs when the system is known to be in the single quantum state. This difiSculty is due to the fact that equation (12 8), on which the equations of the present section are based, becomes invalid at very low temperature. Under such conditions the Boltzmann statistics must be replaced by Einstein-Bose or Fermi-Dirac statistics. 

Since Ludwig Boltzmann (1844-1906) introduced a statistical definition of entropy in 1872, entropy is associated with disorder. The increase of entropy is then described as an increase of disorder, as the destruction of any coherence which may be present in the initial state. This has unfortunately led to the view that the consequences of the Second Law are self-evident or trivial. This is, however, not true even for equilibrium thermodynamics, which leads to highly nontrivial predictions. Anyway, equilibrium thermodynamics covers only a small fraction of our everyday experience. We now understand that we cannot describe Nature around us without an appeal to nonequilibrium situations. The biosphere is maintained in nonequihbrium through the flow of energy coming from the sun, and this flow is itself the result of the nonequilibrium situation of our present state in the universe. 

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

We, shall next consider the statistical definition of entropy in magnetic resonance. According to usual definition introduced by von Neumann, the quantum statistical definition of entropy is... 

In general, it seems more reasonable to suppose that in chemisorption specific sites are involved and that therefore definite potential barriers to lateral motion should be present. The adsorption should therefore obey the statistical thermodynamics of a localized state. On the other hand, the kinetics of adsorption and of catalytic processes will depend greatly on the frequency and nature of such surface jumps as do occur. A film can be fairly mobile in this kinetic sense and yet not be expected to show any significant deviation from the configurational entropy of a localized state. 

Although it is obviously impossible to enumerate all possible configurations for infinite lattices, so long as the values of far separated sites are statistically independent, the average entropy per site can nonetheless be estimated by a limiting procedure. To this end, we first generalize the definitions for the spatial set and spatial measure entropies given above to their respective block-entropy forms. 

The name entropy is used here because of the similarity of Eq. (4-6) to the definition of entropy in statistical mechanics. We shall show later that H(U) is the average number of binary digits per source letter required to represent the source output. 

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

To close this chapter we emphasize that Hie statistical mechanical definition of macroscopic parameters such as temperature and entropy are well designed to describe isentropic equilibrium systems, but are not immediately applicable to the discussion of transport processes where irreversible entropy increase is an essential feature. A macroscopic system through which heat is flowing does not possess a single tempera-... 

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