The Statistical Definition of Entropy

This represents a manifestation of the second driving tendency described in Frame 13, section 13.2 and observed in nature -namely, the tendency to achieve increased disorder during any natural (spontaneous) process with the result that the entropy, S is increased in accordance with the Second Law of Thermodynamics (equation (13.16), Frame 13). 

The occurrence of the reverse effect - namely one in which a previously uniformly coloured volume of water clears itself and reverts to just having a small region (of volume) in which is contained a high concentration of the colorant molecules, whilst the main volume of the water has become completely clear - is never observed in our experience of natural phenomena This kind of paradox was pointed out by Boltzmann. Its rationalisation is based on the fact that  

The driving force (which we have called entropy (Frame 13) and labelled, S is then linked to the achieving of a more probable (i.e. more likely) molecular distribution. 

Boltzmann proposed the general relationship for solids whereby  

If we attempt to interpret the observations with regard to residual entropy in Frame 16, section 16.4 and those features that are not entirely in accord with the Third Law, we see that equation (17.1) represents a statistical interpretation of entropy which gives a reasonable account of these departures from the Third Law as well as giving an entirely consistent account of the Third Law itself. 

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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 ... 

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. 

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. 

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... 

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

The Statistical Rate Theory (SRT) is based on considering the quantum-mechanical transition probability in an isolated many particle system. Assuming that the transport of molecules between the phases at the thermal equilibrium results primarily from single molecular events, the expression for the rate of molecular transport between the two phases 1 and 2 , R 2, was developed by using the first-order perturbation analysis of the Schrodinger equation and the Boltzmann definition of entropy. 

The successful development of the thermodynamics of irreversible phenomena depends on the possibility of an explicit evaluation of the production of entropy, and for this it is necessary to assume that the thermodynamic definition of entropy can be applied equally to systems which are not in equilibrium, that is to states whose mean lifetime is limited. We are thus confronted immediately with the problem of the domain of validity of the thermodynamic treatment of irreversible phenomena, which can be determined only by a comparison of the results of the thermodynamic treatment with those obtained by the use of statistical mechanics. This problem wall be dealt with in more detail in the third volume of this work meanwhile the main conclusions can be summarized as follows. 

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... 

It is to be noted that the above definition of entropy is identical to statistical measures of information, and more discussion on the definition of entropy will be given in Chap. 5. Substituting the equilibrium solution, Eq. (4.34), into Eq. (4.39) and carrying out the momentum integrations leads to (Prob. 4.10)... 

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. 

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. 

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. 

The expressions in Eq. 1 and Eq. 6 are two different definitions of entropy. The first was established by considerations of the behavior of bulk matter and the second by statistical analysis of molecular behavior. To verify that the two definitions are essentially the same we need to show that the entropy changes predicted by Eq. 6 are the same as those deduced from Eq. 1. To do so, we will show that the Boltzmann formula predicts the correct form of the volume dependence of the entropy of an ideal gas (Eq. 3a). More detailed calculations show that the two definitions are consistent with each other in every respect. In the process of developing these ideas, we shall also deepen our understanding of what we mean by disorder. ... 

Traditional thermodynamics gives a clear definition of entropy but unfortunately does not tell us what it is. An idea of the physical nature of entropy can be gained from statistical thermodynamics. Kelvin and Boltzmann recognised diat there was a relationship between entropy and probability (cf., disorder) of a system with the entropy given by... 

The term entropy, which literally means a change within, was first used in 1851 by Rudolf Clausius, one of the formulators of the second law of thermodynamics. A rigorous quantitative definition of entropy involves statistical and probability considerations. However, its nature can be illustrated qualitatively by three simple examples, each demonstrating one aspect of entropy. The key descriptors of entropy are randomness and disorder, manifested in different ways. 

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

The next important thermodynamic function that we must obtain is the entropy S. The statistical thermodynamic definition of entropy is... 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

Entropy is interpreted as the number of microscopic arrangements included in the macroscopic definition of a system. The second law is then used to derive the distribution of molecules and systems over their states. This allows macroscopic state functions to be calculated from microscopic states by statistical methods. 

In practice, diere are an infinite, or at least very large, number of statistically identical models diat can be obtained from a system. If I know that a chromatographic peak consists of two components, I can come up widi any number of ways of fitting the chromatogram all with identical least squares fits to die data. In the absence of further information, a smoother solution is preferable and most definitions of entropy will pick such an answer. 

Similarly, if one is interested in a macroscopic thermodynamic state (i.e., a subset of microstates that corresponds to a macroscopically observable system with bxed mass, volume, and energy), then the corresponding entropy for the thermodynamic state is computed from the number of microstates compatible with the particular macrostate. All of the basic formulae of macroscopic thermodynamics can be obtained from Boltzmann s definition of entropy and a few basic postulates regarding the statistical behavior of ensembles of large numbers of particles. Most notably for our purposes, it is postulated that the probability of a thermodynamic state of a closed isolated system is proportional to 2, the number of associated microstates. As a consequence, closed isolated systems move naturally from thermodynamic states of lower 2 to higher 2. In fact for systems composed of many particles, the likelihood of 2 ever decreasing with time is vanishingly small and the second law of thermodynamics is immediately apparent. 

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