The Statistical Postulate

In discussing classical mechanics in Section 16.3, we noted that if the position and velocity of the molecules were known at some time along with the forces acting upon them, we could then follow their paths and be able to determine their positions and velocities at any time t. The average values could then be calculated. 

Leaving aside the computational problems associated with following such a large number of molecules - of the order of 10 - simultaneous knowledge of their original positions and velocities is impossible by virtue of the uncertainty principle. Any measurement of the position of the molecules will effect their velocity and vice versa. 

Knowledge of the state of the system in complete detail is thus unattainable and this represents one of the bases of the molecular interpretation of the second law (Denbigh, 1981, p.336 see also Section 3.13). It is impossible, therefore, to determine the average values of the properties of a given system by applying Newtonian mechanics to its individual molecules. 

We resort rather to a postulate, in addition to those of mechanics, which is of statistical character and asserts that For a system in internal equilibrium the probability of a given quantum state is a function of its energy only. 

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According to the statistical postulate, the probability of q.s. i with energy , is a function of this energy only ... 

An alternative way of deriving the BET equation is to express the problem in statistical-mechanical rather than kinetic terms. Adsorption is explicitly assumed to be localized the surface is regarded as an array of identical adsorption sites, and each of these sites is assumed to form the base of a stack of sites extending out from the surface each stack is treated as a separate system, i.e. the occupancy of any site is independent of the occupancy of sites in neighbouring stacks—a condition which corresponds to the neglect of lateral interactions in the BET model. The further postulate that in any stack the site in the ith layer can be occupied only if all the underlying sites are already occupied, corresponds to the BET picture in which condensation of molecules to form the ith layer can only take place on to molecules which are present in the (i — l)th layer. 

The earliest hint that physics and information might be more than just casually related actually dates back at least as far as 1871 and the publication of James Clerk Maxwell s Theory of Heat, in which Maxwell introduced what has become known as the paradox of Maxwell s Demon. Maxwell postulated the existence of a hypothetical demon that positions himself by a hole separating two vessels, say A and B. While the vessels start out being at the same temperature, the demon selectively opens the hole only to either pass faster molecules from A to B or to pass slower molecules from B to A. Since this results in a systematic increase in B s temperature and a lowering of A s, it appears as though Maxwell s demon s actions violate the second law of thermodynamics the total entropy of any physical system can only increase, or, for totally reversible processes, remain the same it can never decrease. Maxwell was thus the first to recognize a connection between the thermodynamical properties of a gas (temperature, entropy, etc.) and the statistical properties of its constituent molecules. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The skeptical reader may reasonably ask from where we have obtained the above rules and where is the proof for the relation with thermodynamics and for the meaning ascribed to the individual terms of the PF. The ultimate answer is that there is no proof. Of course, the reader might check the contentions made in this section by reading a specialized text on statistical thermodynamics. He or she will find the proof of what we have said. However, such proof will ultimately be derived from the fundamental postulates of statistical thermodynamics. These are essentially equivalent to the two properties cited above. The fundamental postulates are statements regarding the connection between the PF and thermodynamics on the one hand (the famous Boltzmann equation for entropy), and the probabilities of the states of the system on the other. It just happens that this formulation of the postulates was first proposed for an isolated system—a relatively simple but uninteresting system (from the practical point of view). The reader interested in the subject of this book but not in the foundations of statistical thermodynamics can safely adopt the rules given in this section, trusting that a proof based on some... 

Photolysis of N-methylborazine with CH3OH (e = 100) yielded 67% ortho substituted product while the reaction with HN(CH3)2 yielded over 90% para product. These results are explained by sterically hindered attack of the CH3OH and HN(CH3)2 at the ortho site. A lower energy barrier is also postulated to explain the statistical distribution of products in the CH3OH + N-methylborazine reaction. 

The statistical theory of rubber elasticity predicts that isothermal simple elongation and compression at constant pressure must be accompanied by interchain effects resulting from the volume change on deformation. The correct experimental determination of these effects is difficult because of very small absolute values of the volume changes. These studies are, however, important for understanding the molecular mechanisms of rubber elasticity and checking the validity of the postulates of statistical theory. 

Onsager s principle supplements these postulates and follows from the statistical theory of reversible fluctuations [5]. Onsager s principle states that when the forces and fluxes are chosen so that they are conjugate, the coupling coefficients are... 

The statistical models determined by factorial design can be used as simplified models with the SQP technique. In this work the results obtained through this approach are compared with the results obtained using a rigorous model of the process. Costa et al. (5) determined quadratic models for productivity and % yield as functions of the significant input variables. These equations evaluated productivity and % yield and the SQP technique to determine the optimal values for S0, tr, R, and r. The optimization problem is postulated as follows ... 

One scenario in which thin shear layers and large shear strains can occur in plastic-bonded explosives is at the interface between grains sliding past one another or along closed cracks within a grain. These processes have been postulated by Dienes [48] as the dominant dissipative mechanism for generating hot spots in plastic-bonded explosives, and have been incorporated into the Statistical Crack Mechanics (SCRAM) model for high-explosive response. 

The second postulate is well established by statistical mechanics. 

Thermodynamics is a simple, general, logical science, based on two postulates, the first and second laws of thermodynamics. We have seen in the last chapter how to derive results from these laws, though we have not used them yet in our applications. But we have seen that they are limited. Typical results are like Eq. (5.2) in Chap. II, giving the difference of specific heats of any substance, CP — CV in terms of derivatives which can be found from the equation of state. Thermodynamics can give relations, but it cannot derive the specific heat or equation of state directly. To do that, we must go to the statistical or kinetic methods. Even the second law is simply a postulate, verified because it leads to correct results, but not derived from simpler mechanical principles as far as thermodynamics is concerned. We shall now take up the statistical method, showing how it can lead not only to the equation of state and specific heat, but to an understanding of the second law as well. 

Boltzon Postulate. Maxwell-Boltzmann (MB) statistics predict that all energies are a priori equally likely, and that all particles in the system are physically distinguishable (labeled by some number, or shirt patch, "color", or whatever, or picked up by "tweezers"). These MB particles can be called boltzons. If, however, we remove this distinguishability, then we have indistinguishable "corrected boltzons (CB)" [2], whose statistics become very roughly comparable to the statistics of fermions or bosons (see Problem 5.3.10 below). 

Nuclear Spin Effects on Rotation. There is an interesting effect on the rotational partition function, even for the hydrogen molecule, due to nuclear spin statistics. The Fermi postulate mandates that the overall wavefunction (including all sources of spin) be antisymmetric to all two-particle interchanges. A simple molecule like (1H1)2, made of two electrons (S = 1/2) and two protons (spin 7=1/2), will have two kinds of molecule ... 

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. 

In its conservative estimate, the German Federal Bureau of Statistics postulates 600,000 victims cf. D. Irving, Und Deutschlands Stadte starben nicht, Weltbild Verlag, Augsburg 1989, p. 373 cf. M. Czesany, Europa im Bombenk-... 

This model is not adequate in itself as it contains nothing to describe the actual location x(t) of the particle which is required for a causal interpretation of quantum theory. It is therefore necessary to postulate a particle that takes the form of a highly localized inhomogeneity that moves with the local fluid velocity v(x,t). The inhomogeneity could be of density close to that of the fluid, which is simply being carried along with the local velocity of the fluid. As in any macroscopic fluid random fluctuations are assumed [37] to occur in the Madelung fluid. It is shown that such fluctuations may lead to the statistical result, P = 2. 

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