Boltzmann paradox

The strange thing is that there is no reason to suppose that the one particular motion, which led to the uniform filling of the room, is any more probable than the same motion reversed, which leads to the collection of the gas in one corner of the room. If this is so, why is it that we never observe the air in a room collecting in one particular portion of the room The fact that we never observe some motions of a system, which are inherently just as probable as those we do observe, is called the Boltzmann paradox. 

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 ... 

Criticism of the Stosszahlansatz and its corollaries arose as soon as it was recognized as paradoxical that the completely reversible gas model of the kinetic theory was apparently able to explain irreversible processes, i.e., phenomena whose development shows a definite direction in time. These nonstationary,51 irreversible processes were brought into the center of interest by the //-theorem of Boltzmann. In order to show that every non-Max-wellian distribution always approaches the Maxwell distribution in time, this theorem synthesizes all the special irreversible processes (like heat conduction and... 

This irreversibility paradox arose slightly less than a hundred years ago in a different context. It will be recalled that Boltzmann used plausible physical assumptions to develop a kinetic equation which described the nonequilibrium... 

Perhaps the most interesting and at first paradoxical aspect of Eyring s theory is the lack of a kinetic equation. No Fokker-Planck or Boltzmann equation is introduced or solved to derive the viscosity. Now this feature is also characteristic for a simple model we have studied recently. For this reason it seems to us appropriate to summarize this model here and to compare it in more detail with Eyring s theory of viscosity. 

The recurrence paradox is easy to refute and was done so by Boltzmann. He pointed out that the recurrence... 

The exponentially decreasing term in the Boltzmann expression would seem to favour the very lowest energy states. However, this would lead to the paradoxical situation in which everything in the Universe should be at zero enthalpy. This can be resolved as follows. 

Apparent paradox. It is known that the laws of electrostatics also work with a very small number of entities, one or two, which forbids applying the approximation of large numbers normally invoked in Boltzmann statistics. The Formal Graph theory resolves this paradox by showing that the exponential shape of the capacitive relationship is not relevant to statistics but to a theory of influence between entities, as demonstrated in Chapter 7. 

Boltzmann attempted to resolve this paradox by considering the system particle dynamics from a probability sense. He estimated the Poincare recurrence time for a cubic centimeter of air containing about 10 molecules. His calculated result is that the Poincare recurrence time, i.e., the average time for the system to pass back through the initial state is on the order of the age of the universe ... 

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. 

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