H theorem

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

Pauii W Jr 1928 Uber das H-Theorem vom Anwachsen der Entropie vom Standpunkt der neuen Quantenmechanik Probleme der modernen Physik ed P Debye (Leipzig Hirzei) pp 30-45... 

Assuming fj, < 1/2, this solution implies a monotonic approach to equilibrium with time. From a purely statistical point of view, this is certainly correct the difference in number between the two different balls decreases exponentially toward a state in which neither color is preferred. In this sense, the solution is consistent with the spirit of Boltzman s H-theorem, expressing as it does the idea of motion towards disorder. But the equation is also very clearly wrong. It is wrong because it is obviously inconsistent with the fundamental properties of the system it violates both the system s reversibility and periodicity. While we know that the system eventually returns to its initial state, for example, this possibility is precluded by equation 8.142. As we now show, the problem rests with equation 8.141, which must be given a statistical interpretation. 

Boltzman s H-Theorem Let us consider a binary elastic collision of two hard-spheres in more detail. Using the same notation as above, so that v, V2 represent the velocities of the incoming spheres and v, V2 represent the velocities of the outgoing spheres, we have from momentum and energy conservation that... 

The fact that the condition dll/dl = 0 is the same as the condition of detailed-balance, and therefore that equation 9.34 is a necessary condition for the solution of equation 9.33, follows from the proof of Boltzman s H-Theorem ... 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

One may also show that MPC dynamics satisfies an H theorem and that any initial velocity distribution will relax to the Maxwell-Boltzmann distribution [11]. Figure 2 shows simulation results for the velocity distribution function that confirm this result. In the simulation, the particles were initially uniformly distributed in the volume and had the same speed v = 1 but different random directions. After a relatively short transient the distribution function adopts the Maxwell-Boltzmann form shown in the figure. 

These evolution equations form the starting point for the derivation of macroscopic kinetic equations, which we now consider. They also serve as the starting point for the proof of the H theorem. This proof can be found in Ref. 11. 

Irreversible processes are those in which entropy increases. The entropy itself can he regarded as a measure of ihe degree of randomness or disorder of the gas. although it must he recognized that disorder really means a lack ol knowledge about the details of molecular eonhguralions. The equilibrium state represents the maximum possible disorder, the H -theorem implies that a gas which is initially in a nonequilihrium (partly ordered I state will eventually reach equilibrium and then stay there forever il it is not disturbed. 

The year 1872 was the year of the formulation of the famous Boltzmann equation (BE), which is one of the most important equations of statistical physics. One of the remarkable consequences of the BE is the H-theorem. Furthermore, the BE is the basic equation for transport processes in macroscopic systems. 

The subsequent attempts of Maxwell and Boltzmann to obtain a derivation reached their first conclusion in the H-theorem. Before reaching this they had irone through the following stages of development ... 

Boltzmann (1872, as one of the corollaries of the H-theorem)47—The Maxwell-Boltzmann distribution is the only distribution which can maintain its invariance,48 and any other distribution under the influence of collisions finally goes over into the Maxwellian one. 

Subsequently, however, this probability is interpreted to mean either the ratio of time intervals or the relative frequencies in other, very different statistical ensembles. In this way formulation (I ) leads to statement (III) and also to all other further statements which, together, make up the modified formulation of the H-theorem. 

From a logical viewpoint such a procedure is not very satisfactory. Therefore the terminology using probabilities will be eliminated in the subsequent discussions. In this way the modified version of the H-theorem will appear as a sequence of hypothetical statements about a... 

This generates a statistical formulation of the H-theorem through the following assertion, which is again unproved ... 

That H almost always immediately decreases from any higher value (in the sense of the H-theorem). 

Since 1876 numerous papers have called attention to these foundations. In these papers the Boltzmann H-theorem, a central theorem of the kinetic theory of gases, was attacked. Without exception all studies so far published dealing with the connection of mechanics with probability theory grew out of the synthesis of these polemics and of Boltzmann s replies. These discussions will therefore be referred to frequently in our report. 

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