Boltzmann’s theorem

A charged surface and the ions, which neutralize the surface, together create an electric double layer. The distribution of the ions can be evaluated from the Poisson-Boltzmann equation where the ions are treated as point particles and the primitive model is used. Further, all correlations between the ions are neglected, which means that the ions are interacting directly only with the colloids and through an external field given by the average distribution of the small ions. The distribution of the particles are assumed to follow Boltzmann s theorem [11]... 

In summary, the statistical tf-theorem of kinetic theory relates to the Maxwellian velocity distribution function and thermodynamics. Most important, the Boltzmann s //-theorem provides a mechanistic or probabilistic prove for the second law of thermodynamics. In this manner, the //-theorem also relates the thermodynamic entropy quantity to probability concepts. Further details can be found in the standard references [97] [39] [12] [100] [47] [28] [61] [85]. 

Due to the statistical relationship between Hq and S expressed through (2.230), the Boltzmann s //-theorem shows that for a gas that is not in a steady state Hq must decrease and the entropy S will increase accordingly. In accordance with (2.222) we can write ... 

In this case no assumptions are made regarding the radiating system except that it possesses different stationary states of constant energy. From these we select two with the energies Wx and W2 (W1>Wa), and suppose that, when statistical equilibrium exists, atoms in these states are present in the numbers Nj and N2 respectively. Then, by Boltzmann s Theorem... 

Boltzmann s //-theorem raises a number of questions, particularly the central one how can a gas that is described exactly by the reversible laws of mechanics be characterized by a quantity that always decreases Perhaps a non-mechanical assumption was introduced here. If so, this would suggest, although not imply, that Boltzmann s equation might not be a useful description of nature. In fact, though, this equation is so useful... 

The effect of these electric forces is counteracred by the thermal motion of the ions, which gives the liquid charge layer its spatial extension ( diffuse layer). In order to take this thermal equilibrium into account the theory makes the implicit assumption that the average concentration of these ions at a given point can be calculated from the average value of the electric potential at the same point with the aid of Boltzmann s theorem ... 

The derivation of his equation runs as follows. If the number of adsorbed ions per is tZi, the number of available adsorption spots per cm of the wall is A i—fti. The number of available positions in the solution is a less easily definable quantity, and is taken by Stern to be NjM per ctn. Hence, applying Boltzmann s theorem, the thermal distribution equilibrium about the adsorbed layer and the solution is determined by ... 

Here we will prove a stronger version of Boltzmann s //-theorem, which holds for more general gas-wall interactions and which gives a nonequilibrium analog of Clausius s formula... 

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. 

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. 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

According to the energy equipartition theorem of classical physics, the three translational kinetic energy modes each acquire average thermal energy kT (where k = R/NA is Boltzmann s constant),... 

Boltzmann s tombstone in Vienna bears the famous formula 5 = k log W (W = Wahrscheinlichkeit—probability) that was a signature of his audacious concepts. The alternative formula (13.69) (which reduces to Boltzmann s in the limit of equal a priori probabilities pa) was ultimately developed by Gibbs, Shannon, and others in a more general and productive way (see Sidebar 13.4), but the key step of employing probability to trump Newtonian determinism was his. Boltzmann was long identified with efforts to establish the //-theorem and Boltzmann equation within the context of classical mechanics, but each such effort to justify the second law (or existence of atoms) in the strict framework of Newtonian dynamics proved futile. Boltzmann s deep intuition to elevate probability to a primary physical principle therefore played a key role in efforts to find improved foundation for atomic and molecular concepts in the pre-quantum era. 

Boltzmann in his replies developed a somewhat modified and more precise formulation of the //-theorem and of his own ideas on the foundations as well as of the arguments of his opponents. Nowadays this formulation is referred to as statistico-mechanical. 10 Of Boltzmann s conclusions it may be said ... 

This deeper formulation of the question was shelved in Boltzmann s investigations by the soon emerging formulation of the 17-theorem (1872). Boltzmann took up this problem again only to counter Loschmidt s Umkehreinwand (Section 7b) and to obtain a modified formulation of the 17-theorem. He tried to show that, if we consider a motion of unlimited duration, then the Maxwell-Boltzmann distribution very strongly dominates in time over all other distributions, and hence the tendency to approach this particular distribution is quite understandable. 

One can discard the claim that the relatively primitive assumptions about the structure of the gas model also give a correct picture of the phenomena even over very long time intervals. This point of view was, of course, also considered by Boltzmann. He emphasized very early (1871)156 that in the further development of the kinetic theory one has to consider the interaction of the molecules and the ether (i.e., the influence of radiation on the thermal equilibrium). However, in the discussions about the F-theorem, he was right to insist on the first point of view to its final consequences. In this case a reference to, for instance, thermal radiation would easily lead to a premature condemnation of Boltzmann s ideas, as if the increase in entropy for processes during an observable time interval could not be interpreted without invoking radiation. 

Summarizing we oan say Boltzmann s definition of measure is not quite arbitrary in the sense that Liouville s theorem together... 

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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