Boltzmann’s Principle

M.J. Sippl, Boltzmann s principle, knowledge based mean fields and protein folding, J. Comp. Aided Mol. Design 7 (1993), 473-501. 

In Section 7.1 we discussed the thermodynamic condition for a stable mixture given in the Flory-Huggins equation (Eq. 7.1-6), where AS denotes the increase of entropy due to mixing. This equation is based on Boltzmann s principle stating that the entropy of a... 

M. J. Sippl,/. Comput.-Aided Mol. Design, 7,473 (1993). Boltzmann s Principle, Knowledge-... 

Finally we would like to mention the work of Lavenda and coworkers, who approached the same problem by means of a generalization of the kinetic analog of Boltzmann s principle so as to include the Omstein-Uhlenbeck process. They carried out an asymptotic analysis in the limit of high resistance. The condition for the validity of their asymptotic expansion has been proved to be identical to the modified Kramers condition derived by Stratonovich. 

The idea of exchangeable entropy is also supported by Boltzmann s theory based on statistical thermodynamics (see AppendixD.3). That is, Boltzmann s principle can be shown as... 

We know that a microscopic approach to a phase considers it as a collection of molecules-objects whose energies are distributed in accordance with a statistical law. The state of a collection of molecules-objects is constantly changing but, over time, the collection goes through a certain type of distribution in which the molecules-objects may be in a variety of different states. We use the term number of complexions for the number of distributions of molecules-objects between the different states that they are liable to assume. Out of all the possible distributions, there is one that corresponds to the maximum number of complexions. Boltzmann s principle accepts that the number of complexions corresponding to the most probable type of distribution is practically equal to the total number of complexions, and vice versa. The state of the collection is then always that which corresponds to the maximum number of complexions. 

We shall now introduce the probability distribution, function F a) in the a-space. According to Boltzmann s principle this function is given by... 

When all is said and done, probably the best definition of a linear material is simply one that follows Boltzmann s principle. Thus, spring-dashpot models, which are linear, automatically follow Boltzmann s principle. However, it is important not to infer a dependence of the Boltzmann principle on spring-dashpot models. The Boltzmann principle applies to a hnear response regardless of whether it can be described with a spring-dashpot model. All that is needed are experimental GXt) or Jdt) data. Models are used here simply as a matter of convenience to illustrate application of the principle. 

L As is usually smaller than 1, the very high potentials in the double layer, where the applicability of Boltzmann s principle in its simple form is questionable, are eliminated. 

---