Boltzmanns Entropy

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

A point of entry for statistical considerations into thermodynamics is the Boltzmann entropy relationship... 

The entropy of the mixture is calculated from this by the Boltzmann entropy equation, Equation (45). By separately letting TV2 and TV, equal zero, the configurational entropies of the solvent and the solute, respectively, are obtained from Sm,x. Finally, by subtracting S, and S2 from S,mx, an expression is obtained for ASm. 

The elasticity of polymer coils is a well-known phenomenon and is involved in many important mechanical properties of bulk polymers. Stated briefly, it arises from a difference in conformational entropy between stretched and randomly jumbled chains. A statistical theory that counts the number of ways the two conformations can come about can be combined with the Boltzmann entropy equation (Equation (3.45)) to give an expression for the... 

Thus, assuming that the ground level E0 is truly nondegenerate (i.e., there is no alternative of equal energy), the T = 0 microstate is unique, and the corresponding Boltzmann entropy is therefore zero ... 

The number of states in Eq. (30) is given as Defining the Boltzmann entropy as... 

Boltzmann entropy Also called statistical entropy... 

For the trans compound there will only be 3 different orientations. Comparing the Boltzmann entropy calculations for the cis and trans forms cis ... 

Boltzmann entropy hypothesis. The entropy of a system of material particles is proportional to the logarithm of the statistical probability of the distribution. 

In the interpenetrational-plus-compressional domain, the mixing free energy calculated as described above must be supplemented by the elastic free energy. Meier (1967) expressed this by the Boltzmann entropy relationship... 

When this absolute value is required, it may be obtained from the Boltzmann entropy by subtraction of Aln(iV ). It is lower in so far as it is based upon neglect of distinctions which a non-quantum theory might have regarded as valid. 

According to the famous formula of Ludwig Boltzmann, entropy S = kg In Q (E), where S is the number of the states available for the system at energy E. The more states there are. the larger the entropy is. 

Perhaps one of the most important consequences of the Boltzmann equation, discovered by Boltzmann himself, was that it led to a kinetic molecular theory basis for the law of entropy increase. This law is contained within the so-called Boltzmann -theorem. Here we will work directly with the Boltzmann entropy Sb rather than the i/-function used in Boltzmann s original work.1 We define a local Boltzmann entropy Ssfri, t) as... 

P1.5 - Calculate the Boltzmann entropy of N spins 1/2 subject to a static magnetic field Bo. at equilibrium temperature T. 

This result corresponds to Boltzmann s definition of entropy. Obviously, the entropy S increases as the number of accessible states increases. This behavior of the Boltzmann entropy has led to such qualitative definitions of entropy as measme of randomness and measure of... 

Since viscosity measurements of uncharged, weak and strong PEs [60] were successfully modeled with the Kuhn entropy [80], it should theoretically be possible to describe weak PE solutions with it as well. This assumption is supported by the fact that the Kuhn entropy is based on the Boltzmann entropy formula, and attempts to use this approach become inaccurate to determine the ion distribution around the charged groups of strong PEs at low salt concentrations, but not the PE structure [81]. Since weak PEs are not strongly affected by counter ion condensation, these effects pose no problem for the Kuhn entropy approach. Another reason for the infrequent use of the Kuhn entropy is the fact that the Flory approach is much simpler and better known than the Kuhn approach [49, 80, 82]. 

In case of a continuous system, the number of states gE is replaced by the density of states g E), i.e., the number of states per energy. Since a microcanonical energetic macrostate is defined by the ensemble of system configurations (or conformations) with energies E — Ais/2 < E < E + i Ejl, the number of states in this ensemble is given by gE = g E)AE. The correct Boltzmann entropy formula would therefore read pE- -AE/2... 

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