Sign In Sign-Up

Welcome!

Close

Would you like to make this site your homepage? It's fast and easy...

Yes, Please make this my home page!

No Thanks

  Don't show this to me again.

Close

and only recently a new scientific revolution was announced which should lead to the abdication of Boltzmann‘s entropy H_n (1).

It was generally accepted that Boltzmann had never given an exact proof of his H theorem (2), and so, when Shannon 40 years ago published his theory of communication (3) which introduced function H_m as the third postulate, the mathematical elegance of his work led to the hope that the new information theory can replace the controversial hypothesis (4).

There are two forms of the H theorem. One is that (1) is a forever growing function. The weaker form is that (1) is a statistical form of thermodynamical entropy S. This was proofed by practice but Boltzmann‘s explanation of the relation of H and S was not understood, because Boltzmann himself did not take it seriously.

In 1877 (5), he pronounced the quantum hypothesis.

In an isolated system of the ideal gas, there are m quanta of energy partitioned between n molecules. Supposing m>n, there are necessarily multiplicities of energy levels m_k of molecules and thus the system is described by the following sums

n = S_j 1_j = S_k n_k (2)

the index j goes from 1 till n, the index k from 1 till m. Since m can be greater then n, many n_k must be zero.

The generally accepted relation between Boltzmann H_n and Shannon H_m entropies or otherwise, between thermodynamical and informational entropy is the negentropy principle (6). According to it, entropy and information are complementary. Information should decrease entropy. This principle is based on nonphysical arguments connected with the incomplete analysis of the Maxwell‘s demon, which decreases entropy by sorting molecules. It was overlooked that the process can be reversed and then demon increases the entropy by the identical method as it was decreased. Between demon‘s information and entropy of the system does not exist any one to one relation postulated by the negentropy hypothesis (7).

Shannon defined entropy H_m with the logarithm to base 2. This definition need not be connected with Boltzmann‘s H_n at all.

For indexing of m objects by the regular code, mlog₂m binary digits are needed at least. If the set of m objects is divided by some information into n subsets, then only S n_km_k log₂m_k binary digits are sufficient. The difference gives the result which limit is (1) and thus H_m is the direct measure of information (8).

But H_m has also another meaning analogical to H_n.

Identity (3) counts undistinguishable objects. If these objects are ordered into a string as symbols into a text, they are indexed by their natural order and thus they are distinguishable.

It is a different problem and indeed both H_n and H_m are calculated differently. Shannon has given a simple example how his function H_m should be calculated. 4 symbols with probabilities p_a = 1/2, p_b = 1/4, p_c = p_d = 1/8 determine the share of symbols in an infinite string

If we want to calculate H_n, then we must compare n symbols with n particles and their repeating with their energy. We have

If there are two possibilities how to calculate entropy according to equation (1), the question arose which of them is right or if both can be applied, how they are related.

Two operations over a word (a string of symbols) N^T, the substitution e.g. abaced ® rirfuk, and the permutation e.g. abaced ® daceab are joined, if we write the word in the transposed form as a column and symbols j as the unit vectors

The word N has a form of a naive matrix, whose elements are {0, 1}. It is a one-sided stochastical matrix giving NJ = J, where J is unit vector-column.

In each row of naive matrix N is only one unit symbol. There are two possibilities how to interpret these matrices. The unit vectors e_j are translations in the n dimensional lattice and the naive matrix N is a path going from the beginning of the nonnegative lattice to the final point determined by the row vector J^TN, where ^T is the symbol of transposing. All naive matrices with m rows and n columns N (m, n) form the vertices of a plane simplex. This projection stresses the constant sum m, it expresses the law of the conservation of energy.

The system is represented by a n-dimensional vector N^TN as a point in the phase space of energy. Because energy of the isolated system is constant, all representation points lie on a plane orthogonal to the unit diagonal vector I_n . Vectors N^TN are hypotenuses of right triangles with (m/n) I_n as the leg. The n dimensional plane is a (n - 1) dimensional body and we use for it the term the plane simplex. The n dimensional space is the sum of plane simplexes, the n dimensional plane complex.

All representation points available for the system with m quanta of energy and n ideal gas particles are counted by the identity above (4). The left hand side of (4) counts all possible partitions of m undistinguishable objects into n boxes.

The summation on the right side of (4) is made over all partitions of number m into at most n parts. The partitions form on the plane simplex spherical orbits, because the length of vector M does not depend on permutations of its elements. The volume of partition orbits is determined by polynomial coefficients.

Molecules of the ideal gas exchange their energy by collisions. lf the energy is not transposed by a collision of two particles, the system goes on another orbit. At large systems with many simultaneous collisions, we can suppose that fluctuations produced by individual collisions eliminate and that the system remains on one orbit or on a small band of orbits. It is possible to say that it rotates on its n dimensional orbit. Boltzmann connected entropy S with the logarithmic measure of the volume of partition orbits which for n going to infinity is approximated by (1).

The alternative interpretation of naive matrices N (m, n) is the cube simplex in m dimensional space. The cube has edges < o, n - 1> and j-th index e_j determines the distance (j - 1) from the beginning on axe i. This maps each string of a plane simplex as a point with natural coordinates (zero including) in a cube. The cube is obtained from the plane complex by truncating all points with values m_k greater than (n - 1).

The permutations are formalized by unit permutation matrices P_m acting on a matrix M from the left and substitutions are formalized by unit permutation matrices P_n acting on a matrix M from the right. Jointly we have P_mMP_n or using the symmetric group S_n substitutions and permutations are equivalent operations forming the group S_mM(m,n)S_n containing all possible partition orbits of M. If m = n, then one of these orbits is the symmetrical group S_n itself, with partition 1ⁿ.

The identity counting all naive matrices N (m, n) with m rows and n columns or corresponding points of the m dimensional cube is

The sum is made over all partitions of the number m into n dimensional vectors which can have zero elements. Terms n!/Ő 0_k!ⁿ⁽⁰⁾ must be calculated.

If matrices N are projected into the space of columns using the unit projection operator J or into the space of rows, using the transposed unit operator J^T‚ a part of the original symmetry is lost.

Both symmetries may be formally separated by defining the corresponding quadratical forms P_nN^TNP_n, and P_mNN^TP_m

The diagonal matrices N^TN (or projections J^TN) are counted by identity (4). It counts all partition orbits in the cube simplex.

Other quadratical forms NN^T are permuted block-diagonal matrices with blocks J_mjJ_mj^T. They are counted by Bell polynomials

where (n-n₀) = 1 n₀ = const, and S(m,n) are Stirling numbers of the second kind (10). There are m! possible symmetrical permutations of a quadratical matrix NN^T. The permutations inside blacks J_mjJ_mj^T are ineffective, which gives m_k!^nk terms of the denominator, also permutations as blocks as the same size are ineffective, which gives nk! terms. The partial sums of Bell polynomials with constant number as the nonzero elements are known to give Stirling numbers of the second kind. Using the definition of Stirling numbers of the second kind and the falling factorial in the form n!/n_o!, we obtain from (5) the identity (7), without terms m₀!ⁿ⁰ = 1 .This offers its additional proof.

The polynomial coefficient for m permutations is known as the Brillouin measure. Its logarithm gives H_m similarly as H_n was obtained from the polynomial coefficient for n permutations.

There is an open question how to interpret H_m in physics. The interpretation of H_n is clear. If the representation point of an isolated system can go through all points as the plane simplex, its average state is given by the left hand side as identity (4), it is described by the Bose-Einstein distribution. If the system is large and it is not spliced into independent ensembles, we can observe it on the largest orbit or its neighbor orbits for sufficient long time. Then its entropy is given by a term of the right hand side as (4).

If the temperature of the system is T= 0, there is only one element of symmetry, the identity and H_m = 0. At higher temperatures, the system moves in the phase space to higher planes with more and longer orbits. There are not taken into account changes of the volume V, and thus H_n cannot be replaced by any other function, as Guy tried (1). We can suppose that n molecules are moving on a lattice with v cells and thus there are (v!/n!(v-n)!) possible additional arrangements of the system at the constant volume V. If we want to consider the effect of the volume on entropy of the ideal gas, we must add another term to H_n, H_m could be applied only at the nonideal gas which molecules are not pointlike.

There is possible to distinguish, at least partly, energy increments of different parts of molecules as it will be shown later. Usually (1) is applied and no distinction is made between Hn and H_m.

There is an exception. The principle of increasing mixing character (11-14) is based explicitly on H_m. According to it, the time development of a Gibbs (microcanonical) ensemble of isolated systems proceeds in such a way that the mixing character increases monotonically. There are some partition orbits which are not comparable by their mixing character (e.g. 3,3,0 and 4,1,1). It is not possible to go from one orbit on the other by one collision of two particles, there are necessary at least two collisions and the principle of the increasing mixing order excludes accessibility of both orbits, it deforms the ideal space by its restrictions.

But the vector space as defined till now is not yet complete. It does not contain e.g. sums and differences of naive matrices N. They form incidence matrices G of graphs. For graphs many special functions (16, 17) are defined. They are known are as information indices. Both functions H_n and H_m were found distinct from them on a special class of graphs, star forests.

S n!/Ő n_k!m_k!^n(k) = n!

where the sum is made over all partitions of n into cycles of m_k length, S m_k = n.

Similarly, formula (4) can be used for counting indexed trees. The Pruefer algorithm (17) connects all strings of (n-2) symbols from a set of n symbols with the indexed trees with n vertices. For these strings we use (5) as

where m_k are valences of the vertices.

If we apply this formula e.g. for octanes, we must remember that carbons and hydrogens can not be permuted, even after a correction we get all isomers on one orbit.

A somewhat better result is obtained, if we consider only the carbon atoms. In such a case 7 partitions are derived:

Also in this case, however, some isomers are counted together, e.g. 2-methyl heptanes, 3-methyl heptanes, 4-methyl heptanes and 3-ethyl hexanes on the partition orbit 3 2⁴ 1³.

These results can be explained by the fact that using (6) for counting trees, we do not distinguish the structure of the off-diagonal elements of the adjacency matrix and their symmetry, which is the symmetry of a quadratical form G^TG.

The known methods for counting trees (17) are better because they determine the product of symmetries of trees more exactly. H_n and H_m are not universal measures and the range of their possible applications is limited.