Please login first
Are Shannon entropy and Residual entropy synonyms?
1  Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602


At temperatures close to absolute zero, individual symmetric or asymmetric molecules that comprise a thermodynamic system organize themselves into crystals that we can comprehend as macromolecules. Crystals can be ideal (perfect crystals) if they consist of monotonous arrays of aligned symmetric or asymmetric molecules. Unideal (imperfect crystals) comprize nonmonotonic arrays of asymmetric molecules. Therefore, nonmonotonous arrays contain an information code and a certain amount of information, I. The disorder of such a thermodynamic-intofmation system is characetrized by Shannon entropy, H, in informatics and by residual entropy, S0 or R0, in thermodynamics. Both quantities are based on the statistical coin model and in both cases probabilities are determined by the Gaussian normal distribution. However, the probabilities in thermodynamic entropy, S, are described by the Boltzmann distribution. Thus, we see that a thermodynamic system possesses three kinds of entropy: thermodynamic entropy, residual entropy and Shannon entropy. The relationship of these three entropies, however, still remains unclear. In order to shed more light on this problem, a comparative analysis of the three entropies was done. Thermodynamic entropy of an ideal crystal was calculated by the Boltzmann entropy equation, treating it as one large macromolecule. Shannon entropy and residual entropy were found through the Shannon equation and Boltzmann entropy equation, respectively. In the calculation of Shannon entropy, the constant in the Shannon equation was set equal to the Boltzmann constant, in order for it to be comparable with thermodynamic and residual entropy. Entropies of real gasses were also analyzed. Thermodynamic entropy of real gasses was calculated from the high temperature ideal limit to the condensation temperature using van der Waals, Berthelot and Redlich-Kwong models. Our analysis leads us to conclusion that the residual entropy, S0 or R0, of a closed thermodynamic system that contains asymmetric particles (i.e. CO, H2O, N2O) aligned in nonmonotonic series is equal to its Shannon entropy, H. So the information theory and classical thermodynamics have at least one strong common point – the equality of Shannon entropy and residual entropy, H = R0.

Keywords: Thermodynamic entropy, Shannon entropy, Residual entropy, symmetric/asymmetric particles, monotonic/non-monotonic series.