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Macroscopic mechanical properties of polymers are determined by theirmicroscopic molecular chain distribution. Due to randomness of these molecularchains, probability theory has been used to find their micro-states and energydistribution. In this paper, aided by central limit theorem and mixed Bayesrule, we showed that entropy elasticity based on Gaussian distribution isquestionable. By releasing freely jointed chain assumption, we found that thereis energy redistribution when each bond of a molecular chain changes itslength. Therefore, we have to change Gaussian distribution used in polymerelasticity to Maxwell-Boltzmann distribution. Since Maxwell-Boltzmanndistribution is only a good energy description for gas molecules, we found amathematical path to change Maxwell-Boltzmann distribution to Fermi-Diracdistribution based on molecular chain structures. Because a molecular chain canbe viewed as many monomers glued by covalent electrons, Fermi-Diracdistribution describes the probability of covalent electron occupancy inmicro-states for solids such as polymers. Mathematical form of Fermi-Diracdistribution is logistic function. Mathematical simplicity and beauty ofFermi-Dirac distribution make many hard mechanics problems easy to understand.Generalized logistic function or Fermi-Dirac distribution function was able tounderstand many polymer mechanics problems such as viscoelasticity [1],viscoplasticity [2], shear band and necking [3], and ultrasonic bonding [4].

On Polymer Statistical Mechanics: From Gaussian Distribution to Maxwell-Boltzmann Distribution to Fermi-Dirac Distribution