The reason there is no absolute zero of entropy is rooted in the fundamental principles of statistical mechanics and the nature of microstates and macrostates. Entropy is a measure of the number of possible microstates that correspond to a given macrostate of a system. The macrostate describes the macroscopic properties of the system, such as its energy, volume, and particle numbers, while the microstate describes the specific arrangement and distribution of individual particles and their energies.
In a system with a fixed total energy, such as an isolated system, the microstates that correspond to different macrostates are not equally probable. The distribution of microstates among macrostates is determined by the principle of maximum entropy, also known as the principle of equal a priori probabilities. According to this principle, when the system is in equilibrium, the macrostate with the largest number of corresponding microstates is the most probable one.
The entropy of a system is related to the logarithm of the number of microstates associated with a given macrostate. As the system approaches equilibrium, the probability distribution of microstates becomes increasingly uniform among the macrostates that are accessible to the system at a given energy. However, even in equilibrium, the microstates are not distributed equally among the macrostates. Instead, the distribution follows the Boltzmann distribution, which gives exponentially decreasing probabilities for macrostates with higher energy or less probable configurations.
The absence of an absolute zero of entropy arises from the fact that there is no restriction on the energy levels that a system can have. As long as a system has energy levels available, there will always be microstates associated with those energy levels, and thus the system will have non-zero entropy. To reach a state where all microstates have equal probabilities and maximum entropy, the system would need to exhaust all its energy levels and reach an energy level called the ground state. However, achieving this state would require cooling the system to absolute zero temperature, which is unattainable according to the third law of thermodynamics.
In summary, the absence of an absolute zero of entropy is a consequence of the statistical nature of microstates and macrostates, the principle of maximum entropy, and the possibility of having multiple energy levels available to a system.