Entropy and distribution of probability

Example 1.3 Entropy and distribution of probability Entropy is a state function. Its foundation is macroscopic and directly related to macroscopic changes. Such changes are mostly irreversible and time asymmetric. Contrary to this, the laws of classical and quantum mechanics are time symmetric, so that a change between states 1 and 2 is reversible. On the other hand, macroscopic and microscopic changes are related in a way that, for example, an irreversible change of heat flow is a direct consequence of the collision of particles that is described by the laws of mechanics. Boltzmann showed that the entropy of a macroscopic state is proportional to the number of configurations fl of microscopic states a system can have 

Consider a system with two chambers containing a total number of particles n. The total number of possibilities for distributing the particles between the two chambers is fl, which is the total number of distinct microstates with nl number of particles in chamber 1 and n2 number of particles in chamber 2 

Equation (1.77) shows that disorganization and randomness increase entropy, while organization and ordering decrease it, and equilibrium states have the maximum value of fl. In the above system, fl reaches its maximum value when , = n2. In parallel, the increase in entropy corresponds to the increase in the number of microscopic states or states with higher probability. The concept of entropy as a measure of organized structures is attracting scientists from diverse fields such as physics, biology, and communication and information systems. 

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