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An equivalent definition of entropy is the expected value of the self-information of a variable. [1] Two bits of entropy: In the case of two fair coin tosses, the information entropy in bits is the base-2 logarithm of the number of possible outcomes — with two coins there are four possible outcomes, and two bits of entropy. Generally ...
Despite the foregoing, there is a difference between the two quantities. The information entropy Η can be calculated for any probability distribution (if the "message" is taken to be that the event i which had probability p i occurred, out of the space of the events possible), while the thermodynamic entropy S refers to thermodynamic probabilities p i specifically.
The Shannon entropy (in nats) is: = = = and if entropy is measured in units of per nat, then the entropy is given by: = which is the Boltzmann entropy formula, where is the Boltzmann constant, which may be interpreted as the thermodynamic entropy per nat.
In the context of entropy, "perfect internal disorder" has often been regarded as describing thermodynamic equilibrium, but since the thermodynamic concept is so far from everyday thinking, the use of the term in physics and chemistry has caused much confusion and misunderstanding.
Boltzmann's equation—carved on his gravestone. [1]In statistical mechanics, Boltzmann's equation (also known as the Boltzmann–Planck equation) is a probability equation relating the entropy, also written as , of an ideal gas to the multiplicity (commonly denoted as or ), the number of real microstates corresponding to the gas's macrostate:
Thermodynamic entropy is equal to the Boltzmann constant times the information entropy expressed in nats. The information entropy expressed with the unit shannon (Sh) is equal to the number of yes–no questions that need to be answered in order to determine the microstate from the macrostate. The concepts of "disorder" and "spreading" can be ...
In this context, either an information-theoretical measure, such as functional clusters (Gerald Edelman and Giulio Tononi's functional clustering model and dynamic core hypothesis (DCH) [47]) or effective information (Tononi's integrated information theory (IIT) of consciousness [48] [49] [50]), is defined (on the basis of a reentrant process ...
The maximum entropy principle: For a closed system with fixed internal energy (i.e. an isolated system), the entropy is maximized at equilibrium. The minimum energy principle: For a closed system with fixed entropy, the total energy is minimized at equilibrium.