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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.
In classical thermodynamics, entropy is defined in terms of macroscopic measurements and makes no reference to any probability distribution, which is central to the definition of information entropy. The connection between thermodynamics and what is now known as information theory was first made by Ludwig Boltzmann and expressed by his equation:
The mutual information is used to learn the structure of Bayesian networks/dynamic Bayesian networks, which is thought to explain the causal relationship between random variables, as exemplified by the GlobalMIT toolkit: [37] learning the globally optimal dynamic Bayesian network with the Mutual Information Test criterion.
An information diagram is a type of Venn diagram used in information theory to illustrate relationships among Shannon's basic measures of information: entropy, joint entropy, conditional entropy and mutual information. [1] [2] Information
For an open thermodynamic system in which heat and work are transferred by paths separate from the paths for transfer of matter, using this generic balance equation, with respect to the rate of change with time of the extensive quantity entropy , the entropy balance equation is: [53] [54] [note 1] = = ˙ ^ + ˙ + ˙ where = ˙ ^ is the net rate ...
To do this, one must acknowledge the difference between the measured entropy of a system—which depends only on its macrostate (its volume, temperature etc.)—and its information entropy, [6] which is the amount of information (number of computer bits) needed to describe the exact microstate of the system.
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 ...
The Gaussian or normal probability distribution plays an important role in the relationship between variance and entropy: it is a problem of the calculus of variations to show that this distribution maximizes entropy for a given variance, and at the same time minimizes the variance for a given entropy.