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The concept entropy was first developed by German physicist Rudolf Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In statistical mechanics, entropy is formulated as a statistical property using probability theory.
The statistical definition of entropy defines it in terms of the statistics of the motions of the microscopic constituents of a system — modelled at first classically, e.g. Newtonian particles constituting a gas, and later quantum-mechanically (photons, phonons, spins, etc.). The two approaches form a consistent, unified view of the same ...
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 ...
Thus the definitions of entropy in statistical mechanics (The Gibbs entropy formula = ) and in classical thermodynamics (=, and the fundamental thermodynamic relation) are equivalent for microcanonical ensemble, and statistical ensembles describing a thermodynamic system in equilibrium with a reservoir, such as the canonical ensemble, grand ...
In statistical mechanics, the entropy of the system is given by Ludwig Boltzmann's equation: = where S is the thermodynamic entropy, W is the number of microstates that may yield the macrostate, and is the Boltzmann constant.
Intuitively, the entropy H X of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X when only its distribution is known. The entropy of a source that emits a sequence of N symbols that are independent and identically distributed (iid) is N ⋅ H bits (per message of N symbols).
The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble , in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles .
The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information).