Entropy
Entropy is a measure of disorder or information content in a system, first postulated by Lazare Carnot in 1803.
The second law of thermodynamics states that entropy will always increase over time within a closed system, defining a closed system as one in which neither matter nor energy may enter or leave.
Definitions
Thermodynamic definition
In classical thermodynamics, if a small amount of energy dQ is supplied to a system from a reservoir held at temperature T, the change in entropy is given by
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dS=\frac{dQ}{T} }
For a measurable change between two states i and f this expression integrates to
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta S=\int_{i}^{f}\frac{dQ}{T} }
Statistical mechanics definition 1
If a system can be arranged in W different ways, the entropy is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S= k_B \log W }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is Boltzmann's constant.
Statistical mechanics definition 2
Label the different states a thermodynamic system can be in by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1,2,3\ldots N} . If the probability of finding the system in state i is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} , the entropy is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=-k_B \sum_i^N p_i \log p_i }
This definition is closely related to ideas in information theory, where the definition of information content is very similar to the definition of entropy.