View source for Thermodynamic Entropy

'''Thermodynamic entropy''' is the macroscopic quantity S that measures a physical system's irreversible spread across available microstates, formally defined by Clausius (1865) as the ratio of reversibly exchanged heat to absolute temperature (dS = δQ_rev / T) and reinterpreted by Boltzmann (1877) as the logarithm of the number of microstates consistent with a given macrostate: S = k_B ln W. These two definitions are equivalent but illuminate different things: Clausius entropy is operational (it tells you what to measure), Boltzmann entropy is explanatory (it tells you what entropy is).

Thermodynamic entropy is not to be confused with [[Information theory|Shannon entropy]], though the mathematical forms are identical up to a constant. Shannon entropy measures uncertainty about the outcome of a random variable; thermodynamic entropy measures the information that would be required to specify a physical system's microstate given its macrostate. The connection is not merely formal — [[Landauer Principle|Landauer's principle]] establishes that erasing one bit of information must increase thermodynamic entropy by at least k_B ln 2, creating a hard bridge between the informational and physical quantities.

The second law of thermodynamics asserts that entropy in a closed system never decreases — a statement that Boltzmann showed is statistical rather than absolute: entropy decrease is overwhelmingly improbable, not forbidden. This statistical character is the source of [[Statistical Mechanics|statistical mechanics']] deepest puzzle: why did the universe begin in an anomalously low-entropy state, and what exactly is the connection between entropy increase and the [[Arrow of Time|arrow of time]]?

[[Category:Physics]]
[[Category:Science]]
[[Category:Mathematics]]