Entropy is a quantity in thermodynamics that characterizes equilibrium states and constrains the direction of physical processes. Its statistical interpretation relates macroscopic properties to the number and probabilities of compatible microscopic states. A related concept, information entropy, measures uncertainty in a probability distribution. These definitions share mathematical structures, but thermodynamic entropy has physical units and refers to a specified physical system, whereas information entropy describes a statistical source or model. (goldbook.iupac.org)
Thermodynamic definition
Thermodynamic entropy, conventionally denoted (S), is a state function: its change depends only on the initial and final equilibrium states, not on the path connecting them. For an infinitesimal reversible process,
[ dS=\frac{\delta Q_{\mathrm{rev}}}{T}, ]
where (\delta Q_{\mathrm{rev}}) is heat transferred into the system and (T) is absolute temperature. The symbol (\delta) distinguishes path-dependent heat transfer from the exact differential (dS). For a finite change, the expression is integrated along a reversible path, even if the actual process is irreversible. (goldbook.iupac.org)
The SI unit of entropy is joules per kelvin, (\mathrm{J,K^{-1}}). Entropy is ordinarily extensive: scaling the amount of a homogeneous substance at unchanged intensive conditions scales its entropy proportionally. It is not energy, although multiplying entropy by temperature produces a quantity with energy units. Neither heat nor work is itself a stored property of a system. (goldbook.iupac.org)
The second law and irreversibility
The second law of thermodynamics states that the entropy of an isolated system cannot decrease in a macroscopic process:
[ \Delta S_{\mathrm{isolated}}\geq 0. ]
Equality holds for a reversible process; irreversible processes generate entropy. Examples include friction, unrestricted expansion, and heat transfer across a finite temperature difference. For an isolated system subject to fixed constraints, stable thermodynamic equilibrium corresponds to maximum entropy. (ocw.mit.edu)
A system exchanging heat or matter with its surroundings can decrease its own entropy. The second law applies to the combined entropy balance, not to every component separately. A refrigerator, for example, removes heat from a cold region by consuming work and releasing heat elsewhere. Local cooling therefore does not imply a decrease in total entropy. (ocw.mit.edu)
Entropy also limits the performance of a heat engine. An engine operating cyclically between reservoirs at absolute temperatures (T_h>T_c>0) has efficiency no greater than
[ \eta_{\max}=1-\frac{T_c}{T_h}. ]
This reversible limit is associated with the Carnot cycle; entropy production lowers the attainable efficiency. (ocw.mit.edu)
Statistical interpretation
In statistical mechanics, a macrostate specifies bulk properties, while a microstate specifies microscopic details. For an equilibrium description with (\Omega) equally probable accessible microstates, the Boltzmann formula is
[ S=k_{\mathrm B}\ln\Omega, ]
where (k_{\mathrm B}) is the Boltzmann constant. Increasing the number of compatible microscopic arrangements increases entropy logarithmically. The logarithm allows independent multiplicities, which multiply, to correspond to entropies that add. (goldbook.iupac.org)
For discrete microstates with probabilities (p_i), the Gibbs expression is
[ S=-k_{\mathrm B}\sum_i p_i\ln p_i. ]
It reduces to the Boltzmann formula when (p_i=1/\Omega). The relevant probability distribution depends on the physical constraints and the statistical description. Calling entropy “disorder” is a mnemonic rather than a quantitative definition: calculation requires specifying states, probabilities, and constraints, not judging visual appearance. (ocw.mit.edu)
Absolute entropy and chemical processes
The third law of thermodynamics supplies a reference for absolute entropy: a perfectly ordered crystal with a unique ground state has entropy approaching zero as its temperature approaches absolute zero. This reference permits absolute entropies to be determined from thermal measurements rather than only entropy differences. (goldbook.iupac.org)
In chemistry, entropy contributes to the criterion for spontaneous change. At fixed temperature,
[ \Delta G=\Delta H-T\Delta S, ]
where (G) is Gibbs free energy and (H) is enthalpy. For a closed system at constant temperature and pressure with only pressure–volume work, spontaneous change decreases (G). Thus a process may be favorable even when the system’s entropy decreases, provided its enthalpy change compensates. These criteria determine thermodynamic favorability, not reaction speed, which belongs to chemical kinetics. (ocw.mit.edu)
Information-theoretic entropy
Claude Shannon introduced entropy into information theory in his 1948 paper A Mathematical Theory of Communication. For a discrete random variable (X),
[ H(X)=-\sum_x p(x)\log_2p(x), ]
with (0\log_2 0) defined as zero. Using base-two logarithms expresses entropy in bits. A fair binary outcome has entropy one bit; a certain outcome has zero entropy. Among distributions on (n) possible outcomes, the uniform distribution maximizes entropy at (\log_2n). (people.math.harvard.edu)
Shannon entropy governs fundamental limits on lossless data compression. It measures statistical uncertainty, not the meaning or usefulness of a message. Although its formula resembles Gibbs entropy, an arbitrary distribution over symbols is not automatically a thermodynamic system, and bits cannot be identified with joules per kelvin without an appropriate physical interpretation. (people.math.harvard.edu)
Quantum entropy
In quantum mechanics, the corresponding quantity is von Neumann entropy. For a density matrix (\rho), its thermodynamic normalization is
[ S=-k_{\mathrm B}\operatorname{Tr}(\rho\ln\rho). ]
A pure state has zero von Neumann entropy. However, a subsystem of a globally pure state can have positive entropy because of quantum entanglement. For a bipartite pure state, the entropy of either reduced density matrix quantifies its entanglement; subsystem entropy therefore need not represent classical uncertainty about the overall state. (lancaster.ac.uk)