Neglecting correlations between the different possible states (or, more generally, neglecting statistical dependencies between states) will lead to an overestimate of the entropy. In every ensemble, the equilibrium configuration of the system is dictated by the maximization of the entropy of the union of the system and its reservoir, according to the second law of thermodynamics (see the statistical mechanics article). Each statistical ensemble (micro-canonical, canonical, grand-canonical, etc.) describes a different configuration of the system's exchanges with the outside, from an isolated system to a system that can exchange one more quantity with a reservoir, like energy, volume or molecules. The set of microstates on which the sum is to be done is called a statistical ensemble. Other definitions assume that the system is in thermal equilibrium, either as an isolated system, or as a system in exchange with its surroundings. This definition remains valid even when the system is far away from equilibrium. The quantity is a physical constant known as Boltzmann's constant, which, like the entropy, has units of heat capacity. In the thermodynamic limit, the fluctuation of the macroscopic quantities from their average values becomes negligible so this reproduces the definition of entropy from classical thermodynamics, given above. If the changes are sufficiently slow, so that the system remains in the same microscopic state, but the state slowly (and reversibly) changes, then в€‘ i (dE i) p i is the expectation value of the work done on the system through this reversible process, dw rev.Д«ut from the first law of thermodynamics, ОґE = Оґw +Оґq. Now, в€‘ i d (E i p i) is the expectation value of the change in the total energy of the system. Where we have twice used the conservation of probability, в€‘ dp i=0. For a classical system (i.e., a collection of classical particles) with a discrete set of microstates, if is the energy of microstate i, and is its probability that it occurs during the system's fluctuations, then the entropy of the system isД®ntropy changes for systems in a canonical stateДЄ system with a well-defined temperature, i.e., one in thermal equilibrium with a thermal reservoir, has a probability of being in a microstate i given by Boltzmann's distribution.Д¬hanges in the entropy caused by changes in the external constraints are then given by: The entropy is given by the Gibbs entropy formula, named after J. So the entropy is defined over two different levels of description of the given system.

The macroscopic state of the system is defined by a distribution on the microstates that are accessible to a system in the course of its thermal fluctuations.

Change in entropy formula