The principle of maximum entropy states, that an isolated system at thermodynamic equilibrium is in the highest entropy possible, given its conditions. This also applies to isolated composite systems. Lets say there is a composite system kept in a stable state with some type of internal constraints, which keep the subsystems from interacting with one another. A simple constraint could be a fixed adiabatic wall separating two halves of a room with different temperatures and volumes. When the constraint is relaxed (allowed to change) then the system will automatically evolve into a new equilibrium state. The principle of maximum entropy tells us that this new equilibrium state will be the one with the highest possible entropy given the new conditions.

Symbolically if there is an isolated composite system Σc containing a number of subsystems Σj which are kept from interacting with each other via internal constraints. When one of the constraints X is relaxed, the resulting uninhibited change of state will alter the constraint from it's initial value Xi to a new equilibrium value Xe. It is important to note that any process during this time is non-quasistatic and irreversible because of the systems isolation. If Xi is not equal to Xe there is an increase in entropy in Σc due to the second law. If Xi = Xe, then nothing happens.

According to Gibbs' fundamental relations the entropy of a thermodynamic state can be written as a function of its extensive variables, one of which is the variable constraint X, so Sc = Sc(X). The second law states that

Sc(Xi) < Sc(Xe)
The principle of maximum entropy says that the allowable equilibrium states are when there is a maximum in the function of Sc(X). :
dSc/dX = 0
d2Sc/dX2 < 0
The limitations of the principle of maximum entropy has are that:
• it only applies to equilibrium states, any changes which occur while the system is not equilibrium are not described.
• it doesn't describe what the equilibrium position will be in a system which undergoes reversible processes which do not increase entropy. For that you need to apply the principle of minimum energy.

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