In statistical thermodynamics, maximum entropy is the state of an isolated ideal gas phase system in which the entropy can no longer increase. [1] In other words, entropy reaches its “maximum” possible value when the system finally reaches its dominant macrostate; when that happens, the system has reached the equilibrium state. This state is defined as the thermodynamic system in its largest macrostate. In 1944, for living systems, Austrian physicist physicist Erwin Schrödinger incorrectly equated the state of maximum entropy with death. [2]

Heterogeneous systems
In more complex systems, i.e. ones that are not composed of ideal gas phase particles, namely any material system, one must study both the entropy and the energy of the system to determine the criteria of stability and equilibrium. [3]

References
1. Lee, Jooh, C. (2002). Thermal Physics - Entropy and Free Energies. (pg. 18). London: World Scientific.
2. Schrödinger, Erwin. (1944). What is Life? (ch. 6 “Order, Disorder, and Entropy). pgs. 67-75 Cambridge: Cambridge University Press.
3. Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy, III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878.