In thermodynamics, Gibbsian thermodynamics is a common term that refers to three papers published by American engineer Willard Gibbs, specifically Graphical Methods in the Thermodynamics of Fluids (1873), A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces (1873), and On the Equilibrium of Heterogeneous Substances (1875-1878). 
All-in-all, this set of publications is an extension of the German physicist Rudolf Clausius’ 1865 thermodynamics textbook Mechanical Theory of Heat, particularly his formulations of energy and entropy, to more varied situations, such as particle flow into or out of the system, gravitations effects, coexistence of phases, stability of fluids, critical phases, geometrical illustrations, phases of dissipated energies, gas mixtures with convertible components, equilibriums of stressed solids, capillarity, liquid films, as well as the effect of electromotive forces. 
The term “Gibbsian thermodynamics” seems to have become common in the 1950s, likely following the publications of Thermodynamics and the Free Energy of Chemical Substances (1923), by American physical chemists Gilbert Lewis and Merle Randall, and Modern Thermodynamics by the Methods of Willard Gibbs (1933) by English physical chemist Edward Guggenheim.  The first clear distinction of the term “Gibbsian thermodynamics”, as a subset of equilibrium (or classical thermodynamics), seems to have been made in 1966 by American physicist Laszlo Tisza, of the MIT school of thermodynamics.  Specifically, Tisza wrote a section in his Generalized Thermodynamics textbook titled Gibbsian thermostatics and used the term Gibbsian thermodynamics ubiquitously, as version of thermodynamics that takes a physical-chemical look inside the “black box” thermodynamic system of Clausius and Kelvin. 
● Prigoginean thermodynamics
1. (a) Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids", Transactions of the Connecticut Academy, I. pp. 309-342, April-May. (b) Gibbs, J. Willard. (1873). "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces", Transactions of the Connecticut Academy, II. pp.382-404, Dec.
(c) 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.
2. Gibbs, Willard. (1878). “Abstract of On the Equilibrium of Heterogeneous Substances”, American Journal of Science, 3 ser., vol. XVI, pgs. 441-58, Dec.
3. (a) Journal of Chemical Education, 1956, pg. 356.
(b) Journal of the American Chemical Society, 1960, pg. 1264., vol. 82, Mar.-Apr.
4. Firoozabadi, Abbas. (1999). Thermodynamics of Hydrocarbon Reservoirs, (pg. 1). McGraw-Hill Professional.
5. Tisza, Laszlo. (1966). Generalized Thermodynamics. Cambridge, (section: 2.3: Gibbsian thermostatics, pgs 38-48, pg. 104). Massachusetts: The MIT Press.