Gibbs then travelled to Germany, where in Heidelberg he learned about the work of Rudolf Clausius, Hermann von Helmholtz, and Gustav Kirchhoff in physics, and Robert Bunsen in chemistry.

Back in New Haven, Gibbs published several long monographs.

Gibbs focused on five thermodynamical variables, volume, pressure, temperature, energy, and entropy. He showed that any three of these are independent variables, from which one can deduce the other two.

In 1873, his first monograph introduced new diagrams relating thermodynamical quantities to one another. In Graphical Methods in the Thermodynamics of Fluids, Gibbs explored various two-dimensional planar graphs showing two of these independent variables to exhibit thermodynamic properties. He first cites the success of the pressure-volume graphs that are most often used to illustrate the thermodynamics of the Carnot Cycle. Émile Clapeyron had in 1934 first drawn such diagrams. Clausius had also used them in 1865.

In the Carnot Cycle, the path 1>2 is an isothermal at the higher (source) temperature T₁. Path 2>3 is usually called adiabatic, though Gibbs prefers isoentropic. Path 3>4 is isothermal at the lower (sink) temperature, and the return path 4>1 is also isoentropic.

Since James Watt combined a pressure gauge with a volume indicator on his steam engine, engineers had graphed the relations between pressure and volume, whose product is the work done. Each point in the pressure-volume graph represents the state of the system, the integral around the curve is the work done by the system (the gray area in the figure), perfect for calculating the efficiency of steam engines or any heat engine.

Gibbs argues that plotting entropy and temperature as the two coordinates is preferable to pressure-volume for many purposes. We believe it is the most intuitive graph as a teaching tool when explaining the relationship between energy and work available. One can look at this graph and visualize directly the maximum theoretical work that can be done by an engine working between a high temperature source and a low temperature sink.

Gibbs wrote in 1873...

It is worthy of notice that the simplest form of a perfect thermodynamic engine, so often described in treatises on thermodynamics, is represented in the entropy-temperature diagram by a figure of extreme simplicity, viz: a rectangle of which the sides are parallel to the co-ordinate axes.
Thus in figure 3, the circuit ABCD may represent the series of states through which the fluid is made to pass in such an engine, the included area representing the work done, while the area ABFE represents the heat received from the heater at the highest temperature AE, and the area CDEF represents the heat transmitted to the cooler at the lowest temperature DE.

"Graphical Methods in the Thermodynamics of Fluids," in The Scientific Papers of J. Willard Gibbs, p.10

The method in which the co-ordinates represent volume and pressure has a certain advantage in the simple and elementary character of the notions upon which it is based, and its analogy with Watt’s indicator has doubtless contributed to render it popular. On the other hand, a method involving the notion of entropy, the very existence of which depends upon the second law of thermodynamics, will doubtless seem to many far-fetched, and may repel beginners as obscure and difficult of comprehension. This inconvenience is perhaps more than counter-balanced by the advantages of a method which makes the second law of thermodynamics so prominent, and gives it so clear and elementary an expression.

In Gibbs entropy-temperature graph, the path A>B is easily seen to be isothermal at the higher (source) temperature T_H. Path B>C is easily seen isoentropic. Path C>D is isothermal at the lower (sink) temperature T_C, and the return path 4>1 is also isoentropic. Gibbs says the circuit ABCD represents the series of states through which the fluid is made to pass, the included area W representing the work done, while the area ABFE represents the heat received from the heater at the highest temperature AE, and the area CDEF represents the waste heat transmitted to the cooler at the lowest temperature DE.

The fact, that the different states of a fluid can be represented by the positions of a point in a plane, so that the ordinates shall represent the temperatures, and the heat received or given out by the fluid shall be represented by the area bounded by the line representing the states through which the body passes, the ordinates drawn through the extreme points of this line, and the axis of abscissas,—this fact, clumsy as its expression in words may be, is one which presents a clear image to the eye, and which the mind can readily grasp and retain. It is, however, nothing more nor less than a geometrical expression of the second law of thermodynamics in its application to fluids, in a form exceedingly convenient for use, and from which the analytical expression of the same law can, if desired, be at once obtained. If, then, it is more important for purposes of instruction and the like to familiarize the learner with the second law, than to defer its statement as long as possible, the use of the entropy-temperature diagram may serve a useful purpose in the popularizing of this science

"Graphical Methods in the Thermodynamics of Fluids," in The Scientific Papers of J. Willard Gibbs, p.11

For example, let it be required to find the greatest amount of mechanical work which can be obtained from a given quantity of a certain substance in a given initial state, without increasing its total volume or allowing heat to pass to or from external bodies, except such as at the close of the processes are left in their initial condition. This has been called the available energy of the body. The initial state of the body is supposed to be such that the body can be made to pass from it to states of dissipated energy by reversible processes.

"Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces," in The Scientific Papers of J. Willard Gibbs, pp. 49-50

Gibbs does not give us another graphical representation of these kinds of energy, but we can broadly identify "available" energy with area ABCD, labelled W, the work done above, and "dissipated" energy with CDEF, the waste heat sent to the low temperate sink. Note that heat from the high temperature source is ABFE, the sum of ABCD and CDEF. In familiar modern terminology, the heat, or original energy content, is transformed into work and waste energy

Today we call the energy available to do work the Gibbs "Free Energy."

So what is the connection between available "free" energy and information structures? Clearly, in a state of thermal equilibrium there is nothing of the "order" we associate with information. Equilibrium is the ultimate "disorder."

In our explanation of the two-step cosmic creation process, the second step is exporting positive entropy away from the newly formed information structure. In the first step, available energy, or work, is essential for arranging the particles into a structure, usually one among many possible arrangements.

In the early universe, the arrangement is controlled by quantum cooperative phenomena with electrostatic attractive and nuclear repulsive forces. In the subsequent billions of years, the formation of planets, star, and galaxies is controlled by gravitational forces. Work is done by these forces as structural components are pulled together. The new configurations cannot be stable entities unless positive entropy is radiated away to satisfy the second law.

These information creation processes do not directly resemble the thermodynamic engines that Gibbs is discussing, with their obvious source of energy and heat sinks. But we can see the earliest universe as a cosmic source of high energy particles and radiation. We can locate the cold sink inside the universe, since there is no "outside." The sink for waste energy is the expanding space itself. Today we can see clearly across vast empty space to the uniform cosmic microwave background radiation at under three degrees Kelvin.

The resemblance to a thermodynamic engine is easier to see for our Earth. The hot source is our Sun, whose radiation leaves the sun at a temperature of thousands of degrees. When it reaches Earth, its energy content temperature is only hundreds of degrees, and when it is thermalized by the planet, it is radiated away from the dark side of Earth into the night sky,

Erwin Schrödinger described the Sun as the source of "negative entropy" on which "life feeds." He did not know how the Sun itself could get so far from equilibrium to be the source of available energy. That we explain by the expansion of the universe.

The cosmological and astrophysical "engines" are doing work not by extracting available energy from a hot gas or liquid and dumping waste energy as a material stream. They are doing work with forces that are action-at-a-distance. They are exporting their positive entropy by radiating it away to the empty space appearing between the information structures.

Gibbs' third monograph, in 1876, "On the Equilibrium of Heterogeneous Substances," began with Clausius' great first and second laws of thermodynamics in two simple sentences

"Die Energie der Welt ist constant. Die Entropie der Welt strebt einen Maximum zu."
"The energy of the world is constant. The entropy of the world tends towards a maximum."

Gibbs' "great memoir," as Lewis and Randall called it in 1923, contains a brief but careful explanation of what later writers called the "Gibbs Paradox" (pp.163-165). E. T. Jaynes said in 1992 it came as a "shock" that Gibbs' explanation had been missed by textbook writers for 80 years? This passage also includes probably the most famous quote about the idea of spontaneous entropy decrease, one cited in hundreds of textbooks, starting with Boltzmann in 1898, and including Lewis and Randall's chapter 8, Entropy and Probability.

Gibbs wrote...

we may easily calculate the increase of entropy which takes place when two different gases are mixed by diffusion, at a constant temperature and pressure. Let us suppose that the quantities of the gases are such that each occupies initially one half of the total volume. If we denote this volume by V, the increase of entropy will be...

(PV/T)log2

It is noticeable that the value of this expression does not depend upon the kinds of gas which are concerned, if the quantities are such as has been supposed, except that the gases which are mixed must be of different kinds. If we should bring into contact two masses of the same kind of gas, they would also mix, but there would be no increase of entropy. But in regard to the relation which this case bears to the preceding, we must bear in mind the following considerations. When we say that when two different gases mix by diffusion, as we have supposed, the energy of the whole remains constant, and the entropy receives a certain increase, we mean that the gases could be separated and brought to the same volume and temperature which they had at first by means of certain changes in external bodies, for example, by the passage of a certain amount of heat from a warmer to a colder body. But when we say that when two gas-masses of the same kind are mixed under similar circumstances there is no change of energy or entropy, we do not mean that the gases which have been mixed can be separated without change to external bodies. On the contrary, the separation of the gases is entirely impossible. We call the energy and entropy of the gas-masses when mixed the same as when they were unmixed, because we do not recognize any difference in the substance of the two masses. So when gases of different kinds are mixed, if we ask what changes in external bodies are necessary to bring the system to its original state, we do not mean a state in which each particle shall occupy more or less exactly the same position as at some previous epoch, but only a state which shall be undistinguishable from the previous one in its sensible properties. It is to states of systems thus incompletely defined that the problems of thermodynamics relate. But if such considerations explain why the mixture of gas-masses of the same kind stands on a different footing from the mixture of gas-masses of different kinds, the fact is not less significant that the increase of entropy due to the mixture of gases of different kinds, in such a case as we have supposed, is independent of the nature of the gases.

Now we may without violence to the general laws of gases which are embodied in our equations suppose other gases to exist than such as actually do exist, and there does not appear to be any limit to the resemblance which there might be between two such kinds of gas. But the increase of entropy due to the mixing of given volumes of the gases at a given temperature and pressure would be independent of the degree of similarity or dissimilarity between them. We might also imagine the case of two gases which should be absolutely identical in all the properties (sensible and molecular) which come into play while they exist as gases either pure or mixed with each other, but which should differ in respect to the attractions between their atoms and the atoms of some other substances, and therefore in their tendency to combine with such substances. In the mixture of such gases by diffusion an increase of entropy would take place, although the process of mixture, dynamically considered, might be absolutely identical in its minutest details (even with respect to the precise path of each atom) with processes which might take place without any increase of entropy. In such respects, entropy stands strongly contrasted with energy. Again, when such gases have been mixed, there is no more impossibility of the separation of the two kinds of molecules in virtue of their ordinary motions in the gaseous mass without any especial external influence, than there is of the separation of a homogeneous gas into the same two parts into which it has once been divided, after these have once been mixed. In other words, the impossibility of an uncompensated decrease of entropy seems to be reduced to improbability.

("Equilibrium of Heterogeneous Substances," in The Scientific Papers of J. Willard Gibbs, pp. 166-167)

Ludwig Boltzmann must have thought this passage extremely important. He used the last line as the opening quotation for the second volume of his Lectures on Gas Theory, "the impossibility of an uncompensated decrease of entropy seems to be reduced to improbability."

It was Gibbs' short text Principles in Statistical Mechanics published the year before his death in 1903 that brought him the most fame. In it, he coined the term phase space, phase volume, his phase rule and the name for his field - statistical mechanics. Earlier he named the chemical potential and the statistical ensemble. He showed how his graph planes can become the surfaces of three-dimensional objects that identify "phase changes," between gases and liquids, and between liquids and solids.

Gibbs "Paradox"

Gibbs paradox and its resolutions (a great list of references, but links are all dead)

Wikipedia

Stanford Encyclopedia of Philosophy