Prigogine’s Explanation of His Method
Now let’s follow Prigogine’s method for accounting for entropy production.
The second principle of thermodynamics postulates the existence of a function of state, called entropy (from the Greek εν τρωπη meaning “evolution”) which possesses the following properties:[i]
Prigogine says that entropy is an “extensive” property. This means that the entropy in a given region is grows as the size of the region is extended.
a) The entropy of the system is an extensive property. If a system consists of several parts, therefore the total entropy is equal to the sum of the entropies of each part.
b) The change of entropy dS can be split into two parts. Denoting by dₑS the flow of entropy due to interactions with the exterior, and by dᵢS the contribution due to changes inside the system, we have dS = dₑS + dᵢS. The entropy increase dᵢS due to changes inside the system is never negative. It is zero when the system undergoes reversible changes only, but it is positive if the system is subject to irreversible processes as well.
dᵢS = 0 (reversible processes)
dᵢS > 0 (irreversible processes)
In this chapter we shall calculate explicit expressions for the entropy production of some important irreversible processes and also the entropy flow related to exchanges of matter and energy with the external environment.[ii]
The analysis and measurement of the dynamics of properties like state variables is the domain of physics, the most precise of the natural sciences. When we call a science precise, we mean that its theoretical predictions agree precisely with experimental measurements. The Second Law of Thermodynamics says that in all processes without exception, the total amount of entropy or disorder never decreases. We can calculate amounts of entropy theoretically and compare the amounts accurately with measurements in controlled physical and chemical experiments.
Returning to Prigogine’s exposition, we note that he speaks loosely of the “flow of entropy.” What he means by a flow of entropy is a change of entropy inside a region enclosed within an imaginary bubble due to influences from the external region. These influences may be the movement of material or the flow of energy into or out of the region inside the bubble.
Prigogine says that, in the case of isolated systems, there is no such thing as “flow of entropy” into or out of the internal region. That is the meaning of saying that a system is isolated.
For isolated systems, there is no flow of entropy so that dS = dₑS + dᵢS and dᵢS ≥ 0 reduce to dS = dᵢS ≥ 0 (isolated system)[iii]
Next, Prigogine uses the production of an increase of entropy as the definition of an irreversible process.
For isolated systems, this relation is equivalent to the classical statement that entropy can never decrease, so that in this case the behavior of the entropy function provides a criterion that enables us to detect the presence of irreversible processes…. The only general criterion of irreversibility is given by the entropy production according to dᵢS = 0 (reversible processes) and dᵢS > 0 (irreversible processes).[iv]
Prigogine then sets up equations for two systems, one inside the other.
The boundary Prigogine draws may be all around the Earth, separating it from the Sun and outer space, or it may be just outside the skin of a living organism, separating it from the exterior world. In either of these two cases, what Prigogine calls “the global system containing both” is the rest of the universe. Using the definition of “the universe” as “everything physical that there is,” Prigogine says that the universe is isolated, meaning that the universe does not have the flow of energy or the transport of material into or out of it.
Suppose we enclose a system which we shall denote by I, inside a larger system II, so that the global system containing both I and II is isolated. In both parts, I and II, some irreversible processes may take place. The classical statement of the Second Law of Thermodynamics would be dS = dSᴵ + dSᴵᴵ ≥ 0. Applying now dᵢS = 0 (reversible processes) and dᵢS > 0 (irreversible processes) to each part separately, we shall postulate here that dᵢSᴵ ≥ 0, dᵢSᴵᴵ ≥ 0.[v]
Prigogine then denies that there can be any “physical situation” (a place where thermodynamic processes are operating) in which the entropy of any region enclosed within an imaginary boundary can decrease because of a sufficiently large increase of entropy outside the region.
A physical situation such that dᵢSᴵ > 0, dᵢSᴵᴵ < 0 with d (Sᴵ + Sᴵᴵ) > 0 is excluded. We can therefore say that “absorption” of entropy in one part, compensated by a sufficient “production” in another part of the system is prohibited.[vi]
Notice that there is no such thing as “‘absorption’ of entropy in one part, compensated by a sufficient ‘production’ in another part of the system.” Prigogine specifically states that a sufficiently large increase in entropy in the Sun and the rest of the universe cannot compensate for an entropy reduction on Earth. The entropy produced on the Earth cannot flow back to the Sun.
This formulation implies that in every macroscopic region of the system the entropy production due to the irreversible processes is positive. The term macroscopic region refers to any region containing a number of molecules sufficiently large for microscopic fluctuations to be negligible. Interference of irreversible processes is only possible when they occur in the same region of the system. Such a formulation may be called a “local” formulation of the second law in contrast to the “global” formulation of classical thermodynamics. Its value lies in the fact that it permits a much closer analysis of irreversible processes and, as such, it will constitute the central postulate on which this book is based. This postulate will have to be justified eventually by considerations based on statistical mechanics.
It is interesting to note that the splitting of the entropy change into two terms dᵢS and dₑS permits an easy discussion of the difference between closed and open systems as will be shown below. Clearly, this difference has to appear in the term dₑS that, for open systems, must contain terms due to the exchange of matter.[vii]
[i] Prigogine, Ilya, Introduction to Thermodynamics of Irreversible Processes (New York: John Wiley & Sons, 1967), p. 15
[ii] Prigogine, Ilya, op. cit., pp. 15–16.
[iii] Ibid. p. 16.
[iv] Ibid. pp. 16–17. Emphasis is Prigogine’s.
[v] Ibid. p. 17.
[vi] Ibid. p. 17.
[vii] Ibid. pp. 17–18.
The second principle of thermodynamics postulates the existence of a function of state, called entropy (from the Greek εν τρωπη meaning “evolution”) which possesses the following properties:[i]
Prigogine says that entropy is an “extensive” property. This means that the entropy in a given region is grows as the size of the region is extended.
a) The entropy of the system is an extensive property. If a system consists of several parts, therefore the total entropy is equal to the sum of the entropies of each part.
b) The change of entropy dS can be split into two parts. Denoting by dₑS the flow of entropy due to interactions with the exterior, and by dᵢS the contribution due to changes inside the system, we have dS = dₑS + dᵢS. The entropy increase dᵢS due to changes inside the system is never negative. It is zero when the system undergoes reversible changes only, but it is positive if the system is subject to irreversible processes as well.
dᵢS = 0 (reversible processes)
dᵢS > 0 (irreversible processes)
In this chapter we shall calculate explicit expressions for the entropy production of some important irreversible processes and also the entropy flow related to exchanges of matter and energy with the external environment.[ii]
The analysis and measurement of the dynamics of properties like state variables is the domain of physics, the most precise of the natural sciences. When we call a science precise, we mean that its theoretical predictions agree precisely with experimental measurements. The Second Law of Thermodynamics says that in all processes without exception, the total amount of entropy or disorder never decreases. We can calculate amounts of entropy theoretically and compare the amounts accurately with measurements in controlled physical and chemical experiments.
Returning to Prigogine’s exposition, we note that he speaks loosely of the “flow of entropy.” What he means by a flow of entropy is a change of entropy inside a region enclosed within an imaginary bubble due to influences from the external region. These influences may be the movement of material or the flow of energy into or out of the region inside the bubble.
Prigogine says that, in the case of isolated systems, there is no such thing as “flow of entropy” into or out of the internal region. That is the meaning of saying that a system is isolated.
For isolated systems, there is no flow of entropy so that dS = dₑS + dᵢS and dᵢS ≥ 0 reduce to dS = dᵢS ≥ 0 (isolated system)[iii]
Next, Prigogine uses the production of an increase of entropy as the definition of an irreversible process.
For isolated systems, this relation is equivalent to the classical statement that entropy can never decrease, so that in this case the behavior of the entropy function provides a criterion that enables us to detect the presence of irreversible processes…. The only general criterion of irreversibility is given by the entropy production according to dᵢS = 0 (reversible processes) and dᵢS > 0 (irreversible processes).[iv]
Prigogine then sets up equations for two systems, one inside the other.
The boundary Prigogine draws may be all around the Earth, separating it from the Sun and outer space, or it may be just outside the skin of a living organism, separating it from the exterior world. In either of these two cases, what Prigogine calls “the global system containing both” is the rest of the universe. Using the definition of “the universe” as “everything physical that there is,” Prigogine says that the universe is isolated, meaning that the universe does not have the flow of energy or the transport of material into or out of it.
Suppose we enclose a system which we shall denote by I, inside a larger system II, so that the global system containing both I and II is isolated. In both parts, I and II, some irreversible processes may take place. The classical statement of the Second Law of Thermodynamics would be dS = dSᴵ + dSᴵᴵ ≥ 0. Applying now dᵢS = 0 (reversible processes) and dᵢS > 0 (irreversible processes) to each part separately, we shall postulate here that dᵢSᴵ ≥ 0, dᵢSᴵᴵ ≥ 0.[v]
Prigogine then denies that there can be any “physical situation” (a place where thermodynamic processes are operating) in which the entropy of any region enclosed within an imaginary boundary can decrease because of a sufficiently large increase of entropy outside the region.
A physical situation such that dᵢSᴵ > 0, dᵢSᴵᴵ < 0 with d (Sᴵ + Sᴵᴵ) > 0 is excluded. We can therefore say that “absorption” of entropy in one part, compensated by a sufficient “production” in another part of the system is prohibited.[vi]
Notice that there is no such thing as “‘absorption’ of entropy in one part, compensated by a sufficient ‘production’ in another part of the system.” Prigogine specifically states that a sufficiently large increase in entropy in the Sun and the rest of the universe cannot compensate for an entropy reduction on Earth. The entropy produced on the Earth cannot flow back to the Sun.
This formulation implies that in every macroscopic region of the system the entropy production due to the irreversible processes is positive. The term macroscopic region refers to any region containing a number of molecules sufficiently large for microscopic fluctuations to be negligible. Interference of irreversible processes is only possible when they occur in the same region of the system. Such a formulation may be called a “local” formulation of the second law in contrast to the “global” formulation of classical thermodynamics. Its value lies in the fact that it permits a much closer analysis of irreversible processes and, as such, it will constitute the central postulate on which this book is based. This postulate will have to be justified eventually by considerations based on statistical mechanics.
It is interesting to note that the splitting of the entropy change into two terms dᵢS and dₑS permits an easy discussion of the difference between closed and open systems as will be shown below. Clearly, this difference has to appear in the term dₑS that, for open systems, must contain terms due to the exchange of matter.[vii]
[i] Prigogine, Ilya, Introduction to Thermodynamics of Irreversible Processes (New York: John Wiley & Sons, 1967), p. 15
[ii] Prigogine, Ilya, op. cit., pp. 15–16.
[iii] Ibid. p. 16.
[iv] Ibid. pp. 16–17. Emphasis is Prigogine’s.
[v] Ibid. p. 17.
[vi] Ibid. p. 17.
[vii] Ibid. pp. 17–18.