### The modern second law of thermodynamics

The second law is one of the most popular laws of nature, because it is often discussed in popular science treatises and educative videos, but what is the more correct and general formulation of this law?

According to P. W. Bridgman (1946 Nobel Prize in Physics) "There have been nearly as many formulations of the second law as there have been discussions of it".

We can find many historical verbal statements of the law, from when the science of thermodynamics was developing in the 19th century. Some examples:

"A transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature is impossible."

"It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects."

"It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat reservoir."

All these first statements were finally generalized to the following statement.

"The entropy change for an irreversible process occurring in an isolated system is greater than or equal to zero, with the equal sign applying to the limiting case of a reversible process."

Or in a formal language

$\Delta S \geq 0$

with $$S$$ the entropy of the thermodynamic system under study. This is the version of the second law that we can find in many standard but outdated textbooks. It is equation 9.1 in Swendsen [1]. However, this version is only valid for isolated systems and no system, except the universe is isolated. It is true that some systems (think of a thermos of coffee) are so well insulated from the environment that they can be considered isolated for all practical purposes, at least for short periods of time, but what about those systems that cannot be considered isolated even as a rough approximation? In this case, the above inequality should be changed to

$\Delta S_{univ} = \Delta S + \Delta S_{sur} \geq 0$

where $$S$$ continues to be the entropy of the system under study and $$S_{sur}$$ the entropy of the rest of the universe. This is the first equation of chapter 3 in Kaufman [2].

This version of the second law is very popular, but invalid for two reasons. The first reason is that the law of entropy increase is only valid when the energy is constant, and according to the current cosmological model the energy of the universe is not conserved [3]. If energy is not conserved, entropy cannot be said to increase. The second reason why $$\Delta S_{univ} \geq 0$$ is not valid is that it imposes only a global inequality. Let us see this in more detail.

Consider the following figure that shows the evolution of entropy for some isolated system. Both evolutions guide the system from an initial state with a given entropy to a final state with higher entropy.

Both evolutions are compatible with the classic inequality $$\Delta S \geq 0$$ but only one of the evolutions is observed in nature. The classical inequality is therefore invalid.

For those two reasons, thermodynamicians developed an improved version of the second law in the early 20th century. The second difficulty with the classic inequality can be solved if we replace the total change with a differential $$\mathrm{d} S \geq 0$$, but before explaining how to solve the first difficulty I must introduce a special notation because I am aware that most Americans are not familiar with the Brussels school of thermodynamics.

Ilya Prigogine initiated the modern formalism of thermodynamics by expressing changes in entropy as a sum of two parts [4]:

$\mathrm{d} S = \mathrm{d}_i S + \mathrm{d}_e S$

where $$\mathrm{d}_e S$$ is the entropy change due to exchanges of matter and energy with the exterior of the system and $$\mathrm{d}_i S$$ is the entropy change produced by irreversible processes in the interior of the system. The quantity $$\mathrm{d}_e S$$ can be positive, negative, or zero, but $$\mathrm{d}_i S$$ can only be equal to or greater than zero.

Dissipative structure maintained by the production of entropy and the flow of entropy into the surroundings.

Therefore, the modern form of the second law is

$\mathrm{d}_i S \geq 0$

This new expression is valid for open, closed, and isolated systems, and can be applied to the universe as a whole despite the fact that the consensus cosmological model does not conserve energy. This modern expression also predicts the correct monotonic approach to equilibrium because it is a differential form.

The modern expression is much better but has still a flaw. For example, if we assume that the entire system is divided into two subsystems,

$\mathrm{d}_i S = \mathrm{d}_i S_1 + \mathrm{d}_i S_2 \geq 0$

where $$\mathrm{d}_i S_1$$ and $$\mathrm{d}_i S_2$$ are the entropy productions in each of the subsystems, the second law $$\mathrm{d}_i S \geq 0$$ is compatible, for example with the nonphysical situation $$\mathrm{d}_i S_1 \gt 0$$ and $$\mathrm{d}_i S_2 \lt 0$$, but $$(\mathrm{d}_i S_1 + \mathrm{d}_i S_2) \geq 0$$.

The second law has to be not only local in time (differential form), but also local in space. The production of entropy in each part of a thermodynamic system has to be positive. For our example above, this means that $$\mathrm{d}_i S_1 \geq 0$$ and $$\mathrm{d}_i S_2 \geq 0$$. This reasoning produces the final version of the second law.

We begin by dividing the Prigogine expression for the entropy change by the time differential

$\frac{\mathrm{d} S}{\mathrm{d} t} = \frac{\mathrm{d}_i S}{\mathrm{d} t} + \frac{\mathrm{d}_e S}{\mathrm{d} t}$

If $$s = s(\boldsymbol{r},t)$$ is the entropy density, then the entropy of a system with volume $$V$$ can be written as $$S = \int s \mathrm{d} V$$, and its total rate of change is

$\frac{\mathrm{d} S}{\mathrm{d} t} = \int \frac{\partial s}{\partial t} \mathrm{d} V$

The rate of production of entropy can be written as

$\frac{\mathrm{d}_i S}{\mathrm{d} t} = \int \sigma \mathrm{d} V$

where $$\sigma$$ is the amount of entropy produced per unit volume per unit time. The entropy change due to exchanges of matter and energy with the surroundings can be written as

$\frac{\mathrm{d}_e S}{\mathrm{d} t} = -\int \boldsymbol{J} \cdot \mathrm{d} \boldsymbol{A}$

where $$\boldsymbol{J}$$ is the rate of change of entropy through the unit area, and using the Gauss theorem, we can rewrite this change as

$\frac{\mathrm{d}_e S}{\mathrm{d} t} = -\int (\nabla \cdot \boldsymbol{J}) \mathrm{d} V$

Finally, equating the total rate of change of the entropy of the system with the rates due to production and exchanges with the surroundings, we obtain

$\int \frac{\partial s}{\partial t} \mathrm{d} V = \int \sigma \mathrm{d} V - \int (\nabla \cdot \boldsymbol{J}) \mathrm{d} V$

and, since this expression is valid for arbitrary values of the volume, the following equality holds

$\frac{\partial s}{\partial t} = \sigma -\nabla \cdot \boldsymbol{J}$

This is often called a balance equation in the modern literature in thermodynamics. The second law of thermodynamics is then

$\sigma \geq 0$

This modern statement about the production of entropy density is stronger than the classical statement that the entropy of an isolated system can only increase or remain unchanged. $$\sigma \geq 0$$ is the modern statement of the second law used in the development of extended thermodynamics [5] and in advanced engineering applications.

## REFERENCES AND NOTES

1. An introduction to statistical mechanics and thermodynamics - 2nd ed 2020: Oxford University Press; Oxford. Swendsen, Robert H.
2. Principles of Thermodynamics 2002: Marcel Dekker, Inc. New York. Kaufman, Myron
3. I am pretty sure that the current cosmological model is invalid, but most physicists, astronomers, and cosmologists disagree with me. I will not explain my own model here, I will simply emphasize that they are applying a version of the second law that is not valid for the current cosmological model and that all the consequences that derive from that application are nonphysical.
4. Introduction to modern thermodynamics 2008: John Wiley & Sons Ltd, Chichester. Kondepudi, Dilip.
5. Extended irreversible thermodynamics - 4th ed 2010: Springer Science+Business Media B.V., New York. Jou, David; Casas-Vázquez, José; Lebon, Georgy.