A Philosophical Investigation of Entropy

            It depends on who you are talking to because even within a language there are different communities of language users that refer to different ideas using the same words. And even if they are referring to the same concept, it might have a different connotation or purpose. Note, the Christian use of the word entropy, it does not have to agree with the scientific use of the word entropy in order to have meaning within its own group of language users. But what we are interested in is not the meaning of the word entropy as defined within a group, but the meaning of the concept of entropy beyond language, as a fundamental law of the universe.

            Penrose brings up an interesting point about the second law of thermodynamics; it is given in the form of an inequality. When we think of laws, we think of Newton’s first law F=ma, or the conservation of energy and momentum, where we can predict with precision how objects will behave. But what does it mean that the entropy if a system will always be greater than or equal to one? What is the process that causes the continuous generation of entropy?

            Let’s first take a look at how entropy is used in language. In the study of thermodynamics, entropy is defined as “a measure of molecular disorder.” A system of fluids has an associated entropy value assigned to it. Just as the total internal energy of a system can be determined, we can also determine the amount of entropy a system has. When we see the use of entropy in language (such as in the Thermodynamics textbook I am using this semester), we see that entropy is something that can be generated, created, transferred, that there can be a change in entropy, etc. Entropy sounds like something tangible similar to energy (of course we cannot create energy, but we can generate energy through heat and work). But what exactly is being generated, what is being transferred? This is what I believe to be the problem causing the confusion in understanding entropy.

            Penrose attempts to explain entropy in the book Cycles of Time, but sadly he does not explain any of the terms that he is using so that the only people who would understand what he is saying already need no explanation. But from reading through his exposition of entropy, it seems that at the heart of it, entropy is not a difficult concept, and I will attempt to demystify what entropy is, through the words of Penrose. He defines entropy in terms of coarse-graining regions and phase spaces. I am not exactly sure what he means by this, but I will try to summarize the gist of what I think he is saying. He seems to be giving us graphical representation of the statistical behavior of many particle systems. Each degree of freedom in the system is given an entropic dimension n, so that the entropy of a system can be depicted as having trillions of dimensions for each of its degree of freedom. As an infinitesimal slice of time passes, dt, the possibilities of each degree of freedom increases (not in the sense that the system actually gains more possible futures, but in the sense that we cannot possibly know exactly the state of every degree of freedom), so that the overall volume of entropy increases. With each increase in volume, the entropy of the system is surrounded by possible futures of even larger volumes and also smaller volumes, but the larger volume entropies far outnumber the smaller volume entropic states by some large exponential amount. Again, when I say possible futures, I am saying it in a statistical sense, in a deterministic sense, there is only one possible future, but there is no way this can be determined because of the sheer number of degrees of freedom. I hope I am interpreting Penrose correctly, but the basic picture seems to be that entropy is a statistical statement about a system, not some abstract essence that is in a system.

            However, the way entropy is used in language paints a different of what entropy is. We get an understanding of concepts by the way it is used in language. Most of the time, this kind of linguistic analysis allows us to understand a concept. For example, we know that running is a type of action just by its grammatical structure. It is a gerund that follows the existential verb “be,” which tells us it is an action done by something. But I think the problem with entropy is that entropy is defined as disorder. When we state that entropy is generated in a system, it sounds as if some physical property like internal energy or pressure has increased in the system because of some physical cause, but in reality, nothing in the system is increasing. When we say there is an increase in entropy, it is the number of possibilities that we can be determined probabilistically from a system that is increased. This has nothing to do with a system being more chaotic or that a system has gained free will, but simply that there are so many billions and trillions of particles each with hundreds of possible configurations, there is no way to determine with 100% precision its future states. Entropy seems to be a concept formed from a statistical fact about systems with many particles.

            Imagine that there is one hydrogen atom in a box, and there was a supercomputer measuring every possible property of that atom, position, velocity, direction of motion, temperature, and etc. Would it make sense to use entropy for this system? How would we define disorder in this case? Is the atom less disorderly in the center of the box or in the top-left corner of the box? It does not really make sense because the notion of entropy is not useful in this case. What if we put two atoms in a box? Then the statistical probabilities of the two atoms increase exponentially by the number of degrees of freedom they have, but with a supercomputer, everything about the system is known and it is unnecessary to introduce the concept of entropy. Entropy only because useful when there are so many particles in a system that we cannot possibly keep track of what each individual particle is doing, so we take its possible state as a statistical whole.

            So what is entropy? I believe that entropy is a concept that came from a thermodynamics context so that thermodynamics calculation could be made, but it should be defined in a statistical or a probabilistic sense rather than in terms of disorder. Looking over our physics textbook, I think it does an excellent job explaining entropy and it is very different from the definition and explanation given by my thermodynamics textbook.