The Nature of Consciousness: Consciousness, Life and Meaning

Entropy: The Curse of Irreversibility

The single biggest change in scientific thinking may have nothing to do with Relativity and Quantum theories: it may well be the discovery that some processes are not symmetric in time. Before the discovery of the second law of Thermodynamics, all laws were symmetric in time, and change could always be bi-directional. Any formula had an equal sign that meant one can switch the two sides at will. We could always replay the history of the universe backwards. Entropy changed all that.

Entropy was "discovered" around 1850 by the German physicist Rudolf Clausius in the process of revising the laws proposed by the French engineer Sadi Carnot, that would become the foundation of Thermodynamics. The first law of Thermodynamics is basically the law of conservation of energy: energy can never be created or destroyed, it can only be transformed. The second law (originally formulated by William Thompson “Kelvin” in 1852) states that any transformation has an energetic cost: this "cost" of transforming energy Clausius called "entropy" (which is numerically obtained by dividing heat by temperature). Natural processes generate entropy. Entropy explains why heat flows spontaneously from hot to cold bodies, but the opposite never occurs: “useful” energy can be lost in entropy, not viceversa. There can never be an isolated process that results in a transfer of energy from a cold body to a hotter body: it is just a feature of our universe. In a sense, entropy measures how useful energy is. Before entropy is created, all energy is useful (e.g., the explosion in the combustion engine). Afterwards, some energy has become largely useless (e.g. heat, noise, motion).

Work is possible only when there is a difference in energy concentration because energy can only spontaneously move from higher concentration to lower concentration (e.g. from higher temperature to lower temperature). Work de facto reduces that difference. Eventually that difference does not exist anymore and work is no longer possible: the system has reached the state of equilibrium.

The total amount of energy in the universe is constant: it has always been what it is and will always be what it is. However, that energy is changing form, from usable to unusable. We can use less and less of the energy of the universe.

The first law talks about the quantity of energy, while the second law talks about the quality of such energy. Energy is always conserved, but something happens to it that causes it to “deteriorate”. Entropy measures the amount of energy that has deteriorated (is not available anymore for further work).

Clausius summarized the situation like this: the energy of the universe is constant, the entropy of the universe is increasing.

In the 1870s, the German physicist Ludwig von Boltzmann tried to deduce entropy from the motion of gas particles, i.e. from dynamic laws that are reversible in nature. Basically, Boltzmann tried to prove that entropy (and therefore irreversibility) is an illusion, that matter at the microscopic level is fundamentally reversible. Convinced that bodies are made of a large number of elementary particles, Boltzmann used statistics and probability theory to summarize their behavior, since it would be impossible to describe each particle’s motion and their innumerable interactions. He noticed that many different configurations (microstates) of those particles could lead to the same external appearance (macrostate) of the system as a whole.

Boltzmann ended up with a statistical definition of entropy to characterize the fact that many different microscopic states of a system result in the same macroscopic state: the entropy of a macrostate is the logarithm of the number of microstates that can implement that macrostate. Intuitively, the law of entropy originates from a statistical trend: a system tends to evolve towards the macrostates with high entropy, i.e. macrostates that correspond to large numbers of microstates; basically from rare configurations towards more likely configurations.

Boltzmann’s implicit assumption was that every microstate is equally probable. The other implicit assumption of the second law is that the universe started in a state of low entropy. That creates the fundamental asymmetry that we recognize as the arrow of time: entropy tends to increase because it is a lot easier to increase than decrease, and that is because the beginning of the story was at low entropy. For example, we assume a low-entropy past when we trust our memories of it: our memories (including photographs, videos, diaries) could have been created in a myriad of ways, but the low-entropy explanation is that they reflect what really happened.

Boltzmann’s definition emphasized that entropy turns out to be also a measure of “disorder” in a system: an ordered system has fewer microstates corresponding to a given macrostate.

The second law of Thermodynamics is an inequality: it states that entropy can never decrease. Indirectly, this law states that transformation processes cannot be run backward, cannot be "undone". Young people can age, but old people cannot rejuvenate. Buildings do not improve over the years: they decay. Scrambled eggs cannot be unscrambled and dissolved sugar cubes cannot be recomposed. The universe must evolve in the direction of higher and higher entropy. Some things are irreversible.

Newton’s equations are symmetric in time, which means that theoretically the same process can run backwards. It is the second law of Thermodynamics which makes it impossible to go back in time, that introduces an “arrow” of time.

The universe as a whole is proceeding towards its unavoidable fate: the “heat death”, i.e. the state of maximum entropy, in which no heat flow is possible, which means that temperature is constant everywhere, which means that there is no energy available to produce more heat, which means that all energy in the universe is in the form of heat. (An escape from the heat death would be possible if the energy in the universe were infinite).

Scientists were (and still are) puzzled by the fact that irreversibility (the law of entropy) had been deduced from reversibility (basically, Newton's laws). Mechanical phenomena tend to be reversible in time, whereas thermodynamic phenomena tend to be irreversible in time. Since a thermodynamic phenomenon is made of many mechanical phenomena, the paradox is how can an irreversible process arise from many reversible processes? It is weird that irreversibility should arise from the behavior of molecules which, if taken individually, obey physical laws that are reversible. We can keep track of the motion of each single particle in a gas, and then undo it. But we cannot undo the macroscopic consequence of the motion of thousands of such particles in a gas.

If one filmed the behavior of each particle of a gas as the gas moves from non-equilibrium to equilibrium, and then played the film backwards, the film would be perfectly consistent with the laws of Mechanics. In practice, though, systems never spontaneously move from equilibrium to non-equilibrium: the film is perfectly feasible, but in practice it is never made.

The only reason one could find was probabilistic, not mechanical: the probability of low-entropy macrostates is smaller, by definition, than the probability of high-entropy macrostates, so the universe tends to proceed towards higher entropy. Thus the second law seems to express the tendency of systems to transition from less probable states (states that can be realized by few microstates) to more probable states (states that can be realized by many microstates). Basically, there are more ways to be disorderly than to be orderly.

And one can rephrase the same idea in terms of equilibrium: since equilibrium states are states that correspond to the maximum number of microstates, it is unlikely that a system moves to a state of non-equilibrium, and likely that it moves to a state of equilibrium.

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