Classical thermodynamics has only a handful of laws, of which the most fundamental are the first and second. The first says that energy is always conserved; the second law says that heat always flows from hot to cold. More commonly this is expressed in terms of entropy, which must increase overall in any process of change. Entropy is loosely equated with disorder, but the Austrian physicist Ludwig Boltzmann formulated it more rigorously as a quantity related to the total number of microstates a system has: how many equivalent ways its particles can be arranged.
The second law appears to show why change happens in the first place. At the level of individual particles, the classical laws of motion can be reversed in time. But the second law implies that change must happen in a way that increases entropy. This directionality is widely considered to impose an arrow of time. In this view, time seems to flow from past to future because the universe began — for reasons not fully understood or agreed on — in a low-entropy state and is heading toward one of ever higher entropy. The implication is that eventually heat will be spread completely uniformly and there will be no driving force for further change — a depressing prospect that scientists of the mid-19th century called the heat death of the universe.
Boltzmann’s microscopic description of entropy seems to explain this directionality. Many-particle systems that are more disordered and have higher entropy vastly outnumber ordered, lower-entropy states, so molecular interactions are much more likely to end up producing them. The second law seems then to be just about statistics: It’s a law of large numbers. In this view, there’s no fundamental reason why entropy can’t decrease — why, for example, all the air molecules in your room can’t congregate by chance in one corner. It’s just extremely unlikely.
Yet this probabilistic statistical physics leaves some questions hanging. It directs us toward the most probable microstates in a whole ensemble of possible states and forces us to be content with taking averages across that ensemble.
But the laws of classical physics are deterministic — they allow only a single outcome for any starting point. Where, then, can that hypothetical ensemble of states enter the picture at all, if only one outcome is ever possible?
David Deutsch, a physicist at Oxford, has for several years been seeking to avoid this dilemma by developing a theory of (as he puts it) “a world in which probability and randomness are totally absent from physical processes.” His project, on which Marletto is now collaborating, is called constructor theory. It aims to establish not just which processes probably can and can’t happen, but which are possible and which are forbidden outright.
Constructor theory aims to express all of physics in terms of statements about possible and impossible transformations. It echoes the way thermodynamics itself began, in that it considers change in the world as something produced by “machines” (constructors) that work in a cyclic fashion, following a pattern like that of the famous Carnot cycle, proposed in the 19th century to describe how engines perform work. The constructor is rather like a catalyst, facilitating a process and being returned to its original state at the end.
“Say you have a transformation like building a house out of bricks,” said Marletto. “You can think of a number of different machines that can achieve this, to different accuracies. All of these machines are constructors, working in a cycle” — they return to their original state when the house is built.
But just because a machine for conducting a certain task might exist, that doesn’t mean it can also undo the task. A machine for building a house might not be capable of dismantling it. This makes the operation of the constructor different from the operation of the dynamical laws of motion describing the movements of the bricks, which are reversible.
The reason for the irreversibility, said Marletto, is that for most complex tasks, a constructor is geared to a given environment. It requires some specific information from the environment relevant to completing that task. But the reverse task will begin with a different environment, so the same constructor won’t necessarily work. “The machine is specific to the environment it is working on,” she said.
Recently, Marletto, working with the quantum theorist Vlatko Vedral at Oxford and colleagues in Italy, showed that constructor theory does identify processes that are irreversible in this sense — even though everything happens according to quantum mechanical laws that are themselves perfectly reversible. “We show that there are some transformations for which you can find a constructor for one direction but not the other,” she said.
The researchers considered a transformation involving the states of quantum bits (qubits), which can exist in one of two states or in a combination, or superposition, of both. In their model, a single qubit B may be transformed from some initial, perfectly known state B1 to a target state B2 when it interacts with other qubits by moving past a row of them one qubit at a time. This interaction entangles the qubits: Their properties become interdependent, so that you can’t fully characterize one of the qubits unless you look at all the others too.
As the number of qubits in the row gets very large, it becomes possible to bring B into state B2 as accurately as you like, said Marletto. The process of sequential interactions of B with the row of qubits constitutes a constructor-like machine that transforms B1 to B2. In principle you can also undo the process, turning B2 back to B1, by sending B back along the row.
But what if, having done the transformation once, you try to reuse the array of qubits for the same process with a fresh B? Marletto and colleagues showed that if the number of qubits in the row is not very large and you use the same row repeatedly, the array becomes less and less able to produce the transformation from B1 to B2. But crucially, the theory also predicts that the row becomes even less able to do the reverse transformation from B2 to B1. The researchers have confirmed this prediction experimentally using photons for B and a fiber optic circuit to simulate a row of three qubits.
“You can approximate the constructor arbitrarily well in one direction but not the other,” Marletto said. There’s an asymmetry to the transformation, just like the one imposed by the second law. This is because the transformation takes the system from a so-called pure quantum state (B1) to a mixed one (B2, which is entangled with the row). A pure state is one for which we know all there is to be known about it. But when two objects are entangled, you can’t fully specify one of them without knowing everything about the other too. The fact is that it’s easier to go from a pure quantum state to a mixed state than vice versa — because the information in the pure state gets spread out by entanglement and is hard to recover. It’s comparable to trying to re-form a droplet of ink once it has dispersed in water, a process in which the irreversibility is imposed by the second law.
So here the irreversibility is “just a consequence of the way the system dynamically evolves,” said Marletto. There’s no statistical aspect to it. Irreversibility is not just the most probable outcome but the inevitable one, governed by the quantum interactions of the components. “Our conjecture,” said Marletto, “is that thermodynamic irreversibility might stem from this.”
Demon in the Machine
There’s another way of thinking about the second law, though, that was first devised by James Clerk Maxwell, the Scottish scientist who pioneered the statistical view of thermodynamics along with Boltzmann. Without quite realizing it, Maxwell connected the thermodynamic law to the issue of information.
Maxwell was troubled by the theological implications of a cosmic heat death and of an inexorable rule of change that seemed to undermine free will. So in 1867 he sought a way to “pick a hole” in the second law. In his hypothetical scenario, a microscopic being (later, to his annoyance, called a demon) turns “useless” heat back into a resource for doing work. Maxwell had previously shown that in a gas at thermal equilibrium there is a distribution of molecular energies. Some molecules are “hotter” than others — they are moving faster and have more energy. But they are all mixed at random so there appears to be no way to make use of those differences.
Enter Maxwell’s demon. It divides the compartment of gas in two, then installs a frictionless trapdoor between them. The demon lets the hot molecules moving about the compartments pass through the trapdoor in one direction but not the other. Eventually the demon has a hot gas on one side and a cooler one on the other, and it can exploit the temperature gradient to drive some machine.
The demon has used information about the motions of molecules to apparently undermine the second law. Information is thus a resource that, just like a barrel of oil, can be used to do work. But as this information is hidden from us at the macroscopic scale, we can’t exploit it. It’s this ignorance of the microstates that compels classical thermodynamics to speak of averages and ensembles.
Almost a century later, physicists proved that Maxwell’s demon doesn’t subvert the second law in the long term, because the information it gathers must be stored somewhere, and any finite memory must eventually be wiped to make room for more. In 1961 the physicist Rolf Landauer showed that this erasure of information can never be accomplished without dissipating some minimal amount of heat, thus raising the entropy of the surroundings. So the second law is only postponed, not broken.
The informational perspective on the second law is now being recast as a quantum problem. That’s partly because of the perception that quantum mechanics is a more fundamental description — Maxwell’s demon treats the gas particles as classical billiard balls, essentially. But it also reflects the burgeoning interest in quantum information theory itself. We can do things with information using quantum principles that we can’t do classically. In particular, entanglement of particles enables information about them to be spread around and manipulated in nonclassical ways.
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