The second law of thermodynamics underlies nearly everything. But is it inviolable?

Two centuries on, scientists are still searching for a proof of its universal validity

Art of four eggs, from left to right, getting progressively more cracked. In the far right egg, it's cracked to the point the yolk is slipping out.

Once an egg is cracked, you can’t put it back together, a phenomenon that may be explained by the second law of thermodynamics.

Sam Green

In real life, laws are broken all the time. Besides your everyday criminals, there are scammers and fraudsters, politicians and mobsters, corporations and nations that regard laws as suggestions rather than restrictions.

It’s not that way in physics.

For centuries, physicists have been identifying laws of nature that are invariably unbreakable. Those laws govern matter, motion, electricity and gravity, and nearly every other known physical process. Nature’s laws are at the root of everything from the weather to nuclear weaponry.

Most of those laws are pretty well understood, at least by the experts who study and use them. But one remains mysterious.

It is widely cited as inviolable and acclaimed as applicable to everything. It guides the functioning of machines, life and the universe as a whole. Yet scientists cannot settle on one clear way of expressing it, and its underlying foundation has evaded explanation — attempts to prove it rigorously have failed. It’s known as the second law of thermodynamics. Or quite commonly, just the Second Law.

In common (oversimplified) terms, the Second Law asserts that heat flows from hot to cold. Or that doing work always produces waste heat. Or that order succumbs to disorder. Its technical definition has been more difficult to phrase, despite many attempts. As 20th century physicist Percy Bridgman once wrote, “There have been nearly as many formulations of the second law as there have been discussions of it.”

This month, the Second Law celebrates its 200th birthday. It emerged from the efforts of French engineer Sadi Carnot to figure out the physics of steam engines. It became the bedrock of understanding the role of heat in all natural processes. But not right away. Two decades passed before physicists began to realize the significance of Carnot’s discovery.

“If your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.”

Arthur Stanley Eddington, astrophysicist

By the early 20th century, though, the Second Law was recognized in the eyes of some as the premier law of physical science. It holds “the supreme position among the laws of Nature,” British astrophysicist Arthur Stanley Eddington declared in the 1920s. “If your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.”

In the two centuries since its birth, the Second Law has proved equally valuable for technological progress and fundamental science. It underlies everyday processes from cooling coffee to air conditioning and heating. It explains the physics of energy production in power plants and energy consumption in cars. It’s essential to understanding chemical reactions. It forecasts the “heat death of the universe” and plays a key role in answering why time flows in one direction.

But in the beginning, it was all about how to build a better steam engine.

The birth of the second law of thermodynamics

Nicolas Léonard Sadi Carnot was born in 1796, son of a well-known French engineer and government official named Lazare Carnot. It was a turbulent time in France, and Sadi’s father soon found himself on the wrong side of prevailing politics. Lazare went into exile in Switzerland (and later Germany), while Sadi’s mother took her baby to a small town in northern France. Eventually power in France shifted and Lazare returned, aided by a previous relationship with Napoleon Bonaparte. (At one time, Napoleon’s wife even babysat little Sadi.)

A biography written by Sadi’s younger brother, Hippolyte, described him as of delicate constitution, compensated for by vigorous exercise. He was energetic but something of a loner, reserved almost to the point of rudeness. But from a young age, he also exhibited intense intellectual curiosity, ultimately maturing into undeniable genius.

By age 16, Sadi was ready to begin higher education in Paris at the famed École Polytechnique (having already been well-trained by his father in math, physics and languages). Subsequent education included mechanics, chemistry and military engineering. During this time, he began writing scientific papers (that have not survived).

After service as a military engineer, Carnot moved back to Paris in 1819 to focus on science. He attended further college courses, including one dealing with steam engines, amplifying his longtime interest in industrial and engineering processes. Soon he began composing a treatise on the physics of heat engines in which he, for the first time, deduced the underlying scientific principles governing the production of useful energy from heat. Published on June 12, 1824, Carnot’s Reflections on the Motive Power of Heat marked the world’s first glimpse of the second law of thermodynamics.

Through his studies of heat engines, French engineer Sadi Carnot introduced the second law of thermodynamics in 1824. It would take another two decades for physicists to recognize his work’s significance.Courtesy of Science History Institute

“He was able to successfully show that there was a theoretical maximum efficiency for a heat engine, that depended only on the temperatures of its hot and cold reservoirs of heat,” computer scientist Stephen Wolfram wrote in a recent survey of the Second Law’s history. “In the setup Carnot constructed he basically ended up introducing the Second Law.”

Carnot had studied the steam engine’s use in 18th century England and its role in powering the Industrial Revolution. Steam engines had become the dominant machines in society, with enormous importance for industry and commerce. “They seem destined to produce a great revolution in the civilized world,” Carnot observed. “Already the steam-engine works our mines, impels our ships, excavates our ports and our rivers, forges iron, fashions wood, grinds grain, spins and weaves our cloths….”

Despite the societal importance of steam engines, Carnot noted, not much was known of the physical principles governing their conversion of heat into work. “Their theory is very little understood,” he wrote, “and the attempts to improve them are still directed almost by chance.” Improving steam engines, he decided, required a more general understanding of heat, apart from any particular properties of steam itself. So he investigated how all heat engines worked regardless of what substance was used to carry the heat.

In those days, heat was commonly believed to be a fluid substance, called caloric, that flowed between bodies. Carnot adopted that view and traced the flow of caloric in an idealized engine consisting of a cylinder and piston, a boiler and a condenser. An appropriate fluid (say water) could be converted to steam in the boiler, the steam could expand in the cylinder to drive the piston (doing work), and the steam could be restored to liquid water in the condenser.

Carnot’s key insight was that heat produced motion for doing work by dropping from a high temperature to a lower temperature (in the case of steam engines, from the boiler to the condenser). “The production of motive power is then due in steam-engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body,” he wrote.

His evaluation of this process, now known as the Carnot cycle, held the key to calculating the maximum efficiency possible for any engine — that is, how much work could be produced from the heat. And it turned out that you can never transform all the heat into work, a major consequence of the Second Law.

Carnot’s belief in caloric, of course, was erroneous. Heat is a manifestation of the motion of molecules. Nevertheless his findings remained correct — the Second Law applies no matter what substance an engine uses or what the actual underlying nature of heat is. Maybe that’s what Einstein had in mind when he called thermodynamics the scientific achievement most likely to stand firm as further advances rewrote humankind’s knowledge of the cosmos.

Within the realm of applicability of its basic concepts, Einstein wrote, thermodynamics “is the only physical theory of universal content concerning which I am convinced … will never be overthrown.”

The Second Law predicts the heat death of the universe

Although Carnot’s book received at least one positive review (in the French periodical Revue Encyclopédique), it went largely unnoticed by the scientific world. Carnot published no more and died of cholera in 1832. Two years later, though, French engineer Émile Clapeyron wrote a paper summarizing Carnot’s work, making it accessible to a broader audience. A decade later, British physicist William Thomson — later to become Lord Kelvin — encountered Clapeyron’s paper; Kelvin soon established that the core of Carnot’s conclusions survived unscathed even when the caloric theory was replaced by the new realization that heat was actually the motion of molecules.

Around the same time, German physicist Rudolf Clausius formulated an early explicit statement of the Second Law: An isolated machine, without external input, cannot convey heat from one body to another at a higher temperature. Independently, Kelvin soon issued a similar conclusion: No part of matter could do work by cooling itself below the temperature of the coldest surrounding objects. Each statement could be deduced from the other, so Kelvin’s and Clausius’ views were equivalent expressions of the Second Law.

Two side-by-side photographs of Lord Kelvin (left) and Rudolf Clausius (right)
In the decades after Sadi Carnot’s death, physicists Lord Kelvin (left) and Rudolf Clausius (right) came up with their own but equivalent ways of expressing the Second Law.From left: T. & R. Annan & Sons/Scottish National Portrait Gallery; Armin Kübelbeck/Wikimedia Commons

It became known as the Second Law because during this time, other work had established the law of conservation of energy, designated the first law of thermodynamics. Conservation of energy merely meant that the amount of energy involved in physical processes remained constant (in other words, energy could be neither created nor destroyed). But the Second Law was more complicated. Total energy stays the same but it cannot all be converted to work — some of that energy is dissipated as waste heat, useless for doing any more work.

“There is an absolute waste of mechanical energy available to man when heat is allowed to pass from one body to another at a lower temperature,” Kelvin wrote.

Kelvin recognized that this dissipation of energy into waste heat suggested a bleak future for the universe. Citing Kelvin, German physicist Hermann von Helmholtz later observed that eventually all the useful energy would become useless. Everything in the cosmos would then converge on the same temperature. With no temperature differences, no further work could be performed and all natural processes would cease. “In short,” von Helmholtz declared, “the universe from that time forward would be condemned to a state of eternal rest.”

Fortunately, this “heat death of the universe” would not arrive until eons into the future.

In the meantime, Clausius introduced the concept of entropy to quantify the transformation of useful energy into useless waste heat — providing yet another way of expressing the Second Law. If the First Law can be stated as “the energy of the universe is constant,” he wrote in 1865, then the Second Law could be stated as “the entropy of the universe tends to a maximum.”

Entropy, roughly, means disorder. Left to itself, an orderly system will degenerate into a disorderly mess. More technically, temperature differences in a system will tend to equalize until the system reaches equilibrium, at a constant temperature.

From another perspective, entropy refers to how probable the state of a system is. A low-entropy, ordered system is in a state of low probability, because there are many more ways to be disordered than ordered. Messier states, with higher entropy, are much more probable. So entropy is always likely to increase — or at least stay the same in systems where molecular motion has already reached equilibrium.

Bringing probability into the picture suggested that it would be impossible to prove the Second Law from analyzing the motions of individual molecules. It was necessary instead to study statistical measures that described large numbers of molecules in motion. Work along these lines by physicists James Clerk Maxwell, Ludwig Boltzmann and J. Willard Gibbs generated the science of statistical mechanics, the math describing large-scale properties of matter based on the statistical interactions of its molecules.

Maxwell concluded that the Second Law itself must possess merely statistical validity, meaning it was true only because it described the most probable of processes. In other words, it was not impossible (though unlikely) that cold could flow to hot. He illustrated his point by inventing a hypothetical little guy (he called it a “being”; Kelvin called it a demon) that could operate a tiny door between two chambers of gas. By allowing only slow molecules to pass one way and fast molecules the other, the demon could make one chamber hotter and the other colder, violating the Second Law.

But in the 1960s, IBM physicist Rolf Landauer showed that erasing information inevitably produces waste heat. Later his IBM colleague Charles Bennett pointed out that a Maxwell demon would need to record molecular velocities in order to know when to open and shut the door. Without an infinite memory, the demon would eventually have to erase those records, preserving the Second Law.

Another enigmatic issue emerging from studies of the Second Law involved its connection to the direction of time.

Laws governing molecular motion do not distinguish between future and past — a video of molecular collisions running backward shows them observing the same laws as a video moving forward. Yet in real life, unlike science fiction stories, time always flows forward.

It seems logical to suggest that time’s arrow was aimed by the Second Law’s requirement of increasing entropy. But the Second Law cannot explain why entropy in the universe has not already reached a maximum. Many scientists today believe time’s arrow cannot be explained by the Second Law alone (SN: 4/1/14), but also must have something to do with the origin of the universe and its expansion following the Big Bang. For some reason, entropy must have been low at the beginning, but why remains a mystery.

The Second Law hasn’t been rigorously proved

In his history of the Second Law, Wolfram recounts the many past efforts to provide the Second Law with a firm mathematical foundation. None have succeeded. “By the end of the 1800s … the Second Law began to often be treated as an almost-mathematically-proven necessary law of physics,” Wolfram wrote. But there were always weak links in the mathematical chain of reasoning. Despite the common belief that “somehow it must all have been worked out,” he commented, his survey showed that “no, it hasn’t all been worked out.”

Some recent efforts to verify the Second Law invoke Landauer’s emphasis on erasing information, which links the Second Law to information theory. In a recent paper, Shintaro Minagawa of Nagoya University in Japan and colleagues assert that merging the Second Law with information theory can secure the law’s foundation.

“The second law of information thermodynamics,” they write, “can now be considered a universally valid law of physics.”

In another information-related approach, Wolfram concludes that the Second Law’s confirmation can be found in principles governing computation. The Second Law’s basis, he says, is rooted in the fact that simple computational rules can produce elaborately complicated results, a principle he calls computational irreducibility.

Whether the Second Law is in fact universally true remains unsettled. Perhaps resolving that question will require a better definition of the law itself.

While many researchers have sought proofs of the Second Law, others have repeatedly challenged it with attempts to contradict its universal validity (SN: 3/8/16; SN: 7/17/17). But a 2020 review in the journal Entropy concludes that no such challenges to the Second Law have yet succeeded. “In fact, all resolved challengers’ paradoxes and misleading violations of the Second Law to date have been resolved in favor of the Second Law and never against,” wrote thermodynamicist Milivoje M. Kostic of Northern Illinois University in DeKalb. “We are still to witness a single, still open Second Law violation, to be confirmed.”

Yet whether the Second Law is in fact universally true remains unsettled. Perhaps resolving that question will require a better definition of the law itself. Variations of Clausius’ statement that entropy tends to a maximum are often given as the Second Law’s definition. But the physicist Richard Feynman found that unsatisfactory. He preferred “a process whose only net result is to take heat from a reservoir and convert it to work is impossible.”

When the Second Law was born, Carnot simply described it without defining it. Perhaps he knew it was too soon. He did, after all, realize that the future would bring new insights into the nature of heat. In unpublished work preserved in personal papers, he deduced the equivalence between heat and mechanical motion — the essence of what would become the first law of thermodynamics. And he foresaw that the caloric theory would probably turn out to be wrong. He cited experimental facts “tending to destroy” caloric theory. “Heat is simply motive power, or rather motion which has changed form,” he wrote. “It is a movement among the particles of bodies.”

Carnot planned to do experiments testing these ideas, but death intervened, one of nature’s two (along with taxes) inviolable certainties. Maybe the Second Law is a third.

But whether the Second Law is inviolable or not, it will forever be true that human laws will be a lot easier to break.

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