In May 1959 the physicist and novelist C.P. Snow delivered a lecture with the title The Two Cultures, which provoked widespread controversy. The response of the prominent literary critic F.R. Leavis was typical of the other side of the argument; he said bluntly that there was only one culture: his. Snow suggested that the sciences and the humanities had lost touch with each other, and argued that this was making it very difficult to solve the world’s problems. We see the same today with climate change denial and attacks on evolution. The motivation may be different, but cultural barriers help such nonsense to thrive – though it is politics that drives it.
Snow was particularly unhappy about what he saw as declining standards of education, saying:
A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics, the law of entropy. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: ‘Have you read a work of Shakespeare’s?’
Perhaps he sensed he was asking too much – many qualified scientists can’t state the second law of thermodynamics. So he later added:
I now believe that if I had asked an even simpler question – such as, What do you mean by mass, or acceleration, which is the scientific equivalent of saying, ‘Can you read?’ – not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their Neolithic ancestors would have had.
Taking Snow literally, my aim in this chapter is to take us out of the Neolithic age. The word ‘thermodynamics’ contains a clue: it appears to mean the dynamics of heat. Can heat be dynamic? Yes: heat can flow. It can move from one location to another, from one object to another. Go outside on a winter’s day and you soon feel cold. Fourier had written down the first serious model of heat flow, Chapter 9 and done some beautiful mathematics. But the main reason scientists were becoming interested in heat flow was a newfangled and highly profitable item of technology: the steam engine.
There is an oft-repeated story of James Watt as a boy, sitting in his mother’s kitchen watching boiling steam lift the lid off a kettle, and his sudden flash of inspiration: steam can perform work. So, when he grew up, he invented the steam engine. It’s inspirational stuff, but like many such tales this one is just hot air. Watt didn’t invent the steam engine, and he didn’t learn about the power of steam until he was an adult. The story’s conclusion about the power of steam is true, but even in Watt’s day it was old hat.
Around 50 BC the Roman architect and engineer Vitruvius described a machine called an aeolipile in his De Architectura (‘On Architecture’), and the Greek mathematician and engineer Hero of Alexandria built one a century later. It was a hollow sphere with some water inside, and two tubes poked out, bent at an angle as in Figure 46. Heat the sphere and the water turns to steam, escapes through the ends of the tubes, and the reaction makes the sphere spin. It was the first steam engine, and it proved that steam could do work, but Hero did nothing with it beyond entertaining people. He did make a similar machine using hot air in an enclosed chamber to pull a rope that opened the doors of a temple. This machine had a practical application, producing a religious miracle, but it wasn’t a steam engine.
Watt learned that steam could be a source of power in 1762 when he was 26 years old. He didn’t discover it watching a kettle: his friend John Robison, a professor of natural philosophy at the University of Edinburgh, told him about it. But practical steam power was much older. Its discovery is often credited to the Italian engineer and architect Giovanni Branca, whose Le Machine (‘Machine’) of 1629 contained 63 woodcuts of mechanical gadgets. One shows a paddlewheel that would spin on its axle when steam from a pipe collided with its vanes. Branca speculated that this machine might be useful for grinding flour, lifting water, and cutting up wood, but it was probably never built. It was more of a thought experiment, a mechanical pipedream like Leonardo da Vinci’s flying machine.
In any case, Branca was anticipated by Taqi al-Din Muhammad ibn Ma’ruf al-Shami al-Asadi, who lived around 1550 in the Ottoman Empire and was widely held to be the greatest scientist of his age. His achievements are impressive. He worked in everything from astrology to zoology, including clock-making, medicine, philosophy, and theology, and he wrote over 90 books. In his 1551 Al-turuq al-samiyya fi al-alat al-ruhaniyya (‘The Sublime Methods of Spiritual Machines’), al-Din described a primitive steam turbine, saying that it could be used to turn roasting meat on a spit.
The first truly practical steam engine was a water pump invented by Thomas Savery in 1698. The first to make commercial profits, built by Thomas Newcomen in 1712, triggered the Industrial Revolution. But Newcomen’s engine was very inefficient. Watt’s contribution was to introduce a separate condenser for the steam, reducing heat loss. Developed using money provided by the entrepreneur Matthew Bolton, this new type of engine used only a quarter as much coal, leading to huge savings. Boulton and Watt’s machine went into production in 1775, more than 220 years after al-Din’s book. By 1776, three were up and running: one in a coal mine at Tipton, one in a Shropshire ironworks, and one in London.
Steam engines performed a variety of industrial tasks, but by far the commonest was pumping water from mines. It cost a lot of money to develop a mine, but as the upper layers became worked out and operators were forced to dig deeper into the ground, they hit the water table. It was worth spending quite a lot of money to pump the water out, since the alternative was to close the mine and start again somewhere else – and that might not even be feasible. But no one wanted to pay more than they had to, so a manufacturer who could design and build a more efficient steam engine would corner the market. So the basic question of how efficient a steam engine could be cried out for attention. Its answer did more than just describe the limits to steam engines: it created a new branch of physics, whose applications were almost boundless. The new physics shed light on everything from gases to the structure of the entire universe; it applied not just to the dead matter of physics and chemistry, but perhaps also to the complex processes of life itself. It was called thermodynamics: the motion of heat. And, just as the law of conservation of energy in mechanics ruled out mechanical perpetual motion machines, the laws of thermodynamics ruled out similar machines using heat.
One of those laws, the first law of thermodynamics, revealed a new form of energy associated with heat, and extended the law of conservation of energy (Chapter 3) into the new realm of heat engines. Another, without any previous precedent, showed that some potential ways to exchange heat, which did not conflict with conservation of energy, were nevertheless impossible because they would have to create order from disorder. This was the second law of thermodynamics.
Thermodynamics is the mathematical physics of gases. It explains how large-scale features like temperature and pressure arise from the way the gas molecules interact. The subject began with a series of laws of nature relating temperature, pressure, and volume. This version is called classical thermodynamics, and did not involve molecules – at that time few scientists believed in them. Later, the gas laws were underpinned by a further layer of explanation, based on a simple mathematical model explicitly involving molecules. The gas molecules were thought of as tiny spheres that bounced off each other like perfectly elastic billiard balls, with no energy being lost in the collision. Although molecules are not spherical, this model proved to be remarkably effective. It is called the kinetic theory of gases, and it led to experimental proof that molecules exist.
The early gas laws emerged in fits and starts over a period of nearly fifty years, and are mainly attributed to the Irish physicist and chemist Robert Boyle, the French mathematician and balloon pioneer Jacques Alexandre César Charles, and the French physicist and chemist Joseph Louis Gay-Lussac. However, many of the discoveries were made by others. In 1834, the French engineer and physicist Émile Clapeyron combined all of these laws into one, the ideal gas law, which we now write as
pV = RT
Here p is pressure, V is volume, T is the temperature, and R is a constant. The equation states that pressure times volume is proportional to temperature. It took a lot of work with many different gases to confirm each separate law, and Clapeyron’s overall synthesis, experimentally. The word ‘ideal’ appears because real gases do not obey the law in all circumstances, especially at high pressures where interatomic forces come into play. But the ideal version was good enough for designing steam engines.
Thermodynamics is encapsulated in a number of more general laws, not reliant on the precise form of the gas law. However, it does require there to be some such law, because temperature, pressure, and volume are not independent. There has to be some relation between them, but it doesn’t greatly matter what.
The first law of thermodynamics stems from the mechanical law of conservation of energy. In Chapter 3 we saw that there are two distinct kinds of energy in classical mechanics: kinetic energy, determined by mass and speed, and potential energy, determined by the effect of forces such as gravity. Neither of these types of energy is conserved on its own. If you drop a ball, it speeds up, thereby gaining kinetic energy. It also falls, losing potential energy. Newton’s second law of motion implies that these two changes cancel each other out exactly, so the total energy does not change during the motion.
However, this is not the full story. If you put a book on a table and give it a push, its potential energy doesn’t change provided the table is horizontal. But its speed does change: after an initial increase produced by the force with which you pushed it, the book quickly slows down and comes to rest. So its kinetic energy starts at a nonzero initial value just after the push, and then drops to zero. The total energy therefore also decreases, so energy is not conserved. Where has it gone? Why did the book stop? According to Newton’s first law, the book should continue to move, unless some force opposes it. That force is friction between the book and the table. But what is friction?
Friction occurs when rough surfaces rub together. The rough surface of the book has bits that stick out slightly. These come into contact with parts of the table that also stick out slightly. The book pushes against the table, and the table, obeying Newton’s third law, resists. This creates a force that opposes the motion of the book, so it slows down and loses energy. So where does the energy go? Perhaps conservation simply does not apply. Alternatively, the energy is still lurking somewhere, unnoticed. And that’s what the first law of thermodynamics tells us: the missing energy appears as heat. Both book and table heat up slightly. Humans have known that friction creates heat even since some bright spark discovered how to rub two sticks together and start a fire. If you slide down a rope too fast, your hands get rope burns from the friction. There were plenty of clues. The first law of thermodynamics states that heat is a form of energy, and energy – thus extended – is conserved in thermodynamic processes.
The first law of thermodynamics places limits on what you can do with a heat engine. The amount of kinetic energy that you can get out, in the form of motion, cannot be more than the amount of energy you put in as heat. But it turned out that there is a further restriction on how efficiently a heat engine can convert heat energy into kinetic energy; not just the practical point that some of the energy always gets lost, but a theoretical limit that prevents all of the heat energy being converted to motion. Only some of it, the ‘free’ energy, can be so converted. The second law of thermodynamics turned this idea into a general principle, but it will take a while before we get to that. The limitation was discovered by Nicolas Léonard Sadi Carnot in 1824, in a simple model of how a steam engine works: the Carnot cycle.
To understand the Carnot cycle it is important to distinguish between heat and temperature. In everyday life, we say that something is hot if its temperature is high, and so confuse the two concepts. In classical thermodynamics, neither concept is straightforward. Temperature is a property of a fluid, but heat makes sense only as a measure of the transfer of energy between fluids, and is not an intrinsic property of the state (that is, the temperature, pressure, and volume) of the fluid. In the kinetic theory, the temperature of a fluid is the average kinetic energy of its molecules, and the amount of heat transferred between fluids is the change in the total kinetic energy of their molecules. In a sense heat is a bit like potential energy, which is defined relative to an arbitrary reference height; this introduces an arbitrary constant, so ‘the’ potential energy of a body is not uniquely defined. But when the body changes height, the difference in potential energies is the same whatever reference height is used, because the constant cancels out. In short, heat measures changes, but temperature measures states. The two are linked: heat transfer is possible only when the fluids concerned have different temperatures, and then it is transferred from the hotter one to the cooler one. This is often called the Zeroth law of thermodynamics because logically it precedes the first law, but historically it was recognised later.
Temperature can be measured using a thermometer, which exploits the expansion of a fluid, such as mercury, caused by increased temperature. Heat can be measured by using its relation to temperature. In a standard test fluid, such as water, every 1-degree rise in temperature of 1 gram of fluid corresponds to a fixed increase in the heat content. This amount is called the specific heat of the fluid, which in water is 1 calorie per gram per degree Celsius. Note that heat increase is a change, not a state, as required by the definition of heat.
We can visualise the Carnot cycle by thinking of a chamber containing gas, with a movable piston at one end. The cycle has four steps:
1Heat the gas so rapidly that its temperature doesn’t change. It expands, performing work on the piston.
2Allow the gas to expand further, reducing the pressure. The gas cools.
3Compress the gas so rapidly that its temperature doesn’t change. The piston now performs work on the gas.
4Allow the gas to expand further, increasing the pressure. The gas returns to its original temperature.
In a Carnot cycle, the heat introduced in the first step transfers kinetic energy to the piston, allowing the piston to do work. The quantity of energy transferred can be calculated in terms of the amount of heat introduced and the temperature difference between the gas and its surroundings. Carnot’s theorem proves that in principle a Carnot cycle is the most efficient way to convert heat into work. This places a stringent limit on the efficiency of any heat engine, and in particular on a steam engine.
In a diagram showing the pressure and volume of the gas, a Carnot cycle looks like Figure 47 (left). The German physicist and mathematician Rudolf Clausius discovered a simpler way to visualise the cycle, Figure 47 (right). Now the two axes are temperature and a new and fundamental quantity called entropy. In these coordinates, the cycle becomes a rectangle, and the amount of work performed is just the area of the rectangle.
Fig 47 Carnot cycle. Left: In terms of pressure and volume. right: In terms of temperature and entropy.
Entropy is like heat: it is defined in terms of a change of state, not a state as such. Suppose that a fluid in some initial state changes to a new state. Then the difference in entropy between the two states is the total change in the quantity ‘heat divided by temperature’. In symbols, for a small step along the path between the two states, entropy S is related to heat q and temperature T by the differential equation dS=dq/T. The change in entropy is the change in heat per unit temperature. A large change of state can be represented as a series of small ones, so we add up all these small changes in entropy to get the overall change of entropy. Calculus tells us that the way to do this is to use an integral.1
Having defined entropy, the second law of thermodynamics is very simple. It states that in any physically feasible thermodynamic process, the entropy of an isolated system must always increase.2 In symbols, dS≥0. For example, suppose we divide a room with a movable partition, put oxygen on one side of the partition and nitrogen on the other. Each gas has a particular entropy, relative to some initial reference state. Now remove the partition, allowing the gases to mix. The combined system also has a particular entropy, relative to the same initial reference states. And the entropy of the combined system is always greater than the sum of the entropies of the two separate gases.
Classical thermodynamics is phenomenological: it describes what you can measure, but it’s not based on any coherent theory of the processes involved. That step came next with the kinetic theory of gases, pioneered by Daniel Bernoulli in 1738. This theory provides a physical explanation of pressure, temperature, the gas laws, and that mysterious quantity entropy. The basic idea – highly controversial at the time – is that a gas consists of a large number of identical molecules, which bounce around in space and occasionally collide with each other. Being a gas means that the molecules are not too tightly packed, so any given molecule spends a lot of its time travelling through the vacuum of space at a constant speed in a straight line. (I say ‘vacuum’ even though we’re discussing a gas, because that’s what the space between molecules consists of.) Since molecules, though tiny, have nonzero size, occasionally two of them will collide. Kinetic theory makes the simplifying assumption that they bounce like two colliding billiard balls, and that these balls are perfectly elastic, so no energy is lost in the collision. Among other things, this implies that the molecules keep bouncing forever.
When Bernoulli first suggested the model, the law of conservation of energy was not established and perfect elasticity seemed unlikely. The theory gradually won support from a small number of scientists, who developed their own versions and added various new ideas, but their work was almost universally ignored. The German chemist and physicist August Krönig wrote a book on the topic in 1856, simplifying the physics by not allowing the molecules to rotate. Clausius removed this simplification a year later. He claimed he had arrived at his results independently, and is now ranked as one of the first significant founders of kinetic theory. He proposed one of the key concepts of the theory, the mean free path of a molecule: how far it travels, on average, between successive collisions.
Both König and Clausius deduced the ideal gas law from kinetic theory. The three key variables are volume, pressure, and temperature. Volume is determined by the vessel that contains the gas, it sets ‘boundary conditions’ that affect how the gas behaves, but is not a feature of the gas as such. Pressure is the average force (per square unit of area) exerted by the molecules of the gas when they collide with the walls of the vessel. This depends on how many molecules are inside the vessel, and how fast they are moving. (They don’t all move at the same speed.) Most interesting is temperature. This also depends on how fast the gas molecules are moving, and it is proportional to the average kinetic energy of the molecules. Deducing Boyle’s law, the special case of the ideal gas law for constant temperature, is especially straightforward. At a fixed temperature, the distribution of velocities doesn’t change, so pressure is determined by how many molecules hit the wall. If you reduce the volume, the number of molecules per cubic unit of space goes up, and the chance of any molecule hitting the wall goes up as well. Smaller volume means denser gas means more molecules hitting the wall, and this argument can be made quantitative. Similar but more complicated arguments produce the ideal gas law in all its glory as long as the molecules aren’t squashed too tightly together. So now there was a deeper theoretical basis for Boyle’s law, based on the theory of molecules.
Maxwell was inspired by Clausius’s work, and in 1859 he placed kinetic theory on mathematical foundations by writing down a formula for the probability that a molecule will travel with a given speed. It is based on the normal distribution or bell curve (Chapter 7). Maxwell’s formula seems to have been the first instance of a physical law based on probabilities. He was followed by the Austrian physicist Ludwig Boltzmann, who developed the same formula, now called the Maxwell–Boltzmann distribution. Boltzmann reinterpreted thermodynamics in terms of the kinetic theory of gases, founding what is now called statistical mechanics. In particular, he came up with a new interpretation of entropy, relating the thermodynamic concept to a statistical feature of the molecules in the gas.
The traditional thermodynamic quantities, such as temperature, pressure, heat, and entropy, all refer to large-scale average properties of the gas. However, the fine structure consists of lots of molecules whizzing around and bumping into each other. The same large-scale state can arise from innumerable different small-scale states, because minor differences on the small scale average out. Boltzmann therefore distinguished macrostates and microstates of the system: large-scale averages and the actual states of the molecules. Using this, he showed that entropy, a macrostate, can be interpreted as a statistical feature of microstates. He expressed this in the equation
S = k log W
Here S is the entropy of the system, W is the number of distinct microstates that can give rise to the overall macrostate, and k is a constant. It is now called Boltzmann’s constant, and its value is 1.38 × 10–23 joules per degree kelvin.
It is this formula that motivates the interpretation of entropy as disorder. The idea is that fewer microstates correspond to an ordered macrostate than to a disordered one, and we can understand why by thinking about a pack of cards. For simplicity, suppose that we have just six cards, marked 2, 3, 4, J, Q, K. Put them in two separate piles, with the low-value cards in one pile and the court cards in the other. This is an ordered arrangement. In fact, it retains traces of order if you shuffle each pile, but keep the piles separate, because however you do this, the low-value cards are all in one pile and the court cards are in the other. However, if you shuffle both piles together, the two types of card can become mixed, with arrangements like 4QK2J3. Intuitively, these mixed-up arrangements are more disordered.
Let’s see how this relates to Boltzmann’s formula. There are 36 ways to arrange the cards in their two piles: six for each pile. But there are 720 ways (6! = 1 × 2 × 3 × 4 × 5 × 6) to arrange all six cards in order. The type of ordering of the cards that we allow – two piles or one – is analogous to the macrostate of a thermodynamic system. The exact order is the microstate. The more ordered macrostate has 36 microstates, the less ordered one has 720. So the more microstates there are, the less ordered the corresponding macrostate becomes. Since logarithms get bigger when the numbers do, the greater the logarithm of the number of microstates, the more disordered the macrostate becomes. Here
log 36 = 3.58 log 720 = 6.58
These are effectively the entropies of the two macrostates. Boltzmann’s constant just scales the values to fit the thermodynamic formalism when we’re dealing with gases.
The two piles of cards are like two non-interacting thermodynamic states, such as a box with a partition separating two gases. Their individual entropies are each log 6, so the total entropy is 2 log 6, which equals log 36. So the logarithm makes entropy additive for non-interacting systems: to get the entropy of the combined (but not yet interacting) system, add the separate entropies. If we now let the systems interact (remove the partition) the entropy increases to log 720.
The more cards there are, the more pronounced this effect becomes. Split a standard pack of 52 playing cards into two piles, with all the red cards in one pile and all the black cards in the other. This arrangement can occur in (26!)2 ways, which is about 1.62 × 1053. Shuffling both piles together we get 52! microstates, roughly 8.07 × 1067. The logarithms are 122.52 and 156.36 respectively, and again the second is larger.
Boltzmann’s ideas were not received with great acclaim. At a technical level, thermodynamics was beset with difficult conceptual issues. One was the precise meaning of ‘microstate’. The position and velocity of a molecule are continuous variables, able to take on infinitely many values, but Boltzmann needed a finite number of microstates in order to count how many there were and then take the logarithm. So these variables had to be ‘coarse-grained’ in some manner, by splitting the continuum of possible values into finitely many very small intervals. Another issue, more philosophical in nature, was the arrow of time – an apparent conflict between the time-reversible dynamics of microstates and the one-way time of macrostates, determined by entropy increase. The two issues are related, as we will shortly see.
The biggest obstacle to the theory’s acceptance, however, was the idea that matter is made from extremely tiny particles, atoms. This concept, and the word atom, which means ‘indivisible’, goes back to ancient Greece, but even around 1900 the majority of physicists did not believe that matter is made from atoms. So they didn’t believe in molecules, either, and a theory of gases based on them was obviously nonsense. Maxwell, Boltzmann, and other pioneers of kinetic theory were convinced that molecules and atoms were real, but to the skeptics, atomic theory was just a convenient way to picture matter. No atoms had ever been observed, so there was no scientific evidence that they existed. Molecules, specific combinations of atoms, were similarly controversial. Yes, atomic theory fitted all sorts of experimental data in chemistry, but that was not proof that atoms existed.
One of the things that finally convinced most objectors was the use of kinetic theory to make predictions about Brownian motion. This effect was discovered by a Scottish botanist, Robert Brown.3 He pioneered the use of the microscope, discovering, among other things, the existence of the nucleus of a cell, now known to be the repository of its genetic information. In 1827 Brown was looking through his microscope at pollen grains in a fluid, and he spotted even tinier particles that had been ejected by the pollen. These tiny particles jiggled around in a random manner, and at first Brown wondered if they were some diminutive form of life. However, his experiments showed the same effect in particles derived from non-living matter, so whatever caused the jiggling, it didn’t have to be alive. At the time, no one knew what caused this effect. We now know that the particles ejected by the pollen were organelles, tiny subsystems of the cell with specific functions; in this case, to manufacture starch and fats. And we interpret their random jiggles as evidence for the theory that matter is made from atoms.
The link to atoms comes from mathematical models of Brownian motion, which first turned up in statistical work of the Danish astronomer and actuary Thorvald Thiele in 1880. The big advance was made by Einstein in 1905 and the Polish scientist Marian Smoluchowski in 1906. They independently proposed a physical explanation of Brownian motion: atoms of the fluid in which the particles were floating were randomly bumping into the particles and giving them tiny kicks. On this basis, Einstein used a mathematical model to make quantitative predictions about the statistics of the motion, which were confirmed by Jean Baptiste Perrin in 1908–9.
Boltzmann committed suicide in 1906 – just when the scientific world was starting to appreciate that the basis of his theory was real.
In Boltzmann’s formulation of thermodynamics, molecules in a gas are analogous to cards in a pack, and the natural dynamics of the molecules is analogous to shuffling. Suppose that at some moment all the oxygen molecules in a room are concentrated at one end, and all the nitrogen molecules are at the other. This is an ordered thermodynamic state, like two separate piles of cards. After a very short period, however, random collisions will mix all the molecules together, more or less uniformly throughout the room, like shuffling the cards. We’ve just seen that this process typically causes entropy to increase. This is the orthodox picture of the relentless increase of entropy, and it is the standard interpretation of the second law: ‘the amount of disorder in the universe steadily increases’. I’m pretty sure that this characterisation of the second law would have satisfied Snow if anyone had offered it. In this form, one dramatic consequence of the second law is the scenario of the ‘heat death of the universe’, in which the entire universe will eventually become a lukewarm gas with no interesting structure whatsoever.
Entropy, and the mathematical formalism that goes with it, provides an excellent model for many things. It explains why heat engines can only reach a particular level of efficiency, which prevents engineers wasting valuable time and money looking for a mare’s nest. That’s not just true of Victorian steam engines, it applies to modern car engines as well. Engine design is one of the practical areas that has benefited from knowing the laws of thermodynamics. Refrigerators are another. They use chemical reactions to transfer heat out of the food in the fridge. It has to go somewhere: you can often feel the heat rising from the outside of the fridge’s motor housing. The same goes for air-conditioning. Power generation is another application. In a coal, gas, or nuclear power station, what it initially generated is heat. The heat creates steam, which drives a turbine. The turbine, following principles that go back to Faraday, turns motion into electricity.
The second law of thermodynamics also governs the amount of energy we can hope to extract from renewable resources, such as wind and waves. Climate change has added new urgency to this question, because renewable energy sources produce less carbon dioxide than conventional ones. Even nuclear power has a big carbon footprint, because the fuel has to be made, transported, and stored when it is no longer useful but still radioactive. As I write there is a simmering debate about the maximum amount of energy that we can extract from the ocean and the atmosphere without causing the kinds of change that we are hoping to avoid. It is based on thermodynamic estimates of the amount of free energy in those natural systems. This is an important issue: if renewables in principle cannot supply the energy we need, then we have to look elsewhere. Solar panels, which extract energy directly from sunlight, are not directly affected by the thermodynamic limits, but even those involve manufacturing processes and so on. At the moment, the case that such limits are a serious obstacle relies on some sweeping simplifications, and even if they are correct, the calculations do not rule out renewables as a source for most of the world’s power. But it’s worth remembering that similarly broad calculations about carbon dioxide production, performed in the 1950s, have proved surprisingly accurate as a predictor of global warming.
The second law works brilliantly in its original context, the behaviour of gases, but it seems to conflict with the rich complexities of our planet, in particular, life. It seems to rule out the complexity and organisation exhibited by living systems. So the second law is sometimes invoked to attack Darwinian evolution. However, the physics of steam engines is not particularly appropriate to the study of life. In the kinetic theory of gases, the forces that act between the molecules are short-range (active only when the molecules collide) and repulsive (they bounce). But most of the forces of nature aren’t like that. For example, gravity acts at enormous distances, and it is attractive. The expansion of the universe away from the Big Bang has not smeared matter out into a uniform gas. Instead, the matter has formed into clumps – planets, stars, galaxies, supergalactic clusters. . . The forces that hold molecules together are also attractive–except at very short distances where they become repulsive, which stops the molecule collapsing – but their effective range is fairly short. For systems such as these, the thermodynamic model of independent subsystems whose interactions switch on but not off is simply irrelevant. The features of thermodynamics either don’t apply, or are so long-term that they don’t model anything interesting.
The laws of thermodynamics, then, underlie many things that we take for granted. And the interpretation of entropy as ‘disorder’ helps us to understand those laws and gain an intuitive feeling for their physical basis. However, there are occasions when interpreting entropy as disorder seems to lead to paradoxes. This is a more philosophical realm of discourse – and it’s fascinating.
One of the deep mysteries of physics is time’s arrow. Time seems to flow in one particular direction. However, it seems logically and mathematically possible for time to flow backwards instead – a possibility exploited by books such as Martin Amis’s Time’s Arrow, the much earlier novel Counter-Clock World by Philip K. Dick, and the BBC television series Red Dwarf, whose protagonists memorably drank beer and engaged in a bar brawl in reverse time. So why can’t time flow the other way? At first sight, thermodynamics offers a simple explanation for the arrow of time: it is the direction of entropy increase. Thermodynamic processes are irreversible: oxygen and nitrogen will spontaneously mix, but not spontaneously unmix.
There is a puzzle here, however, because any classical mechanical system, such as the molecules in a room, is time-reversible. If you keep shuffling a pack of cards at random, then eventually it will get back to its original order. In the mathematical equations, if at some instant the velocities of all particles are simultaneously reversed, then the system will retrace its steps, back-to-front in time. The entire universe can bounce, obeying the same equations in both directions. So why do we never see an egg unscrambling?
The usual thermodynamic answer is: a scrambled egg is more disordered than an unscrambled one, entropy increases, and that’s the way time flows. But there’s a subtler reason why eggs don’t unscramble: the universe is very, very unlikely to bounce in the required manner. The probability of that happening is ridiculously small. So the discrepancy between entropy increase and time–reversibility comes from the initial conditions, not the equations. The equations for moving molecules are time-reversible, but the initial conditions are not. When we reverse time, we must use ‘initial’ conditions given by the final state of the forward-time motion.
The most important distinction here is between symmetry of equations and symmetry of their solutions. The equations for bouncing molecules have time-reversal symmetry, but individual solutions can have a definite arrow of time. The most you can deduce about a solution, from time-reversibility of the equation, is that there must also exist another solution that is the time-reversal of the first. If Alice throws a ball to Bob, the time-reversed solution has Bob throwing a ball to Alice. Similarly, since the equations of mechanics allow a vase to fall to the ground and smash into a thousand pieces, they must also allow a solution in which a thousand shards of glass mysteriously move together, assemble themselves into an intact vase, and leap into the air.
There’s clearly something funny going on here, and it repays investigation. We don’t have a problem with Bob and Alice tossing a ball either way. We see such things every day. But we don’t see a smashed vase putting itself back together. We don’t see an egg unscrambling.
Suppose we smash a vase and film the result. We start with a simple, ordered state – an intact vase. It falls to the floor, where the impact breaks it into pieces and propels those pieces all over the floor. They slow down and come to a halt. It all looks entirely normal. Now play the movie backwards. Bits of glass, which just happen to be the right shape to fit together, are lying on the floor. Spontaneously, they start to move. They move at just the right speed, and in just the right direction, to meet. They assemble into a vase, which heads skywards. That doesn’t seem right.
In fact, as described, it’s not right. Several laws of mechanics appear to be violated, among them conservation of momentum and conservation of energy. Stationary masses can’t suddenly move. A vase can’t gain energy from nowhere and leap into the air.
Ah, yes. . . but that’s because we’re not looking carefully enough. The vase didn’t leap into the air of its own accord. The floor started to vibrate, and the vibrations came together to give the vase a sharp kick into the air. The bits of glass were similarly impelled to move by incoming waves of vibration of the floor. If we trace those vibrations back, they spread out, and seem to die down. Eventually friction dissipates all movement. . . Oh, yes, friction. What happens to kinetic energy when there’s friction? It turns into heat. So we’ve missed some details of the time-reversed scenario. Momentum and energy do balance, but the missing amounts come from the floor losing heat.
In principle, we could set up a forward-time system to mimic the time-reversed vase. We just have to arrange for molecules in the floor to collide in just the right way to release some of their heat as motion of the floor, kick the pieces of glass in just the right way, then hurl the vase into the air. The point is not that this is impossible in principle: if it were, time-reversibility would fail. But it’s impossible in practice, because there is no way to control that many molecules that precisely.
This, too, is an issue about boundary conditions – in this case, initial conditions. The initial conditions for the vase-smashing experiment are easy to implement, and the apparatus is easy to acquire. It’s all very robust, too: use another vase, drop it from a different height. . . much the same will happen. The vase-assembling experiment, in contrast, requires extraordinarily precise control of gazillions of individual molecules and exquisitely carefully made pieces of glass. Without all that control equipment disturbing a single molecule. That’s why we can’t actually do it.
However, notice how we’re thinking here: we’re focusing on initial conditions. That sets up an arrow of time: the rest of the action comes later than the start. If we looked at the vase-smashing experiment’s final conditions, right down to the molecular level, they would be so complex that no one in their right mind would even consider trying to replicate them.
The mathematics of entropy fudges out these very small scale considerations. It allows vibrations to die away but not to increase. It allows friction to turn into heat but not heat to turn into friction. The discrepancy between the second law of thermodynamics and microscopic reversibility arises from coarse-graining, the modelling assumptions made when passing from a detailed molecular description to a statistical one. These assumptions implicitly specify an arrow of time: large-scale disturbances are allowed to die down below the perceptible level as time passes, but small-scale disturbances are not allowed to follow the time-reversed scenario. Once the dynamics passes through this temporal trapdoor, it’s not allowed to come back.
If entropy always increases, how did the chicken ever create the ordered egg to begin with? A common explanation, advanced by the Austrian physicist Erwin Schrödinger in 1944 in a brief and charming book What is Life?, is that living systems somehow borrow order from their environment, and pay it back by making the environment even more disordered than it would otherwise have been. This extra order corresponds to ‘negative entropy’, which the chicken can use to make an egg without violating the second law. In Chapter 15 we will see that negative entropy can, in appropriate circumstances, be thought of as information, and it is often claimed that the chicken accesses information – provided by its DNA, for example – to obtain the necessary negative entropy. However, the identification of information with negative entropy makes sense only in very specific contexts, and the activities of living creatures are not one of them. Organisms create order through the processes that they carry out, but those processes are not thermodynamic. Chickens don’t access some storehouse of order to make the thermodynamic books balance: they use processes for which a thermodynamic model is inappropriate, and throw the books away because they don’t apply.
The scenario in which an egg is created by borrowing entropy would be appropriate if the process that the chicken used were the time-reversal of an egg breaking up into its constituent molecules. At first sight this is vaguely plausible, because the molecules that eventually form the egg are scattered throughout the environment; they come together in the chicken, where biochemical processes put them together in an ordered manner to form the egg. However, there is a difference in the initial conditions. If you went round beforehand labelling molecules in the chicken’s environment, to say ‘this one will end up in the egg at such and such a location’, you would in effect be creating initial conditions as complex and unlikely as those for unscrambling an egg. But that’s not how the chicken operates. Some molecules happen to end up in the egg and are conceptually labelled as part of it after the process is complete. Other molecules could have done the same job – one molecule of calcium carbonate is just as good for making a shell as any other. So the chicken is not creating order from disorder. The order is assigned to the end result of the egg-making process – like shuffling a pack of cards into a random order and then numbering them 1, 2, 3, and so on with a felt-tipped pen. Amazing – they’re in numerical order!
To be sure, the egg looks more ordered than its ingredients, even if we take account of this difference in initial conditions. But that’s because the process that makes an egg is not thermodynamic. Many physical processes do, in effect, unscramble eggs. An example is the way minerals dissolved in water can create stalactites and stalagmites in caves. If we specified the exact form of stalactite we wanted, ahead of time, we’d be in the same position as someone trying to unsmash a vase. But if we’re willing to settle for any old stalactite, we get one: order from disorder. Those two terms are often used in a sloppy way. What matters are what kind of order and what kind of disorder. That said, I still don’t expect to see an egg unscrambling. There is no feasible way to set up the necessary initial conditions. The best we can do is turn the scrambled egg into chickenfeed and wait for the bird to lay a new one.
In fact, there is a reason why we wouldn’t see an egg unscrambling, even if the world did run backwards. Because we and our memories are part of the system that is being reversed, we wouldn’t be sure which way time was ‘really’ running. Our sense of the flow of time is produced by memories, physico-chemical patterns in the brain. In conventional language, the brain stores records of the past but not of the future. Imagine making a series of snapshots of the brain watching an egg being scrambled, along with its memories of the process. At one stage the brain remembers a cold, unscrambled egg, and some of its history when taken from the fridge and put into the saucepan. At another stage it remembers having whisked the egg with a fork, and having moved it from the fridge to the saucepan.
If we now run the entire universe in reverse, we reverse the order in which those memories occur, in ‘real’ time. But we don’t reverse the ordering of a given memory in the brain. At the start (in reversed time) of the process that unscrambles the egg, the brain does not remember the ‘past’ of that egg – how it emerged from a mouth on to a spoon, was unwhisked, gradually building up a complete egg . . . Instead, the record in the brain at that moment is one in which it remembers having cracked open an egg, along with the process of moving it from the fridge to the saucepan and scrambling it. But this memory is exactly the same as one of the records in the forward-time scenario. The same goes for all the other memory snapshots. Our perception of the world depends on what we observe now, and what memories our brain holds, now. In a time-reversed universe, we would in effect remember the future, not the past.
The paradoxes of time-reversibility and entropy are not problems about the real world. They are problems about the assumptions we make when we try to model it.
