3 The irreversible Universe

3.1 Thermodynamics and entropy

'Science owes more to the steam engine than the steam engine owes to Science.'
L. J. Henderson (1917)

From the time of Newton till the end of the nineteenth century the development of physics consisted essentially of the refinement and extension of the mechanical view of the Universe. There were many stages in this process but one of the most interesting came towards its end with the realization that the cosmic clockwork was inevitably unwinding and running down. The source of this realization was the development of thermodynamics.

The first half of the nineteenth century was a period of great economic and industrial growth. The steam engine, invented in the previous century, was becoming increasingly common in locomotives, mines and factories; power was becoming available on demand. A major priority for engineers was to produce more efficient engines, in order to deliver more useful power for less expenditure on fuel. Thermodynamics emerged as a study of the basic principles determining energy flows and the efficiency of machines.

This may seem like a big idea in engineering rather than a big idea in physics. Certainly, thermodynamics is important to engineers, and continues to guide the design of engines of all sorts, but thermodynamics is just as important to physicists. It explains a wealth of natural phenomena, from the freezing of water to the evaporation of a black hole, and casts light on concepts like temperature, heat and spontaneous processes, which do not fit naturally into the Newtonian world-view.

It is still instructive to return to the origins of the subject. Speaking very roughly, a steam engine is a device which uses fuel to convert water into steam and uses the resulting expansion in volume to drive a piston. The kinetic energy of the piston is exploited using a variety of mechanical devices - gears, drive belts, camshafts and so on, but thermodynamics concentrates on the early stages of the process, where heat is used to create kinetic energy.
figure 1.13, A steam engineFigure 1.13 A steam engine, in which energy stored in coal is used to create heat to vaporize water. The resulting increase in volume drives a piston, so allowing useful work to be done.
To begin with there was much dispute about the nature of heat. Many people thought of it as a sort of fluid which could flow from one body to another. Eventually, it became clear that no such fluid exists and that heat is best defined as energy transferred because of a temperature difference. This scientific definition of the word 'heat' is slightly different from everyday usage, so it may help to consider a specific example. Think of a hot steak (veggie-burger, if you prefer) resting on a cold plate. The steak cools down and the plate warms up as energy flows from the steak to the plate. The energy transferred in this way is called heat. By contrast, work is energy transferred by non-thermal means. For example, if you rub the plate vigorously with a cloth, the energy of the plate will increase and it will get slightly warmer. But, this energy transfer is not caused by a temperature difference between the plate and the cloth, so energy transferred by rubbing is classified as work rather than heat.

In general, the total energy gained by a system (such as the plate) is the sum of the heat and the work transferred to it. It is worth emphasizing that heat and work are not themselves properties of a system. We cannot examine a plate and deduce that it has received so much energy from heat and so much energy from work. All that counts is that the plate has a total amount of energy, and that any increase in this energy is the sum of the heat and work transferred to the plate. This understanding of heat, work and energy is incorporated in the first law of thermodynamics.
First law of thermodynamics
When all types of energy transfer, including work and heat, are taken into account, the energy of an isolated system remains constant
From a modern perspective, we can see that this is just another way of stating the law of conservation of energy with the explicit recognition of heat as a quantity of energy to be included, alongside work, in any energy audit. Inventors should take note: an engine may convert energy from one form to another, but it cannot produce energy from nothing. The kinetic energy of the piston of a steam engine, for instance, has been paid for in advance by the heat transferred to the steam.

Given this modern understanding of heat as energy transferred in a particular way, you might wonder why we bother to distinguish between heat and work at all. The reason is that heat can be used to define another important quantity: entropy.

We cannot define entropy properly in this introductory survey. In very broad terms you can think of entropy as a measure of 'disorder' - the random motion of molecules in steam corresponds to more disorder, and hence more entropy, than the more orderly motion of molecules in ice. Interestingly enough, there is a connection between entropy and heat: whenever heat is transferred to a body, the entropy of that body increases. In the simplest case, if a small amount of heat Q is transferred gently to a body, whilst the temperature of the body is T, the entropy of the body increases by Q/T.

The term entropy was deliberately chosen to be reminiscent of energy, though the differences between the two quantities are just as important as their similarities. Entropy and energy are similar in that an isolated body may be said to have a certain 'entropy content' just as it may be said to have a certain 'energy content'. However, while the first law of thermodynamics ensures that the energy of an isolated system is always conserved, the second law of thermodynamics makes a slightly weaker assertion about entropy:
Second law of thermodynamics
The total entropy of an isolated system cannot decrease; it may (and generally does) increase
The requirement that the total entropy should not decrease has the effect of ruling out enormous numbers of processes that are perfectly consistent with energy conservation.

When heat flows between a steak and a plate there is no violation of energy conservation; the energy lost by the steak is gained by the plate. However, conservation of energy does not explain why the heat always flows from the hot steak to the cold plate; this is where the second law of thermodynamics comes in. Suppose the steak is at temperature T, the plate is at a slightly lower temperature 0.95T, and that a small amount of heat Q is transferred from the steak to the plate.
you will never see heat flowing spontaneously from a cold body to a hotter one
Then the entropy of the steak decreases by Q/T while the entropy of the plate increases by Q/0.95T. It is easy to see that the entropy lost by the steak is smaller than the entropy gained by the plate, so the total entropy of the Universe has increased; this process is therefore consistent with the second law of thermodynamics. If, on the other hand, heat Q had flowed from the cold plate to the hot steak, the entropy lost by the plate (Q/0.95T) would have been greater than the entropy gained by the steak (Q/T), and the total entropy of the Universe would have decreased. This violates the second law of thermodynamics, so we can be sure that the process is impossible. Heat flow is said to be an irreversible process - you will never see heat flowing spontaneously from a cold body to a hotter one.

Whenever energy is transferred or transformed, the final entropy of the Universe must be at least as high as the initial entropy. This usually means that heat flows are required to ensure that the total entropy does not decrease. Inventors should again take note. In most engines, heat is an unwanted by-product: the real aim is to transfer energy as work, perhaps to propel a vehicle or lift a weight. Since part of the energy initially stored in the fuel is inevitably wasted as heat, only a fraction is left to do useful work. Thus, thermodynamics imposes fundamental limits on the efficiency of engines. Fortunately, it also suggests ways of increasing efficiency, explaining for example, why a diesel engine is likely to be more efficient than a petrol engine, a topic we will return to in Classical physics of matter, book 4.

3.2 Equilibrium and irreversibility

As the science of thermodynamics developed beyond its industrial roots, two powerful ideas came to the fore - equilibrium and irreversibility. These ideas were already implicit in studies of heat. You have already seen that heat flow from a hot steak to a cold plate is an irreversible process. The effect of this process is to cool down the hot steak and warm up the cold plate, leading to a more uniform distribution of temperature. The heat transfer continues until a state of equilibrium is reached, characterized by a completely uniform temperature.

Understanding the conditions needed for equilibrium, and the irreversible processes that drive systems towards equilibrium, has deep consequences throughout the sciences. For example, under normal conditions, the equilibrium state of carbon is graphite, rather than diamond. Fortunately, the processes that restore equilibrium are very slow in this case, so diamonds do not perceptibly turn into graphite. But, under some rather extreme conditions, diamond is the equilibrium state rather than graphite, and this fact can be used to create new diamonds from soot. More generally, thermodynamics determines which states of matter are in equilibrium under any given set of conditions.

Entropy and the second law of thermodynamics provide the key to understanding equilibrium. An isolated system, free from all other influences, may undergo various spontaneous changes, some of which will increase its entropy. If the total entropy increases during a process, as it usually does, the process is irreversible - it is impossible to return to the starting point, leaving no other traces, since that would require a decrease in the total entropy, which is impossible. Once the entropy has increased, it cannot decrease again. An isolated system therefore approaches a state in which the entropy has the highest possible value. This is a state of equilibrium. In equilibrium, the entropy of the system cannot increase (because it is already at a maximum) and it cannot decrease (because that would violate the second law of thermodynamics). The only changes allowed are those in which the entropy remains constant.

This equilibrium can be disturbed if the system is allowed to interact with its surroundings. The entropy of the system may then decrease, provided the entropy of the surroundings increases by at least as much, ensuring that there is no decrease in the entropy of the Universe as a whole.

If we start with a system that is close to, but has not quite reached, equilibrium, thermodynamics can suggest which processes will increase the entropy and lead towards equilibrium. Heat transfers are one source of entropy changes, but there are others. If you take two different gases and allow them to mix together in a flask that is so well insulated that no heat can be transferred to or from the flask, the entropy of the mixture turns out to be greater than the entropy of the two separate gases. That is why the mixing is an irreversible process. Once mixed, the gases will not spontaneously separate. Similar considerations explain why a dropped glass can shatter into a thousand fragments, but a thousand fragments will never spontaneously form themselves into a glass. Also, an egg can be made into an omelette, but an omelette will not make itself into an egg. There is an 'arrow of time' that points from the past to the future, and tomorrow will be different from today.

Figure 1.14 Some examples of irreversibility
figure 1.14 (a) a smashed glass(a) a smashed glass
(b) an omelettefigure 1.14 (b) an omelette

If these ideas are correct, the Universe must be inescapably and irreversibly approaching a state in which its entropy has the highest possible value. This will be a state of equilibrium for the Universe as a whole, where all the fuel will have been expended and the temperature will be uniform, leaving no prospect of generating heat flows and extracting useful work. In a phrase made popular in the 1930s by the Cambridge cosmologist Sir Arthur Eddington, the Universe is said to be approaching a final state of 'heat death'. In this sense, the clockwork of the Newtonian Universe is running down.

3.3 Statistical mechanics

You saw earlier that very strong claims were made for Newtonian mechanics. Many regarded it as a basic framework that would underlie all scientific explanations. It is therefore natural to ask about the relationship between Newtonian mechanics and thermodynamics:

Do they contradict one another?

Are they separate aspects of the truth?

Can thermodynamics be derived from Newtonian mechanics?

These are not easy questions. Thermodynamics was specifically designed to deal with concepts like temperature, heat and entropy which had no clear Newtonian interpretation. The gulf between the two subjects can be illustrated by taking, say, a glass of water in a state of equilibrium. We now know that this contains an enormous number of molecules (roughly 10 to the power of 24 ), each feeling electrical forces due to other molecules and moving rapidly around, colliding with other molecules in the liquid and the glass. The Newtonian world-view would require us to keep track of each and every molecule, building up an immensely complicated and detailed description. Of course, this is utterly beyond our powers. Even if it were possible, the results would provide little or no insight. It would be like looking at a painting under a microscope when its true significance is only apparent from a distance of a few metres. Thermodynamics adopts a more practical viewpoint. Rather than tracking each water molecule in detail, it uses just a few well-chosen variables - including energy, volume, pressure, temperature and entropy - to characterize the state of the water as a whole. The amazing thing is that this works. The thermodynamic description is massively incomplete, yet it is sufficient to make useful predictions.

There is a special branch of physics, called statistical mechanics, which attempts to bridge the gap between descriptions on the scale of molecules and thermodynamics. It recognizes that our knowledge of a complicated system, such as a glass of water, is inevitably incomplete so we are essentially reduced to making guesses. This may seem to be a terrible weakness, but statistical mechanics actually turns it into an advantage. It replaces precise knowledge of the motion of molecules by probabilities indicating how the molecules are likely to move, on average. It then goes on to estimate the probability of measuring a particular pressure, energy or entropy in the system as a whole.
Figure 1.15 Ludwig Boltzmann
(1844-1906)

The statistical interpretation of thermodynamics was brought to fruition by Austrian physicist

figure 1.15
This is rather like the trick pulled by opinion pollsters when they predict the result of a general election without knowing how every individual in the country intends to vote. Pollsters have a mixed reputation, but the calculations of statistical mechanics are much more clear cut. They turn out to provide predictions that are overwhelmingly likely to happen - so much so, that they appear to be laws of Nature. The second law of thermodynamics is a case in point. From the viewpoint of statistical mechanics, the entropy of the Universe is not bound to increase, it is just overwhelmingly likely to do so. Perhaps 'heat death' will not be the end after all. After countless years of dull uniformity, a very unlikely (but possible) new fluctuation may occur with a lower than maximum entropy, and interesting things will start to happen again.