# Second law of thermodynamics

In physics, the second law of thermodynamics, in its many forms, is a statement about the quality and direction of energy flow, and it is closely related to the concept of entropy.

This law and its derivatives, such as the law of friction, define the arrow of time: most other physical laws are time-reversal invariant.

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## General description

The first law of thermodynamics states that one form of energy, e.g. kinetic, potential, electrical energy, thermal,... can be converted into another without loss. The second law states that thermal energy, or heat, is special among the types of energies: all the forms of energy can be converted into heat, but in a way that is not reversible; it is not possible to convert the heat back fully in its original form. In other words, heat is a form of energy of lower quality.

What makes heat so special? According to the kinetic theory, heat is due to the random movement of atoms and molecules, so it looks much like kinetic energy. The difference is that those movements of atoms and molecules that produce heat cannot be observed or predicted, while all the other forms of energy are the result of some orderly movement of particles. The second law says that the amount of random movement, i.e. the entropy, can only increase in a closed system, i.e. that we cannot put this randomness in order without some external influence. (Some systems, for example living cells, spontaneously become structured when they receive energy from the outside - see dissipative structures.)

The following example illustrates the law. When a stone falls on earth, its kinetic energy is converted into heat, i.e. it becomes random movements of earth particles. The second law says that this random movement will never become ordered again. For example, the random movement will never become synchronized to throw the stone back in the air: the heat energy will not revert to the original kinetic energy.

Yet, there is one thing predictable about heat: it flows from hot to cold bodies. This can be used to convert some heat into mechanical energy, using a Carnot heat engine. The cycle stops when both bodies reach the same temperature: it can be shown that the amount of random movements has not decreased in the process.

The second law of thermodynamics is important to engineers because it provides a way to determine the quality, as well as the amount of degradation of energy during a process. It is also used to determine the theoretical upper limits for the performance of many commonly used engineering systems like refrigerators, internal combustion engines, and chemical reactors.

Many claimed perpetual motion machines would, to function, have to violate the second law of thermodynamics. In general, such machines seem to generate energy from "nowhere". In all cases, when examined, the machine has some hidden mechanism for drawing energy in from the outside. One example of this would be a device that can do work such as pumping water, simply by taking energy from the air. Such claims are labelled "perpetual motion machines of the second kind".

## History

The first theory on the conversion of heat into mechanical work is due to Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of the process depends on the difference of temperature between the hot and cold bodies.

Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law in 1850, in this form: heat does not spontaneously flow from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat in vogue at the time, which considered heat as a liquid. From there he was able to infer the law of Sadi Carnot and the definition of entropy (1865).

Established in the 19th century, the Kelvin-Planck statement of the second law of thermodynamics says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.

The second law of thermodynamics is a law about macroscopic irreversibility. Boltzmann had first investigated the link with microscopic reversibility. He has given an explanation by means of Statistical mechanics, for dilute gases (in the zero density limit where the Ideal gas equation of state holds) in his H-theorem. He did not derive the second law of thermodynamics from mechanics alone, but also from the probability arguments. His idea was to write an equation of motion for the probability that a single particle has a particular position and momentum at a particular time. One of the terms in this equation accounts for how the single particle distribution changes through collisions of pairs of particles. This rate depends of the probability of pairs of particles. Boltzmann introduced the assumption of "molecular chaos" to reduce this pair probability to a product of single particle probabilities. From the resulting "Boltzmann Equation" he derived his famous H-theorem which implies that on average the entropy of an ideal gas can only increase.

The assumption of molecular chaos in fact violates time reversal symmetry. It assumes that particle momenta are uncorrelated BEFORE collisions. If you replace this assumption with "anti-molecular chaos" namely that particle momenta are uncorrelated AFTER collision, then you can derive an anti-Boltzmann equation and an anti-H-Theorem which implies entropy decreases on average. Thus we see that in reality Boltzmann did not succeed in solving Loschmidt's paradox. The molecular chaos assumption is the key element that introduces the arrow of time.

The Ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

In 1871, James Clerk Maxwell proposed a thought experiment that challenged the second law. It is now called Maxwell's demon and is an example of the importance of observability in discussing the second law (see the article for details).

In Quantum mechanics, the ergodicity approach can also be used. However, there is an alternative explanation, which involves Quantum collapse - it is a straightforward result that quantum measurement increases entropy of the ensemble. Thus, second law of thermodynamics is intimately related to quantum measurement theory and quantum collapse - and none of them is completely understood.

## Derivation of the Second Law from Time Reversible Mechanics

The paradox of how time reversible dynamics can lead to time irreversible behaviour as summarised in the Second Law of Thermodynamics (Loschmidt's paradox) has finally been resolved with the proof of the Fluctuation theorem FT - first proposed heuristically by Evans Cohen and Morriss in 1993 and first proved by Evans and Searles [Phys Rev E50, 1645(1994)]. The Fluctuation Theorem generalises the Second Law to small systems observed for short times. It applies to all systems which obey Classical mechanics (i.e. Newtonian dynamics), regardless of density. A quantum version of the Theorem is also now known.

The Fluctuation Theorem shows that in small systems entropy can sometimes be consumed rather than produced, but as the system size or the observation time gets longer, the probability of entropy consumption (rather than production) decreases exponentially. In the large system limit the conventional Second Law is obtained. The Evans and Searles proof of the Fluctuation Theorem is quite elementary - see Evans and Searles, Adv Phys, 51, 1529(2002). It uses the exact time reversible equations of motion - as embodied in the Liouville equation - and computes the probability of time averages of entropy production from a given initial distribution of molecular states.

This use of initial (rather than final) states, is consistent with the Law of Causality which is taken as axiomatic. Causality implies that the probability of events can be computed from the probability of preceding events - that causes determine effects. It is thus seen that the Second Law of thermodynamics is a result of the Axiom of Causality. If the Universe was anti-causal (that effects determine their causes - that for example, electric currents begin to change BEFORE the applied voltage is changed!), then entropy could only decrease. In an anticausal Universe not only do currents start to flow BEFORE voltages are applied but the anti Fluctuation Theorem proves that those currents would flow AGAINST the direction of the applied voltage. The Law, or Axiom, of Causality cannot be proved. It is an often unrecognized Law of physics every bit as fundamental and unproveable as the laws of quantum or classical mechanics.

Quantitative predictions of the Fluctuation theorem were confirmed in laboratory experiments in 2002 by Wang et al. [Phys Rev Lett, 89, 050601(2002)] and later by Carberry et al, [Phys Rev Lett, 92, 140601(2004)]. The Fluctuation Theorem has important applications in nanotechnology. These experiments confirm the predictions of the Fluctuation theorem, that as times increase, macroscopic Second Law behaviour is approached EXPONENTIALLY as the averaging time increases.

## Other

Flanders and Swann produced a particularly insightful explanation of thermodynamics, which was popular in the 1950's and 1960's. Their work, entitled First and Second Law is unique in having been set to music.

The Second Law is exhibited (coarsely) by a box of electrical cables. Cables added from time to time tangle, inside the 'closed system' (cables in a box) by adding and then removing cables. The best way to untangle them is to start by taking the cables out of the box and placing them stretched out. The cables in a closed system (the box) will never untangle, but giving them some extra space starts the process of untangling (by going outside the closed system).

In the book, Learning to See the Timeless Infinite Universe (http://everythingforever.com), Gevin Giorbran describes a flaw in the semantics of the second law of thermodynamics by revealing that general classifications of order and disorder do not exist in nature, but rather two directions of order exist in opposition to one another. Giorbran explains that all systems evolve from grouping order to symmetry order, showing that the order of one type is the disorder of the other.

## Evolution, creationism and the second law

Creationists often claim that biological evolution violates the second law of thermodynamics, asserting that the law states that entropy spontaneously increases. However, the second law of thermodynamics applies only to a closed system, which the surface of the Earth is not since it receives megajoules per second of energy from the Sun. The vast majority of living organisms derive their energy from the sun, either primarily (as in the case of plants) or secondarily (in the case of organisms which eat other organisms). Solar energy is thus used by biological organisms to maintain and increase their complexity. For instance, a fully-grown tree is more complex than a sapling; the energy that enables its increase in complexity comes from the sun.

Even where solar energy is not available, such as at the bottom of the ocean, alternative energy sources such as hydrothermal vents can be used by living organisms. Some biologists have speculated that the earliest living organisms may have originated in or around such vents (see Origin of life for more). In this instance, the Earth could be considered to be operating as a closed system, as a finite amount of energy is being dissipated into the local environment. However, the second law of thermodynamics permits localised complexity within a closed system. The amount of energy within the system remains the same overall – so that it makes no overall difference whether geothermal energy is absorbed by living or non-living objects – but the distribution of that energy within the system may vary.

The universe as a whole may be considered an isolated system, so that its overall disorder should be constantly increasing even as local disorder may decrease due to an influx of energy. The cosmic microwave background radiation provides a universe-wide example of this: although the amount of energy within the universe is finite, its local distribution is uneven. Some creationists nonetheless claim that entropy never decreases locally, but this cannot be true as it would preclude such phenomena as the formation of snowflakes or galaxies.

## Quotes including the second law

• "Nothing in life is certain except death, taxes and the second law of thermodynamics. All three are processes in which useful or accessible forms of some quantity, such as energy or money, are transformed into useless, inaccessible forms of the same quantity. That is not to say that these three processes don't have fringe benefits: taxes pay for roads and schools; the second law of thermodynamics drives cars, computers and metabolism; and death, at the very least, opens up tenured faculty positions"---Seth Lloyd, writing in Nature 430, 971 (26 August 2004); doi:10.1038/430971a
• "If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations, then so much the worse for Maxwell's equations. And if your theory contradicts the facts, well, sometimes these experimentalists make mistakes. But 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 Eddington
• "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 response was cold: it was also negative."---C.P. Snow Rede Lecture in 1959 entitled "The Two Cultures and the Scientific Revolution".

### "Evolution does not violate the second law"

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