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Thermodynamics
The classical Carnot heat engine
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The most concise statement of the second law of thermodynamics states that the total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value.

General description

In a general sense, the second law says that the differences between systems in contact with each other tend to even out. Pressure differences, density differences, and particularly temperature differences, all tend to equalize if given the opportunity. This means that an isolated system will eventually come to have a uniform temperature. A thermodynamic engine is an engine that provides useful work from the difference in temperature of two bodies. Since any thermodynamic engine requires such a temperature difference, it follows that no useful work can be derived from an isolated system in equilibrium, there must always be energy fed from the outside. The second law is often quoted as the reason that we cannot build perpetual motion machines.

The second law of thermodynamics have been stated in several ways. Succinctly, the second law of thermodynamics can be stated as:

• It is impossible to obtain a process such that the unique effect is the subtraction of a positive heat from a reservoir and the production of a positive work;
• A system operating in contact with a thermal reservoir cannot produce positive work in its surroundings (Kelvin);
• A system operating in a cycle cannot produce a positive heat flow from a colder body to a hotter body (Clausius)
• The entropy of a closed system will not decrease for any sustained period of time (see Maxwell's demon)

A downside to this last description is that it requires an understanding of the concept of entropy. There are, however, consequences of the second law that are understandable without a full understanding of entropy. These are described in the first section below.

The second law can be described mathematically as:

${\displaystyle {\frac {dS}{dt}}\geq 0}$

where S is the entropy and the equality sign holds only when the entropy is at its maximum (equilibrium) value.

A common misconception is that the second law means that entropy never ever decreases - but the second law is only a tendency, hence, it is only means that it is highly unlikely that entropy will decrease in a closed system at any given instant.

History

The first theory on the conversion of heat into mechanical work is due to Nicolas Léonard 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 popular 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 first investigated the link with microscopic reversibility. In his H-theorem he gave an explanation, by means of statistical mechanics, for dilute gases in the zero density limit where the ideal gas equation of state holds. He derived the second law of thermodynamics not 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

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The paradox of how time reversible dynamics can lead to time irreversible behavior as summarized 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 Denis Evans and Debra Searles (Evans & Searles, 1994). The Fluctuation Theorem generalizes 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 (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 were 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. and later by Carberry et al, . The Fluctuation Theorem has important applications in nanotechnology. These experiments confirm the predictions of the Fluctuation theorem, that as times increase, macroscopic Second Law behavior is approached exponentially as the averaging time increases.

Complex systems and the Second Law

It is occasionally perceived, sometimes in the context of debates about evolution and creationism, that the Second Law is incompatible with self-organisation or the coming into existence of complex systems. As considered further in the article on self-organisation, this is a misconception: it is not correct.

In fact, as hot systems cool down in accordance with the second law, it is not unusual for them to undergo spontaneous symmetry breaking, i.e. for structure to spontaneously appear as their cooling passes a critical temperature. Complex structures also spontaneously appear in systems where there is steady flow of energy from a high temperature input source to a low temperature external sink. It is conjectured that such systems tend to evolve into complex, critically unstable "edge of chaos" arrangements which very nearly maximise the rate of energy degradation (the rate of entropy production).

The above discussion is not meant to be a comprehensive synopsis of all claims that have ever been asserted involving the 2nd law. However, no exposition of such creationist claims have survived analysis in scientific peer-reviewed fora of primary research for thermodynamics and biology. As such, no further discussion of this is needed in a scientific article of wikipedia - which is a forum for secondary research.

Miscellanea

Flanders and Swann produced a setting of a statement of the Second Law of Thermodynamics to music, called First and Second Law.

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).

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"---Sir 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".
• "The second law of thermodynamics is, without a doubt, one of the most perfect laws in physics. Any reproducible violation of it, however small, would bring the discoverer great riches as well as a trip to Stockholm. The world’s energy problems would be solved at one stroke. It is not possible to find any other law (except, perhaps, for super selection rules such as charge conservation) for which a proposed violation would bring more skepticism than this one. Not even Maxwell’s laws of electricity or Newton’s law of gravitation are so sacrosanct, for each has measurable corrections coming from quantum effects or general relativity. The law has caught the attention of poets and philosophers and has been called the greatest scientific achievement of the nineteenth century. Engels disliked it, for it supported opposition to dialectical materialism, while Pope Pius XII regarded it as proving the existence of a higher being."---Ivan P. Bazarov 1964, in his textbook Thermodynamics

See also

• Entropy
• Final Anthropic Principle
• First law of thermodynamics
• Laws of thermodynamics
• Statistical mechanics
• MaxEnt thermodynamics - Bayesian view of entropy and the Second Law

References

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External links

• Philosophy of Statistical Mechanics Article by Lawrence Sklar for the Stanford Encyclopedia of Philosophy
• The evolution of Carnot's principle, by E.T. Jaynes, in G. J. Erickson and C. R. Smith (eds.) Maximum-Entropy and Bayesian Methods in Science and Engineering vol. 1, p. 267 (1988).
• 75+ Variations of the 2nd Law
• A Student's approach to the second law and thermodynamics
• Entropy and the second law of thermodynamics
• The Second Law of Thermodynamics
• Thermodynamics, Gravity and Human Economics