Second law of thermodynamics 1

Second law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature,pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamicequilibrium, the law deduced the principle of the increase of entropy and explains the phenomenon of irreversibilityin nature. The second law declares the impossibility of machines that generate usable energy from the abundantinternal energy of nature by processes called perpetual motion of the second kind.The second law may be expressed in many specific ways, but the first formulation is credited to the French scientistSadi Carnot in 1824 (see Timeline of thermodynamics). The law is usually stated in physical terms of impossibleprocesses. In classical thermodynamics, the second law is a basic postulate applicable to any system involvingmeasurable heat transfer, while in statistical thermodynamics, the second law is a consequence of unitarity inquantum theory. In classical thermodynamics, the second law defines the concept of thermodynamic entropy, whilein statistical mechanics entropy is defined from information theory, known as the Shannon entropy.

DescriptionThe first law of thermodynamics provides the basic definition of thermodynamic energy, also called internal energy,associated with all thermodynamic systems, but unknown in mechanics, and states the rule of conservation of energyin nature.However, the concept of energy in the first law does not account for the observation that natural processes have apreferred direction of progress. For example, spontaneously, heat always flows to regions of lower temperature,never to regions of higher temperature without external work being performed on the system. The first law iscompletely symmetrical with respect to the initial and final states of an evolving system. The key concept for theexplanation of this phenomenon through the second law of thermodynamics is the definition of a new physicalproperty, the entropy.A change in the entropy (S) of a system is the infinitesimal transfer of heat (Q) to a closed system driving areversible process, divided by the equilibrium temperature (T) of the system.[1]

The entropy of an isolated system that is in equilibrium is constant and has reached its maximum value.Empirical temperature and its scale is usually defined on the principles of thermodynamics equilibrium by the zerothlaw of thermodynamics.[2] However, based on the entropy, the second law permits a definition of the absolute,thermodynamic temperature, which has its null point at absolute zero.[3]

The second law of thermodynamics may be expressed in many specific ways,[4] the most prominent classicalstatements[3] being the statement by Rudolph Clausius (1850), the formulation by Lord Kelvin (1851), and thedefinition in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law ingeneral physical terms citing the impossibility of certain processes. They have been shown to be equivalent.

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Clausius statementGerman scientist Rudolf Clausius is credited with the first formulation of the second law, now known as the Clausiusstatement:[4]

No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body ofhigher temperature.[5]

Spontaneously, heat cannot flow from cold regions to hot regions without external work being performed on thesystem, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows fromcold to hot, but only when forced by an external agent, a compressor.

Kelvin statementLord Kelvin expressed the second law in another form. The Kelvin statement expresses it as follows:[4]

No process is possible in which the sole result is the absorption of heat from a reservoir and its completeconversion into work.

This means it is impossible to extract energy by heat from a high-temperature energy source and then convert all ofthe energy into work. At least some of the energy must be passed on to heat a low-temperature energy sink. Thus, aheat engine with 100% efficiency is thermodynamically impossible. This also means that it is impossible to buildsolar panels that generate electricity solely from the infrared band of the electromagnetic spectrum withoutconsideration of the temperature on the other side of the panel (as is the case with conventional solar panels thatoperate in the visible spectrum).Note that it is possible to convert heat completely into work, such as the isothermal expansion of ideal gas. However,such a process has an additional result. In the case of the isothermal expansion, the volume of the gas increases andnever goes back without outside interference.

Principle of CarathéodoryConstantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statementof the second law is known as the Principle of Carathéodory, which may be formulated as follows:[6]

In every neighborhood of any state S of an adiabatically isolated system there are states inaccessible from S.[7]

With this formulation he described the concept of adiabatic accessibility for the first time and provided thefoundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics.

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Equivalence of the statements

Derive Kelvin Statement from Clausius Statement

Suppose there is an engine violating the Kelvin statement: i.e.,one that drains heat and converts it completely intowork in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the graph.The net and sole effect of this newly created engine consisting of the two engines mentioned is transferring heat

from the cooler reservoir to the hotter one, which violates the Clausius statement. Thus the

Kelvin statement implies the Clausius statement. We can prove in a similar manner that the Clausius statementimplies the Kelvin statement, and hence the two are equivalent.

Corollaries

Perpetual motion of the second kindPrior to the establishment of the Second Law, many people who were interested in inventing a perpetual motionmachine had tried to circumvent the restrictions of First Law of Thermodynamics by extracting the massive internalenergy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of thesecond kind". The second law declared the impossibility of such machines.

Carnot theoremCarnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiencysolely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:• All irreversible heat engines between two heat reservoirs are less efficient than a Carnot engine operating between

the same reservoirs.• All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating

between the same reservoirs.In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot however further postulated that some caloric is lost, not being converted to mechanical work. Hence no real heat engine could realise the Carnot cycle's reversibility

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and was condemned to be less efficient.Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insightinto the second law.

Clausius theoremThe Clausius theorem (1854) states that in a cyclic process

The equality holds in the reversible case[8] and the '>' is in the irreversible case. The reversible case is used tointroduce the state function entropy. This is because in cyclic processes the variation of a state function is zero.

Thermodynamic temperatureFor an arbitrary heat engine, the efficiency is:

where A is the work done per cycle. Thus the efficiency depends only on qC/qH.Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.Thus, any reversible heat engine operating between temperatures T1 and T2 must have the same efficiency, that is tosay, the effiency is the function of temperatures only:

In addition, a reversible heat engine operating between temperatures T1 and T3 must have the same efficiency as oneconsisting of two cycles, one between T1 and another (intermediate) temperature T2, and the second between T2andT3. This can only be the case if

Now consider the case where is a fixed reference temperature: the temperature of the triple point of water. Thenfor any T2 and T3,

Therefore if thermodynamic temperature is defined by

then the function f, viewed as a function of thermodynamic temperature, is simply

and the reference temperature T1 will have the value 273.16. (Of course any reference temperature and any positivenumerical value could be used—the choice here corresponds to the Kelvin scale.)

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EntropyAccording to the Clausius equality, for a reversible process

That means the line integral is path independent.

So we can define a state function S called entropy, which satisfies

With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolutevalue, we need the Third Law of Thermodynamics, which states that S=0 at absolute zero for perfect crystals.For any irreversible process, since entropy is a state function, we can always connect the initial and terminal statuswith an imaginary reversible process and integrating on that path to calculate the difference in entropy.Now reverse the reversible process and combine it with the said irreversible process. Applying Clausius inequalityon this loop,

Thus,

where the equality holds if the transformation is reversible.

Notice that if the process is an adiabatic process, then , so .

Available useful workAn important and revealing idealized special case is to consider applying the Second Law to the scenario of anisolated system (called the total system or universe), made up of two parts: a sub-system of interest, and thesub-system's surroundings. These surroundings are imagined to be so large that they can be considered as anunlimited heat reservoir at temperature TR and pressure PR — so that no matter how much heat is transferred to (orfrom) the sub-system, the temperature of the surroundings will remain TR; and no matter how much the volume ofthe sub-system expands (or contracts), the pressure of the surroundings will remain PR.Whatever changes to dS and dSR occur in the entropies of the sub-system and the surroundings individually,according to the Second Law the entropy Stot of the isolated total system must not decrease:

According to the First Law of Thermodynamics, the change dU in the internal energy of the sub-system is the sum ofthe heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energyentering the sub-system d ∑μiRNi, so that:

where μiR are the chemical potentials of chemical species in the external surroundings.Now the heat leaving the reservoir and entering the sub-system is

where we have first used the definition of entropy in classical thermodynamics (alternatively, in statisticalthermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then theSecond Law inequality from above.It therefore follows that any net work δw done by the sub-system must obey

Second law of thermodynamics 6

It is useful to separate the work δw done by the subsystem into the useful work δwu that can be done by thesub-system, over and beyond the work pR dV done merely by the sub-system expanding against the surroundingexternal pressure, giving the following relation for the useful work that can be done:

It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called theavailability or exergy X of the subsystem,

The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem,and an unlimited temperature and pressure reservoir with which it is in contact,

i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in thesubsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be lessthan or equal to zero.In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, thenthe Second Law predicts a decrease in X for an irreversible process and no change for a reversible process.

Is equivalent to This expression together with the associated reference state permits a design engineer working at the macroscopicscale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropychange in a total isolated system. (Also, see process engineer). Those changes have already been considered by theassumption that the system under consideration can reach equilibrium with the reference state without altering thereference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may alsobe found (See second law efficiency.)This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systemsecology, and other disciplines.to the formal energy created may not be supportive but it can be created from oneform to another form ..it can be also called as conservation of energy .,

HistoryThe first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. Hewas the first to realize correctly that the efficiency of this conversion depends on the difference of temperaturebetween an engine and its environment.Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was thefirst to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies.While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which consideredheat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for anydevice that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This wasshown to be equivalent to the statement of Clausius.The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the timespent in some region of the phase space of microstates with the same energy is proportional to the volume of thisregion, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says thattime average and average over the statistical ensemble are the same.

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It has been shown that not only classical systems but also quantum mechanical ones tend to maximize their entropyover time. Thus the second law follows, given initial conditions with low entropy. More precisely, it has been shownthat the local von Neumann entropy is at its maximum value with a very high probability.[9] The result is valid for alarge class of isolated quantum systems (e.g. a gas in a container). While the full system is pure and therefore doesnot have any entropy, the entanglement between gas and container gives rise to an increase of the local entropy ofthe gas. This result is one of the most important achievements of quantum thermodynamics.Today, much effort in the field is attempting to understand why the initial conditions early in the universe were thoseof low entropy,[10] [11] as this is seen as the origin of the second law (see below).

Informal descriptionsThe second law can be stated in various succinct ways, including:• It is impossible to produce work in the surroundings using a cyclic process connected to a single heat reservoir

(Kelvin, 1851).• It is impossible to carry out a cyclic process using an engine connected to two heat reservoirs that will have as its

only effect the transfer of a quantity of heat from the low-temperature reservoir to the high-temperature reservoir(Clausius, 1854).

• If thermodynamic work is to be done at a finite rate, free energy must be expended.[12]

Mathematical descriptionsIn 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in themechanical theory of heat" in the following form:[13]

where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved ina cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels ofthis definition, that same year, the most famous version of the second law was read in a presentation at thePhilosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:

The entropy of the universe tends to a maximum.This statement is the best-known phrasing of the second law. Moreover, owing to the general broadness of theterminology used here, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, to whichthis statement applies, many people take this simple statement to mean that the second law of thermodynamicsapplies virtually to every subject imaginable. This, of course, is not true; this statement is only a simplified versionof a more complex description.In terms of time variation, the mathematical statement of the second law for an isolated system undergoing anarbitrary transformation is:

whereS is the entropy andt is time.

Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law

Second law of thermodynamics 8

must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the orderof 1/√N where N is the number of particles in the system. For everyday (macroscopic) situations, the probability thatthe second law will be violated is practically zero. However, for systems with a small number of particles,thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted bythe second law. Classical thermodynamic theory does not deal with these statistical variations.

Derivation from statistical mechanicsIn statistical mechanics,the Second Law is not a postulate, rather it is a consequence of the fundamental postulate,also known as the equal prior probability postulate, so long as one is clear that simple probability arguments areapplied only to the future, while for the past there are auxiliary sources of information which tell us that it was lowentropy. The first part of the second law, which states that the entropy of a thermally isolated system can onlyincrease is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy tosystems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount ofenergy of is:

where is the number of quantum states in a small interval between and . Here is amacroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on thechoice of . However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specificentropy (entropy per unit volume or per unit mass) does not depend on .Suppose we have an isolated system whose macroscopic state is specified by a number of variables. Thesemacroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then willdepend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certainposition), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium willbe such that is maximized as that is the most probable situation in equilibrium.If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached,the fact the variable will adjust itself so that is maximized, implies that the entropy will have increased or it willhave stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).The entropy of a system that is not in equilibrium can be defined as:

see here. Here the is the probabilities for the system to be found in the states labeled by the subscript j. Inthermal equilibrium the probabilities for states inside the energy interval are all equal to , and in that casethe general definition coincides with the previous definition of S that applies to the case of thermal equilibrium.Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right afterwe do this, there are a number of accessible microstates, but equilibrium has not yet been reached, so the actualprobabilities of the system being in some accessible state are not yet equal to the prior probability of . We havealready seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to theprevious equilibrium state. Boltzmann's H-theorem, however, proves that the entropy will increase continuously as afunction of time during the intermediate out of equilibrium state.

Second law of thermodynamics 9

Derivation of the entropy change for reversible processesThe second part of the Second Law states that the entropy change of a system undergoing a reversible process isgiven by:

where the temperature is defined as:

See here for the justification for this definition. Suppose that the system has some external parameter, x, that can bechanged. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem ofquantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in thesame energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.The generalized force, X, corresponding to the external variable x is defined such that is the work performedby the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalizedforce for a system known to be in energy eigenstate is given by:

Since the system can be in any energy eigenstate within an interval of , we define the generalized force for thesystem as the expectation value of the above expression:

To evaluate the average, we partition the energy eigenstates by counting how many of them have a value for

within a range between and . Calling this number , we have:

The average defining the generalized force can now be written:

We can relate this to the derivative of the entropy w.r.t. x at constant energy E as follows. Suppose we change x to x+ dx. Then will change because the energy eigenstates depend on x, causing energy eigenstates to move into

or out of the range between and . Let's focus again on the energy eigenstates for which lies

within the range between and . Since these energy eigenstates increase in energy by Y dx, all suchenergy eigenstates that are in the interval ranging from E - Y dx to E move from below E to above E. There are

such energy eigenstates. If , all these energy eigenstates will move into the range between andand contribute to an increase in . The number of energy eigenstates that move from below

to above is, of course, given by . The difference

is thus the net contribution to the increase in . Note that if Y dx is larger than there will be the energyeigenstates that move from below E to above . They are counted in both and ,therefore the above expression is also valid in that case.Expressing the above expression as a derivative w.r.t. E and summing over Y yields the expression:

Second law of thermodynamics 10

The logarithmic derivative of w.r.t. x is thus given by:

The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inversesystem size and will thus vanishes in the thermodynamic limit. We have thus found that:

Combining this with

Gives:

Derivation for systems described by the canonical ensembleIf a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probabilitydistribution over the energy eigenvalues are given by the canonical ensemble:

Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partitionfunction. We now consider an infinitesimal reversible change in the temperature and in the external parameters onwhich the energy levels depend. It follows from the general formula for the entropy:

that

Inserting the formula for for the canonical ensemble in here gives:

General derivation from unitarity of quantum mechanicsThe time development operator in quantum theory is unitary, because the Hamiltonian is hermitian. Consequentlythe transition probability matrix is doubly stochastic, which implies the Second Law of Thermodynamics.[14] [15]

This derivation is quite general, based on the Shannon entropy, and does not require any assumptions beyondunitarity, which is universally accepted. It is a consequence of the irreversibility or singular nature of the generaltransition matrix.

Second law of thermodynamics 11

Non-equilibrium statesStatistically it is possible for a system to achieve moments of non-equilibrium. In such statistically unlikely eventswhere hot particles "steal" the energy of cold particles enough that the cold side gets colder and the hot side getshotter, for an instant. Such events have been observed at a small enough scale where the likelihood of such a thinghappening is significant.[16] The physics involved in such an event is described by the fluctuation theorem.

Controversies

Maxwell's demonJames Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gasat equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demonguards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, thedemon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increasedwhile in A they will have slowed down on average. Since average molecular speed corresponds to temperature, thetemperature decreases in A and increases in B, contrary to the second law of thermodynamics.One of the most famous responses to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin.Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed,and that the act of acquiring information would require an expenditure of energy. But later exceptions were found.

Loschmidt's paradoxLoschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deducean irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all knownlow-level fundamental physical processes at odds with any attempt to infer from them the second law ofthermodynamics which describes the behavior of macroscopic systems. Both of these are well-accepted principles inphysics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.One approach to handling Loschmidt's paradox is the fluctuation theorem, proved by Denis Evans and DebraSearles, which gives a numerical estimate of the probability that a system away from equilibrium will have a certainchange in entropy over a certain amount of time. The theorem is proved with the exact time reversible dynamicalequations of motion and the Axiom of Causality. The fluctuation theorem is proved utilizing the fact that dynamics istime reversible. Quantitative predictions of this theorem have been confirmed in laboratory experiments at theAustralian National University conducted by Edith M. Sevick et al. using optical tweezers apparatus.

Gibbs paradoxIn statistical mechanics, a simple derivation of the entropy of an ideal gas based on the Boltzmann distribution yieldsan expression for the entropy which is not extensive (is not proportional to the amount of gas in question). This leadsto an apparent paradox known as the Gibbs paradox, allowing, for instance, the entropy of closed systems todecrease, violating the second law of thermodynamics.The paradox is averted by recognizing that the identity of the particles does not influence the entropy. In theconventional explanation, this is associated with an indistinguishability of the particles associated with quantummechanics. However, a growing number of papers now take the perspective that it is merely the definition of entropythat is changed to ignore particle permutation (and thereby avert the paradox). The resulting equation for the entropy(of a classical ideal gas) is extensive, and is known as the Sackur-Tetrode equation.

Second law of thermodynamics 12

Poincaré recurrence theoremThe Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state veryclose to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence, which is ofthe order of .[17] The result applies to physical systems in which energy is conserved. TheRecurrence theorem apparently contradicts the Second law of thermodynamics, which says that large dynamicalsystems evolve irreversibly towards the state with higher entropy, so that if one starts with a low-entropy state, thesystem will never return to it. There are many possible ways to resolve this paradox, but none of them is universallyaccepted. The most typical argument is that for thermodynamical systems like an ideal gas in a box, recurrence timeis so large that for all practical purposes it is infinite.

Heat death of the universeAccording to the second law, the entropy of any isolated system, such as the entire universe, never decreases. If theentropy of the universe has a maximum upper bound then when this bound is reached the universe has nothermodynamic free energy to sustain motion or life, that is, the heat death is reached.

QuotesThe law that entropy always increases holds, I think, the supreme position among the laws of Nature. Ifsomeone 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. If it is found to be contradicted by observation — well, theseexperimentalists do bungle things sometimes. But if your theory is found to be against the second law ofthermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.—Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)The tendency for entropy to increase in isolated systems is expressed in the second law of thermodynamics —perhaps the most pessimistic and amoral formulation in all human thought.—Gregory Hill and Kerry Thornley, Principia Discordia (1965)There are almost as many formulations of the second law as there have been discussions of it.—Philosopher / Physicist P.W. Bridgman, (1941)

Notes[1] A. Bejan, (2006). 'Advanced Engineering Thermodynamics', Wiley. ISBN 0-471-67763-9[2] J. S. Dugdale (1996, 1998). Entropy and its Physical Meaning. Tayler & Francis. p. 13. ISBN 9-7484-0569-0. "This law is the basis of

temperature."[3] E. H. Lieb, J. Yngvason (1999). "The Physics and Mathematics of the Second Law of Thermodynamics". Physics Reports 310: 1–96.

arXiv:cond-mat/9708200. Bibcode 1999PhR...310....1L. doi:10.1016/S0370-1573(98)00082-9.[4] "Concept and Statements of the Second Law" (http:/ / web. mit. edu/ 16. unified/ www/ FALL/ thermodynamics/ notes/ node37. html).

web.mit.edu. . Retrieved 2010-10-07.[5] Translated from German original: Es gibt keine Zustandsänderung, deren einziges Ergebnis die Übertragung von Wärme von einem Körper

niederer auf einen Körper höherer Temperatur ist.[6] C. Caratheodory (1909). "Untersuchungen über die Grundlagen der Thermodynamik". Mathematische Annalen 67: 363. "Axiom II: In jeder

beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungennicht beliebig approximiert werden können."

[7] H.A. Buchdahl (1966). The Concepts of Classical Thermodynamics. Cambridge University Press. p. 68.[8] Clausius theorem (http:/ / scienceworld. wolfram. com/ physics/ ClausiusTheorem. html) at Wolfram Research[9] Gemmer, Jochen; Otte, Alexander; Mahler, Günter (2001). "Quantum Approach to a Derivation of the Second Law of Thermodynamics".

Physical Review Letters 86 (10): 1927–1930. arXiv:quant-ph/0101140. Bibcode 2001PhRvL..86.1927G. doi:10.1103/PhysRevLett.86.1927.PMID 11289822

[10] Carroll; Jennifer Chen (2335). "Does Inflation Provide Natural Initial Conditions for the Universe?". Gen.Rel.Grav. () ; InternationalJournal of Modern Physics D D14 () -2340 37 (2005): 1671–1674. arXiv:gr-qc/0505037. Bibcode 2005GReGr..37.1671C.doi:10.1007/s10714-005-0148-2.

Second law of thermodynamics 13

[11] Wald, R (2006). "The arrow of time and the initial conditions of the universe". Studies in History and Philosophy of Science Part B: Studiesin History and Philosophy of Modern Physics 37 (3): 394–398. doi:10.1016/j.shpsb.2006.03.005.

[12] Stoner (2000). "Inquiries into the Nature of Free Energy and Entropy in Respect to Biochemical Thermodynamics". Entropy 2 (3): 106–141.arXiv:physics/0004055. Bibcode 2000Entrp...2..106S. doi:10.3390/e2030106.

[13] Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies.London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.

[14] Hugh Everett, "Theory of the Universal Wavefunction" (http:/ / www. pbs. org/ wgbh/ nova/ manyworlds/ pdf/ dissertation. pdf), Thesis,Princeton University, (1956, 1973), Appendix I, pp 121 ff, in particular equation (4.4) at the top of page 127, and the statement on page 29 that"it is known that the [Shannon] entropy [...] is a monotone increasing function of the time."

[15] Bryce Seligman DeWitt, R. Neill Graham, eds, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics,Princeton University Press (1973), ISBN 0-691-08131-X Contains Everett's thesis: The Theory of the Universal Wavefunction, pp 3-140.

[16] Wang, G.; Sevick, E.; Mittag, Emil; Searles, Debra; Evans, Denis (2002). "Experimental Demonstration of Violations of the Second Law ofThermodynamics for Small Systems and Short Time Scales". Physical Review Letters 89 (5). Bibcode 2002PhRvL..89e0601W.doi:10.1103/PhysRevLett.89.050601.

[17] L. Dyson, J. Lindesay and L. Susskind, Is There Really a de Sitter/CFT Duality, JHEP 0208, 45 (2002)

References

Further reading• Goldstein, Martin, and Inge F., 1993. The Refrigerator and the Universe. Harvard Univ. Press. Chpts. 4-9 contain

an introduction to the Second Law, one a bit less technical than this entry. ISBN 978-0-674-75324-2• Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. Maxwell's Demon 2 : Entropy, classical and quantum

information, computing. Bristol UK; Philadelphia PA: Institute of Physics. ISBN 978-0-585-49237-7• Halliwell, J.J. et al. (1994). Physical Origins of Time Asymmetry. Cambridge. ISBN 0-521-56837-4.(technical).• Iftime (2010). "Complex aspects of gravity". arXiv:1001.4571 [physics.gen-ph].• Carnot, Sadi; Thurston, Robert Henry (editor and translator) (1890). Reflections on the Motive Power of Heat and

on Machines Fitted to Develop That Power. New York: J. Wiley & Sons. ( full text of 1897 ed.) (http:/ / books.google. com/ books?id=tgdJAAAAIAAJ)) ( html (http:/ / www. history. rochester. edu/ steam/ carnot/ 1943/ ))

External links• Stanford Encyclopedia of Philosophy: " Philosophy of Statistical Mechanics (http:/ / plato. stanford. edu/ entries/

statphys-statmech/ )" -- by Lawrence Sklar.• Second law of thermodynamics (http:/ / web. mit. edu/ 16. unified/ www/ FALL/ thermodynamics/ notes/ node30.

html) in the MIT Course Unified Thermodinamics and Propulsion (http:/ / web. mit. edu/ 16. unified/ www/FALL/ thermodynamics/ notes/ notes. html) from Prof. Z. S. Spakovszky

• E.T. Jaynes, 1988, " The evolution of Carnot's principle, (http:/ / bayes. wustl. edu/ etj/ articles/ ccarnot. pdf)" inG. J. Erickson and C. R. Smith (eds.)Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol1, p. 267.

• Website devoted to the Second Law. (http:/ / www. secondlaw. com/ )

Article Sources and Contributors 14

Article Sources and ContributorsSecond law of thermodynamics  Source: http://en.wikipedia.org/w/index.php?oldid=465970387  Contributors: 2over0, ABF, AC+79 3888, AP Shinobi, AThing, Abb3w, Acroterion, Aeternium,Aeusoes1, Ahoerstemeier, Aircorn, Akamad, Alai, Ale jrb, Alfredwongpuhk, AnandaDaldal, Andyparkins, Anonymous Dissident, Antandrus, Antixt, AppleJuggler, Aquillion, Arjun S Ariyil,Arjun r acharya, ArnoldReinhold, Arthena, Arthur Rubin, Ashley Y, Astrobradley, AugPi, Aunt Entropy, AwesomeMachine, BF6-NJITWILL, Bbanerje, Bduke, Ben Rogers, Bluecheese333,Bmord, Bobby1011, Bobo192, Bonaparte, BorgQueen, Brianhe, Caltas, CambridgeBayWeather, Canadian-Bacon, CarbonCopy, Cdh1001, ChXu, Chemeditor, Chinasaur, ChrisChiasson,ChrisNoe, Cirejcon, ComaVN, Complexica, Count Iblis, Cpcjr, Crash Underride, Crosbiesmith, Crowsnest, Curps, Cutler, Cyp, CzarB, DJ Clayworth, DMZ, DVD R W, Da500063, Daa89563,Daarznieks, Dan Gluck, DanielCD, Dantecubed, Dav4is, Dave souza, David Shear, David spector, Dawn Bard, Denevans, Desp, Dhollm, Dieseldrinker, Dna-webmaster, Doanison, Dominic,Dreadstar, Duncharris, EPM, EdJohnston, Edcolins, Egbertus, Emilio Juanatey, Ems2715, Enormousdude, Enviroboy, Eroica, Euyyn, Evil Monkey, Eyu100, Favonian, FeloniousMonk, Fluentaphasia, Flying Jazz, Fredrik, Fresheneesz, Frisettes, Funhistory, FunnyMan3595, GSlicer, Gaius Cornelius, Gandalf61, Gatewayofintrigue, George100, Gerbrant, Giftlite, Gilliam, Gonfer,Grstain, Gökhan, H-J-Niemann, Hadal, Hairy Dude, Hbent, Headbomb, Helix84, Henning Makholm, Heqwm, Hjlim, Hubbardaie, Hweimer, Ian.thomson, IceKarma, Ignignot, Ilmari Karonen,Ilyanep, Infinity0, Ivan Bajlo, Ixfd64, J.delanoy, JMS Old Al, Jab843, Jacob2718, Jarash, Jason One, JayEsJay, Jdpipe, Jebba, Jeffq, Jeronimo, Jewk, Jheald, Jim62sch, Jimbomonkey, Jittat,Jncraton, JodyB, John254, Johnstone, JorisvS, Josedanielc, Jossi, Jovianeye, Jtir, Jucati, K, KTC, Karn, Karol Langner, Kbdank71, Kbrose, Kenan82, Keta, KillerChihuahua, Klangenfurt,Kmarinas86, Knowledgeum, Koavf, Kommando797, Laughitup2, Lbrewer42, LeBofSportif, Lea phys, LeaveSleaves, LiDaobing, Lilmy13, LonelyBeacon, Lorenzarius, Lsommerer, MER-C,Madmardigan53, Maebmij, Marc Girod, Mbarbier, Mbweissman, McGeddon, Meeples, Meno25, Mgiganteus1, Mgrierson, Miaow Miaow, Michael C Price, Michael Hardy, Michaeladenner,Michaelbusch, Miftime, Miguel de Servet, Mikaduki, Mikiemike, Mineralogy, MrArt, Musaran, N12345n, Naji Khaleel, Nanog, Neptunius, Netheril96, Neutrality, Nk, Nonsuch, Nrsmith,Number 0, Oleg Alexandrov, Omegatron, PAR, Palinurus, Panzi, PaulLowrance, Pauli133, Pcarbonn, Peyre, Phys, Physical Chemist, PigFlu Oink, Pjacobi, Postdlf, Ppithermo, Profero, Ptbptb,Ragesoss, Ratiocinate, Ravikiran r, Rdsmith4, Reddi, Rhettballew, Rhobite, Rich Farmbrough, Rifleman 82, Ring0, Rjwilmsi, Roadrunner, Robinh, Romanm, SCZenz, Sadi Carnot, Sam Hocevar,San Diablo, Savarona1, ScienceGuy, Seqsea, Sholto Maud, Sietse Snel, Sin.pecado, Snoid Headly, Snoyes, Speaker to Lampposts, Spk ben, Srleffler, Stephenb, SteveCoast, Stirling Newberry,Stootoon, Subversive.sound, SunCreator, Tantalate, Tb, Tercer, Terse, The Anome, The Gnome, Theresa knott, ThorinMuglindir, Tide rolls, Timc, Time traveller, Tisthammerw, TobiasBergemann, Trevor MacInnis, Tubbyspencer, Vanished user 03, VanishedUser314159, Verrai, Vh mby, WAS 4.250, Wafulz, Wavelength, Wavesmikey, WebDrake, WikiPidi, Wikimol,Wisemove, Wkussmaul, Wndl42, Wtshymanski, XJamRastafire, Yamamoto Ichiro, Yappy2bhere, Zchenyu, Zebas, Zenosparadox, Zmanish, 607 ,کاشف عقیل anonymous edits

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