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Second law of thermodynamicsFrom Wikipedia, the free encyclopedia

The second law of thermodynamics is an expression of the tendency that overtime, differences in temperature, pressure, and chemical potential equilibrate inan isolated physical system. From the state of thermodynamic equilibrium, thelaw deduced the principle of the increase of entropy and explains thephenomenon of irreversibility in nature. The second law declares the impossibilityof machines that generate usable energy from the abundant internal energy ofnature by processes called perpetual motion of the second kind.

The second law may be expressed in many specific ways, but the first formulationis credited to the French scientist Sadi Carnot in 1824 (see Timeline ofthermodynamics). The law is usually stated in physical terms of impossibleprocesses. In classical thermodynamics, the second law is a basic postulateapplicable to any system involving measurable heat transfer, while in statisticalthermodynamics, the second law is a consequence of unitarity in quantum theory.In classical thermodynamics, the second law defines the concept ofthermodynamic entropy, while in statistical mechanics entropy is defined frominformation theory, known as the Shannon entropy.

Contents

1 Description1.1 Clausiusstatement1.2 Kelvinstatement1.3 Principle ofCarathéodory1.4 Equivalenceof thestatements

2 Corollaries2.1 Perpetualmotion of thesecond kind2.2 Carnottheorem2.3 Clausiustheorem2.4

Branches

Laws

Systems

Second law of thermodynamics - Wikipedia, the fr... file:///media/TOSHIBA/Second_law_of_thermodyn...

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Isobaric · Isochoric · IsothermalAdiabatic · Isentropic · Isenthalpic

Quasistatic · PolytropicFree expansion

Reversibility · IrreversibilityEndoreversibility

Cycles:Heat engines · Heat pumps

Thermal efficiency

Property diagramsIntensive and extensive properties

State functions:Temperature / Entropy (intro.) †

Pressure / Volume †Chemical potential / Particle no. †

(† Conjugate variables)Vapor quality

Reduced properties

Process functions:Work · Heat

Specific heat capacity c =T

N

Compressibility β = −1

V

Thermal expansion α =1

VProperty database

Carnot's theoremClausius theorem

Fundamental relationIdeal gas law

Maxwell relations

Thermodynamictemperature2.5 Entropy2.6 Availableuseful work

3 History3.1 Informaldescriptions3.2Mathematicaldescriptions

4 Derivation fromstatistical mechanics

4.1 Derivationof the entropychange forreversibleprocesses4.2 Derivationfor systemsdescribed bythe canonicalensemble4.3 Generalderivation fromunitarity ofquantummechanics

5 Non-equilibriumstates6 Controversies

6.1 Maxwell'sdemon6.2 Loschmidt'sparadox6.3 Gibbsparadox6.4 Poincarérecurrencetheorem6.5 Heat deathof the universe

7 Quotes

System properties

Material properties

Equations


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Table of thermodynamic equations

Free energy · Free entropy

Internal energy U(S,V)Enthalpy H(S,p) = U + pVHelmholtz free energy A(T,V) = U − TSGibbs free energy G(T,p) = H − TS

Philosophy:Entropy and time · Entropy and life

Brownian ratchetMaxwell's demon

Heat death paradoxLoschmidt's paradox

Synergetics

History:General · Heat · Entropy · Gas laws

Perpetual motionTheories:

Caloric theory · Vis vivaTheory of heat

Mechanical equivalent of heatMotive powerPublications:

"An Experimental Enquiry Concerning ... Heat""On the Equilibrium of Heterogeneous Substances"

"Reflections on theMotive Power of Fire"

Timelines of:Thermodynamics · Heat engines

Art:Maxwell's thermodynamic surface

Education:Entropy as energy dispersal

Daniel Bernoulli

8 Notes9 See also10 References11 Further reading12 External links

Description

The first law of thermodynamicsprovides the basic definition ofthermodynamic energy, alsocalled internal energy,associated with allthermodynamic systems, butunknown in mechanics, andstates the rule of conservation ofenergy in nature.

However, the concept of energyin the first law does not accountfor the observation that naturalprocesses have a preferreddirection of progress. Forexample, spontaneously, heatalways flows to regions of lowertemperature, never to regions ofhigher temperature withoutexternal work being performedon the system. The first law iscompletely symmetrical withrespect to the initial and finalstates of an evolving system. Thekey concept for the explanationof this phenomenon through thesecond law of thermodynamics isthe definition of a new physicalproperty, the entropy.

A change in the entropy (S) of asystem is the infinitesimaltransfer of heat (Q) to a closedsystem driving a reversibleprocess, divided by the

Potentials

History and culture

Scientists


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Sadi CarnotBenoît Paul Émile Clapeyron

Rudolf ClausiusHermann von HelmholtzConstantin Carathéodory

Pierre DuhemJosiah Willard GibbsJames Prescott JouleJames Clerk Maxwell

Julius Robert von MayerWilliam RankineJohn Smeaton

Georg Ernst StahlBenjamin Thompson

William Thomson, 1st Baron KelvinJohn James Waterston

equilibrium temperature (T) ofthe system.[1]

The entropy of an isolatedsystem that is in equilibrium isconstant and has reached itsmaximum value.

Empirical temperature and itsscale is usually defined on theprinciples of thermodynamicsequilibrium by the zeroth law ofthermodynamics.[2] However,based on the entropy, the secondlaw permits a definition of theabsolute, thermodynamictemperature, 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 classical statements[3] being the statement by RudolphClausius (1850), the formulation by Lord Kelvin (1851), and the definition inaxiomatic thermodynamics by Constantin Carathéodory (1909). These statementscast the law in general physical terms citing the impossibility of certainprocesses. They have been shown to be equivalent.

Clausius statement

German scientist Rudolf Clausius is credited with the first formulation of thesecond law, now known as the Clausius statement:[4]

No process is possible whose sole result is the transfer of heat from a body oflower temperature to a body of higher temperature.[note 1]

Spontaneously, heat cannot flow from cold regions to hot regions without externalwork being performed on the system, which is evident from ordinary experienceof refrigeration, for example. In a refrigerator, heat flows from cold to hot, butonly when forced by an external agent, a compressor.

Kelvin statement

Lord Kelvin expressed the second law in another form. The Kelvin statementexpresses it as follows:[4]


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No process is possible in which the sole result is the absorption of heat from areservoir and its complete conversion into work.

This means it is impossible to extract energy by heat from a high-temperatureenergy source and then convert all of the energy into work. At least some of theenergy must be passed on to heat a low-temperature energy sink. Thus, a heatengine with 100% efficiency is thermodynamically impossible. This also meansthat it is impossible to build solar panels that generate electricity solely from theinfrared band of the electromagnetic spectrum without consideration of thetemperature on the other side of the panel (as is the case with conventional solarpanels that operate in the visible spectrum).

Note that it is possible to convert heat completely into work, such as theisothermal expansion of ideal gas. However, such a process has an additionalresult. In the case of the isothermal expansion, the volume of the gas increasesand never goes back without outside interference.

Principle of Carathéodory

Constantin Carathéodory formulated thermodynamics on a purely mathematicalaxiomatic foundation. His statement of the second law is known as the Principle ofCarathéodory, which may be formulated as follows:[5]

In every neighborhood of any state S of an adiabatically isolated system thereare states inaccessible from S.[6]

With this formulation he described the concept of adiabatic accessibility for thefirst time and provided the foundation for a new subfield of classicalthermodynamics, often called geometrical thermodynamics.

Equivalence of the statements

Suppose there is an engine violating the Kelvin statement: i.e.,one that drainsheat and converts it completely into work in a cyclic fashion without any otherresult. Now pair it with a reversed Carnot engine as shown by the graph. The netand 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 statementimplies the Clausius statement. We can prove in a similar manner that theClausius statement implies the Kelvin statement, and hence the two areequivalent.

Corollaries


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Derive Kelvin Statement from ClausiusStatement

Perpetual motion of the secondkind

Main article: perpetual motion

Prior to the establishment of the SecondLaw, many people who were interested ininventing a perpetual motion machinehad tried to circumvent the restrictionsof First Law of Thermodynamics byextracting the massive internal energy ofthe environment as the power of themachine. Such a machine is called a"perpetual motion machine of the secondkind". The second law declared theimpossibility of such machines.

Carnot theorem

Carnot's theorem (1824) is a principle that limits the maximum efficiency for anypossible engine. The efficiency solely depends on the temperature differencebetween the hot and cold thermal reservoirs. Carnot's theorem states:

All irreversible heat engines between two heat reservoirs are less efficientthan a Carnot engine operating between the same reservoirs.All reversible heat engines between two heat reservoirs are equallyefficient with a Carnot engine operating between the same reservoirs.

In his ideal model, the heat of caloric converted into work could be reinstated byreversing the motion of the cycle, a concept subsequently known asthermodynamic reversibility. Carnot however further postulated that some caloricis lost, not being converted to mechanical work. Hence no real heat engine couldrealise the Carnot cycle's reversibility and was condemned to be less efficient.

Though formulated in terms of caloric (see the obsolete caloric theory), ratherthan entropy, this was an early insight into the second law.

Clausius theorem

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

The equality holds in the reversible case[7] and the '<' is in the irreversible case.


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The reversible case is used to introduce the state function entropy. This isbecause in cyclic processes the variation of a state function is zero.

Thermodynamic temperature

Main article: Thermodynamic temperature

For 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 sameheat reservoirs are equally efficient. Thus, any reversible heat engine operatingbetween temperatures T1 and T2 must have the same efficiency, that is to say, the

effiency is the function of temperatures only:

In addition, a reversible heat engine operating between temperatures T1 and T3must have the same efficiency as one consisting of two cycles, one between T1and another (intermediate) temperature T2, and the second between T2 andT3.This can only be the case if

Now consider the case where T1 is a fixed reference temperature: thetemperature of the triple point of water. Then for 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 anyreference temperature and any positive numerical value could be used—the


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choice here corresponds to the Kelvin scale.)

Entropy

Main article: entropy (classical thermodynamics)

According 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 aboveformula. To obtain the absolute value, 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 alwaysconnect the initial and terminal status with an imaginary reversible process andintegrating on that path to calculate the difference in entropy.

Now reverse the reversible process and combine it with the said irreversibleprocess. Applying Clausius inequality on this loop,

Thus,

where the equality holds if the transformation is reversible.

Notice that if the process is an adiabatic process, then δQ = 0, so .

Available useful work

See also: Available useful work (thermodynamics)


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An important and revealing idealized special case is to consider applying theSecond Law to the scenario of an isolated system (called the total system oruniverse), made up of two parts: a sub-system of interest, and the sub-system'ssurroundings. These surroundings are imagined to be so large that they can beconsidered as an unlimited heat reservoir at temperature TR and pressure PR —so that no matter how much heat is transferred to (or from) the sub-system, thetemperature of the surroundings will remain TR; and no matter how much thevolume of the sub-system expands (or contracts), the pressure of thesurroundings will remain PR.

Whatever changes to dS and dSR occur in the entropies of the sub-system and thesurroundings individually, according to the Second Law the entropy Stot of theisolated total system must not decrease:

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

where μiR are the chemical potentials of chemical species in the externalsurroundings.

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 statistical thermodynamics, the relation between entropychange, temperature and absorbed heat can be derived); and then the SecondLaw inequality from above.

It therefore follows that any net work δw done by the sub-system must obey

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


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

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

i.e. the change in the subsystem's exergy plus the useful work done by thesubsystem (or, the change in the subsystem's exergy less any work, additional tothat done by the pressure reservoir, done on the system) must be less than orequal to zero.

In sum, if a proper infinite-reservoir-like reference state is chosen as the systemsurroundings in the real world, then the Second Law predicts a decrease in X foran irreversible process and no change for a reversible process.

Is equivalent to

This expression together with the associated reference state permits a designengineer working at the macroscopic scale (above the thermodynamic limit) toutilize the Second Law without directly measuring or considering entropy changein a total isolated system. (Also, see process engineer). Those changes havealready been considered by the assumption that the system under considerationcan reach equilibrium with the reference state without altering the referencestate. An efficiency for a process or collection of processes that compares it to thereversible ideal may also be found (See second law efficiency.)

This approach to the Second Law is widely utilized in engineering practice,environmental accounting, systems ecology, and other disciplines.to the formalenergy created may not be supportive but it can be created from one form toanother form ..it can be also called as conservation of energy .,

History

See also: History of entropy

The first theory of the conversion of heat into mechanical work is due to NicolasLéonard Sadi Carnot in 1824. He was the first to realize correctly that theefficiency of this conversion depends on the difference of temperature between anengine and its environment.

Recognizing the significance of James Prescott Joule's work on the conservation of


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energy, Rudolf Clausius was the first to formulate the second law during 1850, inthis form: heat does not flow spontaneously from cold to hot bodies. Whilecommon knowledge now, this was contrary to the caloric theory of heat popular atthe time, which considered heat as a fluid. From there he was able to infer theprinciple of Sadi Carnot and the definition of entropy (1865).

Established during the 19th century, the Kelvin-Planck statement of the SecondLaw says, "It is impossible for any device that operates on a cycle to receive heatfrom a single reservoir and produce a net amount of work." This was shown to beequivalent to the statement of Clausius.

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 ofmicrostates with the same energy is proportional to the volume of this region, i.e.that all accessible microstates are equally probable over a long period of time.Equivalently, it says that time average and average over the statistical ensembleare the same.

It has been shown that not only classical systems but also quantum mechanicalones tend to maximize their entropy over time. Thus the second law follows, giveninitial conditions with low entropy. More precisely, it has been shown that thelocal von Neumann entropy is at its maximum value with a very highprobability.[8] The result is valid for a large class of isolated quantum systems(e.g. a gas in a container). While the full system is pure and therefore does nothave any entropy, the entanglement between gas and container gives rise to anincrease of the local entropy of the gas. This result is one of the most importantachievements of quantum thermodynamics.

Today, much effort in the field is attempting to understand why the initialconditions early in the universe were those of low entropy,[9][10] as this is seen asthe origin of the second law (see below).

Informal descriptions

The second law can be stated in various succinct ways, including:

It is impossible to produce work in the surroundings using a cyclicprocess connected to a single heat reservoir (Kelvin, 1851).It is impossible to carry out a cyclic process using an engine connected totwo heat reservoirs that will have as its only effect the transfer of aquantity of heat from the low-temperature reservoir to thehigh-temperature reservoir (Clausius, 1854).If thermodynamic work is to be done at a finite rate, free energy must beexpended.[11]


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Mathematical descriptions

In 1856, the German physicist Rudolf Clausius stated what he called the "secondfundamental theorem in the mechanical theory of heat" in the following form:[12]

where Q is heat, T is temperature and N is the "equivalence-value" of alluncompensated transformations involved in a cyclical process. Later, in 1865,Clausius would come to define "equivalence-value" as entropy. On the heels of thisdefinition, that same year, the most famous version of the second law was read ina presentation at the Philosophical Society of Zurich on April 24, in which, in theend 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 tothe general broadness of the terminology used here, e.g. universe, as well as lackof specific conditions, e.g. open, closed, or isolated, to which this statementapplies, many people take this simple statement to mean that the second law ofthermodynamics applies virtually to every subject imaginable. This, of course, isnot true; this statement is only a simplified version of a more complex description.

In terms of time variation, the mathematical statement of the second law for anisolated system undergoing an arbitrary transformation is:

where

S is the entropy andt is time.

Statistical mechanics gives an explanation for the second law by postulating thata material is composed of atoms and molecules which are in constant motion. Aparticular set of positions and velocities for each particle in the system is called amicrostate of the system and because of the constant motion, the system isconstantly changing its microstate. Statistical mechanics postulates that, inequilibrium, 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 thesecond law must hold in a statistical sense. That is, the second law will hold onaverage, with a statistical variation on the order of 1/√N where N is the number ofparticles in the system. For everyday (macroscopic) situations, the probabilitythat the second law will be violated is practically zero. However, for systems with


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a small number of particles, thermodynamic parameters, including the entropy,may show significant statistical deviations from that predicted by the second law.Classical thermodynamic theory does not deal with these statistical variations.

Derivation from statistical mechanics

In statistical mechanics,the Second Law is not a postulate, rather it is aconsequence of the fundamental postulate, also known as the equal priorprobability postulate, so long as one is clear that simple probability argumentsare applied only to the future, while for the past there are auxiliary sources ofinformation which tell us that it was low entropy. The first part of the second law,which states that the entropy of a thermally isolated system can only increase is atrivial consequence of the equal prior probability postulate, if we restrict thenotion of the entropy to systems in thermal equilibrium. The entropy of anisolated system in thermal equilibrium containing an amount of energy of E is:

where is the number of quantum states in a small interval between E andE + δE. Here δE is a macroscopically small energy interval that is kept fixed.Strictly speaking this means that the entropy depends on the choice of δE.However, in the thermodynamic limit (i.e. in the limit of infinitely large systemsize), the specific entropy (entropy per unit volume or per unit mass) does notdepend on δE.

Suppose we have an isolated system whose macroscopic state is specified by anumber of variables. These macroscopic variables can, e.g., refer to the totalvolume, the positions of pistons in the system, etc. Then Ω will depend on thevalues of these variables. If a variable is not fixed, (e.g. we do not clamp a pistonin a certain position), then because all the accessible states are equally likely inequilibrium, the free variable in equilibrium will be such that Ω is maximized asthat is the most probable situation in equilibrium.

If the variable was initially fixed to some value then upon release and when thenew equilibrium has been reached, the fact the variable will adjust itself so that Ωis maximized, implies that the entropy will have increased or it will have stayedthe same (if the value at which the variable was fixed happened to be theequilibrium value).

The entropy of a system that is not in equilibrium can be defined as:

see here. Here the Pj is the probabilities for the system to be found in the states


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labeled by the subscript j. In thermal equilibrium the probabilities for statesinside the energy interval δE are all equal to 1 / Ω, and in that case the generaldefinition coincides with the previous definition of S that applies to the case ofthermal equilibrium.

Suppose we start from an equilibrium situation and we suddenly remove aconstraint on a variable. Then right after we do this, there are a number Ω ofaccessible microstates, but equilibrium has not yet been reached, so the actualprobabilities of the system being in some accessible state are not yet equal to theprior probability of 1 / Ω. We have already seen that in the final equilibrium state,the entropy will have increased or have stayed the same relative to the previousequilibrium state. Boltzmann's H-theorem, however, proves that the entropy willincrease continuously as a function of time during the intermediate out ofequilibrium state.

Derivation of the entropy change for reversible processes

The second part of the Second Law states that the entropy change of a systemundergoing a reversible process is given by:

where the temperature is defined as:

See here for the justification for this definition. Suppose that the system has someexternal parameter, x, that can be changed. In general, the energy eigenstates ofthe system will depend on x. According to the adiabatic theorem of quantummechanics, in the limit of an infinitely slow change of the system's Hamiltonian,the system will stay in the same energy eigenstate and thus change its energyaccording to the change in energy of the energy eigenstate it is in.

The generalized force, X, corresponding to the external variable x is defined suchthat Xdx is the work performed by the system if x is increased by an amount dx.E.g., if x is the volume, then X is the pressure. The generalized force for a systemknown to be in energy eigenstate Er is given by:

Since the system can be in any energy eigenstate within an interval of δE, wedefine the generalized force for the system as the expectation value of the aboveexpression:


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To evaluate the average, we partition the energy eigenstates by counting

how many of them have a value for within a range between Y and Y + δY.

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 asfollows. Suppose we change x to x + dx. Then will change because theenergy eigenstates depend on x, causing energy eigenstates to move into or out ofthe range between E and E + δE. Let's focus again on the energy eigenstates for

which lies within the range between Y and Y + δY. Since these energy

eigenstates increase in energy by Y dx, all such energy eigenstates that are in theinterval 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 intothe range between E and E + δE and contribute to an increase in Ω. Thenumber of energy eigenstates that move from below E + δE to above E + δE is,of course, given by . The difference

is thus the net contribution to the increase in Ω. Note that if Y dx is larger thanδE there will be the energy eigenstates that move from below E to aboveE + δE. They are counted in both and , therefore the aboveexpression is also valid in that case.

Expressing the above expression as a derivative w.r.t. E and summing over Yyields the expression:


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

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

Combining this with

Gives:

Derivation for systems described by the canonical ensemble

If a system is in thermal contact with a heat bath at some temperature T then, inequilibrium, the probability distribution over the energy eigenvalues are given bythe canonical ensemble:

Here Z is a factor that normalizes the sum of all the probabilities to 1, thisfunction is known as the partition function. We now consider an infinitesimalreversible change in the temperature and in the external parameters on which theenergy levels depend. It follows from the general formula for the entropy:

that


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

General derivation from unitarity of quantum mechanics

The time development operator in quantum theory is unitary, because theHamiltonian is hermitian. Consequently the transition probability matrix is doublystochastic, which implies the Second Law of Thermodynamics.[13][14] Thisderivation is quite general, based on the Shannon entropy, and does not requireany assumptions beyond unitarity, which is universally accepted. It is aconsequence of the irreversibility or singular nature of the general transitionmatrix.

Non-equilibrium states

Statistically it is possible for a system to achieve moments of non-equilibrium. Insuch statistically unlikely events where hot particles "steal" the energy of coldparticles enough that the cold side gets colder and the hot side gets hotter, for aninstant. Such events have been observed at a small enough scale where thelikelihood of such a thing happening is significant.[15] The physics involved insuch an event is described by the fluctuation theorem.

Controversies

Maxwell's demon

Main article: Maxwell's demon

James Clerk Maxwell imagined one container divided into two parts, A and B.Both parts are filled with the same gas at equal temperatures and placed next toeach other. Observing the molecules on both sides, an imaginary demon guards atrapdoor between the two parts. When a faster-than-average molecule from A fliestowards the trapdoor, the demon opens it, and the molecule will fly from A to B.The average speed of the molecules in B will have increased while in A they willhave slowed down on average. Since average molecular speed corresponds totemperature, the temperature decreases in A and increases in B, contrary to thesecond law of thermodynamics.

One of the most famous responses to this question was suggested in 1929 by Leó


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Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell'sdemon would need to have some means of measuring molecular speed, and thatthe act of acquiring information would require an expenditure of energy. But laterexceptions were found.

Loschmidt's paradox

Loschmidt's paradox, also known as the reversibility paradox, is the objection thatit should not be possible to deduce an irreversible process from time-symmetricdynamics. This puts the time reversal symmetry of (almost) all known low-levelfundamental physical processes at odds with any attempt to infer from them thesecond law of thermodynamics which describes the behavior of macroscopicsystems. Both of these are well-accepted principles in physics, with soundobservational and theoretical support, yet they seem to be in conflict; hence theparadox.

One approach to handling Loschmidt's paradox is the fluctuation theorem, provedby Denis Evans and Debra Searles, which gives a numerical estimate of theprobability that a system away from equilibrium will have a certain change inentropy over a certain amount of time. The theorem is proved with the exact timereversible dynamical equations of motion and the Axiom of Causality. Thefluctuation theorem is proved utilizing the fact that dynamics is time reversible.Quantitative predictions of this theorem have been confirmed in laboratoryexperiments at the Australian National University conducted by Edith M. Sevicket al. using optical tweezers apparatus.

Gibbs paradox

Main article: Gibbs paradox

In statistical mechanics, a simple derivation of the entropy of an ideal gas basedon the Boltzmann distribution yields an expression for the entropy which is notextensive (is not proportional to the amount of gas in question). This leads to anapparent paradox known as the Gibbs paradox, allowing, for instance, the entropyof closed systems to decrease, violating the second law of thermodynamics.

The paradox is averted by recognizing that the identity of the particles does notinfluence the entropy. In the conventional explanation, this is associated with anindistinguishability of the particles associated with quantum mechanics. However,a growing number of papers now take the perspective that it is merely thedefinition of entropy that is changed to ignore particle permutation (and therebyavert the paradox). The resulting equation for the entropy (of a classical ideal gas)is extensive, and is known as the Sackur-Tetrode equation.

Poincaré recurrence theorem


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The Poincaré recurrence theorem states that certain systems will, after asufficiently long time, return to a state very close to the initial state. The Poincarérecurrence time is the length of time elapsed until the recurrence, which is of theorder of .[16] The result applies to physical systems in which energy isconserved. The Recurrence theorem apparently contradicts the Second law ofthermodynamics, which says that large dynamical systems evolve irreversiblytowards the state with higher entropy, so that if one starts with a low-entropystate, the system will never return to it. There are many possible ways to resolvethis paradox, but none of them is universally accepted[citation needed]. The mosttypical argument is that for thermodynamical systems like an ideal gas in a box,recurrence time is so large that for all practical purposes it is infinite.

Heat death of the universe

Main article: Heat death of the universe

According to the second law, the entropy of any isolated system, such as theentire universe, never decreases. If the entropy of the universe has a maximumupper bound then when this bound is reached the universe has no thermodynamicfree energy to sustain motion or life, that is, the heat death is reached.

Quotes

The law that entropy always increases holds, I think, the supremeposition among the laws of Nature. If someone points out to you thatyour pet theory of the universe is in disagreement with Maxwell'sequations — then so much the worse for Maxwell's equations. If it isfound to be contradicted by observation — well, these experimentalistsdo bungle things sometimes. But if your theory is found to be againstthe second law of thermodynamics I can give you no hope; there isnothing 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 inthe second law of thermodynamics — perhaps the most pessimistic andamoral 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 havebeen discussions of it.

—Philosopher / Physicist P.W. Bridgman, (1941)

Notes


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^ Translated from German original: Es gibt keine Zustandsänderung, deren einzigesErgebnis die Übertragung von Wärme von einem Körper niederer auf einen Körperhöherer Temperatur ist.

1.

See also

Clausius–Duhem inequalityConstructal lawEntropyEntropy (arrow of time)Entropy: A New World View[book]First law of thermodynamicsHistory of thermodynamicsSecond-law efficiency

Jarzynski equalityLaws of thermodynamicsMaximum entropythermodynamicsReflections on the MotivePower of Fire [book]Statistical mechanicsThermal diodeRelativistic heat conduction

References

^ A. Bejan, (2006). 'Advanced Engineering Thermodynamics', Wiley. ISBN0-471-67763-9

1.

^ 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."

2.

^ a b E. H. Lieb, J. Yngvason (1999). "The Physics and Mathematics of the SecondLaw of Thermodynamics". Physics Reports 310: 1–96. arXiv:cond-mat/9708200(http://arxiv.org/abs/cond-mat/9708200) . Bibcode 1999PhR...310....1L(http://adsabs.harvard.edu/abs/1999PhR...310....1L) .doi:10.1016/S0370-1573(98)00082-9 (http://dx.doi.org/10.1016%2FS0370-1573%2898%2900082-9) .

3.

^ a b c "Concept and Statements of the Second Law" (http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html) . web.mit.edu. http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html. Retrieved 2010-10-07.

4.

^ C. Caratheodory (1909). "Untersuchungen über die Grundlagen derThermodynamik". Mathematische Annalen 67: 363. "Axiom II: In jeder beliebigenUmgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände,die durch adiabatische Zustandsänderungen nicht beliebig approximiert werdenkönnen."

5.

^ H.A. Buchdahl (1966). The Concepts of Classical Thermodynamics. CambridgeUniversity Press. p. 68.

6.

^ Clausius theorem (http://scienceworld.wolfram.com/physics/ClausiusTheorem.html) at Wolfram Research

7.

^ Gemmer, Jochen; Otte, Alexander; Mahler, Günter (2001). "Quantum Approach to aDerivation of the Second Law of Thermodynamics". Physical Review Letters 86 (10):1927–1930. arXiv:quant-ph/0101140 (http://arxiv.org/abs/quant-ph/0101140) .Bibcode 2001PhRvL..86.1927G (http://adsabs.harvard.edu/abs/2001PhRvL..86.1927G)

8.


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. doi:10.1103/PhysRevLett.86.1927 (http://dx.doi.org/10.1103%2FPhysRevLett.86.1927) . PMID 11289822 (http://www.ncbi.nlm.nih.gov/pubmed/11289822)^ Carroll; Jennifer Chen (2335). "Does Inflation Provide Natural Initial Conditions forthe Universe?". Gen.Rel.Grav. () ; International Journal of Modern Physics D D14 ()-2340 37 (2005): 1671–1674. arXiv:gr-qc/0505037 (http://arxiv.org/abs/gr-qc/0505037) . Bibcode 2005GReGr..37.1671C (http://adsabs.harvard.edu/abs/2005GReGr..37.1671C) . doi:10.1007/s10714-005-0148-2 (http://dx.doi.org/10.1007%2Fs10714-005-0148-2) .

9.

^ Wald, R (2006). "The arrow of time and the initial conditions of the universe".Studies in History and Philosophy of Science Part B: Studies in History andPhilosophy of Modern Physics 37 (3): 394–398. doi:10.1016/j.shpsb.2006.03.005(http://dx.doi.org/10.1016%2Fj.shpsb.2006.03.005) .

10.

^ Stoner (2000). "Inquiries into the Nature of Free Energy and Entropy in Respect toBiochemical Thermodynamics". Entropy 2 (3): 106–141. arXiv:physics/0004055(http://arxiv.org/abs/physics/0004055) . Bibcode 2000Entrp...2..106S(http://adsabs.harvard.edu/abs/2000Entrp...2..106S) . doi:10.3390/e2030106(http://dx.doi.org/10.3390%2Fe2030106) .

11.

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

12.

^ 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, andthe statement on page 29 that "it is known that the [Shannon] entropy [...] is amonotone increasing function of the time."

13.

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

14.

^ Wang, G.; Sevick, E.; Mittag, Emil; Searles, Debra; Evans, Denis (2002)."Experimental Demonstration of Violations of the Second Law of Thermodynamics forSmall Systems and Short Time Scales". Physical Review Letters 89 (5). Bibcode2002PhRvL..89e0601W (http://adsabs.harvard.edu/abs/2002PhRvL..89e0601W) .doi:10.1103/PhysRevLett.89.050601 (http://dx.doi.org/10.1103%2FPhysRevLett.89.050601) .

15.

^ L. Dyson, J. Lindesay and L. Susskind, Is There Really a de Sitter/CFT Duality, JHEP0208, 45 (2002)

16.

Further reading

Goldstein, Martin, and Inge F., 1993. The Refrigerator and the Universe.Harvard Univ. Press. Chpts. 4-9 contain an introduction to the SecondLaw, one a bit less technical than this entry. ISBN 978-0-674-75324-2Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. Maxwell's Demon 2 :Entropy, classical and quantum information, computing. Bristol UK;


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Philadelphia PA: Institute of Physics. ISBN 978-0-585-49237-7Halliwell, 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(http://arxiv.org/abs/1001.4571) [physics.gen-ph (http://arxiv.org/archive/physics.gen-ph) ].Carnot, Sadi; Thurston, Robert Henry (editor and translator) (1890).Reflections on the Motive Power of Heat and on Machines Fitted toDevelop 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 LawrenceSklar.Second law of thermodynamics (http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html) in the MIT Course UnifiedThermodinamics and Propulsion (http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html) from Prof. Z. S. SpakovszkyE.T. Jaynes, 1988, "The evolution of Carnot's principle,(http://bayes.wustl.edu/etj/articles/ccarnot.pdf) " in G. J. Erickson and C.R. Smith (eds.)Maximum-Entropy and Bayesian Methods in Science andEngineering, Vol 1, p. 267.Website devoted to the Second Law. (http://www.secondlaw.com/)

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