second law of thermodynamics - wikipedia, the free encyclopedia

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Second law of thermodynamicsrom Wikipedia, the free encyclopedia

he secondlaw of thermodynamicsstates that in a natural thermodynamic process, there is an increase in theum of the entropies of the participating systems.

he secondlaw is an empirical finding that has been accepted as an axiom of thermodynamic theory.

he law defines the concept of thermodynamic entropy for a thermodynamic system in its own state of internalermodynamic equilibrium. It considers a process in which that state changes, with increases in entropy due tossipation of energy and to dispersal of matter and energy.

he law envisages a compound thermodynamic system that initially has interior walls that constrain transfersithin it. The law then envisages that a process is initiated by a thermodynamic operation that changes those

onstraints, and isolates the compound system from its surroundings, except that an externally imposednchanging force field is allowed to stay subject to the condition that the compound system moves as a wholeithin that field so that in net, there is no transfer of energy as work between the compou nd system and the

urroundings, and finally, eventually, the system is stationary within that field.

uring the process, there may occur chemical reactions, and transfers of matter and of energy. In eachdiabatically separated compartment, the temperature becomes spatially homogeneous, even in the presence ofe externally imposed unchanging external force field. If, between two adiabatically separated compartments,ansfer of energy as work is possible, then it proceeds until the sum of the entropies of the equilibratedompartments is maximum subject to the other constraints. If the externally imposed force field is zero, then thehemical concentrations also become as spatially homogeneous as is allowed by the permeabilities of the interioralls, and by the possibilities of phase separations, which occur so as to maximize the sum of the entropies of the

quilibrated phases subject to the other constraints. Such homogeneity and phase separation is characteristic of

e state of internal thermodynamic equilibrium of a thermodynamic system.[1][2]If the externally imposed force

eld is non-zero, then the chemical concentrations spatially redistribute themselves so as to maximize the sum ofe equilibrated entropies subject to the other constraints and phase separations.

tatistical thermodynamics, classical or quantum, explains the law.

he second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadiarnot in 1824 (see Timeline of thermodynamics).

Contents1 Introduction

2 Various statements of the law

2.1 Carnot's principle

2.2 Clausius statement

2.3 Kelvin statement

2.4 Equivalence of the Clausius and the Kelvin statements

2.5 Planck's proposition2.6 Relation between Kelvin's statement and Planck's proposition

2.7 Planck's statement
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2.8 Principle of Carathodory

2.9 Planck's Principle

2.10 Statement for a system that has a known expression of its internal energy as a function of its

extensive state variables

2.11 Gravitational systems

3 Corollaries

3.1 Perpetual motion of the second kind

3.2 Carnot theorem

3.3 Clausius Inequality

3.4 Thermodynamic temperature

3.5 Entropy

3.6 Energy, available useful work

4 History

4.1 Informal descriptions

4.2 Mathematical descriptions5 Derivation from statistical mechanics

5.1 Derivation of the entropy change for reversible processes

5.2 Derivation for systems described by the canonical ensemble

5.3 General derivation from unitarity of quantum mechanics

6 Non-equilibrium states

7 Arrow of time

8 Controversies

8.1 Maxwell's demon

8.2 Loschmidt's paradox

8.3 Poincar recurrence theorem

9 Quotations

10 See also

11 References

11.1 Bibliography of citations

12 Further reading13 External links

ntroduction

he first law of thermodynamics provides the basic definition of thermodynamic energy, also called internalnergy, associated with all thermodynamic systems, but unknown in classical mechanics, and states the rule of

onservation of energy in nature.[3][4]

he concept of energy in the first law does not, however, account for the observation that natural processes havepreferred direction of progress. The first law is symmetrical with respect to the initial and final states of an

volving system. But the second law asserts that a natural process runs only in one sense, and is not reversible.
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or example, heat always flows spontaneously from hotter to colder bodies, and never the reverse, unlessxternal work is performed on the system. The key concept for the explanation of this phenomenon through the

econd law of thermodynamics is the definition of a new physical quantity, the entropy.[5][6]

or mathematical analysis of processes, entropy is introduced as follows. In a fictive reversible process, anfinitesimal increment in the entropy (dS) of a system results from an infinitesimal transfer of heat (Q) to aosed system divided by the common temperature (T) of the system and the surroundings which supply the

eat.[7]

he zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation ofermal equilibrium have the same temperature, especially that a test body has the same temperature as a

ference thermometric body.[8]For a body in thermal equilibrium with another, there are indefinitely manympirical temperature scales, in general respectively depending on the properties of a particular referenceermometric body. The second law allows a distinguished temperature scale, which defines an absolute,

ermodynamic temperature, independent of the properties of any particular reference thermometric body.[9][10]

Various statements of the law

he second law of thermodynamics may be expressed in many specific ways,[11]the most prominent classical

atements[12]being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and theatement in axiomatic thermodynamics by Constantin Carathodory (1909). These statements cast the law ineneral physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have

een shown to be equivalent.[13]

arnot's principle

he historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of aarnot engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat

nd work transfers are between subsystems that are always in their own internal states of thermodynamicquilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with thefficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heatas seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical

xpression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the

econd law of thermodynamics, and remains valid today. It states

The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of thetwo heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated inthis way is the most efficient possible heat engine using those two

temperatures.[14][15][16][17][18][19][20]

lausius statement

he German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by

xamining the relation between heat transfer and work.[21]His formulation of the second law, which wasublished in German in 1854, is known as the Clausius statement:
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Derive Kelvin Statement from Clausius Statement

Heat can never pass from a colder to a warmer body without some other change, connected

therewith, occurring at the same time.[22]

he statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, thiseans 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.

eat cannot spontaneously flow from cold regions to hot regions without external work being performed on the

ystem, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flowsom cold to hot, but only when forced by an external agent, the refrigeration system.

Kelvin statement

ord Kelvin expressed the second law as

It is impossible, by means of inanimate material agency, to derive mechanical effect from any

portion of matter by cooling it below the temperature of the coldest of the surrounding objects.[23]

quivalence of the Clausius and the Kelvin statements

uppose there is an engine violating the Kelvin statement:e., one that drains heat and converts it completely into work

a cyclic fashion without any other result. Now pair it withreversed Carnot engine as shown by the figure. The net and

ole effect of this newly created engine consisting of the twongines mentioned is transferring heat

from the cooler reservoir to the hotter

ne, which violates the Clausius statement. Thus a violationf the Kelvin statement implies a violation of the Clausiusatement, i.e. the Clausius statement implies the Kelvinatement. We can prove in a similar manner that the Kelvinatement implies the Clausius statement, and hence the twoe equivalent.

lanck's proposition

lanck offered the following proposition as derived directly from experience. This is sometimes regarded as hisatement of the second law, but he regarded it as a starting point for the derivation of the second law.

It is impossible to construct an engine which will work in a complete cycle, and produce no effect

except the raising of a weight and cooling of a heat reservoir.[24][25]

Relation between Kelvin's statement and Planck's proposition

is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law. For example, see. [26]

ne text gives a statement that for all the world looks like Planck's proposition, but attributes it to Kelvin without

ention of Planck.[27]One monograph quotes Planck's proposition as the "Kelvin-Planck" formulation, the text
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aming Kelvin as its author, though it correctly cites Planck in its references. [28]The reader may compare thewo statements quoted just above here.

lanck's statement

lanck stated the second law as follows.

Every process occurring in nature proceeds in the sense in which the sum of the entropies of all

bodies taking part in the process is increased. In the limit,i.e.for reversible processes, the sum ofthe entropies remains unchanged.[29][30][31]

rinciple of Carathodory

onstantin Carathodory formulated thermodynamics on a purely mathematical axiomatic foundation. His

atement of the second law is known as the Principle of Carathodory, which may be formulated as follows: [32]

In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessiblefrom S.[33]

With this formulation, he described the concept of adiabatic accessibility for the first time and provided theundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It followsom Carathodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process

unction, in other words, .[34]

hough it is almost customary in textbooks to say that Carathodory's principle expresses the second law and to

eat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all theontent of the second law, Carathodory's principle needs to be supplemented by Planck's principle, thatochoric work always increases the internal energy of a closed system that was initially in its own internal

ermodynamic equilibrium.[35][36][37][38]

lanck's Principle

n 1926, Max Planck wrote an important paper on the basics of thermodynamics.[37][39]He indicated the principle

The internal energy of a closed system is increased by an adiabatic process, throughout the durationof which, the volume of the system remains constant.[35][36]

his formulation does not mention heat and does not mention temperature, nor even entropy, and does notecessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related

atement is that "Frictional pressure never does positive work."[40]Using a now obsolete form of words, Planck

mself wrote: "The production of heat by friction is irreversible."[41][42]

ot mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the

elvin statement given just above.[43]Nevertheless, this principle of Planck is not actually Planck's preferredatement of the second law, which is quoted above, in a previous sub-section of the present section of thisresent article, and relies on the concept of entropy.
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he link to Kelvin's statement is illustrated by an equivalent statement by Allahverdyan & Nieuwenhuizen,hich they attribute to Kelvin: "No work can be extracted from a closed equilibrium system during a cyclic

ariation of a parameter by an external source."[44][45]

tatement for a system that has a known expression of its internal energy as a function of

s extensive state variables

he second law has been shown to be equivalent to the internal energy Ubeing a weakly convex function, when

ritten as a function of extensive properties (mass, volume, entropy, ...).[46][47]

Gravitational systems

n non-gravitational systems, objects always have positive heat capacity, meaning that the temperature rises withnergy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the sourcemperature is decreased while the sink temperature is increased hence temperature differences tend to diminishver time.

owever, this is not always the case for systems in which the gravitational force is important. The most striking

xamples are black holes, which according to theory have negative heat capacity. The larger the black hole,e more energy it contains, but the lower its temperature. Thus, the supermassive black hole in the center of the

Milky Way is supposed to have a temperature of 1014K, much lower than the cosmic microwave backgroundmperature of 2.7K, but as it absorbs photons of the cosmic microwave background its mass is increasing so thats low temperature further decreases with time.

or this reason, gravitational systems tend towards non-even distribution of mass and energy. The universe inrge scale is importantly a gravitational system, and the second law may therefore not apply to it.

Corollaries

erpetual motion of the second kind

efore the establishment of the Second Law, many people who were interested in inventing a perpetual motionachine had tried to circumvent the restrictions of First Law of Thermodynamics by extracting the massiveternal energy of the environment as the power of the machine. Such a machine is called a "perpetual motionachine of the second kind". The second law declared the impossibility of such machines.

arnot theorem

arnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiencyolely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theoremates:

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.

n his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of theycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated thatome caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot
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ycle's reversibility and was condemned to be less efficient.

hough formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an earlysight into the second law.

lausius Inequality

he Clausius Theorem (1854) states that in a cyclic process

he equality holds in the reversible case[48]and the '



nd the reference temperature T1will have the value 273.16. (Of course any reference temperature and any

ositive numerical value could be usedthe choice here corresponds to the Kelvin scale.)

ntropy

ccording to the Clausius equality, for a reversible process

hat means the line integral is path independent.

o 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 absolutealue, we need the Third Law of Thermodynamics, which states that S=0 at absolute zero for perfect crystals.

or any irreversible process, since entropy is a state function, we can always connect the initial and terminalates with an imaginary reversible process and integrating on that path to calculate the difference in entropy.

ow reverse the reversible process and combine it with the said irreversible process. Applying Clausius

equality on this loop,

hus,

here the equality holds if the transformation is reversible.otice that if the process is an adiabatic process, then , so .

nergy, available useful work

n important and revealing idealized special case is to consider applying the Second Law to the scenario of anolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-

ystem's surroundings. These surroundings are imagined to be so large that they can be considered as annlimitedheat reservoir at temperature TRand pressurePR so that no matter how much heat is transferred to

r from) the sub-system, the temperature of the surroundings will remain TR and no matter how much theolume of the sub-system expands (or contracts), the pressure of the surroundings will remain PR.
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Whatever changes to dSand dSRoccur in the entropies of the sub-system and the surroundings individually,

ccording to the Second Law the entropy Stotof the isolated total system must not decrease:

ccording to the First Law of Thermodynamics, the change dUin the internal energy of the sub-system is theum of the heat qadded to the sub-system, lessany work wdone bythe sub-system,plusany net chemicalnergy entering the sub-system d iRNi, so that:

here iRare the chemical potentials of chemical species in the external surroundings.

ow the heat leaving the reservoir and entering the sub-system is

here we have first used the definition of entropy in classical thermodynamics (alternatively, in statisticalermodynamics, the relation between entropy change, temperature and absorbed heat can be derived) and thene Second Law inequality from above.

therefore follows that any net work wdone by the sub-system must obey

is useful to separate the work wdone by the subsystem into the usefulwork wuthat can be done bythe sub-

ystem, over and beyond the workpRdVdone merely by the sub-system expanding against the surrounding

xternal pressure, giving the following relation for the useful work (exergy) that can be done:

is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called thevailabilityor exergyEof the subsystem,

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

e. the change in the subsystem's exergy plus the useful work done bythe subsystem (or, the change in theubsystem's exergy less any work, additional to that done by the pressure reservoir, done onthe system) must bess than or equal to zero.

n sum, if a proper infinite-reservoir-likereference state is chosen as the system surroundings in the real world,

en the Second Law predicts a decrease inEfor an irreversible process and no change for a reversible process.

Is equivalent to
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Nicolas Lonard Sadi Carnotin the traditional uniform ofa student of the colePolytechnique.

his expression together with the associated reference state permits a design engineer working at theacroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or

onsidering entropy change in a total isolated system. (Also, see process engineer). Those changes have alreadyeen considered by the assumption that the system under consideration can reach equilibrium with the referenceate without altering the reference state. An efficiency for a process or collection of processes that compares it toe reversible ideal may also be found (See second law efficiency.)

his approach to the Second Law is widely utilized in engineering practice, environmental accounting, systemscology, and other disciplines.

History

he first theory of the conversion of heat into mechanical work is due to Nicolasonard Sadi Carnot in 1824. He was the first to realize correctly that thefficiency of this conversion depends on the difference of temperature between anngine and its environment.

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

f energy, Rudolf Clausius was the first to formulate the second law during 1850,this form: heat does not flowspontaneouslyfrom cold to hot bodies. Whileommon knowledge now, this was contrary to the caloric theory of heat popular

the time, which considered heat as a fluid. From there he was able to infer therinciple of Sadi Carnot and the definition of entropy (1865).

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

he ergodic hypothesis is also important for the Boltzmann approach. It says that,ver long periods of time, the time spent in some region of the phase space of microstates with the same energyproportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long

eriod of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

has been shown that not only classical systems but also quantum mechanical ones tend to maximize theirntropy over time. Thus the second law follows, given initial conditions with low entropy. More precisely, it has

een shown that the local von Neumann entropy is at its maximum value with a very high probability.[49]Thesult is valid for a large class of isolated quantum systems (e.g. a gas in a container). While the full system is

ure and therefore does not have any entropy, the entanglement between gas and container gives rise to ancrease of the local entropy of the gas. This result is one of the most important achievements of quantumermodynamics.

oday, much effort in the field is attempting to understand why the initial conditions early in the universe were

ose of low entropy,[50][51]as this is seen as the origin of the second law (see below).

nformal descriptions

he 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).
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Rudolf Clausius

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. (Stoner, 2000) [52]

Mathematical descriptions

n 1856, the German physicist Rudolf Clausius stated what he called the "secondundamental theorem in the mechanical theory of heat" in the following form:[53]

here Qis heat, Tis temperature andNis the "equivalence-value" of allncompensated transformations involved in a cyclical process. Later, in 1865,lausius would come to define "equivalence-value" as entropy. On the heels ofis definition, that same year, the most famous version of the second law was

ad in a presentation at the Philosophical Society of Zurich on April 24, inhich, in the end of his presentation, Clausius concludes:

The entropy of the universe tends to a maximum.

his statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g.niverse, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple

atement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This, ofourse, is not true this statement is only a simplified version of a more extended and precise description.

n terms of time variation, the mathematical statement of the second law for an isolated system undergoing anbitrary transformation is:

here

Sis the entropy of the system and

tis time.

he equality sign holds in the case that only reversible processes take place inside the system. If irreversiblerocesses take place (which is the case in real systems in operation) the >-sign holds. An alternative way ofrmulating of the second law for isolated systems is:

with

ith the sum of the rate of entropy production by all processes inside the system. The advantage of thisrmulation is that it shows the effect of the entropy production. The rate of entropy production is a very

mportant concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient
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mperature it gives the so-called dissipated energy .

he expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but notxchange of matter) is:

with

ere

is the heat flow into the system

is the temperature at the point where the heat enters the system.

heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.

or open systems (also allowing exchange of matter):

with

ere is the flow of entropy into the system associated with the flow of matter entering the system. It shouldot be confused with the time derivative of the entropy. If matter is supplied at several places we have to take thegebraic sum of these contributions.

tatistical mechanics gives an explanation for the second law by postulating that a material is composed of atomsnd molecules which are in constant motion. A particular set of positions and velocities for each particle in theystem is called a microstate of the system and because of the constant motion, the system is constantly changings microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is

qually likely to occur, and when this assumption is made, it leads directly to the conclusion that the second lawust hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the

rder of 1/N whereNis the number of particles in the system. For everyday (macroscopic) situations, therobability that the second law will be violated is practically zero. However, for systems with a small number ofarticles, thermodynamic parameters, including the entropy, may show significant statistical deviations from thatredicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.

Derivation from statistical mechanics

ue to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past,amely that the system is uncorrelated at some time in the past this allows for simple probabilistic treatment.his assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequencef the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though

her scenarios have also been suggested.[54][55][56]

iven these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence ofe fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple

robability arguments are applied only to the future, while for the past there are auxiliary sources of informationhich tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermallyolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restricte notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal

quilibrium containing an amount of energy of is:
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here is the number of quantum states in a small interval between and . Here is aacroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends one choice of . However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the

pecific entropy (entropy per unit volume or per unit mass) does not depend on .

uppose we have an isolated system whose macroscopic state is specified by a number of variables. Theseacroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then will

epend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certainosition), then because all the accessible states are equally likely in equilibrium, the free variable in equilibriumill be such that is maximized as that is the most probable situation in equilibrium.

the variable was initially fixed to some value then upon release and when the new equilibrium has beenached, the fact the variable will adjust itself so that is maximized, implies that the entropy will havecreased or it will have stayed the same (if the value at which the variable was fixed happened to be the

quilibrium value). Suppose we start from an equilibrium situation and we suddenly remove a constraint on aariable. Then right after we do this, there are a number of accessible microstates, but equilibrium has not yet

een reached, so the actual probabilities of the system being in some accessible state are not yet equal to the priorrobability of . We have already seen that in the final equilibrium state, the entropy will have increased orave stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that theuantityHincreases monotonically as a function of time during the intermediate out of equilibrium state.

Derivation of the entropy change for reversible processes

he second part of the Second Law states that the entropy change of a system undergoing a reversible process isven by:

here the temperature is defined as:

ee here for the justification for this definition. Suppose that the system has some external parameter, x, that cane changed. In general, the energy eigenstates of the system will depend on x. According to the adiabaticeorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the systemill stay in the same energy eigenstate and thus change its energy according to the change in energy of the

nergy eigenstate it is in.

he generalized force, X, corresponding to the external variable x is defined such that is the workerformed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. Theeneralized force for a system known to be in energy eigenstate is given by:

ince the system can be in any energy eigenstate within an interval of , we define the generalized force fore system as the expectation value of the above expression:
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o evaluate the average, we partition the energy eigenstates by counting how many of them have a value

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

he 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 xx + dx. Then will change because the energy eigenstates depend on x, causing energy eigenstates to

ove 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,

l such energy eigenstates that are in the interval ranging from E Y dx to E move from below E to above E.here are

uch energy eigenstates. If , all these energy eigenstates will move into the range between and andontribute to an increase in . The number of energy eigenstates that move from below to above

is, of course, given by . The difference

thus the net contribution to the increase in . Note that if Y dx is larger than there will be the energygenstates that move from below E to above . They are counted in both and , therefore the above

xpression is also valid in that case.

xpressing the above expression as a derivative w.r.t. E and summing over Y yields the expression:

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

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

ombining this with



ives:

Derivation for systems described by the canonical ensemble

a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probabilitystribution over the energy eigenvalues are given by the canonical ensemble:

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

at

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

General derivation from unitarity of quantum mechanics

he time development operator in quantum theory is unitary, because the Hamiltonian is hermitian.onsequently, the transition probability matrix is doubly stochastic, which implies the Second Law ofhermodynamics.[57][58]This derivation is quite general, based on the Shannon entropy, and does not require anysumptions beyond unitarity, which is universally accepted. It is a consequenceof the irreversibility or singular

ature of the general transition matrix.

Non-equilibrium states

he theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often notven explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. Ineneral, a region of space containing a physical system at a given time, that may be found in nature, is not inermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or

as ever been truly in exact thermodynamic equilibrium.[59][60]

or purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamicquilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, itan often be very valuable and useful because it makes available the theory of thermodynamics. Elements of thequilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and thatere are so many particles in a system, that its particulate nature can be entirely ignored. Under such an

quilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception,e case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratoryudies of critical states, exceptionally long observation times are needed.
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James Clerk Maxwell

n all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putativeandidate "fluctuation" alters the entropy of the system.

can easily happen that a physical system exhibits internal macroscopic changes that are fast enough tovalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the

articulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to

e abandoned. There is no unqualified general definition of entropy for non-equilibrium states.[61]

on-equilibrium thermodynamics is then appropriate. There are intermediate cases, in which the assumption of

cal thermodynamic equilibrium is a very good approximation,[62][63][64][65]but strictly speaking it is still anpproximation, not theoretically ideal. For non-equilibrium situations in general, it may be useful to consideratistical mechanical definitions of quantities that may be conveniently called 'entropy'. These indeed belong toatistical mechanics, not to macroscopic thermodynamics.

he physics of macroscopically observable fluctuations is beyond the scope of this article.

Arrow of time

he second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction.

he second law has been proposed to supply an explanation of the difference between moving forward and

ackwards in time, such as why the cause precedes the effect (the causal arrow of time).[66]

Controversies

Maxwell's demon

ames Clerk Maxwell imagined one container divided into two parts,AandB.oth parts are filled with the same gas at equal temperatures and placed next to

ach other. Observing the molecules on both sides, an imaginary demon guards aapdoor between the two parts. When a faster-than-average molecule fromAies towards the trapdoor, the demon opens it, and the molecule will fly from A

B. The average speed of the molecules inBwill have increased while inAtheyill have slowed down on average. Since average molecular speed corresponds tomperature, the temperature decreases inAand increases inB, contrary to the

econd law of thermodynamics.

ne of the most famous responses to this question was suggested in 1929 by Lezilrd and later by Lon Brillouin. Szilrd pointed out that a real-life Maxwell'semon would need to have some means of measuring molecular speed, and thate act of acquiring information would require an expenditure of energy.

Maxwell's demon repeatedly alters the permeability of the wall betweenAandB.is therefore performing thermodynamic operations, not just presiding over natural processes.

oschmidt's paradox

oschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible toeduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of nearly allnown low-level fundamental physical processes at odds with any attempt to infer from them the second law of
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Wikiquote has quotationsrelated to:Second law ofthermodynamics

ermodynamics which describes the behavior of macroscopic systems. Both of these are well-acceptedrinciples in physics, with sound observational and theoretical support, yet they seem to be in conflict hence thearadox.

ne proposed resolution of this paradox is as follows. The Loschmidt scenario refers to a strictly isolated systemr to a strictly adiabatically isolated system. Heat and matter transfers are not allowed. The Loschmidt reversalmes are fantastically long, far longer than any laboratory isolation of the required degree of perfection could beaintained in practice. In this sense, the Loschmidt scenario will never be subjected to empirical testing. Also inis sense, the second law, stated for an isolated system, will never be subjected to empirical testing. A system,

upposedly perfectly isolated, in strictly perfect thermodynamic equilibrium, can be observed only once in itsntire life, because the observation must break the isolation. Two observations would be needed to checkmpirically for a change of state, one initial and one final. When transfer of heat or matter are permitted, thequirements of perfection are not so tight. In practical laboratory reality, therefore, the second law can be tested

nly for systems with transfer of heat or matter, and not for isolated systems.

ue to this paradox, derivations of the second law have to make an assumption regarding the past, namely thate system is uncorrelated at some time in the past or, equivalently, that the entropy in the past was lower than ine future. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately

erived from the initial conditions of the Big Bang.[54][67]

oincar recurrence theorem

he Poincar recurrence theorem states that certain systems will, after a sufficiently long time, return to a stateery close to the initial state. The Poincar recurrence time is the length of time elapsed until the recurrence,

hich is of the order of .[68]The result applies to physical systems in which energy is conserved. The Recurrenceeorem apparently contradicts the Second law of thermodynamics, which says that large dynamical systems

volve irreversibly towards the state with higher entropy, so that if one starts with a low-entropy state, the systemill never return to it. There are many possible ways to resolve this paradox, but none of them is universally

ccepted. The most reasonable argument is that for typical thermodynamical systems the recurrence time is sorge (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannotbserve the recurrence.

Quotations

The law that entropy always increases holds, I think, the supremeposition among the laws of Nature. If someone points out to you

that your pet theory of the universe is in disagreement withMaxwell's equations then so much the worse for Maxwell'sequations. If it is found to be contradicted by observation well,these experimentalists do bungle things sometimes. But if yourtheory is found to be against the second law of thermodynamics Ican give you no hope there is nothing for it but to collapse indeepest humiliation.

Sir Arthur Stanley Eddington, The Nature of the Physical

World(1927)

There have been nearly as many formulations of the second law as there have been discussions of it.

Philosopher / Physicist P.W. Bridgman, (1941)
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Clausius is the author of the sibyllic utterance, "The energy of the universe is constant the entropyof the universe tends to a maximum." The objectives of continuum thermomechanics stop far shortof explaining the "universe", but within that theory we may easily derive an explicit statement insome ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finitesize.

Truesdell, C., Muncaster, R.G. (1980).Fundamentals of Maxwell's Kinetic Theory of a Simple

Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York,

ISBN0-12-701350-4, p.17.

ee also

References

ClausiusDuhem inequality

Entropy: A New World View

Fluctuation theorem

History of thermodynamicsJarzynski equality

Laws of thermodynamics

Maximum entropy thermodynamics

Reflections on the Motive Power of Fire

Thermal diode

Relativistic heat conduction

1. ^Planck, M. (1897/1903), p. 3.

2. ^Bailyn, M. (1994), Section 71, pp. 254256.

3. ^Planck, M. (1897/1903), pp. 4041.

4. ^Munster A. (1970), pp. 89, 5051.

5. ^Planck, M. (1897/1903), pp. 79107.

6. ^Bailyn, M. (1994), Section 71, pp. 113154.

7. ^Bailyn, M. (1994), p. 120.

8. ^J. S. Dugdale (1996).Entropy and its Physical Meaning. Taylor & Francis. p. 13. ISBN 0-7484-0569-0. "This law is

the basis of temperature."

9. ^Zemansky, M.W. (1968), pp. 207209.

10. ^Quinn, T.J. (1983), p. 8.

11. ^"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.

12. ^Lieb & Yngvason (1999).

13. ^Rao (2004), p. 213.14. ^Carnot, S. (1824/1986).

15. ^Truesdell, C. (1980), Chapter 5.

16. ^Adkins, C.J. (1968/1983), pp. 5658.
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. ns er, . , p. .

18. ^Kondepudi, D., Prigogine, I. (1998), pp.6775.

19. ^Lebon, G., Jou, D., Casas-Vzquez, J. (2008), p. 10.

20. Êu, B.C. (2002), pp. 3235.

21. ^Clausius (1850).

22. ^Clausius (1854), p. 86.

23. ^Thomson (1851).

24. ^Planck, M. (1897/1903), p. 86.

25. ^Roberts, J.K., Miller, A.R. (1928/1960), p. 319.26. ^ter Haar, D., Wergeland, H. (1966), p. 17.

27. ^Pippard, A.B. (1957/1966), p. 30.

28. ^pek, V., Sheehan, D.P. (2005), p. 3

29. ^Planck, M. (1897/1903), p. 100.

30. ^Planck, M. (1926), p. 463, translation by Uffink, J. (2003), p. 131.

31. ^Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not

cite Planck. This source calls the statement the principle of the increase of entropy.

32. ^Carathodory, C. (1909).33. ^Buchdahl, H.A. (1966), p. 68.

34. ^Sychev, V. V. (1991). The Differential Equations of Thermodynamics (http://www.amazon.com/The-Differential-

Equations-Of-Thermodynamics/dp/1560321210/ref=sr_1_4?ie=UTF8&qid=1353986248&sr=8-4&keywords=Sychev).

Taylor & Francis. ISBN 978-1560321217. Retrieved 2012-11-26.

35. ^ abMnster, A. (1970), p. 45.

36. ^ abLieb & Yngvason (1999), p. 49.

37. ^ abPlanck, M. (1926).

38. ^Buchdahl, H.A. (1966), p. 69.

39. Ûffink, J. (2003), pp. 129132.

40. ^Truesdell, C., Muncaster, R.G. (1980).Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas,

Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN0-12-701350-4, p. 15.

41. ^Planck, M. (1897/1903), p. 81.

42. ^Planck, M. (1926), p. 457, Wikipedia editor's translation.

43. ^Lieb, E.H., Yngvason, J. (2003), p. 149.

44. Âllahverdyan, A.E., Nieuwenhuizen, T.H. (2001). A mathematical theorem as the basis for the second law:

Thomson's formulation applied to equilibrium, http://arxiv.org/abs/cond-mat/0110422

45. ^pek, V., Sheehan, D.P. (2005), p. 1146. ^van Gool, W. Bruggink, J.J.C. (Eds) (1985).Energy and time in the economic and physical sciences. North-

Holland. pp. 4156. ISBN 0444877487.

47. ^Grubbstrm, Robert W. (2007). "An Attempt to Introduce Dynamics Into Generalised Exergy Considerations".

Applied Energy84: 701718. doi:10.1016/j.apenergy.2007.01.003

(https://dx.doi.org/10.1016%2Fj.apenergy.2007.01.003).

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

49. ^Gemmer, Jochen Otte, Alexander Mahler, Gnter (2001). "Quantum Approach to a Derivation of the Second Law of

Thermodynamics". Physical Review Letters86 (10): 19271930. arXiv:quant-ph/0101140 (https://arxiv.org/abs/quant-ph/0101140). Bibcode:2001PhRvL..86.1927G (http://adsabs.harvard.edu/abs/2001PhRvL..86.1927G).

doi:10.1103/PhysRevLett.86.1927 (https://dx.doi.org/10.1103%2FPhysRevLett.86.1927). PMID 11289822

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^ " "
http://en.wikipedia.org/wiki/Max_Planckhttp://en.wikipedia.org/wiki/Max_Planckhttp://en.wikipedia.org/wiki/PubMed_Identifierhttp://en.wikipedia.org/wiki/Special:BookSources/978-1560321217http://dx.doi.org/10.1016%2Fj.apenergy.2007.01.003http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0444877487http://en.wikipedia.org/wiki/Bibcodehttp://arxiv.org/abs/cond-mat/0110422http://en.wikipedia.org/wiki/Brian_Pippardhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://scienceworld.wolfram.com/physics/ClausiusTheorem.htmlhttp://adsabs.harvard.edu/abs/2001PhRvL..86.1927Ghttp://arxiv.org/abs/quant-ph/0101140http://en.wikipedia.org/wiki/Harald_Wergelandhttp://en.wikipedia.org/wiki/Wolfram_Researchhttp://en.wikipedia.org/wiki/Max_Planckhttp://en.wikipedia.org/wiki/Ilya_Prigoginehttp://en.wikipedia.org/wiki/Max_Planckhttp://en.wikipedia.org/wiki/Dirk_ter_Haarhttp://en.wikipedia.org/wiki/ArXivhttp://en.wikipedia.org/wiki/Constantin_Carath%C3%A9odoryhttp://www.amazon.com/The-Differential-Equations-Of-Thermodynamics/dp/1560321210/ref=sr_1_4?ie=UTF8&qid=1353986248&sr=8-4&keywords=Sychevhttp://en.wikipedia.org/wiki/Max_Planckhttp://en.wikipedia.org/wiki/Clifford_Truesdellhttp://en.wikipedia.org/wiki/Physical_Review_Lettershttp://www.ncbi.nlm.nih.gov/pubmed/11289822http://dx.doi.org/10.1103%2FPhysRevLett.86.1927http://en.wikipedia.org/wiki/Max_Planckhttp://en.wikipedia.org/wiki/Digital_object_identifier


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(https://dx.doi.org/10.1007%2Fs10714-005-0148-2).

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Science Part B: Studies in History and Philosophy of Modern Physics37(3): 394398.

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55. ^Greene, Brian (2004). The Fabric of the Cosmos. Alfred A. Knopf. p. 171. ISBN 0-375-41288-3.

56. ^Lebowitz, Joel L. (September 1993). "Boltzmann's Entropy and Time's Arrow"

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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."

58. ^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 ofthe Universal Wavefunction, pp 3140.

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60. ^Callen, H.B. (1960/1985), p. 15.

61. ^Lieb, E.H., Yngvason, J. (2003), p. 190.

62. ^Gyarmati, I. (1967/1970), pp. 4-14.

63. ^Glansdorff, P., Prigogine, I. (1971).

64. ^Mller, I. (1985).

65. ^Mller, I. (2003).66. ^Halliwell, J.J. et al. (1994).Physical Origins of Time Asymmetry. Cambridge. ISBN 0-521-56837-4. chapter 6

67. ^Greene, Brian (2004). The Fabric of the Cosmos. Alfred A. Knopf. p. 161. ISBN 0-375-41288-3.

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

Adkins, C.J. (1968/1983).Equilibrium Thermodynamics, (1st edition 1968), third edition 1983, Cambridge University

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Bailyn, M. (1994).A Survey of Thermodynamics, American Institute of Physics, New York, ISBN 0-88318-797-3.

Buchdahl, H.A. (1966). The Concepts of Classical Thermodynamics, Cambridge University Press, Cambridge UK.
http://arxiv.org/abs/gr-qc/0505037http://en.wikipedia.org/wiki/Special:BookSources/0883187973http://en.wikipedia.org/wiki/Special:BookSources/069108131Xhttp://en.wikipedia.org/wiki/Bibcodehttp://arxiv.org/abs/gr-qc/0505037http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Brian_Greenehttp://en.wikipedia.org/wiki/Princeton_University_Presshttp://en.wikipedia.org/wiki/Special:BookSources/0-375-41288-3http://dx.doi.org/10.1007%2Fs10714-005-0148-2http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz_Prizehttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdfhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.3390%2Fe2030106http://adsabs.harvard.edu/abs/1985PhRvD..32.2489Hhttp://en.wikipedia.org/wiki/Special:BookSources/0521254450http://en.wikipedia.org/wiki/Bibcodehttp://adsabs.harvard.edu/abs/2000Entrp...2..106Shttp://en.wikipedia.org/wiki/JHEPhttp://en.wikipedia.org/w/index.php?title=R._Neill_Graham&action=edit&redlink=1http://en.wikipedia.org/wiki/Bryce_Seligman_DeWitthttp://adsabs.harvard.edu/abs/1993PhT....46i..32Lhttp://en.wikipedia.org/wiki/Hugh_Everetthttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdfhttp://dx.doi.org/10.1103%2FPhysRevD.32.2489http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/International_Journal_of_Modern_Physics_Dhttp://en.wikipedia.org/wiki/Special:BookSources/9780199662760http://en.wikipedia.org/wiki/Special:BookSources/0-375-41288-3http://en.wikipedia.org/wiki/ArXivhttp://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Brian_Greenehttp://dx.doi.org/10.1063%2F1.881363http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://dx.doi.org/10.1016%2Fj.shpsb.2006.03.005http://prd.aps.org/abstract/PRD/v32/i10/p2489_1http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz_Prizehttp://adsabs.harvard.edu/abs/2005GReGr..37.1671Chttp://en.wikipedia.org/wiki/ArXivhttp://arxiv.org/abs/physics/0004055http://en.wikipedia.org/wiki/Herbert_Callenhttp://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Special:BookSources/0-521-56837-4



Callen, H.B. (1960/1985). Thermodynamics and an Introduction to Thermostatistics, (1st edition 1960) 2nd edition

1985, Wiley, New York, ISBN 0-471-86256-8.

pek, V., Sheehan, D.P. (2005). Challenges to the Second Law of Thermodynamics: Theory and Experiment,

Springer, Dordrecht, ISBN 1-4020-3015-0.

C. Carathodory (1909). "Untersuchungen ber die Grundlagen der Thermodynamik" (http://gdz.sub.uni-

goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1).Mathematische

Annalen67: 355386. doi:10.1007/bf01450409 (https://dx.doi.org/10.1007%2Fbf01450409). "Axiom II: In jeder

beliebigen Umgebung eines willkrlich vorgeschriebenen Anfangszustandes gibt es Zustnde, die durch adiabatische

Zustandsnderungen nicht beliebig approximiert werden knnen. (p.363)". A translation may be found here (http://neo-

classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf). Also a mostly reliable translation

is to be found (http://books.google.com.au/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations) at

Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.

Carnot, S. (1824/1986).Reflections on the motive power of fire(http://www.worldcat.org/title/reflections-on-the-

motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-

robert/oclc/812944517&referer=brief_results), Manchester University Press, Manchester UK, ISBN 0719017416. Also

here. (http://www.archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up)

Clausius, R. (1850). "Ueber Die Bewegende Kraft Der Wrme Und Die Gesetze, Welche Sich Daraus Fr DieWrmelehre Selbst Ableiten Lassen" (http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image).Annalen der Physik

79: 368397, 500524. doi:10.1002/andp.18501550403 (https://dx.doi.org/10.1002%2Fandp.18501550403). Retrieved

26 June 2012. Translated into English: Clausius, R. (July 1851). "On the Moving Force of Heat, and the Laws

regarding the Nature of Heat itself which are deducible therefrom"

(https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up). London, Edinburgh and Dublin Philosophical

Magazine and Journal of Science. 4th 2(VIII): 121 102119. Retrieved 26 June 2012.

Clausius, R. (1854). "ber eine vernderte Form des zweiten Hauptsatzes der mechanischen Wrmetheorie"

(http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf). Annalen

der Physik(Poggendoff). xciii: 481. doi:10.1002/andp.18541691202

(https://dx.doi.org/10.1002%2Fandp.18541691202). Retrieved 24 March 2014. Translated into English: Clausius, R.

(July 1856). "On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat"

(http://www.biodiversitylibrary.org/item/20044#page/100/mode/1up). London, Edinburgh and Dublin Philosophical

Magazine and Journal of Science. 4th 2: 86. Retrieved 24 March 2014. Reprinted in: Clausius, R. (1867). The

Mechanical Theory of Heat with its Applications to the Steam Engine and to Physical Properties of Bodies

(http://books.google.com/books?

id=8LIEAAAAYAAJ&printsec=frontcover&dq=editions:PwR_Sbkwa8IC&hl=en&sa=X&ei=h6DgT5WnF46e8gSVvby

nDQ&ved=0CDYQuwUwAA#v=onepage&q&f=false). London: John van Voorst. Retrieved 19 June 2012.Eu, B.C. (2002). Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized

Hydrodynamics, Kluwer Academic Publishers, Dordrecht, ISBN 1402007884.

Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability, and Fluctuations, Wiley-

Interscience, London, 1971, ISBN 0-471-30280-5.

Grandy, W.T., Jr (2008).Entropy and the Time Evolution of Macroscopic Systems

(http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?

cc=au&lang=en&). Oxford University Press. ISBN 978-0-19-954617-6.

Greven, A., Keller, G., Warnecke (editors) (2003). Entropy, Princeton University Press, Princeton NJ, ISBN 0-691-11338-6.

Gyarmati, I. (1967/1970)Non-equilibrium Thermodynamics. Field Theory and Variational Principles, translated by E.

Gyarmati and W.F. Heinz, Springer, New York.

Konde udi D. Pri o ine I. 1998 .Modern Thermod namics: From Heat En ines to Dissi ative Structures John
http://en.wikipedia.org/wiki/Herbert_Callenhttp://www.archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2uphttp://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdfhttp://en.wikipedia.org/wiki/Nicolas_L%C3%A9onard_Sadi_Carnothttp://en.wikipedia.org/wiki/Rudolf_Clausiushttp://dx.doi.org/10.1002%2Fandp.18541691202http://en.wikipedia.org/wiki/Special:BookSources/9780199546176https://archive.org/stream/londonedinburghd02lond#page/1/mode/1uphttp://www.biodiversitylibrary.org/item/20044#page/100/mode/1uphttp://en.wikipedia.org/wiki/Special:BookSources/1402030150http://en.wikipedia.org/wiki/Special:BookSources/0471302805http://books.google.com.au/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundationshttp://en.wikipedia.org/wiki/Special:BookSources/0691113386http://books.google.com/books?id=8LIEAAAAYAAJ&printsec=frontcover&dq=editions:PwR_Sbkwa8IC&hl=en&sa=X&ei=h6DgT5WnF46e8gSVvbynDQ&ved=0CDYQuwUwAA#v=onepage&q&f=falsehttp://en.wikipedia.org/wiki/Ilya_Prigoginehttp://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_resultshttp://en.wikipedia.org/wiki/Constantin_Carath%C3%A9odoryhttp://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdfhttp://en.wikipedia.org/wiki/Special:BookSources/0471862568http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en&http://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1007%2Fbf01450409http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Rudolf_Clausiushttp://en.wikipedia.org/wiki/Special:BookSources/0719017416http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.imagehttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1002%2Fandp.18501550403http://en.wikipedia.org/wiki/Herbert_Callenhttp://en.wikipedia.org/wiki/Rudolf_Clausius



Further reading

Goldstein, Martin, and Inge F., 1993. The Refrigerator and the Universe. Harvard Univ. Press. Chpts. 49

Wiley & Sons, Chichester, ISBN 0471973939.

Lebon, G., Jou, D., Casas-Vzquez, J. (2008). Understanding Non-equilibrium Thermodynamics: Foundations,

Applications, Frontiers, Springer-Verlag, Berlin, e-ISBN 978-3-540-74252-4.

Lieb, E. H. Yngvason, J. (1999). "The Physics and Mathematics of the Second Law of Thermodynamics"

(http://arxiv.org/pdf/cond-mat/9708200v2.pdf).Physics Reports310: 196. arXiv:cond-mat/9708200

(https://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

(https://dx.doi.org/10.1016%2FS0370-1573%2898%2900082-9). Retrieved 24 March 2014.Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp. 147195, Chapter 8 ofEntropy,

Greven, A., Keller, G., Warnecke (editors) (2003).

Mller, I. (1985). Thermodynamics, Pitman, London, ISBN 0-273-08577-8.

Mller, I. (2003). Entropy in Nonequilibrium, pp. 79109, Chapter 5 of Entropy, Greven, A., Keller, G., Warnecke

(editors) (2003).

Mnster, A. (1970), Classical Thermodynamics, translated by E.S. Halberstadt, WileyInterscience, London, ISBN 0-

471-62430-6.

Pippard, A.B. (1957/1966).Elements of Classical Thermodynamics for Advanced Students of Physics, original

publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.

Planck, M. (1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans Green, London, p. 100.

(https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up)

Planck, M. (1926). ber die Begrndung des zweiten Hauptsatzes der Thermodynamik, Sitzungsberichte der

Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse: 453463.

Quinn, T.J. (1983). Temperature, Academic Press, London, ISBN 0-12-569680-9.

Rao, Y.V.C. (2004).An Introduction to thermodynamics (http://books.google.com/books?

id=iYWiCXziWsEC&pg=PA213). Universities Press. p. 213. ISBN 978-81-7371-461-0.

Roberts, J.K., Miller, A.R. (1928/1960).Heat and Thermodynamics, (first edition 1928), fifth edition, Blackie & SonLimited, Glasgow.

ter Haar, D., Wergeland, H. (1966).Elements of Thermodynamics, Addison-Wesley Publishing, Reading MA.

Thomson, W. (1851). "On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent

of a Thermal Unit, and M. Regnault's Observations on Steam"

(http://www.biodiversitylibrary.org/item/126047#page/295/mode/1up). Transactions of the Royal Society of Edinburgh

XX(part II): 261268 289298. Also published in Thomson, W. (December 1852). "On the Dynamical Theory of

Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on

Steam" (https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up).Philos. Mag. 4 IV(22): 13.

Retrieved 25 June 2012.

Truesdell, C. (1980). The Tragicomical History of Thermodynamics 18221854, Springer, New York, ISBN 0387

904034.

Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 ofEntropy, Greven, A., Keller,

G., Warnecke (editors) (2003).

Zemansky, M.W. (1968).Heat and Thermodynamics. An Intermediate Textbook, fifth edition, McGraw-Hill Book

Company, New York.
http://en.wikipedia.org/wiki/Special:BookSources/9783540742524http://en.wikipedia.org/wiki/Max_Planckhttp://dx.doi.org/10.1016%2FS0370-1573%2898%2900082-9http://en.wikipedia.org/wiki/Mark_Zemanskyhttp://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvinhttp://en.wikipedia.org/wiki/Bibcodehttp://books.google.com/books?id=iYWiCXziWsEC&pg=PA213http://en.wikipedia.org/wiki/Special:BookSources/0273085778http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz_Prizehttp://en.wikipedia.org/wiki/Dirk_ter_Haarhttp://en.wikipedia.org/wiki/Harald_Wergelandhttp://adsabs.harvard.edu/abs/1999PhR...310....1Lhttp://en.wikipedia.org/wiki/Special:BookSources/0471624306http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Ilya_Prigoginehttp://en.wikipedia.org/wiki/Brian_Pippardhttp://arxiv.org/abs/cond-mat/9708200http://en.wikipedia.org/wiki/Special:BookSources/0125696809http://arxiv.org/pdf/cond-mat/9708200v2.pdfhttps://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2uphttps://archive.org/stream/londonedinburghp04maga#page/12/mode/2uphttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Clifford_Truesdellhttp://en.wikipedia.org/wiki/ArXivhttp://www.biodiversitylibrary.org/item/126047#page/295/mode/1uphttp://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz_Prizehttp://en.wikipedia.org/wiki/Special:BookSources/978-81-7371-461-0http://en.wikipedia.org/wiki/Max_Planck



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. (1994).Physical Origins of Time Asymmetry. Cambridge. ISBN 0-521-56837-4.(technical).

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/))

Stephen Jay Kline (1999). The Low-Down on Entropy and Interpretive Thermodynamics, La Caada, CA:

DCW Industries. ISBN 1928729010.

Kostic, M.,Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi

Carnot's Ingenious Reasoning to Holistic GeneralizationAIP Conf. Proc. 1411, pp. 327350 doi:

http://dx.doi.org/10.1063/1.3665247. American Institute of Physics, 2011. ISBN 978-0-7354-0985-9.

Abstract at: [1] (http://adsabs.harvard.edu/abs/2011AIPC.1411..327K). Full article (24 pages [2](http://scitation.aip.org/getpdf/servlet/GetPDFServlet?

filetype=pdf&id=APCPCS001411000001000327000001&idtype=cvips&doi=10.1063/1.3665247&prog=n

ormal&bypassSSO=1)), also at [3]

(http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degrad

ation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-

4.pdf).

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 Thermodynamics 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)" in

G. J. Erickson and C. R. Smith (eds.)Maximum-Entropy and Bayesian Methods in Science and

Engineering, Vol 1, p. 267.

Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich

(http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf)

etrieved from "http://en.wikipedia.org/w/index.php?title=Second_law_of_thermodynamics&oldid=645805496"

ategories: Concepts in physics Laws of thermodynamics Non-equilibrium thermodynamics
http://en.wikipedia.org/wiki/Special:BookSources/9780674753242http://en.wikipedia.org/wiki/Special:BookSources/9780674753242http://plato.stanford.edu/entries/statphys-statmech/http://en.wikipedia.org/w/index.php?title=Second_law_of_thermodynamics&oldid=645805496http://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophyhttp://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdfhttp://en.wikipedia.org/wiki/Special:BookSources/9780735409859http://en.wikipedia.org/wiki/Special:BookSources/0-521-56837-4http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.htmlhttp://bayes.wustl.edu/etj/articles/ccarnot.pdfhttp://adsabs.harvard.edu/abs/2011AIPC.1411..327Khttp://en.wikipedia.org/wiki/Category:Laws_of_thermodynamicshttp://books.google.com/books?id=tgdJAAAAIAAJhttp://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.htmlhttp://en.wikipedia.org/wiki/Category:Non-equilibrium_thermodynamicshttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/E.T._Jayneshttp://www.history.rochester.edu/steam/carnot/1943/http://en.wikipedia.org/wiki/Special:BookSources/9780585492377http://en.wikipedia.org/wiki/Category:Concepts_in_physicshttp://en.wikipedia.org/wiki/Robert_Henry_Thurstonhttp://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdfhttp://en.wikipedia.org/wiki/Special:BookSources/1928729010http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=APCPCS001411000001000327000001&idtype=cvips&doi=10.1063/1.3665247&prog=normal&bypassSSO=1http://en.wikipedia.org/wiki/Help:Categoryhttp://en.wikipedia.org/wiki/Institute_of_Physicshttp://dx.doi.org/10.1063/1.3665247


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