Chapter 1:Entropy and the Second Law 9 other versions of entropy to be used.It is noteworthy that,in commenting on the possible limitations of his formulation,Planck explicitly mentions the perpetuum mobile.Evidently,even as thermodynamics begins to mature,the specter of the perpetuum mobile lurks in the background. Gibbs takes a different tack to the second law by avoiding thermodynamic processes,and instead conjoins entropy with equilibrium 25,6: (17)Gibbs For the equilibrium of an isolated system,it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy,the variation of its entropy shall either vanish or be negative. In other words,thermodynamic equilibrium for an isolated system is the state of maximum entropy.Although Gibbs does not refer to this as a statement of the second law,per se,this marimum entropy principle conveys its essential content. The maximum entropy principle [26]has been broadly applied in the sciences,en- gineering economics,information theory-wherever the second law is germane, and even beyond.It has been used to reformulate classical and quantum sta- tistical mechanics 26,27.For instance,starting from it one can derive on the back of an envelope the continuous or discrete Maxwell-Boltzmann distributions, the Planck blackbody radiation formula(and,with suitable approximations,the Rayleigh-Jeans and Wien radiation laws)[24]. Some recent authors have adopted more definitional entropy-based versions [9]: (18)Entropy Properties Every thermodynamic system has two properties (and perhaps others):an intensive one,absolute temper- ature T,that may vary spatially and temporally in the system T(z,t); and an extensive one,entropy S.Together they satisfy the following three conditions: (i)The entropy change dS during time interval dt is the sum of:(a) entropy flow through the boundary of the system deS;and(b)entropy production within the system,diS;that is,ds=des+diS. (ii)Heat fux (not matter fux)through a boundary at uniform tem- perature T results in entropy change des=s. (iii)For reversible processes within the system,dis=0,while for irreversible processes,diS>0. This version is a starting point for some approaches to irreversible thermodynam- ics. While there is no agreement in the scientific community about how best to state the second law,there is general agreement that the current melange of statements, taken en masse,pretty well covers it.This,of course,gives fits to mathematicians, who insist on precision and parsimony.Truesdell [28,6 leads the charge:
Chapter 1: Entropy and the Second Law 9 other versions of entropy to be used. It is noteworthy that, in commenting on the possible limitations of his formulation, Planck explicitly mentions the perpetuum mobile. Evidently, even as thermodynamics begins to mature, the specter of the perpetuum mobile lurks in the background. Gibbs takes a different tack to the second law by avoiding thermodynamic processes, and instead conjoins entropy with equilibrium [25, 6]: (17) Gibbs For the equilibrium of an isolated system, it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy, the variation of its entropy shall either vanish or be negative. In other words, thermodynamic equilibrium for an isolated system is the state of maximum entropy. Although Gibbs does not refer to this as a statement of the second law, per se, this maximum entropy principle conveys its essential content. The maximum entropy principle [26] has been broadly applied in the sciences, engineering economics, information theory — wherever the second law is germane, and even beyond. It has been used to reformulate classical and quantum statistical mechanics [26, 27]. For instance, starting from it one can derive on the back of an envelope the continuous or discrete Maxwell-Boltzmann distributions, the Planck blackbody radiation formula (and, with suitable approximations, the Rayleigh-Jeans and Wien radiation laws) [24]. Some recent authors have adopted more definitional entropy-based versions [9]: (18) Entropy Properties Every thermodynamic system has two properties (and perhaps others): an intensive one, absolute temperature T, that may vary spatially and temporally in the system T(x, t); and an extensive one, entropy S. Together they satisfy the following three conditions: (i) The entropy change dS during time interval dt is the sum of: (a) entropy flow through the boundary of the system deS; and (b) entropy production within the system, diS; that is, dS = deS + diS. (ii) Heat flux (not matter flux) through a boundary at uniform temperature T results in entropy change deS = δQ T . (iii) For reversible processes within the system, diS = 0, while for irreversible processes, diS > 0. This version is a starting point for some approaches to irreversible thermodynamics. While there is no agreement in the scientific community about how best to state the second law, there is general agreement that the current melange of statements, taken en masse, pretty well covers it. This, of course, gives fits to mathematicians, who insist on precision and parsimony. Truesdell [28, 6] leads the charge:
10 Challenges to the Second Law Clausius'verbal statement of the second law makes no sense....All that remains is a Mosaic prohibition;a century of philosophers and journal- ists have acclaimed this commandment;a century of mathematicians have shuddered and averted their eyes from the unclean. Arnold broadens this assessment [29,6: Every mathematician knows it is impossible to understand an elemen- tary course in thermodynamics. In fact,mathematicians have labored to drain this "dismal swamp of obscurity" [28,beginning with Caratheodory [30]and culminating with the recent tour de force by Lieb and Yngvason [31].While both are exemplars of mathematical rigor and logic,both suffer from incomplete generality and questionable applicability to realistic physical systems;in other words,there are doubts about their empirical content. Caratheodory was the first to apply mathematical rigor to thermodynamics [30.He imagines a state space I of all possible equilibrium states of a generic system.I is an n-dimensional manifold with continuous variables and Euclidean topology.Given two arbitrary states s and t,if s can be transformed into t by an adiabatic process,then they satisfy adiabatically accessibility condition,written s <t,and read s precedes t.This is similar to Lieb and Yngvason [31],except that Lieb and Yngvason allow sets of possibly disjoint ordered states,whereas Caratheodory assumes continuous state space and variables.Max Born's simplified version of Caratheodory's second law reads 32: (19a)Caratheodory (Born Version):In every neighborhood of each state (s)there are states (t)that are inaccessible by means of adiabatic changes of state.Symbolically,this is: (付s∈T,∀Us):t∈Us犬t, (1.6) where Us and Ut are open neighborhoods surrounding the states s and t. Caratheodory's originally published version is more precise [30,6]. (19b)Caratheodory Principle In every open neighborhood Us CI of an arbitrarily chosen state s there are states t such that for some open neighborhood Ut of t:all states r within U:cannot be reached adiabatically from s.Symbolically this is: ∀s∈TVUs3t∈Us&3 Ut CUVr∈U:s求r. (1.7) Lieb and Yngvason 31]proceed along similar lines,but work with an set of distinct states,rather than a continuous space of them.For them,the second law is a theorem arising out of the ordering of the states via adiabatic accessibility. Details can be found in 81.3
10 Challenges to the Second Law Clausius’ verbal statement of the second law makes no sense.... All that remains is a Mosaic prohibition; a century of philosophers and journalists have acclaimed this commandment; a century of mathematicians have shuddered and averted their eyes from the unclean. Arnold broadens this assessment [29, 6]: Every mathematician knows it is impossible to understand an elementary course in thermodynamics. In fact, mathematicians have labored to drain this “dismal swamp of obscurity” [28], beginning with Carath´eodory [30] and culminating with the recent tour de force by Lieb and Yngvason [31]. While both are exemplars of mathematical rigor and logic, both suffer from incomplete generality and questionable applicability to realistic physical systems; in other words, there are doubts about their empirical content. Carath´eodory was the first to apply mathematical rigor to thermodynamics [30]. He imagines a state space Γ of all possible equilibrium states of a generic system. Γ is an n-dimensional manifold with continuous variables and Euclidean topology. Given two arbitrary states s and t, if s can be transformed into t by an adiabatic process, then they satisfy adiabatically accessibility condition, written s ≺ t, and read s precedes t. This is similar to Lieb and Yngvason [31], except that Lieb and Yngvason allow sets of possibly disjoint ordered states, whereas Carath´eodory assumes continuous state space and variables. Max Born’s simplified version of Carath´eodory’s second law reads [32]: (19a) Carath´eodory (Born Version): In every neighborhood of each state (s) there are states (t) that are inaccessible by means of adiabatic changes of state. Symbolically, this is: (∀s ∈ Γ, ∀Us) : ∃t ∈ Uss ≺ t, (1.6) where Us and Ut are open neighborhoods surrounding the states s and t. Carath´eodory’s originally published version is more precise [30, 6]. (19b) Carath´eodory Principle In every open neighborhood Us ⊂ Γ of an arbitrarily chosen state s there are states t such that for some open neighborhood Ut of t: all states r within Ut cannot be reached adiabatically from s. Symbolically this is: ∀s ∈ Γ∀Us∃t ∈ Us&∃Ut ⊂ Us∀r ∈ Ut : s ≺ r. (1.7) Lieb and Yngvason [31] proceed along similar lines, but work with an set of distinct states, rather than a continuous space of them. For them, the second law is a theorem arising out of the ordering of the states via adiabatic accessibility. Details can be found in §1.3. / /
Chapter 1:Entropy and the Second Law 11 In connection with analytical microscopic formulations of the second law,the recent work by Allahverdyan and Nieuwenhuizen [33]is noteworthy.They rederive and extend the results of Pusz,Woronowicz 34]and Lenard [35],and provide an analytical proof of the following equilibrium formulation of the Thomson(Kelvin) statement: (20)Thomson (Equilibrium)No work can be extracted from a closed equilibrium system during a cyclic variation of a parameter by an external source. The Allahverdyan-Niewenhuizen (A-N)theorem is proved by rigorous quantum mechanical methods without invoking the time-invariance principle.This makes it superior to previous treatments of the problem.Although significant,it is insuf- ficient to resolve most types of second law challenges,for multiple reasons.First, the A-N theorem applies to equilibrium systems only,whereas the original forms of the second law (Kelvin and Clausius)are strictly nonequilibrium in character and most second law challenges are inherently nonequilibrium in character;thus, the pertinence of the A-N theorem is limited.Second,it assumes that the system considered is isolated,but realistically,no such system exists in Nature.Third, it assumes the Gibbs form of the initial density matrix.While this assumption is natural when temperature is well defined,once finite coupling of the system to a bath is introduced,this assumption can be violated appreciably,especially for systems which purport second law violation (e.g.,[36). The relationships between these various second law formulations are complex, tangled and perhaps impossible to delineate completely,especially given the muzzi- ness with which many of them and their underlying assumptions and definitions are stated.Still,attempts have been made along these lines [2,6,7,9]6.This exercise of tracing the connections between the various formulations has historical, philosophical and scientific value;hopefully,it will help render a more inclusive formulation of the second law in the future. In addition to academic formulations there are also many folksy aphorisms that capture aspects of the law.Many are catchphrases for more formal statements. Although loathe to admit it,most of these are used as primary rules of thumb by working scientists.Most are anonymous;when possible,we try to identify them with academic forms.Among these are: .Disorder tends to increase. (Clausius,Planck) Heat goes from hot to cold. (Clau.sius） There are no perfect heat engines. (Carnot) There are no perfect refrigerators. (Clausius) ·urphy'sLaw(and corollary）(Murphy~l947) 6See,Table I in Uffink [6]and Table II (Appendix A)in Koenig [9]
Chapter 1: Entropy and the Second Law 11 In connection with analytical microscopic formulations of the second law, the recent work by Allahverdyan and Nieuwenhuizen [33] is noteworthy. They rederive and extend the results of Pusz, Woronowicz [34] and Lenard [35], and provide an analytical proof of the following equilibrium formulation of the Thomson (Kelvin) statement: (20) Thomson (Equilibrium) No work can be extracted from a closed equilibrium system during a cyclic variation of a parameter by an external source. The Allahverdyan-Niewenhuizen (A-N) theorem is proved by rigorous quantum mechanical methods without invoking the time-invariance principle. This makes it superior to previous treatments of the problem. Although significant, it is insuf- ficient to resolve most types of second law challenges, for multiple reasons. First, the A-N theorem applies to equilibrium systems only, whereas the original forms of the second law (Kelvin and Clausius) are strictly nonequilibrium in character and most second law challenges are inherently nonequilibrium in character; thus, the pertinence of the A-N theorem is limited. Second, it assumes that the system considered is isolated, but realistically, no such system exists in Nature. Third, it assumes the Gibbs form of the initial density matrix. While this assumption is natural when temperature is well defined, once finite coupling of the system to a bath is introduced, this assumption can be violated appreciably, especially for systems which purport second law violation (e.g., [36]). The relationships between these various second law formulations are complex, tangled and perhaps impossible to delineate completely, especially given the muzziness with which many of them and their underlying assumptions and definitions are stated. Still, attempts have been made along these lines [2, 6, 7, 9] 6. This exercise of tracing the connections between the various formulations has historical, philosophical and scientific value; hopefully, it will help render a more inclusive formulation of the second law in the future. In addition to academic formulations there are also many folksy aphorisms that capture aspects of the law. Many are catchphrases for more formal statements. Although loathe to admit it, most of these are used as primary rules of thumb by working scientists. Most are anonymous; when possible, we try to identify them with academic forms. Among these are: • Disorder tends to increase. (Clausius, Planck) • Heat goes from hot to cold. (Clausius) • There are no perfect heat engines. (Carnot) • There are no perfect refrigerators. (Clausius) • Murphy’s Law (and corollary) (Murphy ∼ 1947) 6See, Table I in Uffink [6] and Table II (Appendix A) in Koenig [9]
12 Challenges to the Second Law 1.If anything can go wrong it will. 2.Situations tend to progress from bad to worse. .A mess expands to fill the space available. The only way to deal with a can of worms is to find a bigger can. ·Laws of Poker in Hell: 1.Poker exists in Hell.(Zeroth Law) 2.You can't win.(First Law) 3.You can't break even.(Second Law) 4.You can't leave the game.(Third Law) Messes don't go away by themselves.(Mom) Perpetual motion machines are impossible.(Nearly everyone) Interestingly,in number,second law aphorisms rival formal statements.Perhaps this is not surprising since the second law began with Carnot and Kelvin as an injunction against perpetual motions machines,which have been scorned publically back to times even before Leonardo da Vinci (~1500).Arguably,most versions of the second law add little to what we already understand intuitively about the dissipative nature of the world;they only confirm and quantify it.As noted by Pirruccello [37: Perhaps we'll find that the second law is rooted in folk wisdom,plati- tudes about life.The second law is ultimately an expression of human disappointment and frustration. For many,the first and best summary of thermodynamics was stated by Clau- sius 150 years ago [4]: 1.Die Energie der Welt ist konstant. 2.Die Entropie der Welt strebt einem Maximum zu. or,in English, 1.The energy of the universe is constant. 2.The entropy of the universe strives toward a maximum. Although our conceptions of energy,entropy and the universe have undergone tremendous change since his time,remarkably,Clausius'summary still rings true today-and perhaps even more so now for having weathered so much. In surveying these many statements,one can get the impression of having stumbled upon a scientific Rorschauch test,wherein the second law becomes a reflection of one's own circumstances,interests and psyche.However,although there is much disagreement on how best to state it,its primordial injunction against perpetuum mobile of the second type generally receives the most support
12 Challenges to the Second Law 1. If anything can go wrong it will. 2. Situations tend to progress from bad to worse. • A mess expands to fill the space available. • The only way to deal with a can of worms is to find a bigger can. • Laws of Poker in Hell: 1. Poker exists in Hell. (Zeroth Law) 2. You can’t win. (First Law) 3. You can’t break even. (Second Law) 4. You can’t leave the game. (Third Law) • Messes don’t go away by themselves. (Mom) • Perpetual motion machines are impossible. (Nearly everyone) Interestingly, in number, second law aphorisms rival formal statements. Perhaps this is not surprising since the second law began with Carnot and Kelvin as an injunction against perpetual motions machines, which have been scorned publically back to times even before Leonardo da Vinci (∼ 1500). Arguably, most versions of the second law add little to what we already understand intuitively about the dissipative nature of the world; they only confirm and quantify it. As noted by Pirruccello [37]: Perhaps we’ll find that the second law is rooted in folk wisdom, platitudes about life. The second law is ultimately an expression of human disappointment and frustration. For many, the first and best summary of thermodynamics was stated by Clausius 150 years ago [4]: 2. Die Entropie der Welt strebt einem Maximum zu. or, in English, 1. The energy of the universe is constant. 2. The entropy of the universe strives toward a maximum. Although our conceptions of energy, entropy and the universe have undergone tremendous change since his time, remarkably, Clausius’ summary still rings true today — and perhaps even more so now for having weathered so much. In surveying these many statements, one can get the impression of having stumbled upon a scientific Rorschauch test, wherein the second law becomes a reflection of one’s own circumstances, interests and psyche. However, although there is much disagreement on how best to state it, its primordial injunction against perpetuum mobile of the second type generally receives the most support 1. Die Energie der Welt ist konstant
Chapter 1:Entropy and the Second Law 13 and the least dissention.It is the gold standard of second law formulations.If the second law is the flesh of thermodynamics,this injunction is its heart. If the second law should be shown to be violable,it would nonetheless remain valid for the vast majority of natural and technological processes.In this case,we propose the following tongue-in-cheek formulation for a post-violation era,should it come to pass: (21)Post-Violation For any spontaneous process the entropy of the universe does not decrease-except when it does. 1.3 Entropy:Twenty-One Varieties The discovery of thermodynamic entropy as a state function is one of the triumphs of nineteenth-century theoretical physics.Inasmuch as the second law is one of the central laws of nature,its handmaiden-entropy-is one of the most central physical concepts.It can pertain to almost any system with more than a few particles,thereby subsuming nearly everything in the universe from nuclei to superclusters of galaxies [38.It is protean,having scores of definitions,not all of which are equivalent or even mutually compatible7.To make matters worse, "perhaps every month someone invents a new one,"[39].Thus,it is not surprising there is considerable controversy surrounding its nature,utility,and meaning.It is fair to say that no one really knows what entropy is. Roughly,entropy is a quantitative macroscopic measure of microscopic disor- der.It is the only major physical quantity predicated and reliant upon wholesale ignorance of the system it describes.This approach is simultaneously its greatest strength and its Achilles heel.On one hand,the computational complexities of even simple dynamical systems often mock the most sophisticated analytic and numerical techniques.In general,the dynamics of n-body systems (n >2)can- not be solved exactly;thus,thermodynamic systems with on the order of a mole of particles (1023)are clearly hopeless,even in a perfectly deterministic Lapla- cian world,sans chaos.Thus,it is both convenient and wise to employ powerful physical assumptions to simplify entropy calculations-e.g.,equal a priori proba- bility,ergodicity,strong mixing,extensivity,random phases,thermodynamic limit. On the other hand,although they have been spectacularly predictive and can be shown to be reasonable for large classes of physical systems,these assumptions are known not to be universally valid.Thus,it is not surprising that no completely satisfactory definition of entropy has been discovered,despite 150 years of effort. Instead,there has emerged a menagerie of different types which,over the decades, have grown increasingly sophisticated both in response to science's deepening un- derstanding of nature's complexity,but also in recognition of entropy's inadequate expression. This section provides a working man's overview of entropy;it focuses on the most pertinent and representative varieties.It will not be exhaustive,nor will 7P.Hanggi claims to have compiled a list of 55 different varieties;here we present roughly 21
Chapter 1: Entropy and the Second Law 13 and the least dissention. It is the gold standard of second law formulations. If the second law is the flesh of thermodynamics, this injunction is its heart. If the second law should be shown to be violable, it would nonetheless remain valid for the vast majority of natural and technological processes. In this case, we propose the following tongue-in-cheek formulation for a post-violation era, should it come to pass: (21) Post-Violation For any spontaneous process the entropy of the universe does not decrease — except when it does. 1.3 Entropy: Twenty-One Varieties The discovery of thermodynamic entropy as a state function is one of the triumphs of nineteenth-century theoretical physics. Inasmuch as the second law is one of the central laws of nature, its handmaiden — entropy — is one of the most central physical concepts. It can pertain to almost any system with more than a few particles, thereby subsuming nearly everything in the universe from nuclei to superclusters of galaxies [38]. It is protean, having scores of definitions, not all of which are equivalent or even mutually compatible7. To make matters worse, “perhaps every month someone invents a new one,” [39]. Thus, it is not surprising there is considerable controversy surrounding its nature, utility, and meaning. It is fair to say that no one really knows what entropy is. Roughly, entropy is a quantitative macroscopic measure of microscopic disorder. It is the only major physical quantity predicated and reliant upon wholesale ignorance of the system it describes. This approach is simultaneously its greatest strength and its Achilles heel. On one hand, the computational complexities of even simple dynamical systems often mock the most sophisticated analytic and numerical techniques. In general, the dynamics of n-body systems (n > 2) cannot be solved exactly; thus, thermodynamic systems with on the order of a mole of particles (1023) are clearly hopeless, even in a perfectly deterministic Laplacian world, sans chaos. Thus, it is both convenient and wise to employ powerful physical assumptions to simplify entropy calculations — e.g., equal a priori probability, ergodicity, strong mixing, extensivity, random phases, thermodynamic limit. On the other hand, although they have been spectacularly predictive and can be shown to be reasonable for large classes of physical systems, these assumptions are known not to be universally valid. Thus, it is not surprising that no completely satisfactory definition of entropy has been discovered, despite 150 years of effort. Instead, there has emerged a menagerie of different types which, over the decades, have grown increasingly sophisticated both in response to science’s deepening understanding of nature’s complexity, but also in recognition of entropy’s inadequate expression. This section provides a working man’s overview of entropy; it focuses on the most pertinent and representative varieties. It will not be exhaustive, nor will 7P. H¨anggi claims to have compiled a list of 55 different varieties; here we present roughly 21