arXiv:1507.02022v1 [cond-mat.stat-mech] 8 Jul 2015 aist ytm ihvr e ere ffedm We freedom. of degrees few very with systems to namics of density non-monotonic with states. systems differences for the oscillators), dramatic harmonic are of system a densities as energy (such monotonic unmeasurable with are systems entropies macroscopic predictions Boltzmann for function and the nondecreasing Gibbs between energy the differences a given of the is While a entropy can- energy. to as of temperature equal negative entropy, of or be definition than not this less With count- states by [4–8]. the obtained is all entropy ing energy Gibbs narrow the a the in while (or interval), of eigenstate energy logarithm an the of degeneracy to corresponds entropy Boltzmann tem- positive for allows only of peratures. entropy function increasing energy. Gibbs an given the is a energy, volume than space less phase logarithm volume is the space the entropy Since phase Gibbs times total constant The the Boltzmann of Gibbs. the to as due should defined one entropy by Boltzmann’s replaced suggest inconsis- be of they be use to Instead, the claim tent. they to which due entropy, arise Boltzmann the temperatures negative and ar- recently that [4] have gued (DH) [5] (HHD) Hilbert en- Dunkel and H¨anggi, and bounded Dunkel Hilbert, However, with systems [1–3]. for ergies abstraction ide- and an alization temperature, negative of concept thermodynamic ∗ [email protected] HadHDas ple hi hoyo thermody- of theory their applied also HHD and DH the levels, energy discrete with systems quantum For the for presented been have arguments Compelling eateto hsc,Ntoa nvriyo igpr,S Singapore, of University National Physics, of Department hnoei atro ento n spsil o inverted for wit e systems We possible connecting analysis. is process the reversible and in and C definition adiabatic used A an are of temperatures equilibrium. matter negative in a whether entropy is Boltzmann one with t along than agrees paramagnet it a that for entropy and thermodynami c as thermodynamic of thermal the their interpretation deduce into of traditional to most brought more with a are disagree of they We favor tempera when possible. the flow are in heat temperatures differences of theory, their direction In the Callen. and Tisza te etrs hi eut ontstsytepostulates the satisfy traditionally not have do that results their properties features, several other aside sets that uedtrie h ieto fha o hl ib temper Gibbs while flow heat of demon a systems. direction give the We determines exist. ture must logically temperatures negative Physics, ture ytm, hs e.E Rev. Phys. systems,” ukladHlet Cnitn hrottsisforbids thermostatistics “Consistent Hilbert, and Dunkel .INTRODUCTION I. rtqeo h ib oueetoyadisimplication its and entropy volume Gibbs the of Critique 10 7(04,adHlet ¨ng,adDne,“Thermodynam Dunkel, H¨anggi, and Hilbert, and (2014), 67 , 90 616(04 aepeetda nsa iwo thermodyna of view unusual an presented have (2014) 062116 , Dtd uy2015) July 8 (Dated: inSegWang Jian-Sheng aisfruae nRf Sc V n considering and XV) (Sec. for 5 thermody- Ref. XIV) the in (Sec. discuss formulated We postulates XIII) namics Callen’s (Sec. states. we entropy temperature of XII), negative Gibbs violation (Sec. the its entropy of of and concavity additivity the the discuss and equivalence ensemble XI) thermo- discussing of (Sec. heat law briefly of second After the direction with dynamics. the consistent is predicts and it transfer cor- since the calculation decisively temperature model is rect two-level temperature a Boltzmann X, de- that Sec. one shows In single application a freedom. the only of criticize with gree we systems IX to Sec. in thermodynamics In Weof contradictions that VIII. to and systems lead invariant. VII model not Sec. adiabatic does of true temperature examples a negative the given not with a is than demonstrate energy less states of and of B) value VI number (Sec. the systems that quantum in demonstrate invariance and temperatures. adiabatic VI) of (Sec. negative classical properties by and the described discuss distributions, then well for We energy are possible inverted we they is for V, discuss one that Sec. engine we than in Carnot larger IV, that, a efficiency Sec. After an no In why has paramagnet. discuss condition a model. a of such entropy Ising that the the out in to point discussed we constraint is and condition III, consistency Sec. A thermodynam- entropy. of and formulation ics the on comment briefly we entropy. Gibbs illustrate neg- of to include predictions and to incorrect thermodynamics, appropriate some in is discus- it temperature these that ative on emphasize expand we to paper, sion this In [9–15]. authors we systems. so large large, to are remarks our fluctuations restrict relative sys- will the to applicable which is for thermodynamics tems that believe not do h raiaino h ae sa olw:I e.II Sec. In follows: as is paper the of organization The several by made been have claims DH’s to Objections ftemdnmc rgnlypooe by proposed originally thermodynamics of Hamiltonian h aebe sue ob re Among true. be to assumed been have ∗ s eso xlctyta ti possible is it that explicitly show We cs. naoe175,Rpbi fSingapore of Republic 117551, ingapore ue ftoojcscno determine cannot objects two of tures eaieaslt eprtrs”Na- temperatures,” absolute negative tainta h otmn tempera- Boltzmann the that stration etos n epeetagmnsin arguments present we and sertions, aoaeo ers’ ruetthat argument Penrose’s on laborate tr osnt o ucetylarge sufficiently for not, does ature elnso lsia thermodynamics classical of lines he nrydsrbtos eadesof regardless distributions, energy ro niewt ffiinylarger efficiency with engine arnot nat hydn htnegative that deny They ontact. H to − H clw nisolated in laws ic spsil,thus possible, is mics 2 the nonequilibrium statistical mechanics of splitting and A. Stability of thermodynamic systems joining of two systems (Sec. XVI). We remark on the non-monotonic dependence of temperature with energy DH and HHD have argued that negative temperature for finite systems and why it disappears (Sec. XVII) for states are unstable or unphysical; any contact with a pos- large systems (or in thermodynamic limit [16]). We con- itive temperature environment will collapse the system clude with some final remarks in Sec. XVIII and XIX. [21]. There seems to be a misunderstanding of thermo- dynamic stability. Any thermal contact of any system with another system at a different temperature takes it through an irreversible process into a different equi- II. THERMODYNAMICS AND ENTROPY librium state. If a system at negative temperature is brought into contact with another system at negative temperature, the final equilibrium will be at an inter- Dunkel and Hilbert [4] began with a statement that mediate tempearature that is also negative. The states the positivity of absolute temperature is a key postulate are stable if the entropy associated with the system is of thermodynamics. DH cited Callen (postulate III, page a concave function of energy. If a negative temperature 28, 2nd edition, [17]) that entropy ‘is a monotonically in- system is in contact with a system of the same negative creasing function of the energy.’ They did not explain temperature, it will happily coexist. why they regard it as a key postulate. The only rea- son we know of is that monotonicity allows the equation S = S(E,V,N) to be inverted to give E = E(S,V,N), from which the usual thermodynamic potentials can be B. Consequences for the definition of entropy obtained by Legendre transforms. This is a matter of convenience, not principle, and Massieu functions can We choose a notation that is largely consistent with produce the same results through Legendre transforms that of DH. The total number of states below energy E of S = S(E,V,N). Apparently, DH rejected most of is Callen’s other postulates of thermodynamics. Ω(E)= 1. (1) The axiomatic formulation of thermodynamics, such as that given by Giles [18], or Lieb and Yngvason [19], does EXj ≤E not feature the concept of temperature until a very late For simplicity, we’ll consider mostly quantum systems stage. So the positivity of temperature is not at all a fun- with a set of discrete energy eigen spectra, Ej , j = 1, 2, damental axiom of thermodynamics. Even energy does . And ω(E)∆ Ω(E) Ω(E ∆) Ω′(E)∆. The not enter into the picture initially. What is fundamental Gibbs··· (volume) entropy≡ is−S = k− ln Ω(≈E), and Boltz- in thermodynamics is the concept of irreversibility. This G B mann, SB = kB ln(ω(E)∆). The small energy ∆ is used concept appears and is an emergent phenomenon only to specify an energy range of microcanonical ensemble of in the macroscopic world; as is well-known, there is no the system. irreversibility in microscopic physical laws. −1 −1 We define TG = ∂SG/∂E and TB = ∂SB/∂E. Whether one follows the traditional approach to ther- We had always thought that the difference between modynamics [20], or the postulatory approach of Gibbs, SB and SG was so small for a macroscopic system that it Tisza, and Callen, or the more precise axiomatic formu- would be impossible to detect, which would imply that lation in the post-modern era, they are all consistent and the choice might be a matter of taste to some extent. This convergent to the same physical theory of thermodynam- is true for systems of particles with unbounded energies. ics. When we say “thermodynamics”, there should not be Now it appears that DH have (inadvertently) provided a disagreement as what we mean by it. But since Hilbert, demonstration that SG is not tenable. H¨anggi, and Dunkel (HHD, [5]) proposal of a thermody- In Refs. [12, 13, 22–25], the definition of entropy, en- namics is substantially different from the usual formula- tropy of macroscopic observables, is based on the joint tion, we need to examine its legitimacy. probability distribution for large, but finite systems. In It is possible to develop statistical mechanics in ex- this particular case, it is equivalent to what DH labelled actly the same way as in the traditional thermodynam- as SB, although the additive constant is expressed differ- ics, where an empirical temperature occupies a promi- ently than their kB ln ∆. Therefore, by construction, the nent place in defining isothermal processes. With the maximum of the sum of the entropies occurs at the max- help of the Carnot cycle, we can define absolute temper- imum probability of a subsystem energy after the two ature, and thus, following Clausius, entropy. This chain subsystems have come to thermal equilibrium. of reasoning puts entropy at the last, so the dispute over This means that – if they insist on taking the small which expression is the correct entropy can be, hopefully, differences seriously – the maximum of their sum of en- resolved. We give some details with such an approach tropies occurs at a non-equilibrium value of the energy of here. Of course, the final result is exactly the same as the subsystems, see Sec. XIV A. If they were to give initial the standard textbook statistical mechanics with Boltz- conditions such that the two subsystems started at the mann entropy. maximum of their total entropy, releasing the constraint 3
(allowing heat transfer), then splitting again would lower thermodynamic variable, i.e., A = J. Then the consis- the entropy, in violation of the second law [12]. tency condition only produces an identity, E = H , for any reasonable expression of S. This is becauseh thei S- dependence gets cancelled between T and ∂S/∂J, since III. CONSISTENT CONDITIONS FOR S(E,J) S(E/J). The consistency condition has no ENTROPY use to judge≡ which entropy expression is correct, at least for the Ising chain. In Sec. IV F below, we also show that We would certainly agree with the importance of DH’s the consistent condition is violated for a paramagnet if Eq.(4), but it would be more appropriate to write it as Gibbs volume entropy is used. Extending the adiabatic invariant and consistency properties of Gibbs volume en- 1 1 dS = dE + a dA , (2) tropy, valid for classical Hamiltonian systems with con- T T j j tinuous energy spectra, to all systems, seems to us an j X over-generalization, and need further justification. where 1 ∂S = , (3) IV. ENTROPY OF A PARAMAGNET FROM T ∂E CLASSICAL THERMODYNAMICS {Aj } Which of the alternative starting points of definitions 1 ∂S for entropy is correct? We address this question in a aj = . (4) T ∂Aj different perspective. Instead of taking a postulatory ap- E,{Ak}k6=j proach to the expression for entropy, we derive it, follow- The volume V should be included in the Aj ’s. It would ing the classical approach to thermodynamics [20]. This not be an appropriate variable for most systems with consists of the following steps: 0) fundamental to sta- bounded energy. tistical mechanics is the equal-a-priori probability and For classical Hamiltonian systems, the argument that microcanonical ensemble. This will be our basis in the TB = TG seems unfounded because the difference is of the following consideration. In order to reproduce thermody- 6 order of 1/N, such as for ideal gas, while fluctuations are namics from statistical mechanics, we work in the ther- of the order 1/√N. If fluctuations are negligible, which modynamic limit (i.e., large system sizes). 1) Build an is usually assumed in thermodynamics, then TB TG is empirical thermometer to realize the zeroth law of ther- negligible. − modynamics. 2) Use the thermometer to determine the The consistency issues have been addressed in the past. equation of state and thus adiabatic and isothermal pro- Boltzmann himself discussed it in what Boltzmann called cesses. 3) Build Carnot cycles to find the universal func- ‘heat theorems’, asking questions of consistency with tion relating the empirical temperature to the Kelvin thermodynamics (see appendix 9A1 of Gallavotti’s text- scale. 4) Define the entropy according to Clausius. This book [26]; see also the discussion in S.-K. Ma, Chap. 23 chain of reasoning puts entropy at the last as an output of [27]). It is not essential that SG satisfies the condition the more mechanical arguments, instead of starting with for all N, but for none with SB, since thermodynamics is the entropy postulates. As a by-product, the concept of well-defined only in the limit N . And in that limit, negative temperature results naturally from the deriva- it does satisfy (Frenkel and Warren→ ∞ [11]), so nothing is tive of entropy with respect to energy, if steps 1) to 4) violated. are not erroneous. To do this in a very general setting is A key feature of the consistent condition, Eq. (6) in obviously difficult. Here we are modest, considering only DH which we write down again here, the simplest possible model systems. In particular, we take a one-dimensional (1D) Ising model as our bath or ∂S ∂H a = T = , (5) thermometer, and a non-interaction paramagnet as the j ∂A − ∂A system (see also Frenkel and Warren, Ref. 11). Our logi- j E j cal conclusion is that the usual textbook entropy formula which the authors do not seem to notice is that when it is sound and implies the existence of negative absolute applies to, say, the one-dimensional (1D) ferromagnetic temperature for systems with bounded energy. Ising model,
N A. Equal a priori probability and microcanonical H = J σ σ , σ = σ , σ = 1, (6) − i i+1 N+1 1 i ± ensemble i=1 X it just produces a “null statement”, as there is no A- We assume that the dynamics is such that energy is variable to use to form the equation. The thermody- conserved. The probability distribution function (for namic equation is simply dE = T dS. Alternatively, one classical systems) or density matrices (for the quantum can use the coupling J as one of the external control case) follows the Liouville or von Neumann equation. 4
However, our Ising model does not have an intrinsic dy- of the equation of state (magnetization) is M = E/h. namics; we assume that the system follows some form In a microcanonical ensemble energy does not fluctuate.− of stochastic dynamics governed by a master equation By the first law of thermodynamics, dE = δQ + δW , (Penrose [28]). We do not need to explicitly specify the we can identify the heat and work as δQ = h dM and dynamics as we are not interested in the actual time- δW = Mdh, respectively. The work is related− to the dependent processes, in other words, we work in the do- part of− internal energy changeable by the control param- main of equilibrium statistical mechanics and quasi-static eter h. From the above, we note that adiabatic lines are processes. The only thing that is required is that the very simple, given by M = const. – horizontal lines in an system evolves in such a way that micro-states with the M v.s. h diagram (see Sec. VI B for further justification). same energy (or other conserved quantities, e.g., mag- However, the isothermal curves are more difficult to spec- netization) will have the same probability, i.e., the mi- ify, as we do not have a real equation of state relating to crocanonical assumption. Another point is that we will the (empirical) temperature yet. work in large size limit (i.e., thermodynamic limit). Sys- tems with few degrees of freedom in isolation belong to the regime of mechanics and cannot possibly behave like D. Measuring the empirical temperature of the a thermodynamic system (e.g., cannot thermalize). This system helps in simplification of our derivations. To find the empirical temperature of the paramagnetic system, we take initially two separate systems of the ther- B. 1D Ising model as a thermometer mometer, characterized by the total energy EB , and the system, characterized by E and h (the sizes NB and N We consider a 1D Ising chain with a periodic bound- are also parameters, but they are considered as fixed con- ary condition without a magnetic field term, with the stants). We then combine them into a new system with Hamiltonian function given by the Hamiltonian Htot = HB + H. As there is no ex- plicit interaction term between the two subsystems in NB B B B B B the Hamiltonian and the dynamics is such that it makes HB = J σ σ , σ = 1, σ = σ . (7) − i i+1 i ± NB +1 1 Htot microcanonical, the effect of combining them is to i=1 X make each subsystem no longer in microcanonical distri- The super-/subscript B indicates the degrees of freedom bution but new marginal distributions dictated by the of the bath (or thermometer) in contrast to the system overall microcanical distribution. We now ask what the expectation value σB σB is. If this value is without the sub- or superscripts. Since we work in micro- h 1 2 iHtot =EB +E canonical ensemble, the (statistical mechanical) system is the same as before making the contact, θ, then we say uniquely characterized by the total energy EB and num- that the system is at the empirical temperature θ. This defines the self-consistent condition for the temperature ber of spin NB. Now we define an empirical temperature θ = σB σB . The average is with respect to the of the paramagnetic system h 1 2 i h· · ·i microcanonical ensemble. It is clear that EB = JNBθ. NB We note that 1 θ 1, so our thermometer− readings 1 B B − ≤ ≤ θ = σi σi+1 . (9) are real numbers from 1 to 1, with negative values corre- NB * i=1 +Htot(σB ,σ)=E +E sponding to the usual− negative temperature. As we’ll see X B later, this temperature scale θ is essentially β =1/(kBT ) This is an equation for θ as the right-hand side also con- mapped to a bounded interval. Unlike the ideal gas ther- tains θ due to E = JN θ. The result will depend on B − B mometer, our Ising thermometer can measure both pos- E and h in the constraints. The equation can be made itive as well as negative temperatures. more explicit as e (i) nB(i) n(N ) E = e (σB ) = i B − , (10) B B nB(i) n(N ) C. Paramagnet working system P i −
where the ‘density of states’P (number of microscopic We build a Carnot engine made of an Ising paramag- states at energy eB(i) = JNB +2Ji, i even) of the net, given by the Hamiltonian 1D Ising is − N B NB! H = h σj , σj = 1. (8) n (i)=2 , i =0, 2, ..., NB. (11) − ± i!(NB i)! j=1 X − We assume N even for simplicity. i denotes the num- Let N and N be the number of spins with plus B + − ber of plus-minus boundaries in a 1D Ising configuration. and minus signs, respectively, then the energy is E = Similarly, the density of states of the paramagnet is hN + hN = hM, and the total magnetization is − + − − M = N+ N−. A thermodynamic state is uniquely spec- N! − n(N−)= n(N+)= , N+ + N− = N, (12) ified by the energy E and the magnetic field h, while one N+!N−! 5
M E. Carnot cycle θ2 θ3 M 1 With the empirical temperature of the system well- θ 1 defined, we can construct Carnot cycles. An example is M 2 given in Figure 1. The two adiabatic curves are charac- terized by M1 and M2 as horizontal lines. The isothermal h curves are labeled by θ1 and θ2 (or equivalently β˜1 and β˜2). They are given by the equation of state for the para- magnet as M = N tanh(βh). (16) An important point is that these family of isothermal FIG. 1. Carnot cycle using paramagnet as substance. curves are identical no matter what empirical tempera- ture scale one uses, thus independent of the temperature measuring devices. Hence, the concept of ‘temperature’ and N = (N E/h)/2, and N = (N + E/h)/2. i and is just a one-parameter family of real numbers labeling + − − N− are related by energy conservation these curves. Positivity of the temperature is not re- quired. As in the usual construction of the Kelvin tem- e (i) (N 2N )h = E + E. (13) perature scale, we compute the heat absorbed at the high B − − − B temperature, Q1 at θ1, and heat released at low temper- Equation (10) would be difficult to evaluation in gen- ature, Q2 at θ2 with the total work, W > 0, done by the eral but great simplification is available in the thermo- system as the area of the loop running clockwise on the dynamic limit, i.e., NB and N are large (but still finite cycle. High and low temperatures are defined according quantities). Then the summation is dominated by the to second law of thermodynamics (not the actually values largest weight, nBn. We find the value i for which the of the parameter θ) - heat flows from high temperature weight is a maximum. Although i and N+ or N− are to low temperature if there is no external intervention. integers, in the large size limit, we can treat them as Then, real numbers. Using the Stirling approximation, ln j! = Q f(θ ) j ln j j, for j sufficiently large, and obtaining the deriva- 2 = 2 . (17) tive of− ln(nB n) with respect to i and setting it to 0, we Q1 f(θ1) obtain the condition ln (i/(N i)) = (J/h) ln(N /N ). B − − + Our job is to find the function f, that depends on our In deriving this, we have assumed self-consistency on the choice of the empirical temperatures, but if we define right-hand side, i.e., E = JN θ, θ =1 2i/N . This B − B − B T = constf(θ), then this new temperature scale, T , the optimal value of i is essentially our temperature of the Kelvin scale, clearly is independent of the makeup of system. Instead of using integers i and N+,N−, we can the thermometer and is universal in the sense that all replace them by more physical quantities, with i by θ, Carnot cycles have exactly the same efficiency formula and N , N by M and N, we find + − η = W/Q1 =1 T2/T1 if temperatures are the same on the Kelvin scale.− Thus, the Kelvin scale is a privileged h 1+ θ N + M ln = ln (14) one over the empirical ones. We now work out explicitly J 1 θ N M − − this function f. Using the formula for heat, we have, or with the equation of state,
M2 M2 M2 h −1 −1 βh˜ = tanh (θ) = tanh (M/N), (15) Q1 = h dM = hM + Mdh J − M1 − M1 M1 Z Z N cosh(β1h2) or M = N tanh(βh˜ ) with θ = tanh(βJ˜ ). Reader famil- = h2M2 + h1M1 + ln − β cosh(β h ) iar with the canonical treatment of 1D Ising model and 1 1 1 = T [S(M ) S(M )], (18) paramagnet will immediately recognize that these results 1 2 − 1 are identical to the usual formulas in canonical ensemble where we define T =1/(k β ), which is just the conven- ˜ 1 B 1 if we identify β with the usual β = 1/(kBT ) (We do tional Kelvin scale, fixing an arbitrary constant relating not use canonical ensemble because that would presume our empirical temperature θ to T , and S is a function ensemble equivalence and defeat the purpose of introduc- defined by ing empirical temperature). Here β˜ is a derived concept, S(M)= k βhM + k N ln cosh(βh) + const characterizing, in an alternative way, the temperature of − B B the system or bath. This simple picture emerges only if 1+ M/N 1+ M/N = k N ln thermodynamic limit is taken. For notational simplicity, − B 2 2 we’ll drop the tilde below but β should be understood as 1 M/Nh 1 M/N + − ln − . (19) constructed above. 2 2 i 6
To obtain the second line, we have used again the equa- does not become dense even for large sizes. If this for- tion of state of the system. We also dropped an arbitrary mula is used for entropy, we can not recover the third law integration constant. We recognize that this is nothing of thermodynamics, as even in the ground state W = 1, but the usual entropy of mixing formula. An important S still depends on the magnetic field h logarithmically. observation is that S does not depend on h explicitly Thus, the prediction by Campisi that the thermodynamic and only on M, which is consistent with the fact that magnetization calculated by the adiabatic lines are M = const. This M-only depen- ∂S/∂h E k T dence also satisfies the consistent condition, Eq. (5). A M = = B , (25) similar calculation can be carried out, and one finds ∂S/∂E − h − h
has a singularity (the last term above) is simply due to Q2 = T2 S(M2) S(M1) , (20) − a wrong application of the Boltzmann principle. We also thus note that W , thus SB, is an even function of the magne- tization M only, and M is an adiabatic invariant. But Q2 T2 = . (21) N! Q1 T1 Ω(E,h)= [(N m)/2]![(N + m)/2]! −hm≤E And the desired f function is X − =Ω+(M)θ(h)+Ω−(M)θ( h) (26) −1 − T = f(θ)=1/(kBβ)= J/ kB tanh (θ) . (22) (m = hN, h(N 2), ,hN; θ(h) Heaviside step func- − +− − − N··· The last step in this argument is to define entropy ac- tion, and Ω +Ω =2 ) depends on M and h separately cording to Clausius, i.e., and experiences a discontinuous jump when h changes sign. Moreover, ω(E,h) = ∂Ω(E,h)/∂E is not an inte- δQ grating factor, since dS = , (23) T dΩ= ωδQ + (Ω+ Ω−)δ(h)dh, (27) or S = M hdM/T + const. As the integrand is a − total differential,− exactly what process or which path to where the heat is δQ = hdM. This prevents Ω being − use is immaterialR — we get exactly the same expression an adiabatic invariant [14]. The appearance of the sec- (e.g., integrating over isothermal curve). Of course, the ond term in Eq.(27) also causes the consistent condition, result is just the standard entropy formula for paramag- Eq.(5) with A = h, violated. net, Eq. (19). By differentiating with respect to E, fixing If we reflect back on our derivation, it is clear we must h (since we have the fundamental thermodynamic rela- have a negative temperature scale of an intensive quan- tion dE = T dS Mdh), we can see that temperature tity, since we have decided to measure the temperature, can take both positive− as well as negative values (when be it positive or negative, by a local property. Boltz- E = hM > 0). mann’s critical insight of connecting thermodynamic en- − tropy with probability and chance is throwing out of win- dow if Gibbs volume entropy is adopted. F. Discussion
Boltzmann’s original definition of entropy is based on V. CARNOT EFFICIENCY GREATER THAN 1 probability. Since each state is equally likely in a mi- crocanonical ensemble, we have SB = kB ln W , where The efficiency of a Carnot cycle when negative tem- W = 1/P is the number of microstates consistent with perature baths are possible is well analyzed by Ramsey, the given constraint, and P is the probability of each Frenkel and Warren, see also Ref. 30. If we define the micro-state. For our magnetic system, Carnot efficiency in the usual way, N! W T W = . (24) η = =1 C , (28) [(N + M)/2]![(N M)/2]! QH − TH − After using the Stirling approximation for the factorial where subcript H means ‘hot’ and C ‘cold’, the efficiency function, valid for large N, we obtained the standard η does get values larger than one. As explained in Ref. 11, entropy mixing formula. this is just a matter of definition. Campisi in Ref. 8 defined a Boltzmann entropy as In Figure 2, we illustration what is possible, what is not S(E,h) = kB ln(W/h) + const. This is not correct, ac- possible, if Clausius’s second law need to be obeyed, i.e., cording to most textbooks (see Nagle, [29], who cited it is not possible for heat flowing from cold bath to hot many popular statistical mechanics books). Replacing bath spontaneously without any other effect. This is the the number of states by the density of states is simply only condition that can show very logically certain pro- not possible as for the Ising model the energy spectrum cesses are not possible. Alternatively and equivalently, 7
φ = -1/TH TH > 0 TH < 0 H sure the total entropy does not decrease. If bath tem- Q > 0 Q < 0 Q > 0 H H H peratures are opposite, case (f), we can use both baths to do work, given an efficiency larger than one, if we still define efficiency as W/Q . The case (e) is a situation W<0 W>0 H that is allowed but is not a reversible process.
Q = Q CH φ = -1/T C < φ T T VI. ADIABATIC INVARIANCE C H C C (a) (b) (c) Since a strong point in promoting the Gibbs volume en- tropy is its adiabatic invariance [4, 8, 31], let us try to un- T > 0 T < 0 T < 0 H H H derstand what is adiabatic invariant. S.-K. Ma, Chap 23 Q > 0 Q > 0 Q > 0 H H H is helpful [27]. In an adiabatic process we change model parameters very slowly so that the system is always in equilibrium - the important statement is we change ac- W>0 W>0 W>0 cording to
Q > 0 Q > 0 Q < 0 ∂H C C C dE = dL, (29) T > 0 T > 0 T > 0 ∂L C C C (d) (e) (f) using Ma’s notation. This gives the adiabatic curves in model parameter space, (E,L). The average is over mi- FIG. 2. Various Carnot cycle configurations with positive and crocanonical ensemble. He proved the adiabatic theorem negative temperatures. using δ-function notation and summing over s notation which is understandable., i.e., (set kB = 1)
S(E,L) = ln Ω, Ω(E,L)= θ E H(s,L) (30) we can argue that the total entropy of the combined sys- − s tem and the baths cannot decrease. Let us use the fol- X lowing sign convention: if the engine performs work to is a constant if dE = ∂H/∂L dL. the environment, W > 0; if the engine absorbs heat from Consider the standardh microcanonicali ensemble for high temperature source, QH > 0; and if it releases heat, classical system defined by the probability density in QC > 0. energy conservation requires QH = QC + W . phase space as ρ = const if E
E M and + − + 2π∆ S = ln Ω(E + ∆) Ω(E) = ln . (35) B − ω We can already see trouble here, as SB does not depend h h on E. The differentials are − − + dE dω dS = , (36) G E − ω (a) (b) and FIG. 3. Adiabatic change in a single spin paramagnet. (a) dω − dSB = . (37) The energy eigenvalues of the + and states with magnetic − ω field h; (b) and associated magnetization eigenvalues. The adiabatic condition means, for the oscillator with a varying frequency and energy (work done by varying frequency equals to the energy increase), coincides with thermodynamic adiabaticity, i.e., no heat transfer. ∂H Penrose [14] gives a counter-example showing that the dE = dω. (38) ∂ω total number of states Ω equal to or smaller than a given energy eigenvalue En is not a true adiabatic invariant. We need to compute the ensemble average explicitly, The reason is in fact quite simple, if there is another which is energy level crossing [32] the current energy En which we focus on, the number of energy levels E E changes 2 m ≤ n ∂H 4 1st quadrad dqdp ωq for fixed n. = lim ∂ω ∆→0 Ω(E + ∆) Ω(E) Consider a paramagnet Hamiltonian viewed as quan- R R − E tum operator, = . (39) ω N Hˆ = hMˆ = h σˆz, (40) If we maintain a finite ∆, the expression is complicated, − − i i=1 but taking the limit makes it very simple. This also X means for the proof of adiabatic invariance, we really whereσ ˆz is the z-component of Pauli matrices. Since need ∆ 0 limit. Now one sees that dE = (E/ω)dω, so i σˆz commutes with Hˆ , the z component of the spin S is an→ adiabatic invariant, while S is not. i G B is a constant of motion from the Heisenberg equation, This adiabaticity is closely connected to the ‘heat the- i¯hdσˆz/dt = [ˆσz, Hˆ ], for any time dependence of h h(t). orem’ or ‘orthodicity’ as discussed in Gallavotti’s book i i In Fig. 3 we demonstrate the situation for a single→ spin. If [26], Chap. 2. The two are just slight reformulate of the we stay in the ground state when h< 0, we have no other same question. Gallavotti explicitly stated that in canon- states below it, so Ω = 1. As we cross h = 0 to the posi- ical ensemble one does not need to take N to infinite, but tive field region, Ω jumps discontinuously to 2. Thus it is one does need thermodynamic limit in micro-canonical not an adiabatic invariant. On the other hand, the mag- ensemble (page 63-65). netization M as well as counting of levels determined by Although entropy must be an adiabatic invariant, due M are (quantum-mechanical) adiabatic invariants. The to Clausius equation dS = δQ/T , there is no reason to fact that the total magnetization is an adiabatic invari- believe that this is the only property of entropy. Adia- ant is fully consistent with earlier discussion in Sec. IV C batic invariants do exist for small mechanical systems, in of the Carnot cycle of paramagnet. the sense demonstrated above, but these systems do not One might suspect that a transition occurs from ‘ ’ obey zero-th law of thermodynamics. state to ‘+’ state if one goes from negative h to positive− h slowly, such as in the Landau-Zener problem. But since there is no couplings between up and down spin states, B. Adiabatic invariance in quantum systems, Penrose’s counter-argument this does not happen. If the system is in a pure quantum eigen state φn of Mˆ , its dynamic evolution is a triv- | i t ial phase change, exp i M h(t′)dt′ φ (0) . Since the In quantum mechanics, one can prove an adiabatic the- h¯ n | n i Hamiltonian is diagonal in the representation when Mˆ is orem which says if one changes the system very slowly R through some model parameters, then if the system is diagonal, it cannot make a transition to another state. in a pure quantum eigenstate, it will stay in this state. One can solve the von Neumann equation exactly, and Thus, when control parameters change, if energy changes finds that the density matrix evolves according to following the instantaneous eigenvalue of the system, it is i t ′ ′ h¯ (Mm−Mn) h(t )dt an adiabatic change. This purely mechanical adiabaticity ρmn(t)= ρmn(0)e 0 , (41) R 9 in the representation for which Mˆ is diagonal. Again we do not evolve. Then we must have find that the diagonal part of the density matrix does not S (E)= S ( E). (45) depend on time, and off-diagonals are oscillatory. This + − − dynamic evolution of the density matrix tells us that if This equation implies if we have a positive temperature the system is initially in a microcanonical ensemble at for a system with Hamiltonian Hˆ at energy E = Tr(ˆρHˆ ), h < 0, it cannot change into a new and different micro- there must exist another system with Hamiltonian Hˆ canonical ensemble at h = 0 occupying all micro-states with a negative temperature at E. If the spectrum of−Hˆ with equal probability. has inversion symmetry, then both− positive and negative It is clear if we start with a microcanonical ensemble temperature can exist in the same system, such as the at ( E, h), we can evolve the system to (E,h) dynam- Ising model. ically.− But− is this an adiabatic process? To answer this The equality of entropies, Eq. (45), and the symme- question we need to ask what is heat and what is work, try argument are presented by Schneider et al [9], using in such a driven process when h(t) varies for an initially canonical distribution and von Neumann entropy expres- equilibrium density matrixρ ˆ. The internal energy of the sion. But as we can see an explicit ensemble and entropy system is E = Tr(Hˆ ρˆ), the differential is definition are not needed for the argument. dE = Tr(Hdˆ ρˆ) Tr(Mˆ ρˆ)dh − = δQ Mdh. (42) − VII. THERMAL EQUILIBRIUM OF AN ISING We interpret the first term associated with distribution MODEL change as heat and second term work. If we say that the change of density matrix is solely from the von Neu- Consider a two-dimensional Ising model, which has a mann dynamics, dρˆ [H,ˆ ρˆ], then the first term is al- bounded spectrum. For energies E > 0, TB < 0, but ways 0. A correct∝ interpretation of change must be DH claim that the correct thermodynamic temperature from re-equilibration of the system at the end of the dy- is TG > 0. namic change. Since the system is already in equilibrium Consider the following two experiments: as given by the microcanonical ensemble (except at the point E = h = 0), there is no need to reequilibrate, and we find that the heat δQ is always 0. Then the adiabatic A. Two Ising models with negative temperatures curves are given by dE = Mdh [33]. − Penrose used a symmetry argument to show negative If 0 >TB,1 >TB,2 when the systems are isolated from temperature must exist in such systems. First of all, each other, the temperatures will change when they are the thermodynamic entropy S is a state function. For a brought into thermal contact. After coming to a new paramagnet, a unique thermodynamic state is specified equilibrium, we will have 0 >TB,1 >TF >TB,2. by E and h, we have S = S(E,h). Since M = E/h If we compare this to an experiment with the same two and M is an adiabatic invariant, it means the adiabatic− systems at T and T , we can see from symmetry | B,1| | B,2| curves are given by E/h = const in the state space. In that the new final equilibrium temperature will be TF . particular, for each pair (E,h), the state ( E, h) is on No paradoxes occur. | | the same adiabatic curve and can be reached− from− each other adiabatically and reversibly, so entropy must be the same, B. An Ising model and simple harmonic oscillators (SHO) S(E,h)= S( E, h). (43) − − Now consider an Ising model at T < 0 and a set Taking the derivative with respect to E, one find B,I of quantum simple harmonic oscillators. The system of 1 1 SHO’s must have TSHO > 0. For simplicity, let ¯hω =2J. = , (44) T (E,h) −T ( E, h) This would make it easy to simulate the model. − − If SB is correct, the temperature of the Ising system i.e., the temperatures of two thermodynamic states are will become positive (higher βI > 0) if the two systems related by an opposite sign. are brought into thermal contact, so that they can ex- The above consideration is a powerful one and very change energy. At the same time, βSHO would decrease general, and certainly not limited to a simple paramag- (higher temperature). net. Given any arbitrary quantum Hamiltonian Hˆ , with DH claim that the true thermodynamic temperature of the only requirement that it has discrete eigen spectrum, the Ising model is TG,I > 0. Test their claim by setting ˆ ˆ we can construct a family of Hamiltonians, Hλ = λH. the temperature of the SHO’s to TSHO = TG,I ˆ ˆ We see that from Eq. (42) that we can connect H+ = H If DH are correct, the Ising system and the SHO’s are to Hˆ− = Hˆ adiabatically, with an arbitrary process at the same temperature, so they should remain in equi- λ(t) that varies− from +1 to 1. This is true for any diag- librium. This is not going to happen. It is both obvious onal initial density matrixρ ˆ−, since the diagonal elements and can be checked by a simple simulation. 10
S/ Nk VIII. AN INTERESTING TOY MODEL ( B )
3 It has occurred to us that we have available a large set of bounded energy models. In Monte Carlo (MC) simulations, it is standard to exclude the momenta, since they can be integrated out exactly. 2 Consider a gas with only configurational degrees of freedom.
H = V (~rj ~rk), (46) − 1 Xj>k where
V (~r)= X exp ~r 2/(2σ2) , (47) −| | 1 3 5 7 9 E/ Nh ν) ( and X and σ are constants. For low positive temperature TB (large βB), the par- FIG. 4. The entropy SG according to DH for harmonic ticles will form a crystal. oscillators with N = 1, 2, 5, 20, 80, and ∞ (from bot- For T = , it will be an ideal gas. tom to top). The limiting curve is given by S/(NkB ) = ∞ (1 + n)ln(1 + n) − n ln n, where n = E/(Nhν) − 1/2. For large TB , with TB < 0, the particles will lie pref- erentially on| top| of each other. This is a state that cannot be realized for positive temperature, although DH claim B. Specific heat of a quantum SHO that it should have a positive thermodynamic tempera- ture TG,r. Now prepare the momentum variables in an equilib- If we’ve read their paper correctly, DH claim that the specific heat of a d = 1 quantum SHO is C = kB . [Three rium state with temperature TG,p = TG,r. According to DH, allowing interactions between the momenta and po- equations after their Eq.(10), but without a number.] sitions should not disturb equilibrium since the kinetic This claim is bizarre. It would be true for a classical and potential degrees of freedom are already at the same SHO, but all quantum systems must satisfy the third temperature, and therefore in thermal equilibrium. law, which requires that C 0 as T 0. The correct expression can be found in any→ textbook.→ It is obvious that the DH claim is wrong, but it would But this is exactly what the oscillator example shows, also be easy to simulate. given a dubious prediction that the quantum oscillator has temperature, hν IX. ONE DEGREE OF FREEDOM ISOLATED kBTG = + E. (49) SYSTEM, OR SMALL SYSTEMS 2 The derivation of this result is questionable [34]. Using DH are in trouble when they discuss small systems, their definition of Ω(E) = Tr θ(E Hˆ ) , the function is especially a single-particle system in one dimension. a series of steps, equally spaced{ both− in} the independent variable E as well as the dependent variable Ω. Taking the logarithm to get SG(E)= kB ln Ω(E) also results in steps, see Fig. 4. Since the step function is a constant for A. Two small classical systems in equilibrium all E except for a set of measure 0 which are the eigen- spectrum of the oscillator, the derivatives are 0 almost Consider two such systems, each of which is a 1D everywhere. Thus the temperature TG is infinite, almost single-particle ideal gas, in thermal equilibrium with each everywhere, just like when the Boltzmann entropy is ap- other, but isolated from the rest of the world. The prob- plied. In passing we also note that SG(E) violated part ability distribution of the energy E1 of the first system of Callen’s postulate III, as it is not a ‘continuous and is differentiable’ function. To get out of this mess, one has to consider systems 1 with N = 1, 2, 3, , oscillators in the macrocanonical P (E1)= . (48) ··· π E1(ET E1) ensemble, not just N = 1. One plots S/Nk vs E/N, − B as in Fig. 4. On this scale, the steps become smaller p This has divergent maxima at E1 = 0 and E1 = ET . To and smaller as N becomes larger and larger, and the demand equipartition seems strange. curves become smoother and smoother. In the N → ∞ 11 limit, the limiting curve is a continuous function, and 1 T derivatives start to make sense. That is, temperature B 0Q N1=1 only make sense in the thermodynamic limit. 0.8 0Q N1=10
0Q N1=100
TG N1=1 T N =10 X. RANDOM REMARKS ON SG AND TG 0.6 G 1 T N =100 2 G 1 /N 2 E Contrary to the authors’ claims, it is SG and TG that 0.4 is inconsistent with thermodynamics. Firstly, let us con- sider a thought experiment with the harmonic oscillator. We can weakly couple it with a heat bath of macroscopic 0.2 in size, then it is clear the oscillator will be in a mixed state described by the canonical distribution. Now we 0 reduce the temperature T of the bath to zero. The os- 0 0.2 0.4 0.6 0.8 1 E /N cillator will be cooled down to a temperature of absolute 1 1 zero and stays in its ground state. Suppose now we turn off the coupling between the oscillator and the bath, then FIG. 5. Continuous energy two-level model. The lines of en- there is good reason to believe that the oscillator will still ergy per particle E1/N1 vs. E2/N2 of two subsystems deter- be in ground state if we turn off the coupling very slowly mined by equality of Boltzmann temperatures, labelled TB , and equality of Gibbs temperatures, labelled TG, as well as (quantum adiabatic theorem). Since the system is now the lines determined by average energies of the combined sys- isolated and the state is in n = 0, it is in a microcanon- tems (lines of zero heat transfer), labelled 0Q, with N1 = 1, ical ensemble with E = hν/2. According to Eq. (49), 10, 100, and N2 = 2N1. then, the temperature of the oscillator becomes hν/kB. It appears then, by a mere change of the mind of the ex- perimentalist that the system was in canonical ensemble the sizes of the systems become larger, we see clearly dis- and now in a microcanoical ensemble, the temperature of tinguished role that Boltzmann temperature can play to the oscillator has been changed dramatically. The experi- determine the direction of heat flow while Gibbs temper- mentalist can do the single oscillator experiment with CO ature can not, for systems with bounded energy spectra. molecule which has a vibrational frequency of 2170 cm−1. In Fig. 5, we present a similar plot to the figure 7 in Ref. 5 Then the associated temperature is TG = 3122K. The but with one additional parameter, the size Nj , j = 1, temperature of the molecule has jumped from 0 K to a 2. Here we consider a two-level system with ‘continuous thousand kelvin just by changing the point of view of his energy’ such that the Boltzmann entropy is given by (set system. One could argue that we are talking about two kB = 1) different temperatures, TB and TG. But thermodynamics has only one kind of temperature. SB = N ln N E ln E (N E ) ln(N E ), (50) j j j − j j − j − j j − j Another example, consider putting a hydrogen atom in with 0 E N , j = 1, 2. Thus, the density of a box, in its ground state it will be extremely hot with a ≤ j ≤ j 5 states is given by ω = exp(SB ), and the integrated TG of order 10 K, while it is cooler in the excited states. j j Ej ′ ′ Another excellent example is given by contacting a finite one Ω(Ej ,Nj ) = 0 ωj (E )dE . The lines marked 0Q spin chain at (the traditional) negative T with an ideal are the lines with no energy exchange on average if sys- R gas, as discussed by Frenkel and Warren [11]. A 3rd ex- tem 1 and 2, having energies E1, E2 with total energy ample is that heat flows from cold to hot, according to E1 + E2 fixed, are combined and allowed to exchange the definition of TG (see Vilar and Rubi, Ref. 10). Heat energy. The 0Q lines approach rapidly the line deter- flow from hot to cold naturally is one form (Clausius) of a mined by the equality of the Boltzmann temperatures, statement of the second law of thermodynamics. If this is T1(E1,N1) = T2(E2,N2) (which is given on this plot as not obeyed, it simply means that the definition or quan- a universal curve E1/N1 = E2/N2)). The 0Q lines do titative measure of the hotness or coldness is wrong, or not approach the lines determined by equality of Gibbs perhaps thermodynamics cannot apply for such systems. temperatures in the region when the Boltzmann temper- We also observe that heat flowing from hot to cold is an atures are negative. Thus, thermodynamics applicable immediate consequence of the Callen’s second and third only to large systems is not “dogmatic”, but is really postulates. This is discussed in Callen’s book, Sec. 2-4 forced upon us, as small systems do not obey thermody- and 2-5. It is clear these discussions apply equally well namic laws. when temperature is negative. The use of Gibbs entropy and temperature gives a In HHD (Ref. 5) it has been argued forcefully that nei- meaningless result in the traditional negative tempera- ther Gibbs temperature nor Boltzmann temperature can ture thermodynamic states, e.g., take a 1D Ising chain determine the direction of heat flow. This is certainly in a microcanonical ensemble. Then, according to DH, true for tiny systems as several examples show, such as SG/N = kB ln 2, and TG = , in the thermodynamic that indicated by Fig. 7 in Ref. 5. However, as soon as limit, for all E > 0; the thermodynamic∞ heat is then 12
T dS = 0 which does not seem to be a useful concept. of entropies of these states. ∞ · We can elaborate a little more for the last point. To perform an adiabatic change, we modulate the spin Take two segments of sufficiently long ferromagnetic Ising coupling J with a time-dependent protocol, J(t). Such chains with periodic boundary conditions, one with a changes do work to the system but no heat is transferred complete antiferromagnetic spin configuration, + + from environment to the system, since the individual de- − − + + + , corresponding to the highest possi- grees of freedom σi are not manipulated. Specifically, we ble− energy,− · · · the− second one with some fraction of the turn J off gradually (or otherwise), and wait sufficiently spins flipped over, forming ++ or pairs, such as long for the system to reequilibrate at the coupling J = 0, + + + + + + . In−− terms of domain and then turn it back to J suddenly. If we start from F walls− of +− and − boundaries,− − − · the · · equal sign pairs form or A, it ends in R. Thus we must have S(F ) S(R), ≤ elementary excitations− of low energy. A microcanoni- S(A) S(R). Can we take the state R and change it ≤ cal Monte Carlo dynamics can be given by flipping spins adiabatically to F or A? If we use the above protocol whose neighbors are opposite in signs, which moves the J(t), we see it is very unlikely if not impossible if N is domain walls around. If we put these segments together, finite. But in the thermodynamic limit R will remains it is clear that the elementary excitations will spread out in R, i.e., F to R or A to R is adiabatic irreversible, so to the whole combined system to maximize the Boltz- S(A) < S(R). This contradicts the assignment of high- mann entropy. But according to DH’s definition of SG, est possible Gibbs entropy to state A as thermodynamic this should not preferably happen, as the system before entropy. and after combination has the same entropy per spin, kB ln 2. The above example shows that the Gibbs entropy vi- XI. ENSEMBLE EQUIVALENCE MEANS olated Carath´eodory’s formulation of second law, which THERMODYNAMICS says one cannot have an entropy function that is a con- stant in any neighborhood of an equilibrium state. It also The authors maintain that ensembles are generally not invalidates Einstein theory for fluctuation. The proba- equivalent (particularly if negative T occurs for systems bility of certain amount of fluctuation, according to Ein- of bounded energies). If so, there would be two sets of stein, is proportional to e∆S/kB , so macroscopic fluctu- thermodynamicses. Clearly, in the traditional sense of ations with entropy difference of order N is extremely thermodynamics, we have only one unique one. So in rare. But if entropy is a constant, then the fluctuations order to be consistent with thermodynamics, different of the type of configurations discussed above is very com- ensembles have to be equivalent [35]. This happens in mon, which is clearly not so. One can verify this just by the thermodynamic limit. putting the system on computer and simulate. We now consider the last example of computer exper- iments. Take again the 1D ferromagnetic Ising chain of XII. ADDITIVITY OF ENTROPY VS N spins with periodic boundary condition and a dynam- POSITIVITY OF TEMPERATURE ics which conserves energy. We consider thermodynamic states and their relationship with respect to adiabatic The authors gave up one fundamental property of ther- changes. Here the ‘adiabaticity’ is in the sense of Lieb modynamic systems, that is, the extensivity and additiv- and Yngvason [19], which does not implies it has to be ity (since the thermostatistical equations are supposed to slow or gentle. We begin with an equilibrium state X be valid for any size N). An example of two-state identi- and end with another equilibrium state Y with the sole cal boson system (is there such a thing in nature?) was effect that work is done to or by the system. The ‘en- shown by the authors that SG is not additive. Why it tropy principle’ states that entropy of the system must is reasonable to insist on positivity of T but to break not decrease, S(X) S(Y ). the additivity? Which property is more fundamental in ≤ We consider three thermodynamic states labelled by thermodynamics? If we insist that additivity of entropy the energies E = JN, the ground state (F), the ran- and other extensive thermodynamic variables must be dom state E = 0− (R), and the anti-ferromagnetically maintained, then a single quantum oscillator system, or ordered state E = +JN (A). The Gibbs en- a particle in a box, for that matter, cannot be viewed tropies are S ( JN) = ln 2, S (0) = ln 2N−1 + as thermodynamic systems, as there is nothing there to G − G N!/((N/2)!)2 N ln 2 (assuming N/2 even), and add. The additivity is clearly needed in order to fulfill ≈ SG(+JN) = N ln 2, and accordingly the Boltzmann the zero-th law of thermodynamics. entropies are SB( JN) = SB(+JN) = ln 2, and S (0) = ln N!/((N/− 2)!)2 N ln 2. We have the rela- B ≈ tion SG( JN) < SG(0) < SG(+JN), and SB( JN) = XIII. CONCAVITY AND SUPER-ADDITIVITY − − SB(+JN) This equation shows that Gibbs entropy is super-addivity and Gibbs and if the system 1 and 2 refer to the same system (but 1,2 (2N) can be difference sizes); it is equivalent to concavity. The S = ln(2 1). (58) G − energies are conserved, E1,2 = E1 + E2. The assumption that internal energy is additive makes sense for macro- The entropy increases, are scopic objects, but not realistic for microscopic systems ∆S = S1,2 S1 S2 = ln 2 > 0, (59) where the interaction is strong comparing to the energies B B − B − B themselves saparately. and Gibbs We note, firstly, that Eq. (51) that HHD has proved is generally a greater than sign ‘>’ not greater or equal 1,2 1 2 N ∆SG = SG SG SG = ln(1 + 1/2 ) > 0. (60) ‘ ’ sign. This is so for most systems where the density of − − states≥ are strictly positive for positive energies, such as an So for both definitions, entropies increase, so “second ideal gas, and also so for most of the parameters if energy law” is obeyed for both. But the actual numerical values is bounded as well. This means putting two systems to- are very different – Boltzmann entropy is a finite amount gether always leads to a strict increase of Gibbs entropy, of 0.69, the Gibbs entropy extremely small for a large and splitting up causes a decrease of entropy. Secondly, system. The temperatures are also very different (us- this equation is not what Planck has in mind. Planck ing approximation T δE/∆S), TB ǫ/ ln 2 = O(1), N ≈ ≈− has been always thinking in terms of two systems and negative; TG =2 ǫ . Which one makes thermody- making them in thermal contact and then putting them namic sense? → ∞ apart (see, Sec.122-126 of Planck’s “Treatise on Thermo- dynamics” [36]). So the proper way to write Planck’s expression of the second law should be XIV. CALLEN’S POSTULATE II S′ + S′ S + S , (52) 1 2 ≥ 1 2 Concavity is a weaker condition than being entropy postulated by Callen, as we cannot see why concavity when the system changed from the non-prime states to implies Callen’s postulate II. This postulate in terms of the primed states adiabatically. Of course, Boltzmann math, means entropy also has the same strictly ‘>’ sign, but if ther- modynamic limit is taken (as it can be identified with S1,2(E) = max S1(E1)+ S2(E E1) , (61) the thermodynamic entropy only when the size is large), E1 − then we will not have problem. 1 2 Consider quantum two-level system (equivalent to a E = E + E is the total energy which is constrained between system 1 and 2, which is also the total energy of paramagnet) with energies 0 and ǫ, respectively, of dis- 1 N the combined system, (1,2). The value E for which the tinguishable particles. H = n ǫ, n =0, 1, ǫ> 0. i=1 i i right-hand side obtains its maximum, E1 , is the value System 1 at highest energy E1 = Nǫ, the Boltzmann max system 1 will taken when the constraint is removed. entropy is, P S1 = ln 1 = 0, (53) B A. A numerical test and Gibbs We use the same non-interacting two-level system 1 N model as in Sec. XIII, i.e., for a single particle the en- SG = ln 2 , (54) ergy is 0 in ground state and ǫ in excited state. We set taken kB = 1. ǫ = 1 for numerical convenience. The entropies are Second system 2 at the second highest energy E2 = N! (N 1)ǫ, one particle must be in the lower energy state, SB(N, E) = ln , 0 E N, (62) the− rest N 1 in high energy state. So E! (N E)! ≤ ≤ − − 2 and SB = ln N, (55) E N! and Gibbs S (N, E) = ln . (63) G k! (N k)! k=0 2 N X − SG = ln(2 1). (56) − Since ǫ = 1, the energy E also takes integer values. If we combine the two systems to get (1, 2), the energy Consider two systems, system 1 with particle number 1,2 is still the second highest, E = (2N 1)ǫ. So the N1, and second system 2 of identical system except the − entropies of combined systems are number of particles is N2 =2N1, twice larger. We set the 1,2 4 energy of the combined system (1, 2) to E = (N1 + 1,2 5 SB = ln(2N), (57) N2). This will make the combined system with negative 14 1 TB. We ask, for what value of E , the right-hand side of bodies are combined. How much it increase? Eq. (61) is a maximum? This table shows the numerical thermodynamics alone cannot tell. answers: The above formulation of thermodynamics may well be correct for small systems, but there is very little pre- 1 1 size N1 Emax from SB Emax from SG % error diction power it can offer. There is not much one can do 5 4 4 0% with it. For example, the formulation has no way to tell 10 8 9 11% us which direction heat flows if two bodies are made in 50 40 43 7.5% thermal contact. There is no numerical relation of the entropy of the combined system (1, 2) with that of sep- 100 80 87 8.7% arated systems 1 and 2, uncoupled. One cannot analyze 500 400 433 8.2% the efficiency of a Carnot engine, as the formulation does 1000 800 867 8.4% not have provision for the thermal coupling/decoupling of the system and the baths. The way to try to understand why thermodynamics Clearly, when the two systems are combined with a par- has to be formulated in such a way is that the authors ticle number 3N , the excitation energies should be dis- 1 trying to fit the bill of Gibbs S . The Gibbs entropy does tributed evenly to the original two systems to maximising G satisfy all the properties listed above [37]. The ‘thermo- entropy, i.e., it should give E1 = (4/5)N , at least for max 1 dynamic laws’ are invented to ‘check’ that S satisfies large enough systems. The Boltzmann entropy predicted G them. Clearly, the argument is logically circular; and this down to small sizes 5 correctly, while Gibbs entropy that S satisfies the thermodynamic laws is a misleading always gives more energy to system 1 and less to system G claim. 2. The right-most column in the table gives the percent- One may argue that we are misrepresenting things, as age errors made in the energy. The deviations do not the above formulation is for isolated systems. Even so, seem to decrease as N goes to infinite (thus it is not 1 why it is useful at all? a 1/N effect). Thus, we have demonstrated numerically that SG violated Callen’s postulate II. The locations of the maxima are associated with the XVI. NONEQUILIBRIUM equality of respective temperatures. An exact calculation STATISTICAL-MECHANICAL ANALYSIS OF can be carried out for the equilibrium values of energy THE COUPLING/DECOUPLING PROCESSES 1 N1 1,2 of system 1 to be E = N1+N2 E in the combined system. The heat exchangeh i between the two subsystems, We give an analysis as what happen if we couple and Q = E1 E1 = 0 if we start at the energies predicted max decouple two identical equilibrium systems, from statis- by theh Boltzmanni− entropy being largest, and Boltzmann tical mechanics point of view and see if entropy increase temperatures being equal. or decrease. But from HHD point of view, this violation of Callen’s a) Consider two identical boxes of ideal gases of macro- postulate II is irrelevant as they rejected Callen’s formu- scopic sizes with exactly the same macroscopic measur- lation of thermodynamics. able properties, i.e., the same thermodynamic state. If there is a partition separating them that can be open or closed, does the total entropy change? Of course not, XV. HHD THERMODYNAMICS according to the usual thermodynamics. In fact this is one of the fundamental axiom in Lieb-Yngvason’s formu- HHD proposed a thermodynamics which we express as lation of thermodynamics (Ref. 19, page 21, axiom (A5), the following: splitting and recombination). The entropy of macro- scopic observables does give zero entropy change by defi- Zeroth law: temperature of the parts is nition [12, 22–25]. This contradicts the HHD second law. the same as the temperature of the whole b) Now we consider systems as small statistical me- when in thermal contact. Note that it is for- chanical systems described by either a classical Hamil- bidden to ask what the temperatures will be tonian dynamics or quantum dynamics for the distribu- if the parts are separated. In fact, the parts tions, ρ. Open or closing of a partition can be modeled get different Gibbs temperatures if one does by a time-dependent external potential. First of all, if so. But thermodynamics alone cannot pre- we start with a product of microcanical distributions, dict them. ρ1ρ2, of two separate systems (let’s call it statistical- First law: there is no dispute on this; we mechanical state P ), it is not possible to evolve into have energy conservation, dE = δQ + δW , a new microcanonical distribution of combined system δQ = T dS. However, even here, T and S are (call it C), simply Liouville’s theorem prevents the value not state functions. of probability ρ to be changed along the dynamic tra- Second law: entropy strictly increase jectory. Also when separating the system, it is also not (generically), i.e., S1,2 > S1 + S2, when two possible to evolved back to a product state, P . The von 15 Neumann entropy is a constant in any events. So the Siii(D) SG(C) < 0, i.e., entropy does not change question about entropy increase or decrease cannot be if Boltzmann− entropy for state C is used and entropy answered in such a framework. still decreases if Gibbs entropy is used. Curiously, C In order to have a Boltzmann-like H-theorem, one need to D is a reversible process or recoverable - removing to use a coarse-grained description [22, 28, 38, 39], which the parition will make the system back to C, so en- essentially lead to: tropy does not change make good sense. (iv) In the c) a third option, a stochastic dynamics of Markov last case we can consider the expression given by [40, 41] chain. In this framework the problem is well-posed. In ρ ln Ω(E ) = ln Ω , where E is the energy of the i i i h i i such a description, we can make transitions from P to C system in micro-state i with probability ρi. With the dynamically which lead to an entropy increase. Still C to Pmicrocanonical constraint, all states have the same en- P is not possible. If we start from the combined system ergy, so it is a constant with the value SG(C); entropy C in a microcanonical distribution, the probability dis- does not change. Unfortunately, it is a constant no mat- tribution ends up to a distribution of energy partitions ter what initial distributions to begin with. So, it cannot of the product microcanonical distributions, let us call describe relaxation towards equilibrium. this state D, the decoupled but still entangled state, so To conclude, if we believe that the opening or closing C D. → a partition is an adiabatic process (possibly performing To think of this with a concrete example, we can still work but no heat exchange), then entropy must not de- take the paramagnet or equivalently the two-level sys- crease. Cases (i)-(iv) show that Gibbs entropy contra- tem, with energy ǫ in the excited state and 0 for the dicts Planck’s formulation of the second law, or at least, ground for each particle. To give a meaning to spatial lo- Gibbs entropy cannot make any definite statement as it cality and partition, we imagine that the spins/particles is not properly defined for state D. The fact that we form a 1D chain. The dynamics is simply to pick two cannot have unambiguous interpretation of entropy in a nearest neighbor particles at random and swap the loca- state D only demonstrates that thermodynamic entropy tions. This is an energy (and magnetization) conserving is a macroscopic concept. The contradictions and ambi- process. Let us consider two identical systems with N guity disappear for large systems. particles each and one excited particle each with total energy E = ǫ. The entropy increases when two systems are combined are, ∆SB = SB(C) SB(P ) = ln(2 1/N), 2 − 2 − and ∆SG = ln (2N + N + 1)/(N + 1) , respectively. For both definitions of entropy, the increase approaches XVII. NON-MONOTONIC DEPENDENCE OF T ln 2 for sufficiently large N. WITH ENERGY E If we split the system from state C by adding parti- tion, it evolves towards D. It is not clear at all how to HHD cited reasons not to use temperature TG or TB to quantify the entropy. We give four possibilities: (i) we judge if heat can flow from one body to another or not. know exactly what the energies are of each subsystems Both of them fail to do so for tiny systems. The reason — such is the case if we simulate the process on com- for this failure is because they insists that thermodynam- puter, and check exactly how many excited particles are ics works for any size N. If one consider only thermo- in one of the partitions. Then total entropy decreases. dynamic limit result (in the sense of omitting any con- (ii) If we are not allowed to measure the energy or our tributions smaller than N for extensive quantities such measurement is not precise, the total entropy still de- as energy and entropy, and omitting quantities of order creases. For example, the total entropy may be defined 1 2 1/N or log(N)/N for intensive quantities such as tem- by Sii(D)= j pj (Sj + Sj ), where pj is the probability perature), thermodynamics works just fine. Both zeroth that subsystem 1 get an energy E1 which can be calcu- P j and second law can be satisfied which define the thermo- lated from the initial microcanonical distribution of sys- dynamic system. tem C. Then, (iii), we can apply the Shannon entropy, The internal energy E is a monotonic function of β ρ ln ρ = ln ρ , over the whole state space. Since for large systems, so increasing temperature always lead− the− probability−h of ai particular microscopic configuration to an increase of the internal energy. This can be seen is theP probability p of getting split into energy E1, times j j easily in the canonical ensemble, since in the canonical the probability in a particular microscopic state of the ensemble the heat capacity measures the fluctuation of joint system, which is equal probable in each subsystem, the energy, we can write, Siii(D)= SB,ii(D)+ pj ln pj . (64) dE 1 − 2 2 j C = = 2 H H , (65) X dT kBT h i − h i This is higher comparing to case (ii) by an amount due to the random selection of energies. For the two par- which is always positive, and one can verify that the heat ticles in a 1D box of size 2N problem, one finds that capacity is always a positive quantity, even for the nega- S (D) S (C) = 0 (this is a general result), and tive Boltzmann temperature states. iii − B 16 XVIII. WHY INSISTS ON SG? XIX. FINAL REMARK If canonical ensemble were used to treat the harmonic By postulate S as the thermodynamic entropy, we got G oscillator, an explicit thermodynamic limit is not neces- 1) positivity of T (but why this is important), 2) concav- sary as the free energy is already a homogeneous func- ity and super-additivity of S . But breaks many things: G tion of particle number even for one and two particles. 1’) ensembles no longer equivalent, even for a paramag- It is clear we are already in the“thermodynamic limit”; net, 2’) temperatures of individual systems and combined the specific one calculated for one particle in a canonical system are not the same, and violating the usual 0-th law ensemble is just one of the many. However, in the mi- of thermodynamics. 3’) Violation of Callen’s postulate II. crocanonical ensemble, situation is different and taking 4’) meaningless thermodynamics when TB < 0, which are the limit is a necessary step in order to have an entropy T = , S = const, heat capacity C = 0. One cannot G G expression that is consistent with thermodynamics. write down∞ a meaningful thermodynamic relation as it is 0. 5’) thermodynamics losses its predicting power in ∞ · the sense of Callen’s postulate II since SG is a constant. ACKNOWLEDGEMENTS In addition, when applied to single quantum degree sys- tems (we believe one cannot do that for both SB and S ), one violates 6’) third law of thermodynamics. The author thanks Robert H. Swendsen for many use- G ful exchanges and even contributions to some of the pas- HHD’s deduction has a logic fallacy: thermodynamic sages in this critique. But the opinions are the author entropy must satisfy zeroth, first, and second law. Gibbs only. He also thanks Peter H¨anggi for an inspiring talk, volume entropy does not satisfy HHD’s version of zeroth although we disagree on most of the issues discussed here. law (average temperatures of subsystems when in ther- Finally, I sincerely apologize to both of them that this mal contact equal the temperature as a whole), when work has caused unnecessary damage on an emotional energy of the system is bounded. Therefore, there is level and damage to our long-lasting friendship. 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