Thermodynamic constraints on fluctuation phenomena
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Thermodynamic constraints on fluctuation phenomena
O J E Maroney ∗ The Centre for Time and The School of PhysicsUniversity of Sydney NSW 2006 Australia andPerimeter Institute for Theoretical Physics31 Caroline St N, Waterloo, ON, N2L 2Y5, Canada (Dated: June 16, 2018)The relationships between reversible Carnot cycles, the absence of perpetual motion machinesand the existence of a non-decreasing, globally unique entropy function forms the starting point ofmany textbook presentations of the foundations of thermodynamics. However, the thermal fluctu-ation phenomena associated with statistical mechanics has been argued to restrict the domain ofvalidity of this basis of the second law of thermodynamics. Here we demonstrate that fluctuationphenomena can be incorporated into the traditional presentation, extending, rather than restricting,the domain of validity of the phenomenologically motivated second law. Consistency conditions leadto constraints upon the possible spectrum of thermal fluctuations. In a special case this uniquelyselects the Gibbs canonical distribution and more generally incorporates the Tsallis distributions.No particular model of microscopic dynamics need be assumed.
PACS numbers: 05.70.-a,05.40.-a
I. INTRODUCTION
The existence of a globally unique entropy as a func-tion of thermodynamic state, which is non-decreasing intime, is one of the central tenets of classical phenomeno-logical thermodynamics[1, 2]. By contrast, the meaningof entropy within the context of statistical mechanicsseems to defy consensus(see [3, 4] for examples). Sincethe start of statistical mechanics there has been con-cern that the existence of fluctuation phenomena leadsto violations of the second law of thermodynamics. Thismay lead to decreases in entropy, the existence of per-petual motion machines or maybe even the inability todefine an entropy at all. Maxwell’s demon representsa persistent strand of thought experiments dedicated toexploring these possibilities[5, 6, 7].Most attempts to construct a second law of thermo-dynamics for statistical mechanics involve one of twostrategies: restrict the domain of validity of the classicalstatement (usually to reliable, continuous processes) soas to exclude fluctuation phenomena; or to attempt toderive a new second law within the domain of statisticalmechanics. Here we investigate the possibility of a thirdapproach: to extend the domain of the phenomenolog-ical second law to include, constrain, and predict theextent of the fluctuation phenomena, which reduces tothe more familiar version if fluctuation phenomena areabsent. We find that such an extension seems, in princi-ple, possible, and that with additional work it is possibleto define an entropy function consistent with this. Somepossible relationships of this fluctuation second law toconventional statistical mechanics can be inferred.The approach of the paper is as follows. Section 2briefly reviews the equivalence of the Kelvin, Clausius ∗ Electronic address: [email protected] and Carnot versions of the second law of thermody-namics. Section 3 then proposes an extension of theKelvin version, to incorporate fluctuation phenomena.Logically equivalent generalisations of the Clausius andCarnot versions are deduced, and some constraints arededuced about the form of the extended second law. Sec-tion 4 reviews the derivation of an entropy function andshows when the existence of a fluctuation entropy func-tion can be deduced. Finally Section 5 considers somerelationships to statistical mechanical entropies, includ-ing the Gibbs and Tsallis[8] entropies.
II. PHENOMENOLOGICAL SECOND LAW
Textbook versions of the Second Law of Thermody-namics (see, for example, [9, 10]), when expressed interms of heat flows and heat baths, take forms such as: • Kelvin : No process is possible whose sole resultis the extraction of heat from a heat bath and itsconversion to work. • Clausius : No process is possible whose sole re-sult is the transfer of heat from one heat bath toanother heat bath at a higher temperature. • Carnot Heat Engine : No heat engine operatingbetween heat baths at temperatures T < T canoperate at an efficiency n E exceeding the efficiencyof a reversible heat engine: n E ≤ n CE = 1 − T T • Carnot Heat Pump : No heat pump operatingbetween heat baths at temperatures T < T canoperate at an efficiency n P exceeding the efficiencyof a reversible heat pump: n P ≤ n CP = T T − T Demonstration of the logical equivalence of each pairof these statements can easily be found in a textbooksuch as [10]. The equivalence is typically proven by the (a) n P = Q p W p ≤ n CP (b) n E = W e Q e ≤ n CE FIG. 1: Reliable Heat Pumps and Engines means of diagrams such as in Figure 2. This diagramshows the combination of heat engine and heat pumpsbeing used to attempt violations of the Kelvin and Clau-sius statements. Figure 2(a) shows that if a heat pumpcan operate with efficiency n p = Q c W p > n CP = Q c W c ,then in combination with a reversible heat engine op-erating at n CE = W c Q c there is a net conversion of W c − W p > n e = W c Q e > n CE = W c Q c can be combined witha reversible heat pump operating at n CP = Q c W c couldtransfer heat Q c − Q e > (a) (b) FIG. 2: Equivalencies of Violations of Second Laws requires a number of usually unstated assumptions, suchas the absence of negative temperatures. In particular,the equivalence requires it to be physically possible toconstruct a reversible heat engine or pump. For exam-ple, if it were not physically possible to build a heatengine whose efficiency could reach that of a theoreti-cal reversible heat engine, then it would not necessarilyfollow that a real heat pump exceeding the Carnot ef-ficiency could violate the Kelvin or Clausius versions of the second law .The problem arises that fluctuation phenomena, suchas Brownian motion, do, in principle, violate all thesestatements of the second law, when probabilistic pro-cesses are allowed. Attempts to define a modified sec-ond law will typically restrict the domain of validity. Itmay be suggested that the second law only applies tothe thermodynamic limit of an infinite number of atoms,where fluctuations become negligible, or it may be sug-gested that the second law only applies to continuous orreliable processes: • No reliable process is possible whose sole result isthe extraction of heat from a heat bath and itsconversion to work. • No process is possible with probability one , whosesole result is the extraction of heat from a heatbath and its conversion to work. • No continuously operating process is possiblewhose sole result is the extraction of heat from aheat bath and its conversion to work. • No process is possible whose sole result is, on av-erage , the extraction of heat from a heat bath andits conversion to work.Restricting the domain of validity in this way, how-ever, proves unable to provide answers to many interest-ing questions about the thermodynamic consequences offluctuation phenomena. Can systems with a finite num-ber of atoms be used to continuously, reliably convertheat to work? If a process can succeed with probabil-ity less than one, how much work can be extracted? Ifa process only operates for a finite amount of time howmuch work can be extracted? Can it be arbitrarily large?Can a process exist which can extract an arbitrarily largequantity of work with probability arbitrarily close to one,while still failing on average due to catastrophic failurewhen it does fail?This can be illustrated by considering a hypotheticalfamily of processes, parameterised by
N >
1. Process N will, with probability 1 − N , generates N units of workfrom heat, but with probability N it requires N units ofwork to be dissipated. The mean work produced is − N , but as N → ∞ arbitrarilylarge amounts of work are produced with probability ar-bitrarily close to one. Even more extreme examples caneasily be constructed. Such a family of processes satisfiesseveral of the restricted laws above, but does not accordwith our experience of fluctuation phenomena. Suppose for all real heat engines n e ≤ n max < n CE . All thatcould be implied would be that the efficiency of real heat pumpswere bounded by n p ≤ n max but n max > n CP . Note that sucha heat pump, with n p > n CP , would not be possible to operatereversibly as a heat engine. III. FLUCTUATIONS AND THE SECOND LAW
In this Section the main argument of the paper willbe explored. Rather than follow the path of the modi-fications in Section II, restricting the domain of validityof the second law so as to exclude fluctuation phenom-ena, it will instead be expanded to include fluctuationphenomena. Fluctuations will be treated as being prob-abilistic processes, occurring with probability less thanone. The modified law should set a constraint upon thesize of fluctuations that can occur, and should reduce tothe fluctuation-free second law when only deterministicprocesses occur.The proposed modification to the phenomenologicalsecond law is based upon nothing more than the obser-vation that the greater the size of the fluctuation, theless probable its occurrence. From this it is proposedthat, for a given size of fluctuation, there is a maximumpossible likelihood of it occurring:There is no cyclic process , whose sole re-sult is the extraction of a quantity of heat, Q , from a heat bath at temperature T , andits conversion to work, which can occur withprobability p , unless: p ≤ f ( Q, T ) (1)where f is a function whose properties will be deducedfrom internal consistency. The definition is such that itis assumed for any given Q and T there exists an ac-tual physical processes which can get arbitrarily close tooccurring with probability f ( Q, T ). If not, then theremust exist a lower value of f that should have been usedinstead.It is possible to immediately note some properties of f :as the function bounds a probability, it cannot becomenegative; it is always possible to dissipate work as heat; ifthere is a process that extracts Q ′ > Q with probability p , then by also dissipating work W = Q ′ − Q , thereis a process that extracts Q with probability p . Theseimmediately constrain the function: f ( Q, T ) ≥ f ( Q, T ) = 1 ∀ Q ≤ f ( Q, T ) ≥ f ( Q ′ , T ) ∀ Q ′ > Q (4)The last condition implies that if f is also a differentiablefunction of Q , then ∂f∂Q ≤ ∀ Q (5) When discussing probabilistic cycles, a cyclic process will meana process which returns to its original state with probability p ,but with probability 1 − p may end up in a different state to itsstarting point. One trivial solution would be: f ( Q, T ) = 0 , ∀ Q > f ( Q, T ) = 1 , ∀ Q wouldimply one could get arbitrarily close to any size of fluc-tuation, at any probability.This is a more restrictive condition than the meanconversion of heat to work over cycle being negative,although it does imply it. The proof of this is straight-forward. If there exists a process which can produce apositive expectation value for production of work overa single cycle, then repeating that cycle a large numberof times produces an expectation value as large as onelikes, with a gaussian spread around that mean. Theprobability that any given quantity of work can be ex-ceeded becomes close to one. Hence any process whichcan produce a positive expectation value for work will,on repeated application, exceed any function f < W work beingextracted from a heat bath at temperature T . (a) (b) FIG. 3: Kelvin and Clausius Fluctuations
The equivalence of Kelvin fluctuations to other kindsof fluctuations will now be demonstrated.
A. Kelvin and Clausius Fluctuations
A Clausius fluctuation, as in Figure 3(b), will denotethe spontaneous transfer of Q work from a heat bath at T to a heat bath at T > T occurring with a maxi-mum probability f C ( Q, T , T ). One way to achieve aClausius fluctuation is given in Figure 4(a), combininga Kelvin fluctuation with a reliable Carnot pump op-erating at efficiency n CP = QW = T T − T . This can oc-cur with probability f ( W, T ), so f C ( Q, T , T ) cannot beless than this: f C ( Q, T , T ) ≥ f ( W, T ) = f ( Qn CP , T ).A Kelvin fluctuation can similarly (Figure 4(b)) be cre-ated from a Clausius fluctuation, by allowing the heat Q from the Clausius fluctuation to drive a reliable Carnotengine at efficiency n CE = WQ = 1 − T T . This implies f C ( Q, T , T ) ≤ f ( W, T ) = f ( Qn CE , T ) and n CE = (a) (b) FIG. 4: Converting Kelvin and Clausius Fluctuations n CP establishes f C ( Q, T , T ) = f (cid:18) Qn CP , T (cid:19) = f ( Qn CE , T )= f (cid:18) Q (cid:18) − T T (cid:19) , T (cid:19) (6) (a) n P = QW > n CP (b) n E = WQ > n CE FIG. 5: Fluctuation Heat Pumps and Engines
B. Kelvin, Clausius and Heat Pump Fluctuations
A fluctuation heat pump (Figure 5(a)) is a heat pumpthat is able to operate with a higher efficiency than areversible Carnot heat pump, but only with a probabil-ity less than one of success. The maximum probabil-ity of success, f P ( W, n P , T , T ) of achieving efficiency n P = QW > n CP can be deduced either from the Kelvinfluctuation law (Figure 6) or the Clausius fluctuation law(Figure 7). In the Figure 6(a), creating a fluctuationpump with efficiency n P = QW > n CP , by augmentingthe behaviour of a regular Carnot pump with a Kelvinfluctuation shows f P ( W, n P , T , T ) ≥ f ( Q , T ) . InFigure 6(b), creating a Kelvin fluctuation of size Q , byextracting the heat pumped by fluctuation heat pumpat efficiency n P = QW > n CP , and using it to drive a (a) (b) FIG. 6: Kelvin Fluctuations and Fluctuation Heat Pumps
Carnot heat engine gives f ( Q , T ) ≥ f P ( W, n P , T , T ).Substituting Q n CP = W ( n P − n CP ) gives f P ( W, n P , T , T ) = f (cid:18) W (cid:18) n P n CP − (cid:19) , T (cid:19) (7)Figure 7(a) augments the Carnot heat pump with a (a) (b) FIG. 7: Clausius Fluctuations and Fluctuation Heat Pumps
Clausius fluctuation of size Q to create a fluctuationpump of efficiency n P = QW > n CP . Now using the workextracted from a Carnot engine to drive a fluctuationheat pump, gives a Clausius fluctuation in Figure 7(b).Combined f P ( W, n P , T , T ) = f C ( Q , T , T ) with Q = W ( n P − n CP ), so f P ( W, n P , T , T ) = f C ( W ( n P − n CP ) , T , T ) (8)It can be easily confirmed that this is consistent with therelationship f C ( Q, T , T ) = f ( Qn CP , T ). C. Kelvin, Clausius and Heat Engine Fluctuations
Similarly, a fluctuation heat engine (Figure 5(b)) isa heat engine that can operate with a higher efficiencythan a reversible Carnot heat engine, but only with aprobability less than one of success.Augmenting a Carnot heat engine with a Kelvin fluc-tuation of size Q , Figure 8(a), creates a fluctuationheat engine, while using the heat pumped by a reg-ular Carnot pump to drive a fluctuation heat engine,Figure 8(b), creates an equivalent Kelvin fluctuation.Giving the maximum probability achievable for a fluc- (a) (b) FIG. 8: Kelvin Fluctuations and Fluctuation Heat Engines tuating heat engine to extract heat Q from a heat bathat temperature T , with efficiency n E = WQ > n CE , de-positing the remainder in a heat bath at temperature T < T as f E ( Q, n E , T , T ), the diagrams quickly yield Q = Q ( n E − n CE ) and the relationship f E ( Q, n E , T , T ) = f ( Q ( n E − n CE ) , T ) (9)Figure 9 provides the equivalent analysis for Clausiusfluctuations, now creating a Clausius fluctuation by driv-ing a regular Carnot pump with the work extracted bya fluctuation heat engine. As Q n CE = Q ( n E − n CE ) (a) (b) FIG. 9: Clausius Fluctuations and Fluctuation Heat Engines f E ( Q, n E , T , T ) = f C (cid:18) Q (cid:18) n E n CE − (cid:19) , T , T (cid:19) (10)Again, this is consistent with the relationship between f C and f . D. Heat Pumps and Engines
It is now possible to compare the expressionsfor f E ( Q, n E , T , T ) and f P ( W, n P , T , T ) directly.This gives f E ( Q, n E , T , T ) = f P ( W, n P , T , T ) if W (cid:16) n P n CP − (cid:17) = Q ( n E − n CE ). To confirm consistencythis can also be derived from the diagrams in Figure10. In Figure 10(a), a fluctuation heat engine, oper- (a) (b) FIG. 10: Fluctuation Heat Pumps and Engines ating at n E = W e Q e improves the efficiency of a Carnotheat pump, by using some of the pumped work to re-turn a higher proportion of the heat into work, to cre-ate a fluctuation heat pump, with efficiency n P = Q p W p .In Figure 10(b), a fluctuation heat pump, with effi-ciency n P = Q p W p improves the efficiency of a Carnotheat engine to create a fluctuation heat engine with ef-ficiency n E = W e Q e . It can readily be confirmed that W (cid:16) n P n CP − (cid:17) = Q ( n E − n CE ). E. Heat and Temperature
There remains six diagrams for fluctuations involvingtwo heat baths. These diagrams determine the relation-ship between Kelvin fluctuations at different tempera-tures. Figure 11 shows how a Kelvin fluctuation canbe converted to an equivalent Kelvin fluctuation at ahigher or lower temperature, by using a Carnot pump orengine. This supplies heat from a second bath to replacethe heat obtained from the fluctuation. The overall pro-cess is then a Kelvin fluctuation from the second heatbath.From Figure 11(a), it can be seen that the probabilityof obtaining a Kelvin fluctuation of size Q at tempera-ture T cannot be less that the probability of obtaininga Kelvin fluctuation of size Q at temperature T , pro-vided Q T = Q T . f ( Q , T ) ≥ f ( Q , T ) (11)Figure 11(b) shows the reverse process, for which f ( Q , T ) ≤ f ( Q , T ), so f ( Q , T ) = f ( Q , T ) when (a) (b) FIG. 11: Kelvin Fluctuations at Different Temperatures Q T = Q T . Writing α = T T this leads to f ( Q, T ) = f ( αQ, αT ). As this must hold for all T and T , and sofor all α f ( Q, T ) = f (cid:18) QT (cid:19) (12)The remaining four diagrams are essentially the sameas the diagrams in Figures 4, 8(b) and 6(b), except theyinvolve a Kelvin fluctuation from the higher temperatureheat bath. Comparison of these processes again leads toEquation 12. F. Fluctuation Friendly Second Law
Combining the result from Section III E, with thosefrom Sections III A to III D, it is now possible to statethe fluctuation compatible generalizations of the formu-lations of the Second Law of Thermodynamics given inSection II • Kelvin : There is no process, whose sole result isthe extraction of a quantity of heat, Q , from a heatbath at temperature T , and its conversion to work,which can occur with probability p , unless: p ≤ f (cid:18) QT (cid:19) • Clausius : There is no process, whose sole resultis the extraction of a quantity of heat, Q , from aheat bath at temperature T , and its transfer toa heat bath at temperature T > T , which canoccur with probability p , unless: p ≤ f (cid:18) Q (cid:18) T − T (cid:19)(cid:19) • Heat Engine : There is no cyclic process, operat-ing solely as a heat engine between heat baths attemperatures T > T , which can extract a quan-tity of heat, Q , from the hotter heat bath, with ef-ficiency n E exceeding that of a reliable, reversible heat engine, n CE , with probability p , unless: p ≤ f (cid:18) QT ( n E − n CE ) (cid:19) • Heat Pump : There is no cyclic process, operat-ing solely as a heat pump between heat baths attemperatures T > T , which can use a quantityof work, W , with efficiency n P exceeding that of areliable, reversible heat engine, n CP , with proba-bility p , unless: p ≤ f (cid:18) WT (cid:18) n P n CP − (cid:19)(cid:19) These four formulations are logically equivalent, inthe same manner that the four formulations of thefluctuation-free second law given in Section II are log-ically equivalent.
G. Kelvin-Clausius inequality.
These four formulations can be expressed in the sameway. Combining a single fluctuation with Carnot pumpsand engines connecting heat baths at multiple temper-atures reveals that there is a more general formulationof the fluctuation laws. Just as all four of the normalphenomenological laws may be seen as special cases ofthe law:There is no process, whose sole result is theextraction of quantities of heat, Q i , fromheat baths at temperatures T i , convertingthe net heat extracted into work, unless: X i Q i T i ≤ Q i , fromheat baths at temperatures T i , convertingthe net heat extracted into work, which canoccur with probability p , unless: p ≤ f X i Q i T i ! (14)The general formulation should make clear the role thatCarnot cycles plays within the derivation of the specificfluctuation laws. Carnot pumps and engines connectinga number of different heat baths are able to reversiblymove heat between them in any combination providedthe net effect is P i Q i T i = 0. Any given fluctuation cantherefore be converted into another fluctuation, involv-ing different heat baths, but which has the same valueof P i Q i T i . H. Combining fluctuations
The next stage is to consider combining fluctuations,by diagrams involving more than one fluctuation. As itturns out, only two diagrams, Figure 12 are required todeduce the general relationship. (a) (b)
FIG. 12: Combining Kelvin Fluctuations
In Figure 12(a) there is a single Kelvin fluctuation re-sulting in Q + Q heat extracted from a heat bath attemperature T . One possible way of this happening is iftwo independent processes occur, each from heat bathsat temperature T , resulting in two separate Kelvin fluc-tuations, extracted Q and Q heat, respectively. Figure12(b) gives a process by which Q + Q can be extracted,so the minimal probability of a Kelvin fluctuation of thatsize cannot be less that the probability of the two inde-pendent fluctuations both occurring: f (cid:18) Q T + Q T (cid:19) ≥ f (cid:18) Q T (cid:19) f (cid:18) Q T (cid:19) (15)As this must happen for all Q , Q , T the fluctuation lawmust satisfy the general functional inequality f ( x + y ) ≥ f ( x ) f ( y ) (16)This leads directly to the general equation f X i Q i T i ! ≥ Y i f (cid:18) Q i T i (cid:19) (17)that would also be deduced from considering diagramswith multiple fluctuations and with Carnot pumps andengines operating between multiple heat baths.This property in itself can be used to demonstratethat, if there exists some x = x > f ( x ) = 0then it must be the case that ∀ x > , f ( x ) = 0, i.e. This may be converted into a more familiar form using F ( x ) = − ln[ f ( x )] to get F ( x ) + F ( y ) ≥ F ( x + y ). In passing, it mayalso be noted that if f ( x ) is differentiable, then it can be shownfrom Equation 16 that f ′ ( x ) ≥ f ( x ) f ′ (0) and f ′′ (0) ≥ f ′ (0) . fluctuations must be possible at all scales, if they arepossible on any scale. Intuitively this should be obvi-ous: provided a small fluctuation can occur with a non-zero probability, p , then accumulating n such fluctua-tions into a fluctuation n times large is always possiblewith probability p n . Any size of fluctuation may occurwith small, but non-zero probability, provided n is largeenough.If it were the case that accumulating small fluctuationswas the optimum process for obtaining a large fluctua-tions, then: f X i Q i T i ! = Y i f (cid:18) Q i T i (cid:19) (18)This requires f ( x + y ) = f ( x ) f ( y ). Provided f is acontinuous function, this has a unique solution: f X i Q i T i ! = e − λ “P i QiTi ” (19)where λ is a universal constant whose value would needdetermining experimentally to be the reciprocal of Boltz-mann’s constant: λ = k − .It is, perhaps, surprising that such a familiar functionwithin statistical mechanics might be obtained from thepurely phenomenological arguments followed here! Un-fortunately, there seems no strong reason to demand thata large fluctuation cannot, in principle, be more probablethan getting an equivalent sized fluctuation through theaccumulation of a large number of small fluctuations.It may, on the arguments considered so far, simply bethe case that large fluctuations can spontaneously oc-cur, with a higher probability.Equation 19 is not the only possibility. The restric-tions on the form of f ( x ) are f ( x ) ≥ f ( x ) = 1 ∀ x ≤ ∂f∂x ≤ ∀ x > f ( x + y ) ≥ f ( x ) f ( y ) ∀ x, y > f ( x ) = 11 + P n a n x n (24)will satisfy all the conditions specified whenever n ! a n ≤ m ! l ! a m a l for all n = m + l . Specific casesinclude:(a) n ! a n = m ! l ! a m a l . This leads to a n = ( a ) n n ! f e ( x ) = e − a x (25)(b) For all n >
1, let a n = 0 f i ( x ) = 11 + a x (26)(c) If some f ( x ) that satisfies the conditions, then g ( x ) = f n ( x ), with n > f q ( x ) = 1(1 + a x ) /a (27)with 0 ≤ a ≤ f l ( x ) = 11 + ln(1 + ax ) (28)can also satisfy the requirements. IV. FLUCTUATIONS AND ENTROPY
In Section III it was shown that the Kelvin-Clausius-Carnot versions of the second law, formulated in termsof cyclic processes and heat baths, can be generalisedin a consistent way to include fluctuation phenomena.However, phenomenological thermodynamics does notbecome genuinely powerful until Equation 13 is used todefine a non-decreasing, global function of state calledentropy. With fluctuations possible, it is clear that anysuch globally defined function of state can decrease withsome probability. In this Section it is shown that it is stillpossible to define a meaningful entropy function, with arelationship to the fluctuation law in Equation 14.
A. Phenomenological Entropy
The Kelvin-Clausius inequality:There is no process, whose sole result is theextraction of quantities of heat, Q i , fromheat baths at temperatures T i , convertingthe net heat extracted into work, unless: X i Q i T i ≤ A , into state B , whileextracting quantities of heat, Q ( AB ) i , from heat bathsat temperatures T i , then there is no process whose soleresult can be to transform state B into state A , whileextracting quantities of heat, Q ( BA ) i , from heat baths attemperatures T i , unless: X i Q ( AB ) i T i + X j Q ( BA ) i T i ≤ set of functions of state, { S θ ( X ) } , which For any two
S, S ′ ∈ { S θ ( X ) } then for any 0 ≤ p ≤
1, it is thecase that pS + (1 − p ) S ′ ∈ { S θ ( X ) } . each satisfy the following condition:If there exists a process, whose sole result isto transform state A into state B , while ex-tracting quantities of heat, Q ( AB ) i , from heatbaths at temperatures T i , then S θ ( A ) ≤ S θ ( B ) − X i Q ( AB ) i T i (30)The functions S θ ( X ) will be referred to as thermody-namic entropies.The expression of the phenomenological second law,in terms of these thermodynamic entropies, isThere exist functions of the thermodynamicstate { S θ ( X ) } , such that for any two thermo-dynamic states A and B , there is no process,whose sole result is to transform state A intostate B , while extracting quantities of heat Q ( AB ) i , from heat baths at temperatures T i ,unless S θ ( A ) ≤ S θ ( B ) − X i Q ( AB ) i T i (31)for all S θ ( X ).In an adiabatic process, no heat is extracted or generatedin any heat bath, so this requires S θ ( A ) ≤ S θ ( B ).As this result must also hold for processes which trans-form B into A , then X i Q ( AB ) i T i ≤ S θ ( B ) − S θ ( A ) ≤ − X i Q ( BA ) i T i (32)This must hold for all processes, so the set { S θ ( X ) } isbounded by the processes which maximise the quantities P i Q ( AB ) i T i and P i Q ( BA ) i T i .If the two states A and B can be connected by a re-versible cycle, then the maximum reached is X i Q ( AB ) i T i + X i Q ( BA ) i T i = 0 (33)in which case the entropy difference between the twostates is fixed to be the same value for all functions in { S θ ( X ) } : S θ ( B ) − S θ ( A ) = X i Q ( AB ) i T i = − X i Q ( BA ) i T i (34)If all states can be connected by reversible cycles, thenthere is a single function, unique up to an additive con-stant. It is important to note that reversibility is re-quired for the uniqueness of the entropy function, but isnot necessary to prove the existence of a non-decreasingset of entropy functions. B. Fluctuation Entropy law
The existence of the fluctuation law does not preventthe derivation of the existence of the thermodynamic en-tropy functions { S θ ( X ) } . Their significance is restrictedto reliable (i.e. probability one) processes. Unfortu-nately it does not immediately follow that a fluctuationlaw can be deduced constraining the probability of a re-duction in thermodynamic entropy.An essential stage in the deduction of a law relatingentropy to fluctuations, is the identification of an appro-priate inequality for closed cycles incorporating any twostates, such as Equation 29, but for cycles involving fluc-tuations. Such an inequality cannot be directly obtainedfrom the fluctuation law.The fluctuation law, Equation (14), implies that, ifthere exists a process, whose sole result is to transformstate A , into state B , while extracting quantities of heat, Q ( AB ) i , from heat baths at temperatures T i , and whichcan occur with probability p AB , then there is no processwhose sole result can be to transform state B into state A , while extracting quantities of heat, Q ( BA ) i , from heatbaths at temperature T i , which can occur with probabil-ity p BA , unless: p AB p BA ≤ f X i Q ( AB ) i T i + X i Q ( BA ) i T i ! (35)Inverting the function gives: f − ( p AB p BA ) ≥ X i Q ( AB ) i T i + X i Q ( BA ) i T i (36)However, the relationship f ( x + y ) ≥ f ( x ) f ( y ), wheninverted, yields f − ( pq ) ≥ f − ( p ) + f − ( q ) (37)and this does not allow the deduction of a suitable in-equality.
1. Reliable Paths
To proceed further, it is necessary to consider reliablepaths between A and B . Let q ( AB ) i be the heat gener- ated in heat baths at temperatures T i , for a process thatcan occur with probability one, and whose sole effect,apart from extracting heat from heat baths and convert-ing them to work, is to transform state A into state B .It follows that there is no process, whose sole result is totransform state B into state A , while extracting quanti-ties of heat Q ( BA ) i from heat baths at temperatures T i ,which can occur with probability p BA , unless f − ( p BA ) ≥ X i q ( AB ) i T i + X i Q ( BA ) i T i (38)Similarly, if q ( BA ) i is the heat generated in heat bathsat temperatures T i , for a process that can occur withprobability one, whose sole effect, apart from extractingheat from heat baths and converting them to work, isto transform state B into state A , then there is no pro-cess, whose sole result is to transform state A into state B , while extracting quantities of heat Q ( AB ) i from heatbaths at temperatures T i , which can occur with proba-bility p AB , unless f − ( p AB ) ≥ X i Q ( AB ) i T i + X i q ( BA ) i T i (39)It is immediately possible to deduce both that X i q ( AB ) i T i + X i q ( BA ) i T i ≤ p AB = 1 or p BA = 1)and that f − ( p AB )+ f − ( p BA ) ≥ X i Q ( AB ) i + q ( BA ) i + q ( AB ) i + Q ( BA ) i T i (41)Equation 40 implies the existence of the thermodynamicentropies { S θ ( X ) } , as before. Equation 41 implies theexistence of a convex set of functions of state { S φ ( X ) } ,which will be called the fluctuation entropies, and whichall satisfy X i Q ( AB ) i + q ( AB ) i T i − f − ( p AB ) ≤ S φ ( B ) − S φ ( A ) ≤ f − ( p BA ) − X i Q ( BA ) i + q ( BA ) i T i (42)In order to narrow down the range of permissible en-tropies, the terms P i q ( AB ) i T i and P i q ( BA ) i T i should each beas large as possible, subject to the constraint of Equation 40.This produces the following entropy fluctuation law:0Let Q ( AB ) i be the heats extracted from heatbath at temperatures T i , by a process, whichoccurs with probability one, whose sole otherresult is to transform state A into state B ,and which maximises the value of P i Q ( AB ) i T i over all such processes.There exists single valued functions of state { S φ ( X ) } , such that, if there exists a processoccurring with probability p , whose sole re-sult is to transform state A into state B ,while extracting quantities of heat, Q ( AB ) i ,from heat baths at temperatures T i , then S φ ( A ) ≤ S φ ( B )+ f − ( p ) − X i Q ( AB ) i T i − X i Q ( AB ) i T i (43)To restrict these to a unique function S φ ( X ) requiresthat there exist cycles for which f − ( p AB )+ f − ( p BA ) = X i Q ( AB ) i + Q ( AB ) i + Q ( BA ) i + Q ( BA ) i T i (44)
2. Reversible Paths
If it is the case that the equality in Equation 40 is met,then Equation 41 takes the form f − ( p AB ) + f − ( p BA ) ≥ X i Q ( AB ) i T i + X i Q ( BA ) i T i (45)and X i Q ( AB ) i T i − f − ( p AB ) ≤ S φ ( B ) − S φ ( A ) ≤ f − ( p BA ) − X i Q ( BA ) i T i (46)This is not sufficient to ensure S φ ( B ) − S φ ( A ) is unique.However, in this case, it is possible to deduce the ex-istence of the globally unique thermodynamic entropyfrom the reliable paths S θ ( B ) − S θ ( A ) = X i Q ( AB ) i T i = − X j Q ( BA ) i T i (47)for which Equations 38 and 39 give X i Q ( AB ) i T i − f − ( p AB ) ≤ S θ ( B ) − S θ ( A ) ≤ f − ( p BA ) − X i Q ( BA ) i T i (48) When dealing with fluctuations, a cycle is a process for whichthe system starts in state A with certainty, reaches the state B with probability p AB , and then the conditional probability forreturning to state A , given that it reached state B , is P BA . and S θ ( X ) ∈ { S φ ( X ) } . If both S θ ( X ) and S φ ( X )are uniquely defined, then P i Q ( AB ) i T i − f − ( p AB ) + P i Q ( BA ) i T i − f − ( p BA ) = 0, in which case S θ ( X ) = S φ ( X ). However, in general, if the thermodynamic en-tropies { S θ ( X ) } are not restricted to a single globallyunique function, then there may exist S θ ( X ) / ∈ { S φ ( X ) } .It is worth noting that Equation 48 implies X i Q ( AB ) i T i − f − ( p AB ) + X i Q ( BA ) i T i − f − ( p BA ) ≤ P i q ( AB ) i T i + P i q ( BA ) i T i <
0, then thisrequires X i Q ( AB ) i T i − f − ( p AB ) + X i Q ( BA ) i T i − f − ( p BA ) > any fluctuations from state A to state B , and vice versa, that can define a uniquefluctuation entropy difference S φ ( B ) − S φ ( A ) when com-bined with a reliable but irreversible cyclic path between A and B , then it must be the case that there are no re-liable, reversible cyclic paths between states A and B .The existence of a globally unique S φ ( X ) that is notsimultaneously a globally unique S θ ( X ) would imply re-liable, reversible processes cannot exist.Reliable, reversible cycles imply an entropy fluctuationlaw: There exists single valued functions of state { S φ ( X ) } , such that, if there exists a cyclicprocess, occurring with probability one, op-erating between states A and states B , witha zero net extraction of heat over the cycle,then for any other process, occurring withprobability p , whose sole result is to trans-form state A into state B , while extractingquantities of heat, Q ( AB ) i , from heat baths attemperatures T i , then S φ ( A ) ≤ S φ ( B ) + f − ( p ) − X i Q ( AB ) i T i (51)and there is a globally unique thermody-namic entropy S θ ( X ) ∈ { S φ ( X ) }
3. Exponential Statistics
Finally, note that if the fluctuation law takes the ex-ponential form discussed in Section III H, then f − ( pq ) = f − ( p ) + f − ( q ) (52)so Equation 36 leads immediately to f − ( p AB ) + f − ( p BA ) ≥ X i Q ( AB ) i T i + X j Q ( BA ) j T j (53)1This gives Equation 48 without needing the existence ofreliable paths. This implies there exists a convex set offluctuation entropies { S η ( X ) } ⊆ { S φ ( X ) } satisfying X i Q ( AB ) i T i − f − ( p AB ) ≤ S η ( B ) − S η ( A ) ≤ f − ( p BA ) − X j Q ( BA ) j T j (54)Uniquely defining an S η ( X ) entropy would require P i Q ( AB ) i T i − f − ( p AB ) + P j Q ( BA ) j T j − f − ( p BA ) = 0, butthis does not necessarily uniquely define either S φ ( X ) or S θ ( X ). In this case, however, if a unique S φ ( X ) doesexist then it is necessarily equal to a unique S θ ( X ), andvice versa. V. FROM FLUCTUATIONS TO STATISTICALMECHANICS
The possible relationship of the fluctuation spectrum f to statistical mechanics will now be briefly explored. Itwill be assumed throughout this Section that a globallyunique entropy S ( X ) = S θ ( X ) = S φ ( X ) can be deter-mined, and only a single heat bath at temperature T will be used. The entropy fluctuation law now takes theform: There exists a single valued function of state S ( X ), such that for any process, occurringwith probability p , whose sole result is totransform state A into state B , while ex-tracting quantities of heat, Q ( AB ) , from heatbaths at temperatures T , then S ( A ) ≤ S ( B ) + f − ( p ) − Q ( AB ) T (55)Suppose the system is in an initial state, with entropy S , internal energy E , and is subject to a process duringwhich it fluctuates to state α with probability p α . Duringthe course of the process, heats Q α are generated in heatbaths at temperatures T and requires work W α to beperformed.By conservation of energy, the internal energy of state α is E α = E + W α − Q α (56)By the entropy fluctuation law, the entropy of state α must obey S ≤ S α + f − ( p α ) − Q α T (57)This equation must hold for each possible fluctuationaway from the initial state, so that S ≤ X α p α (cid:18) S α + f − ( p α ) − Q α T (cid:19) (58)necessarily holds. The form of this constraint is verysuggestive of entropy functions that occur in statisticalmechanics. A. Maximal Fluctuations
The definition of the f function is such that there mustexist some process for which the equality in Equation 57is met: S = S α + f − ( p α ) − Q α T (59) p α = f (cid:18)(cid:18) S − ET (cid:19) − (cid:18) S α − E α − W α T (cid:19)(cid:19) (60)However, there is no guarantee that a single processcan exist which achieves the maximum fluctuation forevery possible outcome. If such a process did exist, then S = X α p α (cid:18) S α + f − ( p α ) − Q α T (cid:19) (61)would hold.This similarity to statistical mechanics is brought evencloser under two conditions:1. If a set of maximal fluctuations occur which donot generate heat, on average, then P α p α Q α T = 0.The entropy formula then becomes: S = X α p α (cid:0) S α + f − ( p α ) (cid:1) (62)2. If a set of maximal fluctuations can take place,without requiring external work to be performed( W α = 0) then: p α = f (cid:18)(cid:18) S − ET (cid:19) − (cid:18) S α − E α T (cid:19)(cid:19) (63)or p α = f ( F − F α T ), where F = T S − E (64) F α = T S α − E α (65) B. Example fluctuation laws
Let us consider the functions from Section III H1. f ( x ) = P n a n x n (a) f e ( x ) = e − a x . This generates the familiarGibbs canonical statistics. f − e ( p ) = − a ln p (66) S = X α p α S α − a p α ln p α (67) p α = 1 Z e e − a F α /T (68)with Z e = e a F/T = P α e − a F α /T f i ( x ) = (1 + a x ) − f − i ( p ) = 1 a (cid:0) p − − (cid:1) (69) S = X α p α S α − a ( N −
1) (70) p α = 1 Z i (1 + a βF α ) − (71)with N the number of distinct states in thesummation, Z i = (1 + a F ) and β = 1 / ( T Z i ).(c) f q ( x ) = (1 + a x ) − /a . This generates statis-tics similar to the Tsallis non-extensive en-tropies. f − q ( p ) = 1 a (cid:0) p − a − (cid:1) (72) S = X α p α S α − a − X α p − a α ! (73) p α = 1 Z q (1 + a βF α ) − /a (74)with Z q = (1 + a F/T ) /a and β =1 / ( T Z a q ).2. The slowly falling function f l ( x ) = (1 + ln(1 + ax )) − yields f − l ( p ) = 1 a (cid:16) e ( p − − − (cid:17) (75) S = X α p α S α + 1 a X α p α e ( p − − ! − a (76) p α = 1 Z l (1 + ln(1 + a βF α ) /Z l ) − (77)with Z l = 1 + ln(1 + a F/T ) and β = e Z l − /T VI. CONCLUSION
Starting from the physical intuition that larger ther-mal fluctuations must be less probable than smaller fluc-tuations, we have suggested a fluctuation law that statesthat for any given size of fluctuation, there is a non-trivial maximum probability of it occurring. This sim-ple suggestion proves surprisingly fruitful. The equiva-lence of the Kelvin, Clausius and Carnot formulations ofthe phenomenological second law of thermodynamics isshown to naturally generalise to the fluctuation law, andfurther constrain it to be of the form:There is no process, whose sole result is theextraction of quantities of heat, Q i , fromheat baths at temperatures T i , convertingthe net heat extracted into work, which canoccur with probability p , unless: p ≤ f X i Q i T i ! (78) with the function f further constrained by the require-ment f X i Q i T i ! ≥ Y i f (cid:18) Q i T i (cid:19) (79)If the underlying dynamics is found to be such that largerfluctuations can only occur through the accumulation ofsmaller fluctuations, then this requires the function tohave the exponential form: f X i Q i T i ! = e − λ “P i QiTi ” (80)It is interesting to note that the phenomenologically mo-tivated approaches of Szilard and of Tisza and Quay[11,12] to statistical mechanics derive the canonical distri-bution by making a similar assumption (see also [13]).We have further shown that the deduction of the exis-tence of a non-decreasing thermodynamic entropy func-tion of state may still be followed, to derive a fluctu-ation entropy function of state. Under a similar kindof circumstance for which the thermodynamic entropycan be deduced to be globally unique, then the fluc-tuation entropy can be deduced to be globally unique.Furthermore, if the thermodynamic and fluctuation en-tropies are both globally unique, then they are neces-sarily identical (up to an additive constant). This holdsout hope that more rigorously axiomatic developmentsof the thermodynamic entropy, such as that of Lieb andYngvason[1], may be generalized in a similar manner toincorporate fluctuation phenomena.Some possible forms of the entropy fluctuation lawhave been investigated. The exponential form naturallyproduces the Gibbs canonical distribution for thermalfluctuations. Non-extensive entropies, such as the Tsal-lis entropy, can also be seen to arise naturally in thisapproach. Further investigation is needed to explore theconsistency of different f functions. In particular, the re-quirement that the mean heat extracted over a cycle isnon-positive, DP i Q i T i E ≤
0, may be expected to furtherconstrain which functions are admissible.
APPENDIX: ENTROPY FUNCTIONS FORIRREVERSIBLE CYCLES
Suppose there exists a path dependant quantity, Ω λAB (a property of a particular path λ , in a state space, fromstate A to state B ) well defined for all paths λ , states A and states B , for which: ∀ λ, λ ′ Ω λAB + Ω λ ′ BA ≥ A toeach B for which the corresponding value of Ω is finite,so that inf λ (cid:2) Ω λAB (cid:3) < ∞ . Then there exists a non-emptyconvex set of functions of state { S ( X ) } , such that for allpaths λ and states A and B : S ( A ) ≤ S ( B ) + Ω λAB (A.2)3 Proof : Define Ω AB = inf λ (cid:2) Ω λAB (cid:3) . So Ω λAB ≥ Ω AB .As Ω BA < ∞ and Ω AB ≥ − Ω BA , then Ω AB > −∞ .By definition, the minimum value of Ω going from A to C cannot be more than the value going from A to C via a path including B :Ω AC ≤ Ω AB + Ω BC (A.3)so Ω AC − Ω AB ≤ Ω BC (A.4)Ω AB − Ω AC ≥ − Ω BC (A.5)Ω AC − Ω BC ≤ Ω AB (A.6)Ω BC − Ω AC ≥ − Ω AB (A.7)Define the set of functions of state { S iY ( X ) } by S + A ( X ) = Ω XA (A.8) S − A ( X ) = − Ω AX (A.9)These are clearly well defined, finite functions of state,and they exist, so the set { S iY ( X ) } is not empty. Notethat as Ω XX = 0:Ω XY = S + Y ( X ) − S + Y ( Y ) (A.10)= S − X ( X ) − S − X ( Y ) (A.11) and S + A ( X ) − S + A ( Y ) = Ω AX − Ω AY (A.12) S − A ( X ) − S − A ( Y ) = − Ω XA + Ω Y A (A.13)It follows that for any A , S + A ( X ) − S + A ( Y ) ≤ Ω XY ≤ Ω λXY (A.14) ≥ − Ω Y X ≥ − Ω λY X (A.15) S − A ( X ) − S − A ( Y ) ≤ Ω XY ≤ Ω λXY (A.16) ≥ − Ω Y X ≥ − Ω λY X (A.17)and it is then easily demonstrated that for any distri-bution P iY w ( iY ) = 1, w ( iY ) ≥
0, that the weightedfunction of state S ( X ) = X iY w ( iY ) S iY ( X ) (A.18)satisfies S ( A ) − S ( B ) ≤ Ω λAB (A.19)as − Ω λY X ≤ − Ω Y X = S + X ( X ) − S + X ( Y ) ≤ S iA ( X ) − S iA ( Y ) ≤ S + Y ( X ) − S + Y ( Y ) = Ω XY ≤ Ω λ ′ XY (A.20)Note, that the set { P iY w ( iY ) S iY ( X ) } does not neces-sarily include all the functions which satisfy the inequal-ity of Eq. (A.2). It only demonstrates the existence ofa non-empty set of such functions.It is now a trivial matter to show from Equa-tion A.20 that, whenever the equality in EquationA.1 can be reached, that all functions in the set { P iY w ( iY ) S iY ( X ) } (indeed, all functions satisfyingEquation A.2) will give the same entropy difference be-tween states A and B . By extension, if the equality inEquation A.1 can be reached for all pairs of states, thenthere is a single function, S ( X ), unique up to an additiveconstant. ACKNOWLEDGMENTS
I would like to thank Harvey Brown, John Norton,Tony Short, Jos Uffink and Steve Weinstein for discus-sions and suggestions that have influenced the develop-ment of this paper, and an anonymous referee for helpfulcomments. Research at the Perimeter Institute for The-oretical Physics is supported in part by the Governmentof Canada through NSERC and by the Province of On-tario through MRI. [1] E. H. Lieb and J. Yngvason, Physics Reports , 1(1999).[2] J. Uffink, Studies in History and Philosophy of ModernPhysics , 305 (2001).[3] D. P. Sheehan, ed., First International Conference onQuantum Limits to the Second Law (American Instituteof Physics, 2000).[4] G. P. Beretta, A. Ghoniem, and G. Hatsopoulos, eds.,
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