Death and resurrection of the zeroth principle of thermodynamics
DDeath and resurrection of the zeroth principle of thermodynamics
Hal M. Haggard and Carlo Rovelli
Centre de Physique Th´eorique de Luminy, Aix-Marseille Universit´e, F-13288 Marseille, EU (Dated: February 5, 2013)The zeroth principle of thermodynamics in the form “temperature is uniform at equilibrium” isnotoriously violated in relativistic gravity. Temperature uniformity is often derived from the maxi-mization of the total number of microstates of two interacting systems under energy exchanges. Herewe discuss a generalized version of this derivation, based on informational notions, which remainsvalid in the general context. The result is based on the observation that the time taken by anysystem to move to a distinguishable (nearly orthogonal) quantum state is a universal quantity thatdepends solely on the temperature. At equilibrium the net information flow between two systemsmust vanish, and this happens when two systems transit the same number of distinguishable statesin the course of their interaction.
I. NON-UNIFORM EQUILIBRIUMTEMPERATURE
According to non-relativistic thermodynamics, a ther-mometer (say, a line of mercury in a glass tube), movedup and down a column of gas at equilibrium in a constantgravitational field, measures a uniform temperature. Butthis prediction is wrong. Relativistic effects make the gaswarmer at the bottom and cooler at the top, by a cor-rection proportional to c − , where c is the speed of light.This is the well known Tolman-Ehrenfest effect, discov-ered in the thirties [1, 2] and later derived in a varietyof different manners [3–11]. The temperatures T and T measured by the same thermometer at two altitudes h and h in a Newtonian potential Φ( h ) are related by theTolman law T (cid:18) h ) c (cid:19) = T (cid:18) h ) c (cid:19) . (1)The general-covariant version of this law reads T | ξ | = constant, (2)where | ξ | is the norm of the timelike Killing field withrespect to which equilibrium is established.A violation of the uniformity of temperature seemscounterintuitive at first, especially if one has in mind adefinition of “temperature” as a label of the equivalenceclasses of all systems in equilibrium with one another.In a relativistic context a physical thermometer does notmeasure this label and we must therefore distinguish twonotions: (i) a quantity T o defined as this label (propor-tional to the constant in (2)), and (ii) the temperature T measured by a standard thermometer.In the micro-canonical framework the entropy S ( E ) isthe logarithm of the number of microstates N ( E ) thathave energy E and T can be identified with the inverseof the derivative of S ( E ), dS ( E ) dE = 1 kT , (3)where k is the Boltzmann constant. The fact that twosystems in equilibrium have the same T can be derived by maximizing the total number of states N = N N under an energy transfer dE between the two. This giveseasily T = T . In the presence of relativistic gravity, thisderivation fails because conservation of energy becomestricky: intuitively speaking, the energy dE reaching theupper system is smaller than the one leaving the lowersystem because “energy weighs”.Is there a more general statistical argument that gov-erns equilibrium in a relativistic context? Can the Tol-man law be derived from a principle generalizing themaximization of the number of microstates, without re-course to specific models of energy transfer, as is com-monly done in the derivations of the Tolman-Ehrenfesteffect?In this paper we show that the answer to these ques-tions is positive, and we provide a generalization of thestatistical derivation of the uniformity of temperature,which remains valid in a relativistic context.The core idea is to focus on histories rather than states .This is in line with the general idea that states at fixedtime are not a convenient handle on general relativisticmechanics, where the notion of process , or history, turnsout to be more useful [12]. Equilibrium in a stationaryspacetime, namely the Tolman law, is our short-term ob-jective, but our long-term aim is understanding equilib-rium in a fully generally covariant context, where thermalenergy can flow also to gravity [13–15], therefore we lookfor a general principle that retains its meaning also inthe absence of a background spacetime.We show in this paper that one can assign an informa-tion content to a history, and two systems are in equi-librium when their interacting histories have the sameinformation content. In this case the net informationflow vanishes, and this is a necessary condition for equi-librium. This generalized principle reduces to standardthermodynamics in the non-relativistic setting, but yieldsthe correct relativistic generalization.This result is based on a key observation: at tempera-ture T , a system transits τ = kT (cid:126) t (4)states in a (proper) time t , in a sense that is made precise a r X i v : . [ g r- q c ] F e b FIG. 1: Typical overlap between ψ (0) and ψ ( t ) as a functionof time. below. The quantity τ was introduced in [13, 14] withdifferent motivations, and called thermal time . Here wefind the physical interpretation of this quantity: it is timemeasured in number of elementary “time steps”, where astep is the characteristic time taken to move to a distin-guishable quantum state. Remarkably, this time step isuniversal at a given temperature. Our main result is thattwo systems are in equilibrium if during their interactionthey cover the same number of time steps. II. THE UNIVERSAL TIME STEP
Consider a conventional hamiltonian system withhamiltonian operator H . Let ψ (0) be the state at timezero and ψ ( t ) its evolution. What is the time scale for ψ ( t ) to become significantly distinct from ψ (0)? The sep-aration of the state from its initial position is given bythe overlap between ψ (0) and ψ ( t ), namely P ( t ) = |(cid:104) ψ (0) | ψ ( t ) (cid:105)| . (5)The typical behavior of P ( t ), for instance in the case ofa semiclassical wave packet, is as in Figure 1. The statebecomes rapidly distinguishable (nearly orthogonal) tothe initial state, in a short time. Let us call t o the char-acteristic decay time for the system self overlap. Whatis its value? The time t o can be estimated by Taylor ex-panding P ( t ) for small times. The first time derivative of P ( t ) clearly vanishes at t = 0 which is a maximum, there-fore we get the time scale from the second derivative. Astraightforward calculation gives d P ( t ) dt = 1 (cid:126) ( (cid:104) H (cid:105) − (cid:104) H (cid:105) ) = (∆ E ) (cid:126) , (6)which implies a characteristic decay time t o = (cid:126) ∆ E , (7)
FIG. 2: A phase space region moves from one cell to next inthe time τ ∼ h/ ∆ E . in accord with the time-energy Heisenberg principle, andwith the fact that energy eigenstates “do not change”. The same conclusion can be reached also in the classi-cal theory. Consider a classical hamiltonian system withphase space Γ and hamiltonian H . For simplicity, saythat the system has a single degree of freedom, so thatΓ is two-dimensional. Let E and E be two (nearby)equal-energy surfaces and γ a line joining the two sur-faces. Consider the motion of γ under the time flow, seeFigure 2. How long does it take for γ to sweep a (small)phase-space volume V ? The answer is easy to find: thevolume of the region R swept by γ is the integral of thesymplectic two form ω = dp ∧ dq and its time derivativeis dV ( t ) dt = ddt (cid:90) R ( t ) ω = (cid:90) γ ω ( X ) , (8)where X is the hamiltonian time flow. This is given bythe Hamilton equations, which can be written in the com-pact form ω ( X ) = − dH. (9)Inserting this in the previous equation gives dV ( t ) dt = − (cid:90) γ dH = E − E ≡ ∆ E. (10)Now consider a small region of phase space, such as theblue region in Figure 2. Say that the volume of thisregion is (cid:126) . How long does it take for this region to becarried along by the dynamics to a new position wherethe overlap with its initial location is negligible? It isclear that the answer is again (7).This is the same result as in the quantum theory: thetime step t o is the time taken generically to move froma state to a distinct (orthogonal) state. Indeed, a semi-classical state can be viewed as related to a Planck-sizecell of the classical phase space, and the decay time ofthe quantum overlap P ( t ) is essentially the time the sys-tem moves from one Planck-size cell to the next. The Notice that this observation provides also a direct meaning tothe time-energy uncertainty principle. argument can be repeated with a bit more labour for asystem with many degrees of freedom. Let us now consider a system in thermal equilibriumwith a thermal bath at temperature T . Its mean energyis going to be kT and the variance of the energy is alsogoing to be kT . Thus we have ∆ E ∼ kT . At a giventemperature T , consider the time step t o = (cid:126) kT . (11)According to the previous discussion, this is the averagetime the system takes to move from a state to the next(distinguishable) state. This average time step is there-fore universal: it depends only on the temperature, andnot on the properties of the system. III. THERMAL TIME, TEMPERATURE ANDTHEIR PHYSICAL MEANING
The dimensionless quantity τ = tt o (12)measures time in units of the time step t o , that is, itestimates the number of distinguishable states the systemhas transited during a given interval. For a system inthermal equilibrium, (11) gives τ = kT (cid:126) t. (13)This same quantity was introduced with different moti-vations in [13, 14] under the name thermal time . It is theparameter of the Tomita flow on the observable algebra,generated by the thermal state. In the classical theory,it is the parameter of the hamiltonian flow of h = − ln ρ ,where ρ is a Gibbs state, in (cid:126) = k = 1 units.The argument in the previous section unveils the phys-ical interpretation of thermal time: thermal time, whichis dimensionless, is simply the number of distinguishablestates a system has transited during an interval. In asense, it is “time counted in natural elementary steps”,which exist because the Heisenberg principle implies aneffective granularity of the phase space.Notice also that temperature is the ratio between ther-mal time and (proper) time [16] T = (cid:126) τkt . (14) If we can diagonalize locally the state and the dynamics inenergy-angle variables ( E n , φ n ), the phase space volume sweptby the boundary of a given region, in a time dt , is dV ∼ (cid:80) n ∆ E n ( V/V n ) dt, where V n is the phase space volume of the n -th degree of freedom. A coherent state has volume (cid:126) in each of itsdegrees of freedom, giving dV ∼ (cid:126) n − (cid:80) n ∆ E n dt ∼ (cid:126) n − ∆ Edt.
Since a phase space cell has volume (cid:126) n , the time taken to moveone cell is again ∼ (cid:126) / ∆ E . Accordingly, in (cid:126) = k = 1 units temperature is measuredin “states per second” and is nothing other than the num-ber of states transited by the system per unit of (proper)time. This is the general informational meaning of tem-perature. A warmer system is a system where individualstates move faster across unit cells of phase space. IV. EQUILIBRIUM BETWEEN HISTORIES
Let us come to the notion of equilibrium. Considertwo systems, System 1 and System 2, that are in inter-action during a certain interval. This interaction can bequite general but should allow the exchange of energy be-tween the two systems. During the interaction intervalthe first system transits N states, and the second N , inthe sense illustrated above. Since an interaction channelis open, each system has access to the information aboutthe states the other has transited through the physicalexchanges of the interaction.The notion of information used here is purely physi-cal, with no relation to semantics, meaning, significance,consciousness, records, storage, or mental, cognitive, ide-alistic or subjectivistic ideas. Information is simply ameasure of a number of states , as is defined in the classictext by Shannon [17].System 2 has access to an amount of information I = log N about System 1, and System 1 has accessto an amount of information I = log N about System2. Let us define the net flow of information in the courseof the interaction as δI = I − I . Equilibrium is by def-inition invariant under time reversal, and therefore anyflow must vanish. It is therefore interesting to postulate that also the information flow δI vanishes at equilibrium.Let us do so, and study the consequences of this assump-tion. That is, we consider the possibility of taking thevanishing of the information flow δI = 0 (15)as a general condition for equilibrium, generalizing themaximization of the number of microstates of the non-relativistic formalism. In the micro-canonical framework equilibrium is characterized bymaximizing entropy, namely the number of micro-states sharinggiven macroscopic values. This is meaningful, e.g. under theergodic hypothesis, according to which time averages can be re-placed by phase-space averages. In other words, if the ergodichypothesis holds, a micro-canonical ensemble is essentially thefamily of states over which the single real individual microstatewanders. What we are doing here is essentially undoing this stepand moving back from phase-space ensembles to actual histories.In the classical theory, there is a measure associated to a space-time volume and not to the length of a history. But in this paperwe have shown that there is also a natural measure associatedto the history of a quantum state. This allows us to backtrackfrom phase-space volume to number of steps along the history.
Let us see what this implies. At equilibrium N = N . (16)Since the rate that states are transited is given by τ andwe assume a fixed interaction interval, the equilibriumconditions also reads τ = τ . (17)Now, consider a non-relativistic context where two sys-tem are in equilibrium states at temperatures T and T , respectively. In the non-relativistic limit, time is auniversal quantity, which we call t . Therefore the con-dition (17) together with (13) implies that t o = (cid:126) /kT has the same value for the two systems and T = T ,which is the standard non-relativistic condition for equi-librium: temperature is uniform at equilibrium. On acurved spacetime, on the other hand, (proper) time is alocal quantity ds that varies from one spatial region toanother. Therefore thermal time is given by dτ = kT (cid:126) ds. (18)In order for equilibrium to exist on a given spacetime,spacetime itself must be stationary, namely have a time-like Killing field ξ , and an equilibrium configuration willbe ξ invariant. Proper time along the orbits of ξ is ds = | ξ | dt where t is an affine parameter for ξ . Thereforethermal time is now dτ = kT (cid:126) | ξ | dt. (19)If two systems located in regions with different | ξ | are inthermal contact for a finite interval ∆ t , then they are inequilibrium if | ξ | T has the same value. This is preciselythe Tolman law (2). Therefore the generalized first prin-ciple (15) gives equality of temperature in the non rela-tivistic case and the Tolman law in the general case.In static coordinates, ds = g ( (cid:126)x ) dt − g ij ( (cid:126)x ) x i x j andthermal time is proportional to coordinate time. TheKilling vector field is ξ = ∂/∂t and | ξ | = √ g . In theNewtonian limit g = 1 + 2Φ /c and we recover (1).Returning to the cylinder of gas in a constant gravita-tional field we see that during a coordinate-time interval∆ t the proper times lapsed in the upper and lower sys-tems are different: identical clocks at different altitudesrun at different rates. But the lower system is hotter,its degrees of freedom move faster in clock time from onestate to the next. This faster motion compensates exactlythe slowing down of proper time, so that upper and lowersystems transit the same number of states during a com-mon interaction interval ∆ t . While a pendulum slowsdown in a deeper gravitational potential, at equilibriumall systems transit from state to state at the same com-mon rate, independent from the gravitational potential.This result provides a simple and intuitive interpretationof the Tolman effect.
V. WIEN’S DISPLACEMENT LAW
The Tolman-Ehrenfest effect is a small relativistic cor-rection, at the surface of the earth ∇ T /T = 10 − cm − ,and is not yet experimentally established. The principleproposed here also provides a mellifluous derivation ofthe well-tested Wien displacement law.Consider an isothermal cavity filled with electromag-netic radiation. A slow, adiabatic expansion of the cavityleaves the radiation in equilibrium throughout the ex-pansion process. During this expansion both the normalmode frequencies and the temperature of the radiationare adjusted. For the mode of frequency ν , the conditionof remaining in equilibrium is that the slow expansiontake place on a time scale much greater than the periodof the mode, i.e. t exp >> t ν ≡ /ν . Hence, the relevantclock for this mode is its period.The condition that this mode be in equilibrium duringthe entire expansion history is that τ = kT (cid:126) t ν = const. (20)or expressed in terms of the mode’s frequency, Tν = const. , (21)which is precisely the general form of Wien’s displace-ment law. This special relativistic application of theprincipal proposed here plays an important role in theastrophysical determination of star temperatures. VI. CONCLUSIONS
We have suggested a generalized statistical principlefor equilibrium in statistical mechanics. We expect thatthis will be of use going towards a genuine foundation forgeneral covariant statistical mechanics.The principle is formulated in terms of histories ratherthan states and expressed in terms of information. Itreads:
Two histories are in equilibrium if the net infor-mation flow between them vanishes, namely if they transitthe same number of states during the interaction period .This is equivalent to saying that the thermal time τ elapsed for the two systems is the same, since thermaltime is the number of states transited, or, equivalently,is (proper) time in t o units, where t o is the (proper) timeneeded for a system to transit to an orthogonal state.The elementary (proper) time step t o is given by (cid:126) /kT and is a universal quantity for all systems at temperature T .In non-relativistic physics, time is universal and theabove principle implies that temperature is uniform atequilibrium. On a curved spacetime, proper time varieslocally and what is constant is the product of tempera-ture and proper time.Temperature admits the informational interpretationas states transited per second, consistent with the factthat in (cid:126) = k = 1 units it has dimension of second − .Temperature is the rate at which systems move from stateto state. ——Thanks to Badr Albanna and Matteo Smerlak for care- fully reading the manuscript and to Alejandro Perez andSimone Speziale for useful discussions. H. M. H. ac-knowledges support from the National Science Founda-tion (NSF) International Research Fellowship Program(IRFP) under Grant No. OISE-1159218. [1] R.C. Tolman, “On the Weight of Heat and Thermal Equi-librium in General Relativity,” Phys. Rev. (1930) 904.[2] R.C. Tolman, P. Ehrenfest, “Temperature Equilibrium ina Static Gravitational Field,” Phys. Rev. (1930) 1791.[3] R.C. Tolman, Relativity, Thermodynamics and Cosmol-ogy.
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