Abstract The problem of how to define and compute temperature fluctuations for a smallsystem in contact with a heat bath is an old one and originates from Einstein’s theory ofBrownian motion. Only for a small enough system does their relative size allow a straight-forward experimental verification. Here we focus on a mesoscopic system in contact with aheat bath at temperature T◦ and provide a self-consistent argument showing as to why, andin what sense, the observable standard deviation of temperature from T◦ equals
√kB/CT◦
where C is the mesoscopic system’s heat capacity. Our argument is based on ergodic decom-position, a simple fact that holds for a system in thermodynamic equilibrium and away fromphase-transition points. In this way we close a line of argument opened by de Haas-Lorentza century ago.
Keywords Temperature fluctuation · Thermal equilibrium · Mesoscopic
1 Introduction
What does a temperature “fluctuation” mean for a small, say mesoscopic (about 100–1,000 nm), system in contact with a heat bath? The system needs to be small enough soas to allow a sensibly precise determination of the energy fluctuations induced by the heatbath. By the definition of heat bath the latter has a temperature T◦ so that all seems well-defined and there is no need to consider fluctuations of the temperature. Nevertheless thequestion of temperature fluctuations is a meaningful one but, as we show here, subtler thanoriginally expected. Somewhat surprisingly, it reappeared on the scene [1–4] after it hadalready been analyzed in fascinating detail at the beginning of last century.
J. Leo van HemmenPhysik Department T35, Technische Universität München, 85747 Garching bei München, Germanye-mail: [email protected]
A. Longtin (B)Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canadae-mail: [email protected]
Temperature Fluctuations in Thermal Equilibrium 1133
Motivated by Einstein’s atomistic theory of Brownian motion [5], developed in conjunc-tion with statistical-mechanical considerations, G.L. de Haas-Lorentz [6] provided a strikingargument leading to the standard deviation 〈(�T )2〉1/2 from the heat bath temperature T◦,
√⟨(�T )2
⟩/T◦ = √
kB/C (1)
where C = d̄Q/dT is the small system’s heat capacity [7], expected to be proportional to theparticle number N , and the angular brackets 〈. . .〉 indicate a canonical average; see below.Equation (1) has often been quoted, already quite early [8, 9], but its derivation has neverbeen reconsidered. In recent times Chui et al. [10] were the first to convincingly show howwell the temperature of a small system can be determined and verify the agreement of its“fluctuation” with the above expression (1).
As we will see, from the very beginning [6] all arguments leading to (1) contain a gapin that �T in (1) is ill-defined. We will close it by noting that we start with a system inthermodynamic equilibrium, specifying �T through �T ≡ C−1�Q with the heat capacityC as a well-defined thermodynamic quantity, quantifying the temperature “fluctuation”, andexploiting ergodicity.
The setup of the present paper is as follows. We start by defining the problem we wantto solve. Then we solve it but cannot do so before clarifying the question of how to quan-tify a “temperature” fluctuation and, in so doing, what thermal equilibrium means once werealize that we face an extra problem in that most systems in thermal equilibrium at T < Tc
are nonergodic. After having sorted out what (1) really means in the context of ergodic de-composition, we deal with the associated problem of timescales. Furthermore, we reviewthe work of de Haas-Lorentz [6] in the new light of our derivation and analyze the mean-ing of thermodynamic uncertainty relations [11, 12] that look like �E�(1/T ) ≥ kB andwith equality sign formally1 give (1). We also show the equivalence of the two approachesand dispense with their key assumption of always taking a Gaussian distribution altogether.Finally, we return to the question we start analyzing in the next section.
Before doing so, it may be worth our while to take a quick look at the more recentliterature on temperature fluctuations so as to put the present issue in a proper historicalperspective. There is no doubt that the triad [1–3] aroused most of the recent interest butthe discussion had already been initiated a bit earlier by Phillies [13], who analyzed theclassical ideal gas in great detail on the basis of the observation that “it is necessary toextend the canonical ensemble to include systems of different temperatures,” a conclusionwe cannot share since the extension, the polythermal ensemble (hence the name), is notneeded. His argument was taken up by Prosper [14], who argued that a new ensemble wasindeed gratuitous and, as Phillies, extensively used conditional probabilities in the style ofBayes. Actually, both are in the realm of Mandelbrot [3, 4] and both interpret �E only as atraditional average in the context of equilibrium statistical mechanics [15], hence their usingconditional probabilities. Balatsky and Zhu [16] employ a “mean” dynamics for the entropyfluctuation �S, a linearized equation in the style of Nyquist together with a simple ansatz,and use what is actually the KMS equilibrium condition [17] to obtain �T for quantumsystems but in so doing avoid answering the question of what �T means. Only Jahnke et al.[18] state the real problem explicitly: “parameters are not observables, so their estimationhas to be done indirectly.” Their ensuing argument presupposes a Gaussian distribution forthe energy and is in a similar vein to that of the thermodynamic “uncertainty” relations
1Formally, �(1/T ) = (�T )/T 2 so that with �E = √kBCT we end up with (1).
1134 J. Leo van Hemmen, A. Longtin
[11, 12]. One may consult Touchette [19] for an excellent mathematical summary of theprevious arguments. All these authors, however, do not refer to the ground-breaking workof de Haas-Lorentz [6]. Advocating ad fontes is yet another goal of the present paper since,as we will see, trying to understand the original work pays off.
2 How to Define Temperature Fluctuations?
We focus on a mesoscopic system that is in thermodynamic equilibrium, in contact witha heat bath at temperature T◦, and so large that we can use thermodynamics. Since theargument of de Haas-Lorentz [6] (in German) is not accessible to most readers, we quicklysummarize it in our own words below and in more detail in Sect. 4.
According to the first law of thermodynamics dE = d̄Q + d̄W , which is energy con-servation; in general, only dE is an exact differential. For a system in contact with a heatbath there is only heat exchange and hence d̄W = 0. Moreover, the system is described bythe canonical ensemble with density function ρ = Z−1 exp(−β◦H) where H is the Hamil-tonian, β◦ = 1/kBT◦ with T◦ as heat-bath temperature and kB as Boltzmann’s constant, andZ = tr exp(−β◦H). There is no proof of this statement; it is just a matter of experience.For the gist of the ensuing argument below it suffices to keep it classical so that Z is aphase-space integral with a (pure) state ω of the system being a point in phase space Ω , i.e.,ω ∈ Ω , where a system configuration ω(t) evolves in time. Only for the sake of clarity andgenerality of exposition do we make a detour to quantum mechanics in Sect. 5.
The probabilistic meaning of ρ = Z−1 exp(−β◦H) is that because of heat exchange theenergy is not conserved but changing steadily so that in general �E ≡ H − 〈H〉 = 0. Notethe definition of �E in terms of the instantaneous H = H(ω(t)). We compare it with 〈H〉,the ensemble average of H , which is independent of time. Since there is heat exchange only,�E = �Q. Though the “real stuff” is energy transfer, we can define �T through �Q =C�T where C is the small, here mesoscopic, system’s heat capacity, a thermodynamicquantity [7], and �T = T −T◦ with T as formal temperature simply defined through �Q =C�T . Why do so?
Though �E(t) is the key physical quantity that depends on the time t through the in-stantaneous system configuration ω(t), we know from statistical mechanics [15] that it is oforder
√N where N scales with the size of the system with, for a mesoscopic one, N ≥ 106
as a typical lower bound, corresponding to a 0.1 µm length scale. �E(t) is therefore notan intensive quantity. Temperature, however, is. Hence it is advantageous to work with tem-perature, artificial as it may look at first sight. Exactly this was de Haas-Lorentz’ [6] brightidea. In passing we note that the present �T is an instantaneous quantity, depending on thetime t . If we want to stress this, we write �T (t).
A mesoscopic system in weak contact with the heat bath evolves under its own dynam-ics. How, then, can we quantify the extent to which the small system’s energy and thus itstemperature fluctuates? We take the time average
limτ→∞ τ−1
∫ τ
0dt (�T )2(t), (2)
where �T (t) = �T (ω(t)) samples specific system configurations ω(t) as they evolve oneafter the other in time under the system’s dynamics. In the context of equilibrium statisti-cal mechanics we follow the system on its way through phase space but weighted by theequilibrium density ρ = Z−1 exp(−β◦H) with β◦ = 1/kBT◦. For finite but large (enough)N the mesoscopic system’s phase space Ω in conjunction with ρ decomposes into disjoint
Temperature Fluctuations in Thermal Equilibrium 1135
ergodic components [20], which in a regular 3-dimensional world with short-range interac-tions are in general separated from each other by free-energy barriers of height ∝ Nα with0.5 < α ≤ 1, a surface term for a volume ∝ N ; cf. Sect. 3. This is the discrete, mechanical,world that the present considerations apply to.
Away from phase-transition points the system lives in one and only one of these compo-nents and its dynamics there is ergodic. On the basis of concrete examples [21, 22] we mayeven expect far more than ergodicity, such as Bernoulli behavior. It may be well to realizethat, though the system can be in only one of the ergodic components, we in general do notknow which one. Nevertheless, as the discussion leading to (4) will show, by “good luck”this does not matter for the problem at hand.
Harmless as (2) may appear, it isn’t. For at least two reasons. First, how to compute thelimit in (2), if it depends on the specific orbit of the system? Most systems are not ergodic;we will address the underlying ergodicity issue in the next Sect. 3. Second, we look at thesmall, mesoscopic, system through a finite time window of width 0 ≤ t ≤ τexp, the experi-mental time scale. In fact, it may be useful to discern three timescales: (i) the experimentalone, τexp, specifying a typical time needed to do an experiment; (ii) the necessarily finitesampling time τsampling needed to approximate (2) in the sense of Monte Carlo-like evalu-ation of the integral by a finite sampling path and, hence, get the limit (4) below; (iii) therelaxation time τrelax needed for an energy fluctuation to return to equilibrium. As we willsee in Sect. 5, we may expect
τrelax � τsampling ≤ τexp. (3)
The relaxation time tells us how long a system needs to equilibrate after a local perturbation.Then, so to speak, the next thermal kick comes, and so on. After many of them, the systemhas sampled enough phase-space configurations belonging to thermal equilibrium and theweighted integral (2) reaches its limit value (4) below. That is, the time elapsed is beyondτsampling � τrelax and we have obtained not only the limit of (2) but also the natural orderindicated in (3).
Following a specific configuration ω(t) during its journey through time is just the physi-cally sensible way of analyzing temperature fluctuations. Crude estimates suffice to convinceus that the relative size of energy fluctuations is small, viz., 1/
√N , so that their dynamics
is that of ‘return to equilibrium’ [23], which can often be computed analytically. We will doso below in Sect. 5 for a typical case.
3 Ergodic Decomposition: Ergodicity for Nonergodic Systems
Having set the perspective we first focus on what the notion of ‘ergodic component’ means.Since Kramers’ [24–27]2 and Yang & Lee’s ground-breaking work [15] we know that, math-ematically, ergodic components can exist as separate thermodynamic phases only after thethermodynamic limit N → ∞. Here, however, N is and remains finite. In case of ergodicity[20] the limit in (2) exists for “almost” every thermodynamic equilibrium configuration ω
and does not depend on ω. That is, the set of all phase-space points where this does nothappen has measure zero and hence does not occur physically.
2See in particular pp. 75–81 of ter Haar [27] and read Kramers’ original paper [25] and/or Dresden’s 1988comments [26] to catch how well Kramers realized the importance of taking the thermodynamic limit.
1136 J. Leo van Hemmen, A. Longtin
A real system is taken to be finite but nevertheless very large, at least mesoscopic, sothat it allows for thermodynamic considerations. As an approximation to an infinite systemit may exhibit a phase transition, say at temperature Tc . Let us suppose that below Tc thereare several ergodic components [20]. For instance, below Tc the 2-dimensional Ising ferro-magnet with nearest-neighbor interactions has two ergodic components μβ,±, one with +and another one with − boundary conditions, and any other equilibrium state μβ is a convexcombination of these two. That is, μβ is a measure on phase space and can be written as aconvex combination μβ = p+μβ,+ + p−μβ,− with p± ≥ 0 and p+ + p− = 1, where the p±are consistent with our a priori knowledge about the system.
Where do the μβ,± live? On disjoint parts Ω± of phase space such that Ω+ ∩ Ω− = ∅,μβ,±(Ω±) = 1 while μβ,+(Ω−) = μβ,−(Ω+) = 0. How much (free) energy does it cost to getfrom Ω+ to Ω− or conversely? For the ground state of our 2-dimensional Ising ferromagnetthis is easy to estimate. We start with a square of N sites with all Ising spins up, i.e., in Ω+belonging to T = 0, and flip all the spins at the left-hand side, a vertical line. That costs anamount of energy ∝ N1/2. We then move this line through the system while flipping spinsuntil we reach the other side and have obtained Ω− with all spins down. Except for the initialspin flip on the left, Ω− has been reached from Ω+ at no cost of energy so that the energybarrier between Ω+ and Ω− scales as N1/2.
As an illustration of why ergodicity is physically relevant we take the 2-dimensional Isingferromagnet with free boundary conditions. In the thermodynamic limit N → ∞ the thermalequilibrium state μβ = p+μβ,+ + p−μβ,− with p± = 1/2. A priori we do not know whichcomponent the system is in once T < Tc but, consistently with our a priori knowledge [28],we can say that with probability 1/2 it lives either in Ω+, the support of μβ,+, or in Ω−, thesupport of μβ,−, but never in both at the same time since Ω+ ∩ Ω− = ∅.
Let us now focus on the Ising spin Si = ±1 at site i. For (almost) every configurationω(t) of the system at time t , Si(t) tells us whether the spin is up or down and mβ =limt→∞ t−1
∫ t
0 dt ′Si(t′) shows us what the magnetization is. “Standard” statistical mechanics
[15] would prescribe our taking ρ = Z−1 exp(−β◦H) and computing 〈Si〉 = tr(ρSi), whichin the present case means = 0, completely contrary to there being a nonzero spontaneousmagnetization mβ,± = 0 below Tc . In reality, though mβ = (mβ,+ +mβ,−)/2 = 0, the systemis either in Ω+, where one finds mβ,+ > 0, or in Ω−, where one finds mβ,− = −mβ,+ < 0.Only the sum of mβ,± vanishes, but the spontaneous magnetization does not because of thedichotomy of the different “ergodic” components Ω± for β > βc . And what holds for thetwo ergodic components of the 2-dimensional Ising ferromagnet below Tc , holds even moreso for systems with many more ergodic components. The total system, in the above exam-ple Ω = Ω+ ∪ Ω−, is never ergodic below Tc so that “standard” formulae are meaningless,unless one’s attention is confined to the specific ergodic component the system is in.
The very same argument is valid for 0 < T < Tc in 3 dimensions. In a 3-dimensionalsystem with short-range interactions, free-energy barriers scale as N2/3 and ergodic compo-nents are evidently stable. Since in general, and in contrast to the Ising model,3 the systemhas a nontrivial dynamics, phase space decomposes into disjoint ergodic components [20].A system in thermodynamic equilibrium lives in one of them, where the dynamical evolu-tion αt is ergodic and, hence, the ergodic theorem holds. So, after all, Boltzmann was right,even though he was wrong and most systems are not at all ergodic for T < Tc . In passing wenote that an analogous but slightly more involved argument holds [20] for quantum systems.
3Whereas even classical Heisenberg spins give rise to a “natural” dynamics, Ising spins do not. One mustgive them one such as Monte Carlo or Metropolis. The dynamics of the infinite classical Heisenberg modelhas been shown to exist by [29], and in full detail in [30].
Temperature Fluctuations in Thermal Equilibrium 1137
Exploiting Ergodicity Returning to (2), we substitute �T = C−1�Q and �Q = �E, ap-ply ergodicity so as to replace the time average by a phase average with respect to thesystem’s ergodic component, and conclude that the time average (2) equals C−2〈(�E)2〉,the canonical mean 〈. . .〉 being taken with respect to the specific ergodic component the sys-tem is in. All components have the same [20] free energy (per particle) f (β) and hence thesame heat capacity C as second derivative of −βf (β) with respect to temperature. Accord-ingly we need not know the specific component the system is in and have a firm basis to goahead and compute the limit in (2). Furthermore, from standard statistical mechanics [15],i.e., a partial differentiation with respect to β , we then directly obtain 〈(�E)2〉 = kBCT 2◦ .Gathering terms we find the observable quantity
limτ→∞ τ−1
∫ τ
0dt (�T )2(t) = (kB/C)T 2
◦ . (4)
This, then, leads to the standard deviation√
kB/CT◦, a result that has hovered through theliterature for more than a century. Of course we need not take the limit τ → ∞ in (4) butcan stick to some finite sampling time τsampling. We will see in Sect. 5 how finite it can be.
To better understand the intricacies of the arguments, we now turn to three, related, ques-tions. First, how did de Haas-Lorentz derive the right-hand side of (4) and what do we learnfrom her reasoning? Second, though we have defined a temperature-fluctuation measurethrough (4), can one determine it experimentally? This concerns both the question of howlarge τ must be, viz., τ > τsampling, to guarantee a good approximation of the right-hand sideof (4), and how small a mesoscopic system can be. That is, third, how large must the systemsize N be so that �Q = C�T ?
4 De Haas-Lorentz’ Arguments in a Nutshell
De Haas-Lorentz [6] analyzes two meso/macroscopic bodies in thermal equilibrium at tem-perature T◦ and being able to exchange heat through metal wires of negligible heat capacity.Let T◦ + θ be system ’s instantaneous temperature as defined above with = 1,2 while C
is the corresponding heat capacity. In exchanging heat, there is no work involved, the totalenergy is conserved, and thus C1θ1 +C2θ2 = 0. A little algebra then gives for the transferredamount of heat between the bodies 1 and 2
d̄Q1 = C1d̄θ1 = Cd̄θ = −C2d̄θ2 = −d̄Q2
where θ = θ1 − θ2 and
C−1 = C−11 + C−1
2 . (5)
Hence the corresponding total entropy change dS = d̄Q/T , an exact differential, equals
dS = Cd̄θ
[1
T◦ + θ1− 1
T◦ + θ2
]= −C
θ1 − θ2
T 2◦d̄θ = −C
θdθ
T 2◦. (6)
Here the second equality is an approximate one following from |θ| � T◦ while d̄θ nowbecomes dθ since the differential is exact with θ as the only remaining variable. Integrat-ing (6) we then find �S = −Cθ2/2T 2◦ . De Haas-Lorentz finishes the argument by formallyapplying the Boltzmann relation �S = kB lnW and interpreting W as a Gaussian probabilitydensity [5] so as to arrive at the mean 〈θ2〉 = (kB/C)T 2◦ . With hindsight, the argument of
1138 J. Leo van Hemmen, A. Longtin
de Haas-Lorentz, who in particular used the probabilistic interpretation of the entropy [5],foreshadowed Onsager’s regression hypothesis for extensive variables [31, 32].
The above argument leading to (6) has never been reproduced but there is ample reason tobelieve that the final result, viz., (1), is correct. Equilibrium statistical mechanics, however,cannot be used to compute the properties of any time-dependent fluctuation of a system’sobservables. In passing we note that the final equation 〈θ2〉 = (kB/C)T 2◦ incorporates thecase of a single system in contact with a heat bath, say 2, since C then reduces to C1 andθ = T − T◦, as before. Nevertheless the angular brackets of 〈θ2〉 cannot indicate a phaseaverage with respect to the canonical ensemble since in the present context the quantityθ2 is not a direct observable. Being in an ergodic component means time average = phaseaverage, which requires following the mesoscopic system during some experimental timeτsampling long enough to guarantee the validity of (4), a dynamical process in accordancewith the experimental situation at hand. Hence the system must also be so large that thethermodynamic relation �Q = C�T makes sense. How large precisely depends on thesystem under consideration.
5 Time Scale of Temperature Fluctuations
How fast, then, do small energy excitations typically return to equilibrium? Let us call thecorresponding relaxation time τrelax. For (4) to be valid we then need τsampling � τrelax. Wecheck them in turn. First τrelax. To this end we turn to a magnetic system. Heat has manyappearances, among others as electromagnetic field with relatively long wavelengths. Herewe envisage our magnetic system irradiated by a magnetic field of suitable (even longer)wavelength so that it gets excited locally. To become concrete, we imagine a quantum spin-1/2 XY chain [33]. Through a Jordan–Wigner transformation its Hamiltonian can be trans-formed into a bilinear fermion one [34],
HΛ = J∑
x,y∈Λ
φ(x − y)a+x ay + [
ψ(x − y)axay − ψ̄(x − y)a+x a+
y
](7)
where a+x and ay denote fermion creation and annihilation operators. We have actually in-
troduced a ν-dimensional notation with Λ ⊂ Zν and assumed φ(−x) = φ̄(x) and ψ(−x) =
−ψ(x) to make HΛ self-adjoint; both φ and ψ are dimensionless. The XY chain is a specialcase with ν = 1 and φ(1) = φ(−1) = ψ(1) = −ψ(−1) = 1, all other values vanishing. Theprefactor J has the dimension of energy, represents a magnetic interaction, and is [35] oforder kBTc for some typical temperature Tc; say, 100 °K.
The dynamics is the Heisenberg one, generated by
AL � A �→ αt (A) = limΛ→∞ exp(itHΛ/h̄)A exp(−itHΛ/h̄). (8)
For local observables A ∈ AL and spin or fermion (CAR) systems with finite-range or fastenough decaying interactions on a lattice such as Z
ν , the limit αt in (8) exists [17, 23, 34,36, 37].
Radiating energy into the system locally, which is what a heat fluctuation is, we can stillwork with the equilibrium state of the larger, formally infinite, system containing it so thatin the Gelfand–Naimark–Segal (GNS) representation [34, 36, 37] induced by an ergodicequilibrium state 〈. . .〉β the system’s dynamics αt in (8) now lives in a suitable Hilbert spaceand is represented, as is usual in quantum mechanics, by a group of unitary operators Ut .
Temperature Fluctuations in Thermal Equilibrium 1139
The intuition behind the GNS representation is appealingly simple. We start with thealgebra of local observables AL, here finite products of fermion creation and annihilationoperators at the same or different sites, and pick two elements A and B ∈AL. Then we definean inner product (A | B) ≡ 〈A∗B〉β with A∗ as the adjoint of A. Checking (A | B) is an innerproduct is trivial, except for (A | A) = 0 ⇒ A = 0 but that follows from a nice property [34,36] of the algebra of (local) observables. So we get something that looks like but is not aHilbert space yet since it is not complete, i.e., not every Cauchy sequence converges. Thereis a standard procedure in mathematics to “complete” it by adding limit points so that weobtain a Hilbert space H. AL is an algebra allowing us not only to add two operators A andB ∈ AL but also to multiply them, the product AB being again in AL. For B ∈ AL but nowconsidered as an element of H, we define a representation π of AL by π(A)B ≡ AB ∈ H.It is not too hard to show [34, 36] that π(A) as representation of A is a bounded operatoron the Hilbert space H and the relation between A and π(A) is 1-1 and norm-preserving; intechnical terms, the algebra of observables is ‘simple’ [37, Thm. IV.1.3]. So we are actuallyback in Hilbert space. There is, however, a subtle difference in that the present Hilbert spaceH stems from thermal equilibrium, here the ergodic component 〈. . .〉β . In a classical context,we could restrict our attention to the support of the thermal-equilibrium measure μβ , whichin general is only a tiny part of phase space. In quantum mechanics with its noncommutingobservables AL, the GNS representation in the Hilbert space H plays a similar role. Whereasin classical physics the ergodic components, if there are more than one, live in disjointparts of phase space, in quantum mechanics the different GNS Hilbert spaces H belong to‘disjoint’ states [38].
Finally, since we focus on an equilibrium state we must have 〈αt (A)〉β = 〈A〉β . With a bitmore work [34, 36] we obtain 〈Aαt(B)〉β = (A∗ | UtB) where Ut = exp(itH/h̄) is a groupof unitary operators with, as in ordinary quantum mechanics, H as “Hamiltonian.” For theinsiders, a few harmless details have been dropped, such as the cyclic and separating vector,which in our case is the unit operator.
Now we only need to evaluate correlation functions of the form 〈Aαt(B)〉β as t → ∞and A and B being finite products of local fermion creation and annihilation operators. Weare then left with two-point correlation functions of the form 〈axαt (a
+y )〉β . The key question
is how fast they converge to zero. Their evaluation is straightforward for a Fermi latticesystem with bilinear Hamiltonian on Z
ν [34] and boils down to computing, as t → ∞,
∫
[−π,π ]νdkf (k) exp
[itJμ(k)/h̄
](9)
where μ(k) is a dimensionless smooth function, of order 1. The exponent stems from theHeisenberg dynamics generated by exp(itH/h̄) and hence has the relevant time constantτrelax = h̄/J . Using the above 100kB we find τrelax = 10−15 s. The limit of (9) as t → ∞ is notcompletely trivial but can be done [39]. In ν dimensions one finds that the integral behaveslike (t/τrelax)
−ν/2. According to Chui et al. [10] τsampling � τrelax and consequently (4) holdsin their case, as they have indeed found experimentally.
Formally, C in (4) means CV but distinguishing Cp and CV is immaterial to most meso-scopic systems so that we will stick to C. On the other hand, away from phase-transitionpoints and being in a specific ergodic component one may expect the system to be far morethan just ergodic [21, 22] so that, once measuring the temperature is meaningful [10] andthe measurement process lasts longer than the required sampling time τsampling, we are done.And if the small system is not in thermal equilibrium, temperature is an ill-defined notionand a discussion of its fluctuation makes no sense.
1140 J. Leo van Hemmen, A. Longtin
6 Thermodynamic Uncertainty Relations
In view of the long history of the notion of temperature “fluctuations” it is worth our whileto tie up our present considerations with so-called thermodynamic uncertainty relations andtheir derivatives [11, 12]. In so doing we will follow Lindhard [11], who considers twoweakly interacting systems 1 and 2 that are in thermal equilibrium with themselves andeach other, and whose total energy Etot is conserved. Because the interaction is weak, theHamiltonian H of the combined system is written as the sum H1 + H2 and the density ofenergy states factorizes, ρ1(E)ρ2(Etot − E) where E refers to the smaller system we focuson. Hence the probability of finding E in an infinitesimal interval dE is given by
Z−1 exp{ln
[ρ1(E)ρ2(Etot − E)
]}dE. (10)
It is assumed that the exponent has a unique, “most probable” maximum at Ep . Since theinverse temperature β1 = 1/kBT1 of system 1 follows [15] from 1/T1 = ∂ ln[ρ1(E)]/∂E andits analog for T2 of system 2, Ep’s existence requires the first derivative of the exponent in(10) with respect to E to vanish and hence the two temperatures T1 and T2 to be equal, say,to T◦. One then stops the Taylor expansion of the exponent after the second-order term [11]and obtains a Gaussian,
(1/σ√
2π) exp[−(E − Ep)2/2σ 2
](11)
where the Gaussian’s variance equals σ 2 = kBT 2◦ C and C is given by (5). Under the va-lidity of (11) one finds 〈H1〉 = Ep . The general result 〈(�E)2〉 = kBCT 2◦ as derived hereis consistent with (11) but needs no Taylor expansion in the exponent and stopping afterthe second-order term so as to get a Gaussian as do Huang [15] and Lindhard [11]. Neitherdoes (4).
Though Eq. (11)’s resemblance to (6) is already striking, we now show that the resultsof Lindhard [11] and de Haas-Lorentz [6] actually agree fully. To this end we note thatthe natural variable in (11) is the energy whereas in (6) it is temperature or, better, thetemperature change θ . We therefore write E = Ep + ε and express (6) in terms of ε =Cθ = ±ε, with ε1 + ε2 = 0,
dS = dε
T◦
[1
1 + ε/C1T◦− 1
1 − ε/C2T◦
]= −εdε/CT 2
◦ (12)
so that �S = −ε2/2CT 2◦ . To estimate the deviation ε as compared to Ep , we note that theformer is O(N1/2), much less than the latter, viz., O(N). Proceeding as de Haas-Lorentzdid with respect to θ in (6) and assuming a Gaussian but now to obtain the distribution of ε
out of (12) we find (11). The rest of the argument, an integration with respect to (11), is atriviality and the final result is exactly the same as after (6): 〈θ2〉 = (kB/C)T 2◦ .
It is tempting to think that there were a “complementarity” between energy and tem-perature [11, 12] à la �E�(1/T ) ≥ kB with kB playing the role of h̄. Though this kind ofargument goes back to Landau and Lifshitz [40], the reasoning is purely formal, effectivelyidentical to that of Einstein [5] and de Haas-Lorentz [6], and leaves us with the problem ofhow to interpret �T . That is, the present arguments are consistent and (4) is an independentrelation intertwining the diverging parts of temperature-fluctuation theory on an experimen-tal timescale.
Temperature Fluctuations in Thermal Equilibrium 1141
7 Is There a Conclusion?
Stepping back for an overview, what can we discern? We have seen that relation (1) is simplya measure of how much, or little, energy is involved in the dynamical process of heat ex-change with a heat bath of temperature T◦. The ensuing energy fluctuations are quantified bymeans of the intensive quantity ‘temperature’ T that has been defined through the system’sheat capacity C in �E = �Q = C�T with �T = T − T◦, an experimentally vindicated[7] relation. Kittel [1] is certainly right in that the heat bath temperature occurring in thesystem’s canonical density ρ = Z−1 exp(−H/kBT◦) does not fluctuate. As a matter of fact,the energy of a small mesoscopic system is the quantity fluctuating in time. Equations (1)and (4) tell us how much. If the system is small enough, this is also accessible to experiment[8–10]. Mandelbrot [3, 4] transformed the temperature-fluctuation problem into a statisticalone. Statistics, however, refers to the beholder’s uncertainty [28] regarding the probabilitydistribution describing the system under consideration and not to the system itself. The lat-ter is what (1) and (4) do, answering the question of the section heading: Telling us how toquantify a small system’s energy fluctuations on a temperature scale as time proceeds.
Acknowledgements A.L. thanks the Humboldt Foundation for a Humboldt Award allowing his stay atthe Technische Universität München. J.L.v.H. gratefully acknowledges the hospitality of the University ofOttawa, where this paper was finished. Both thank Aernout van Enter for constructive criticism.
References
1. Kittel, C.: Temperature fluctuation: an oxymoron. Phys. Today 41(5), 43 (1988)2. Feshbach, H.: Small systems: when does thermodynamics apply? Phys. Today 40(11), 9–11 (1987)3. Mandelbrot, B.B.: Temperature fluctuation: a well-defined and unavoidable notion. Phys. Today 42(1),
71–73 (1989). His statistical sampling argument has been reanalyzed recently by Falcioni et al. [4]4. Falcioni, M., Villamaina, D., Vulpiani, A., Puglisi, A., Sarracino, A.: Estimate of temperature and its
uncertainty in small systems. Am. J. Phys. 79, 777–785 (2011)5. Einstein, A.: Über die Gültigkeitsgrenze des Satzes vom thermodynamischen Gleichgewicht und über
die Möglichkeit einer neuen Bestimmung der Elementarquanta. Ann. Phys. (Leipz.) 22, 569–572 (1907)6. de Haas-Lorentz, G.L.: Die Brownsche Bewegung und einige verwandte Erscheinungen, i.e., ‘Brownian
motion and some related phenomena’. Vieweg, Braunschweig (1913). See in particular pp. 93–95. Thisis the German edition of the author’s Ph.D. thesis obtained the year before with H.A. Lorentz
7. Fermi, E.: Thermodynamics. Prentice Hall, New York (1937). Dover, New York (1956). Heat capacityis an accepted notion antedating the first edition of M. Planck’s classic on thermodynamics (1897)
8. Ornstein, L.S., Milatz, J.M.W.: Accidental deviations in the conduction of heat. Physica (Utrecht) 6,1139–1145 (1939)
9. Milatz, J.M.W., van der Velden, H.A.: Natural limit of measuring radiation with a bolometer. Physica(Utrecht) 10, 369–380 (1943)
10. Chui, T.C.P., Swanson, D.R., Adriaans, M.J., Nissen, J.A., Lipa, J.A.: Temperature fluctuations in thecanonical ensemble. Phys. Rev. Lett. 69, 3006–3008 (1992)
11. Lindhard, J.: “Complementarity” between energy and temperature. In: de Boer, J., Dal, E., Ulfbeck, O.(eds.) The Lesson of Quantum Theory, pp. 99–112. North-Holland, Amsterdam (1986). See in particular,§6
12. Uffink, J., van Lith, J.: Thermodynamic uncertainty relations. Found. Phys. 29, 655–692 (1999)13. Phillies, G.D.J.: The polythermal ensemble: a rigorous interpretation of temperature fluctuations in sta-
tistical mechanics. Am. J. Phys. 52, 629–632 (1984)14. Prosper, H.B.: Temperature fluctuations in a heat bath. Am. J. Phys. 61, 54–58 (1993)15. Huang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987). Sects. 7.2, 9.3 & 9.416. Balatsky, A.V., Zhu, J.-X.: Quantum Nyquist temperature fluctuations. Physica E 18, 341–342 (2003)17. Hugenholtz, N.M.: States and representations in statistical mechanics. In: Streater, R.F. (ed.) Mathemat-
ics of Contemporary Physics, pp. 145–182. Academic, London (1972)18. Jahnke, T., Lanéry, S., Mahler, G.: Operational approach to fluctuations of thermodynamic variables in
finite quantum systems. Phys. Rev. E 83, 011109 (2011)
1142 J. Leo van Hemmen, A. Longtin
19. Touchette, H.: Temperature fluctuations and mixtures of equilibrium states in the canonical ensemble. In:Gell-Mann, M., Tsallis, C. (eds.) Nonextensive Entropy—Interdisciplinary Applications, pp. 159–176.Oxford University Press, Oxford (2004)
20. van Enter, A.C.D., van Hemmen, J.L.: Statistical-mechanical formalism for spin glasses. Phys. Rev. A29, 355–365 (1984). Section II and Appendix. As exposed there in detail, similar arguments hold forquantum systems
21. Lanford, O.E. III, Lebowitz, J.L.: Time evolution and ergodic properties of harmonic systems. In:Springer Lecture Notes in Physics, vol. 38, pp. 144–177 (1975)
22. van Hemmen, J.L.: Dynamics and ergodicity of the infinite harmonic crystal. Phys. Rep. 65, 43–149(1980)
23. Robinson, D.W.: Return to equilibrium. Commun. Math. Phys. 31, 171–189 (1973)24. Heller, G., Kramers, H.A.: Ein klassisches Modell des Ferromagnetikums und seine nachträgliche Quan-
tisierung im Gebiete tiefer Temperaturen. Proc. Roy. Acad. Sci. Amsterdam 37, 378–385 (1934)25. Kramers, H.A.: Zur Theorie des Ferromagnetismus. In: 7e Congrès International du Froid, La Haye-
Amsterdam, Juin (1936). Rapports et Communications. Also in: Commun. Kamerlingh Onnes Lab. Univ.Leiden 22(83), 1–22 (1936)
26. Dresden, M.: Kramers’s contributions to statistical mechanics. Phys. Today 41(9), 26–33 (1988). Onemay consult Dresden for, amongst others, a thoughtful evaluation of Kramers’ paper [25]
27. ter Haar, D.: Master of Modern Physics: the Scientific Contributions of H.A. Kramers. Princeton Uni-versity Press, Princeton (1998)
28. de Finetti, B.: Theory of Probability, vols. I/II. Wiley, London (1974/5)29. Vuillermot, P.A.: Intertwining relations between the dynamics of the infinite classical and quantum
Heisenberg models: a new application of Trotter approximations and of the coherent-state formalism.Lett. Nuovo Cimento 24, 333–338 (1979)
30. Vuillermot, P.A.: Nonlinear dynamics of the infinite classical Heisenberg model: existence proof andclassical limit of the corresponding quantum time evolution. Commun. Math. Phys. 76, 1–26 (1980)
31. Onsager, L.: Reciprocal relations in irreversible processes II. Phys. Rev. 38, 2265–2279 (1931)32. Keizer, J.: Statistical Thermodynamics of Nonequilibrium Processes p. 68. Springer, New York (1987)33. Emch, G.G., Radin, C.: Relaxation of local thermal deviations from equilibrium. J. Math. Phys. 12,
2043–2046 (1971)34. van Hemmen, J.L.: Linear fermion systems, molecular field models, and the KMS condition. Fortschr.
Phys. 26, 397–439 (1978). Sect. II35. Landau, D.P., Binder, K.: A Guide to Monte Carlo Simulations in Statistical Physics, 3rd edn. Cambridge
University Press, Cambridge (2009)36. Sewell, G.L.: Quantum Theory of Collective Phenomena. Oxford University Press, Oxford (1986)37. Simon, B.: The Statistical Mechanics of Lattice Gases, vol. I. Princeton University Press, Princeton
(1993)38. Hepp, K.: Quantum theory of measurement and macroscopic observables. Helv. Phys. Acta 45, 237–248
