Eigenstate-specific temperatures in two-level paramagnetic spin lattices

Though temperature is a ubiquitous concept in the physical and biological sciences, its nature and definition have become subjects of fresh debate in the last three decades in two notable contexts: (1) the Feshbach¹–Kittel²–Mandelbrot³ (FKM) debate regarding the differences between thermodynamic temperatures and the so-called “effective temperatures”² or “temperature estimators”³ for systems in contact with small heat baths² and (2) the debate over the thermodynamic legitimacy of negative temperatures in systems with bounded energy spectra,^4–33 which was recently reignited^4–6,31 by the realization of negative temperatures in optical lattices.^28–32 The FKM^{1–3,34–36} and negative temperature^4–6,27,31 debates both address issues central to a number of important topics: (1) the thermodynamics of small systems;^1–3,34 (2) the thermodynamics of isolated systems;^{12,34,35,37–59} (3) the thermodynamic uncertainty relation (TUR);^{1–3,34–36} and (4) quantum thermodynamics,^40,60 in which the temperatures of individual quantum eigenstates, hereafter designated eigenstate-specific temperatures (ESTs), apply.

In the present study small, two-level paramagnetic spin lattices (PSLs)^{7,9,11–18,36,61–66} are used to address two questions raised in the FKM^1–3 and negative temperature^4–25 debates: (1) “Are the ESTs of PSLs true thermodynamic temperatures or merely good temperature estimators?” and (2) “Are the negative ESTs of population-inverted PSLs thermodynamically legitimate?” We address these questions by characterizing the size- and energy-dependence of four distinct ESTs: (a) continuous and (b) discrete canonical ESTs, which apply to PSLs in equilibrium with heat baths, and (c) continuous and (d) discrete microcanonical ESTs, which apply to isolated PSLs.

The four ESTs share much in common with the conventional canonical temperature $Tcconv$⁠, which is defined as (1) a parameter which is equal to the temperature T_bath of the system’s heat bath^2,67 and (2) the continuous rate of change (⁠$∂$U/$∂$S_c[U])_N,V$$_U* of U with respect to the canonical entropy S_c(U) evaluated at the most-probable energy U*. The continuous canonical and continuous microcanonical ESTs are equal to the continuous (i.e., instantaneous) rates of change of U with respect to S_c(U) and the microcanonical entropy S_μ(U) for the initial eigenstates [N − j, j] in transitions between adjacent PSL eigenstates with spin quantum numbers j and j+1: $Tcj→j+1$ = (⁠$∂$U/$∂$S_c[U])_N,V$$_Uj and $Tμj→j+1$ = (⁠$∂$U/$∂$S_μ[U])_N,V$$_Uj. The discrete canonical and discrete microcanonical ESTs $Tdcj→j+1$ = (ΔU/ΔS_c)_N,V$$_Uj and $Tdμj→j+1$ = (ΔU/ΔS_μ)_N,V$$_Uj are the discrete analogs of the continuous ESTs.

As their name indicates, the ESTs are eigenstate-specific; they are equal to the temperatures of specific, individual eigenstates. They thus constitute a distinct contrast to $Tcconv$ = $Tcj¯→j¯+1$⁠, which is eigenstate-nonspecific^11,40,68 because $j¯$ is equal to the (typically non-integer) average value of j over a Boltzmann distribution of eigenstates. Even so, ESTs can provide meaningful estimates of $Tcconv$ in PSLs, particularly when the population distribution is dominated by a single eigenstate (i.e., when $j¯$ = j), as occurs for microcanonical (i.e., thermally isolated) PSLs,^3,69 PSLs at low temperatures, large PSLs in the N → ∞, thermodynamic limit (TDL),^42,70–73 and PSLs subjected to repetitive magnetization measurements.^{36,68,74–78}

This paper is organized as follows. Background materials are provided in Secs. II A and II B. $Tcconv$ and the four ESTs are derived in Secs. II C and II D. The general properties of the ESTs are detailed in Sec. III A. The spin-permutation antisymmetries (SPAs) characteristic of positive and negative temperature eigenstates are detailed in Sec. III B and the supplementary material. The size- and energy-dependency of the ESTs—with special emphasis on their functional dependence on the spin-down mole fraction X_j = j/N— are detailed in Sec. III C and the supplementary material. We then detail the adherence of the ESTs to the four laws of thermodynamics (Sec. III D), the relationships of the ESTs to Boltzmann distributions and the TUR,^34,35 the relationships of the ESTs to temperature-dependent system energy levels (TDSELs)³³ (Sec. III E), and the FKM debate (Sec. III F).^{1–3,34–36} The implications of the Boltzmann and Gibbs ESTs for the negative temperature debate^4–6,27,31 are detailed in Sec. III G. We detail the implications of the ESTs for experimental temperature measurements in PSLs^{7,13–17,22,54,63–65,79–89} (Sec. III H), the potential utility of ESTs in nanothermometry^{60,83,84,90–102} (Sec. III I), and potential applications of ESTs to systems with long-range interactions^{36,60–62,66–68,74,83,84,92–94,103–106} (Sec. III J). Finally, a number of issues raised by the present study are detailed in Sec. IV.

PSLs are fixed arrays of atoms, ions, or molecules with nuclear^{7,9,11–17,63–65,79,81,107} or electron^{7,9,11–17,63–65,79,87–89,108–110} spin. Two-level PSLs (“PSLs” hereafter) result when each site is comprised of a spin-½ $ℏ$ nucleus, paramagnetic ion, or free radical with a spin-up (magnetic moment μ parallel to external magnetic field H) ground state $↑$ with energy

and a spin-down (μ antiparallel to H) excited state $↓$ with energy

in which g is the nuclear or electron g factor and β is the nuclear or electron (Bohr) magneton.^10,111,112 For convenience we rescale the single particle energies to $u↑=0$ and $u↓=ε$⁠.

PSLs are characterized by three important features. First, because their spin sites are localized, the sites are distinguishable so that the Pauli exclusion principle does not apply.¹² Second, the sites do not interact with each other so that the total internal energy of an N-particle PSL eigenstate [$N↑$⁠, $N↓$] = [N − j, j] is equal to the sum of the individual particle energies,¹² 

Third, because ε ∝ H, the energy spectra of PSLs are discrete in high fields but become continuous in the H → 0 limit.

The thermodynamic temperature T is equal to the rate of change of internal energy U with entropy S,

Hence, when the entropy is analytic in U, ESTs may be obtained via the expression

Using the canonical partition function for a PSL in equilibrium with a heat bath, Kittel¹¹ obtained expressions for mean energy

and the conventional canonical temperature

as functions of the average number of spin-down sites $j¯$ and the average spin-down mole fraction $Xj¯$ = $j¯/N.$ Equation (4) is well behaved for all $j¯$ ≤ N provided $j¯$$≠$ 0 or N/2¹¹³ and for all $Xj¯$ ≤ 1 provided $Xj¯$$≠$ 0 or 0.5.¹¹³ 

The canonical entropy S_c(⁠$Uj¯$⁠) is obtained by integrating dS_c(⁠$Uj¯$⁠) = d$Uj¯$/T_c(⁠$Uj¯$⁠) from 0 to $Uj¯$⁠, yielding the concave-downward^42,70–73 expression

which is well defined provided 0 ≤ $j¯$≤ N.¹¹³ Since it is a function of energy, S_c(U) is microcanonical in character;¹¹ it can thus also be obtained from the microcanonical partition function.^10,112 Even so, S_c(U) differs from the microcanonical entropy S_μ(U) at finite N but converges to S_μ(U) in the TDL^42,70–73 [compare Eqs. (5) and (7)].

For Boltzmann-distributed PSLs, the average number of spin-down lattice sites $j¯=N¯↓$ is typically not an integer j.^36,40 Consequently, $Tcconv$ = $Tcj¯→j¯+1$ is usually not equal to a continuous canonical EST $Tcj→j+1$⁠. The temperature becomes eigenstate-specific when $j¯$ → j. This single eigenstate occupancy condition (SEOC) applies in four scenarios: (i) under microcanonical conditions, in which the PSL is in the single eigenstate it occupied at the moment it was isolated from its heat bath;^3,69 (ii) at low temperatures, in which only the ground eigenstate is occupied; (iii) in the thermodynamic limit (TDL), in which the Boltzmann distribution is dominated by its most-probable eigenstate [N − j*, j*];^42,70–73 and (iv) when PSLs are subjected to repetitive magnetization measurements which narrow the eigenstate distribution.^{36,68,74–78}

Four distinct ESTs may be calculated. The continuous microcanonical EST $Tμj→j+1$ = (⁠$∂$U_j/$∂$S_μ[U_j])_N,V and the continuous canonical EST $Tcj→j+1$ = (⁠$∂$U_j/$∂$S_c[U_j])_N,V are equal to the derivatives of U with respect to S_c and S_μ, respectively. They are thus equal to the tangential slopes at the points on the U vs. S profiles corresponding to the initial eigenstates [N − j, j] in j → j + 1 transitions (see Figs. 1 and 2). The discrete microcanonical EST $Tdμj→j+1$ = $(ΔUj→j+1/ΔSμj→j+1)N,V$ and the discrete canonical EST $Tdcj→j+1$ = $(ΔUj→j+1/ΔScj→j+1)N,V$ are equal to the finite difference slopes between the points on the U vs. S profiles corresponding to the initial [N − j, j] and final [N − j − 1, j + 1] eigenstates in the transitions [see Figs. 2(a)–2(c)].

The transition energy between energetically adjacent eigenstates in PSLs is equal to

The microcanonical entropy S_μ(U_j) of [N − j, j] is equal to

in which the gamma function Γ(j+1) = j! is introduced to make the entropy a continuous function of j.¹¹⁴ Taking the derivative with respect to j and inverting the result yields

in which Ψ₀(x) is the digamma function.^68,115 Eq. (8a) is well defined for all 0 ≤ j ≤ N¹¹³ provided j$≠$N/2. As a function of the spin-down mole fraction, $Tμj→j+1$ is equal to

which is well behaved for all 0 ≤ X_j ≤ 1¹¹³ provided X_j$≠$ 0.5. $Tμj→j+1$ approaches $∓∞$ in the limits as j→(N/2)^± and X_j → 0.5^±.

When $j¯$ is equal to the spin-down quantum number j of an eigenstate [N − j, j], the thermodynamic functions become eigenstate-specific. Under these conditions, eigenstate-specific energy, temperature, and entropy expressions are obtained by substituting j for $j¯$ in Eqs. (3)–(5), yielding

and the concave-downward^42,70–73 entropy expression

Equation (10) is well behaved for all j ≤ N provided j$≠$ 0 or N/2¹¹³ and for all 0 ≤ X_j ≤ 1 provided X_j$≠$ 0 or 0.5.¹¹³ $Tcj→j+1$ approaches zero in the limits as j → 0⁺ and X_j → 0⁺; it approaches $∓∞$ in the limits as j → (N/2)^± and X_j → 0.5^±.

$Tdμj→j+1$ is obtained by combining Eqs. (2a), (6), and (7) to yield

which is well behaved for 0 ≤ j < N¹¹³ and j$≠$ (N − 1)/2 and for 0 ≤ X_j < 1¹¹³ and X_j$≠$ 0.5(1 − 1/N). $Tdμj→j+1$ approaches $∓∞$ in the limits as j → (N − 1)/2^± and X_j → 0.5(1 − 1/N)^±.¹¹⁶ 

$Tdcj→j+1$ is equal to the finite difference ratio

in which ΔU^j→j+1 and $ΔScj→j+1$ are obtained using Eqs. (6) and (11). Equation (13) is well behaved provided 0 ≤ j < N^113,117 and j$≠$ (N − 1)/2 and provided 0 ≤ X_j < 1^113,117 and X_j$≠$ 0.5(1 − 1/N). $Tdcj→j+1$ approach $∓∞$ in the limits as j → (N − 1)/2^±116 and X_j → 0.5(1 − 1/N)^±.¹¹⁶ Equation (13) is to our knowledge the first derivation of discrete canonical temperatures in any context.

Conventional absolute (Kelvin) temperatures $Tcconv$ are (1) positive because the translational entropy of an ideal gas increases monotonically with increasing energy,¹¹⁸ (2) finite because for entropically monotonic systems, infinite temperatures occur only in the limit of infinite energy, (3) continuous because the energetic splittings between the translational energy levels of ideal gases are small,¹¹⁸ and (4) intensive (i.e., independent of N) because typical systems are large (N ≥ 10¹⁸) and because ideal gas particles are non-interacting.^{36,60–62,66–68,74,83,84,92–94,103–106} In contrast to those of ideal gases, the entropies of PSLs^{7–9,11–20,63–65,79} and other energetically bounded systems^{7,21–25,28–32} increase with energy for eigenstates with energies between the ground (U = 0) and median energy (U = Nε/2) but decrease upon further increases in energy. Hence, the ESTs are positive, infinite, and negative for U_j < Nε/2, U_j = Nε/2, and (population-inverted) U_j > Nε/2 eigenstates,^{7–9,11–17,20} respectively (see Figs. 1 and 2).

Negative spin temperature (i.e., population-inverted) PSL eigenstates are populated under two conditions: (i) upon rapid 180° rotation of an external magnetic field, which permutes the spin-up and spin-down states;^8,9,12,15 and (ii) upon repetitive magnetization measurements.^68,74–77 Neither of these conditions involves direct thermal heating: Population inversions can be achieved only through non-thermal means.^{7–9,11–18,21–32,36,61–66}

ESTs change sign upon permuting the spin-up and spin-down spin sites; that is, they manifest spin permutation antisymmetry. The most significant manifestation of spin permutation antisymmetry is the difference in the j values for which the continuous and discrete ESTs become infinite (see Table I and the supplementary material).¹¹⁹ 

Temperature is typically intensive so that T = T(U). Because the energy of PSLs is extensive, non-intensive temperatures T(U,N) occur when the entropy is non-extensive, in which case the N-dependence of the energy numerator in Eq. (2a) is not canceled by a comparable N-dependence in the entropy denominator. Non-intensive temperatures occur in small systems, in which finite-size effects^{36,40,60,68,83,84,92,93} are important, and in systems with long-range interactions.^{36,60–62,66–68,74,83,84,92–94,103–106}

Because the spin sites in PSLs are non-interacting, it follows that non-intensive ESTs in PSLs originate exclusively from finite-size effects. Although these effects are well known,^{36,40,60,68,83,84,92,93} we report here a previously unrecognized non-intensive temperature behavior which depends functionally on X_j in small PSLs.^120,121

The ESTs may be grouped into triads comprised of sets of three types of energetically adjacent (j = aN − 1, aN, and aN + 1) eigenstates: (1) constant-X_j eigenstates, for which X_j = a is constant with increasing N; (2) increasing-X_j eigenstates, for which X_j = a − 1/N increases with increasing N; and (3) decreasing-X_j eigenstates, for which X_j = a + 1/N decreases with increasing N. The constant 0 ≤ a ≤ 1 is specific to a given triad.

Although the ESTs in a given triad converge to the common thermodynamic-limiting value ε/kln[(1 − X_j)/X_j] = ε/kln[(1 − a)/a], the increasing-X_j, constant-X_j, and decreasing-X_j eigenstates within each triad manifest different functional N-dependencies when N < 1000. This heretofore unreported behavior is demonstrated for continuous canonical ESTs in Eqs. (14a)–(14c) and for the discrete canonical, continuous microcanonical, and discrete microcanonical ESTs in the supplementary material.

The continuous canonical ESTs

of constant-X_j eigenstates are converged to ε/kln[(1 − a)/a] for all N [see Fig. 3(b)]; these ESTs are inherently intensive. The continuous canonical ESTs

of increasing-X_j eigenstates are smaller than ε/kln[(1 − a)/a] at small N but ascend to this value as N → ∼1000 [see Fig. 4(a)]. In contrast, the continuous canonical ESTs

of decreasing-X_j eigenstates are larger than ε/kln[(1 − a)/a] at small N and descend to this value as N → ∼1000 [see Figs. 4(b) and 4(c)]. The continuous microcanonical, discrete microcanonical, and discrete canonical ESTs manifest similar behaviors [see Figs. 3(a), 3(c), and 3(d) and the supplementary material]. These unique small N-dependencies suggest that PSLs may prove effective at characterizing the temperatures of systems smaller than those accessible with most currently available nanothermometers^{60,83,84,90–102} (see Sec. III I, Figs. 3 and 4, and the supplementary material).

To be thermodynamically legitimate, ESTs must obey the four laws of thermodynamics. As demonstrated below, this condition applies only when ESTs are intensive.

For small N_A and N_B, the ESTs of two PSLs A and B differ if N_A $≠$ N_B—even when equilibrium (i.e., X_jA = X_jB) conditions apply. Under such conditions, A and B can be brought to the same temperature only by adjusting their respective energetic splittings ε_Α and ε_B. The zeroth law is thus violated by small PSLs; it is obeyed by large PSLs, for which the ESTs are intensive.¹²² 

The first law mandates that the energy change ΔU^j→j+1 = C_V (⁠$TYj+1→j+2$ − $TYj→j+1$⁠) = ε for Y = c, dc, μ, dμ. Using the continuous canonical ESTs of positive (X_j = 0.2) and negative (X_j = 0.8) temperature eigenstates, we find that ΔU = ε for both eigenstates when N ≥ 10³ but that when N = 10, ΔU = 1.411ε for X_j = 0.2 and 0.819ε for X_j = 0.8. The continuous canonical ESTs are thus consistent with the first law when N is large but violate this law when N is small; comparable behavior is predicted for the other three ESTs. The first law is thus violated by small PSLs; it is obeyed by large PSLs, for which the ESTs are intensive.

When “statistical” changes dS_Ω = d(k ln Ω) in the microcanonical entropy are equal to the “thermodynamic” entropy changes dS_q/T = dq/T, the second law is obeyed.¹²³ This property is used here as a criterion for genuine thermodynamic behavior.

As a working system, we assume a 2N-particle microcanonical “super-PSL” A + B comprised of two N-particle “sub-PSLs” A and B which exchange energy with each other but are otherwise thermally isolated. We further assume that the sub-PSLs are initially in hot (A = [0.7N, 0.3N]) and cold (B = [0.9N, 0.1N]) eigenstates and that heat flows from A to B until both sub-PSLs are in their [0.8N, 0.2N] eigenstates. The heat exchange can occur in two ways: (1) a single-step process

in which 0.1N heat quanta ε flow instantaneously and isothermally from A to B; and (2) a sequential, multi-step process

in which 0.1N individual heat quanta ε are successively transferred from A to B and the temperatures of sub-PSLs A and B fall and rise, respectively, with each heat exchange. Since both processes are temperature-independent and share the same initial and final states, the entropy changes are identical for both processes,

The single-step thermodynamic entropy change

is larger than the multi-step thermodynamic entropy change

which at finite N adopts a value lying between those of ΔS_{q/T,single–step} and ΔS_Ω. Because they incorporate “excess” entropy contributions originating from the temperature differences between the PSLs, the thermodynamic entropy changes are both larger than ΔS_Ω.

The entropy increases ΔS_q/T = (–ε/$TcAjA→jA+1$ + ε/$TcBjB→jB+1$⁠) induced by exchanges of individual heat quanta become smaller as the temperatures of the sub-PSLs converge, i.e., as j_A → j_B → j_final = 0.2N. Because the number of heat exchanges in which the values of j_A and j_B are similar increases with increasing N, the average excess entropy per exchange is smallest when N is large. Consequently, ΔS_{q/T,multi–step}/ΔS_Ω → 1 in the TDL: For 10-, 50-, 100-, 500-, and 1000-particle sub-PSLs initially in X_jA = 0.3 and X_jB = 0.1 eigenstates, ΔS_{q/T,multi–step}/ΔS_Ω = 2.081, 1.209, 1.104, 1.020, and 1.010, respectively.¹²³ Hence, ΔS_{q/T,multi–step} → ΔS_Ω in the TDL; under these conditions, the second law is obeyed because the ESTs are intensive. In contrast, since both ΔS_{q/T,single–step} and ΔS_Ω scale with N, ΔS_{q/T,single–step}/ΔS_Ω = 2.086 is constant for all N so that ΔS_{q/T,single–step} does not converge to ΔS_Ω in the TDL.

Three conclusions can be drawn from these results. First, from the standpoint of entropy, thermodynamics and statistical mechanics are equivalent for PSLs in the TDL because ΔS_{q/T,multi–step} → ΔS_Ω as N becomes large. Second, from the standpoint of temperature, statistical mechanics and thermodynamics are equivalent for PSLs in the TDL because the ESTs become intensive at large N. Third, the second law is violated by small PSLs (for which the ESTs are non-intensive) but obeyed by large PSLs (for which the ESTs are intensive).

Since a PSL in its ground eigenstates [N,0] is a perfect spin crystal, the four ESTs should equal zero for this eigenstate when the third law is obeyed. In contrast to this expectation, however, only the continuous canonical EST $Tc0→1$ is equal to 0K for all N; $Tμ0→1$⁠, $Tdμ0→1$⁠, and $Tdc0→1$ are each greater than zero at finite N and approach zero only in the TDL. Hence, when $Tμ0→1$⁠, $Tdμ0→1$⁠, and $Tdc0→1$ are applied, the third law is violated by small PSLs (for which the ESTs are non-intensive) but obeyed by large PSLs (for which the ESTs are intensive; see Table I).^6,33,124,125

ESTs apply to individual eigenstates regardless of their connection—or lack thereof—to a Boltzmann distribution. This lack of a necessary connection of ESTs to Boltzmann distributions makes their similarities to $Tcconv$ both intriguing and useful.

$Tcconv$ is equal to T_bath only for canonical PSLs which have remained in contact with an infinite bath^{34−36,67,105,106,126} for equilibration time scales t_equil long enough for a Boltzmann distribution to be established. Since a new eigenstate is populated with each PSL $↔$ bath energy exchange, many eigenstates are successively occupied so that the energy and temperature fluctuate during equilibration. ESTs specify the temperature during the brief microcanonical intervals t_j $≪$ t_equil between PSL $↔$ bath energy exchanges.

Three conclusions regarding TUR ΔUΔ(1/T) ≥ k follow from the temporal properties of ESTs. First, because the TUR applies only to Boltzmann-distributed systems^{34−36,67,105,106,126} and because the observed energy and temperature values are equal to the energy and the EST of the eigenstate occupied at the time of measurement—and hence change with each successive measurement—ΔU and Δ(1/T) are both non-zero: The energy and temperature measurements both fluctuate so that the TUR is obeyed by small PSLs in equilibrium with infinite baths.^{34−36,67,105,106,126}

Second, since large canonical PSLs in equilibrium with infinite baths are dominated by their most-probable eigenstate [N − j*, j*],^42,70–73 each consecutive energy and temperature measurement yields the same values j*ε and $Tcj*→j*+1$ = $Tcconv$ = T_bath. The observed energy and temperature values are thus both non-fluctuating: ΔU = Δ(1/T) = 0 so that the TUR is violated by large canonical PSLs in equilibrium with infinite baths.^{34−36,67,105,106,126}

Third, since they are rigorously microcanonical, all measurements performed on an isolated PSL yield identical energy and temperature values: ΔU = 0 (by definition) and Δ(1/T) = Δ(1/$Tμj→j+1$⁠) = 0 (because the EST is precise). The TUR is thus violated by microcanonical PSLs of any size.

Significantly, this third conclusion disagrees with that of Mandelbrot,³ who contended that the TUR applies to a single microcanonical eigenstate “extracted” from a canonical system via thermal isolation. According to Mandelbrot, this eigenstate mysteriously retains the uncertainties in energy and reciprocal temperature of the canonical distribution from which it is extracted. Hence, in agreement with Uffink and van Lith,³⁴ we conclude that Mandelbrot’s arguments regarding the TUR for microcanonical systems are “counterfactual.”^127,128

The three conclusions above are undergirded by a single unifying feature: ESTs are thermodynamically accurate and precise only when they are identical to the thermodynamic temperature. The SEOC applies to rigorously microcanonical systems of any size and effectively to large canonical systems in equilibrium with infinite baths.^{34−36,67,72,105,106,126} Since the ESTs converge in the TDL, differences between the ESTs will not be manifest in scenario (iii); scenarios (i), (ii), and (iv) thus provide the most interesting and important test cases for the utility of ESTs and their applications to the TUR (see Sec. II D).

The FKM debate^{1–3,34–36} was initiated by Feshbach,¹ who contended that for small systems, T_system can be usefully approximated provided the range of inverse temperature estimates does not exceed the Δ(1/T) value specified by the TUR. In response, Kittel² contended that T_system is defined only for canonical systems—large or small—in equilibrium with large heat baths, in which case T_system = T_bath = $Tcconv$⁠. Since the large heat capacity of the bath precludes fluctuations in T_bath,^67,126 and since the SEOC is effectively satisfied in large baths,^42,70–73 both T_system and T_bath are non-fluctuating.^{2,34,67,105,106,126} Mandelbrot³ took an intermediate position, in which the actual reciprocal temperature of a canonical system is equal to T_bath and hence does not fluctuate, but that estimations of the reciprocal temperature obtained using estimators k$β^$ = $T^−1$ do fluctuate in accord with the TUR: ΔUΔ(k$β^$⁠) ≥ k.

The ESTs in the present study—which are analogous to Mandelbrot’s temperature estimators—are precise regardless of the PSL size. Though they deviate from T_bath = $Tcconv$⁠, ESTs provide new insights into many of the issues raised by the FKM debate,^{1–3,34–36} as detailed below.

First, Feshbach was only partially correct: The TUR applies to finite canonical systems, for which energy and temperature measurements fluctuate. Even so, the TUR applies only when Boltzmann statistics are in effect—and this condition prevails only for systems which are in equilibrium with infinite baths.^{34−36,67,105,106,126} Feshbach was thus incorrect to assume that the TUR applies to finite canonical systems in contact with finite baths.

Second, Kittel was likewise only partially correct: The TUR is violated in large canonical systems in equilibrium with infinite baths,^{34−36,67,105,106,126} for which energy and temperature measurements are accurate and precise. He was nevertheless incorrect in concluding that T_system is always equal to T_bath for small canonical systems in equilibrium with infinite baths. T_system and T_bath are not necessarily identical unless the energetic splittings ε_bath in the bath are effectively continuous;^{2,34,67,105,106,126} if ε_bath > ε_PSL, small PSLs in contact with infinite baths can violate both the zeroth law and the TUR.^121,129

Third, Mandelbrot was also only partially correct: His temperature estimators obey the TUR under canonical conditions provided the bath is infinite, quasi-continuous, and comprised of non-interacting particles.^{34−36,67,105,106,126} Even so, he was incorrect to conclude that his temperature estimators obey the TUR under microcanonical conditions:^3,34,127,128 The temperatures of microcanonical systems of any size are precise—and hence violate the TUR.

Temperature, entropy, and other statistically derived quantities are thermodynamically legitimate when the predictions of statistical mechanics concur with thermodynamic measurements. It is generally assumed that temperature is thermodynamically legitimate when it is intensive^130,131 and that entropy is thermodynamically legitimate when it is adiabatically invariant (i.e., constant in reversible, adiabatic processes).^4−6,27,132 Because of the intimate relationship between temperature and entropy, these thermodynamic legitimacy requirements raise the following question: “Are intensive temperatures synonymous with adiabatically invariant entropies?” The answer to this question revolves around the form of the entropy—Boltzmann or Gibbs—used to calculate temperature and is partially addressed by the negative temperature debate,^4−6,27,31 as detailed below.

The microcanonical Gibbs entropy

for an eigenstate [N − j, j] is equal to the logarithm of the sum of the microcanonical degeneracies of all eigenstates of energy up to and including [N − j, j]. It is commonly assumed that the Gibbs entropy is an adiabatic invariant for all N.^{4−6,27,132,133} In contrast, the microcanonical Boltzmann entropies S_μBj in Eq. (7) are equal to the degeneracy Ω_j of the [N − j, j] eigenstate. It is commonly assumed that S_μBj is not adiabatically invariant for small N but that it becomes so in the TDL because it converges to S_μGj as N becomes large.^4–6,27

Because S_μBj decreases with increasing energy above the energy median, Boltzmann ESTs of PSLs are negative for population-inverted eigenstates. In contrast, because S_μGj of PSLs increases monotonically with increasing energy for all j, the discrete microcanonical Gibbs ESTs

of PSLs are uniformly positive—even for population-inverted eigenstates.^4,5,27

The thermodynamic legitimacy of Boltzmann entropies and of negative absolute Boltzmann temperatures in PSLs and other energetically bounded systems has recently been challenged, for three reasons: (1) the Gibbs ESTs of PSLs are positive for all eigenstates, whereas the Boltzmann ESTs are negative for population-inverted eigenstates; (2) negative absolute temperatures imply that the Boltzmann populations of population-inverted eigenstates should increase with increasing energy;⁸ and (3) Gibbs entropies are commonly believed to be adiabatically invariant, whereas Boltzmann entropies are not.^4–6,27 Even so, there are strong reasons to believe that negative absolute temperatures are thermodynamically legitimate and that Boltzmann temperatures and entropies are preferable to their Gibbs analogs.

First, the Gibbs ESTs of population-inverted eigenstates grow exponentially with increasing N⁵ and hence manifest no TDL. This super-nonintensive character of the Gibbs ESTs is manifestly non-thermodynamic,^4–6,27 as it violates both the normal notions of hot and cold^5,31 and the zeroth law of thermodynamics.^6,31 In contrast, the Boltzmann ESTs of population-inverted eigenstates are intensive and thermodynamically legitimate in the TDL (see Sec. III C).⁵ Boltzmann ESTs are thus preferable to their Gibbs analogs in PSLs.

Second, Gibbs entropies are not necessarily adiabatically invariant. Based on the N-dependence of the chemical potential, Tavassoli and Montakhab¹³³ have recently demonstrated that neither the Gibbs nor the Boltzmann entropies are adiabatic invariants in any system for any value of N.^5,31,133 Hence, neither S_Gj nor S_Bj give perfect statistical mechanical–thermodynamic equivalence for thermodynamic observables. It is thus not possible to establish a direct correlation between temperature intensity and entropic adiabatic invariance. Even so, Boltzmann ESTs obey the four laws of thermodynamics when they are intensive (see Sec. III D), whereas the Gibbs ESTs of population-inverted eigenstates are non-thermodynamic. Hence, intensity of temperature is a more important criterion for thermodynamic legitimacy than adiabatic invariance of the entropy.

With few exceptions,^5,54,85 temperature has been characterized with $Tcconv$ in previous studies of PSLs.^{7,9,11−18,63−65,79,89} Because PSLs containing more than 10²¹ nuclei of each element were used in the earlier studies,^{13−17,64,65,79,134} these PSLs were in the TDL. Hence, the (unreported) nuclear spin ESTs are effectively identical to the (reported) conventional canonical nuclear spin temperatures. ESTs thus provide no new insights into the temperatures of the large PSLs utilized in previous studies; the principal utility of ESTs lies in their application to studies of small PSLs, which may find application in nanothermometry, as detailed below.

An experimental thermometry setup in which a small PSL-based nanothermometer (PSLnt) can yield accurate measurements of the temperature of a canonical system, provided five conditions are satisfied. First, to ensure that the PSLnt does not perturb the system temperature, the size and heat capacity of the PSLnt must both be small compared to those of the system. Second, to ensure that the bath temperature remains constant during system–bath energy exchanges, the size, and heat capacity of the bath must be large compared to those of the system. Third, the system–bath interaction $Ĥsystem−bath′$ must be large enough to allow the system to equilibrate with the bath but small enough to prevent the bath from perturbing the energies of the system eigenstates;^135,136 i.e., the bath must be weakly coupled to the system. Fourth, the system–PSLnt interaction $Ĥsystem−PSLnt′$ must be large enough to allow the PSLnt to equilibrate with the system but small enough to preclude changes in the energies of the eigenstates of the system and the PSLnt;^135,136 i.e., the PSLnt must be weakly coupled to the system. Fifth, to ensure that it measures the temperature of the system exclusively, with minimal inaccuracies induced by the bath, the PSLnt must be effectively uncoupled from the bath; i.e., $Ĥbath−PSLnt′$ ∼ 0.

Provided the five conditions above are satisfied, the canonical temperatures of both the system and the PSLnt may be characterized following equilibration. Since the net magnetization of a PSLnt is proportional to its energy, the temperature of the PSLnt may be assessed by measuring its net magnetization, ipso facto yielding the system temperature in accord with the zeroth law.

For measuring microcanonical temperatures, we propose a setup utilizing a PSLnt which is similar to the “minimal quantum thermometer” proposed by Dunkel and Hilbert.⁴ Provided the first and fourth conditions above are satisfied, such a setup should yield reliable measurements of the microcanonical Gibbs temperature of a system. Since the microcanonical system is isolated, it will initially be in a single eigenstate. If the PSLnt is first prepared in a (preferably) very low energy state with well-defined initial magnetization by magnetic cooling^68,76,77 and then brought into contact with the system under constrained conditions in which the combined energy of the system and the PSLnt is constant, then upon equilibration the magnetization of the PSLnt will change. The initial microcanonical temperature of the system may then be inferred by noting the change in the net magnetization of the PSLnt.

Our results regarding the impact of N_PSLnt on the four ESTs have two important implications. First, because the four ESTs of a given PSL eigenstate are typically intensive for N ≥ 10³, our results suggest a minimum temperature intensity size limit of N ∼ 10³ particles—significantly smaller than the sizes of most existing magnetic nanoparticle (N ≥ 10⁶),⁹⁸⁻¹⁰² paramagnetic salt (N ≥ 10²⁰),⁹⁵ and optical (N ≥ 10⁶)^96,98 nanothermometers. Assuming a PSLnt must be no more than one-tenth the size of its target system, PSLnts could yield reliable intensive temperatures for systems as small as 10⁴ particles, potentially resulting in a 1000–fold reduction in the size of target systems accessible with the smallest currently available magnetic nanothermometers.⁹⁸⁻¹⁰² Second, because the X_j-dependent deviations of the ESTs from their thermodynamic-limiting values are monotonic with decreasing N, reliable—albeit non-intensive—estimates of temperature may be attained with PSLnts containing fewer than 1000 particles (see Sec. III C and Figs. 3 and 4).

Since the spins of PSLs are non-interacting, it follows that the deviations of the ESTs of small PSLs from their thermodynamic-limiting values originate exclusively from finite-size effects^{36,40,60,68,83,84,92,93} (see Sec. III C). Since inter-particle interactions also give rise to non-intensive temperatures,^{36,60−62,66−68,74,83,84,92−94,103−106} any contrasts between the temperatures of PSLs and those of comparably sized systems with long-range spin–spin interactions will provide new insights into the relative impact of finite-size effects and inter-particle interactions on temperature non-intensity.

Four types of eigenstate-specific temperatures—continuous canonical, continuous microcanonical, discrete canonical, and discrete microcanonical—have been derived for two-level paramagnetic spin lattices. To our knowledge, this study constitutes the first detailed application of continuous microcanonical and discrete canonical ESTs to PSLs.¹³⁷ 

Our results lead us to conclude the following. First, the Boltzmann ESTs of small (N ≤ 10³) PSLs deviate from their thermodynamic-limiting values in previously unreported ways which differ depending on whether the spin-down mole fraction X_j increases, decreases, or remains constant with increasing N. Because these X_j-dependencies are monotonic in N, PSL-based nanothermometers can in principle provide meaningful temperature estimates for systems containing fewer than 10³ particles.

Second, although the four Boltzmann ESTs of PSLs are not true thermodynamic temperatures, they are useful temperature estimators for the full eigenstate spectrum of small PSLs. Gibbs ESTs are also useful temperature estimators—but only for positive temperature eigenstates; for population-inverted eigenstates, Boltzmann ESTs provide reliable temperature estimates, whereas the Gibbs ESTs do not.

Third, the thermodynamic uncertainty relation^34,35 is violated by microcanonical PSLs of any size and by large canonical PSLs in equilibrium with infinite baths; it is obeyed only by finite canonical PSLs in equilibrium with infinite baths.^{34−36,67,105,106,126} Temperature measurements are thus non-fluctuating in microcanonical PSLs of any size, non-fluctuating and thermodynamically accurate in large canonical PSLs,^{42,70−73,126} and fluctuating and thermodynamically approximate in finite canonical PSLs.

Fourth, intensity of temperature is a more important criterion for genuine thermodynamic behavior than adiabatic invariance of the entropy.^5,133

Collectively, our results suggest that ESTs will provide insights into a number of important current topics, including (1) the relative impacts of finite-size effects^{36,40,60,68,83,84,92,93} and long-range interactions^{36,60−62,66−68,74,83,84,92−94,103−106} on thermostatistical behavior, (2) the eigenstate thermalization hypothesis,^138,139 (3) the potential impact of temperature-dependent system energy levels on the thermodynamic uncertainty relation,^33,121,129 and (4) nanothermometry.^{60,83,84,90−102}

See supplementary material for details regarding (A) the spin-permutation antisymmetries of the ESTs¹¹⁹ and (B) the impact of spin-down mole fraction X_j on the continuous microcanonical, discrete canonical, and discrete microcanonical ESTs of constant-X_j, increasing-X_j, and decreasing-X_j eigenstates.^120,121

The authors thank R. Ananthoji and R. N. Karingithi for contributions to this research during its earliest stages, Mr. T. J. Masthay, Mr. T. M. Masthay, Dr. J. E. Adams, Dr. V. A. Benin, Dr. R. L. Berney, Dr. C. J. Cairns, Dr. G. S. Crosson, Dr. R. Lustig, Dr. P. G. Nelson, Dr. J. M. Standard, and Dr. M. Usman for helpful discussions, and J. T. Allen, D. A. Bucher, M. E. Griffin, M. E. Kelleher, A. Khan, J. M. Mabrouk, K. P. Mayrand, J. B. McGregor, T. Pair, R. J. Provost, R. G. Raspberry, M. L. Rudolph, J. A. Schatz, M. R. Short, T. S. Sirls, and L. K. White for editorial assistance. M.B.M. thanks the National Science Foundation (No. NSF-EPS-0132295) and the Howard Hughes Medical Institute (Undergraduate Biological Sciences Education Initiative Year 2000 Award) for partial funding of this research.