The cooling of boiling water all the way down to freezing, by thermally connecting it to a thermal bath held at ambient temperature without external intervention, would be quite unexpected. We describe the equivalent of a “thermal inductor,” composed of a Peltier element and an electric inductance, which can drive the temperature difference between two bodies to change sign by imposing inertia on the heat flowing between them, and enable continuing heat transfer from the chilling body to its warmer counterpart without the need of an external driving force. We demonstrate its operation in an experiment and show that the process can pass through a series of quasi-equilibrium states while fully complying with the second law of thermodynamics. This thermal inductor extends the analogy between electrical and thermal circuits and could serve, with further progress in thermoelectric materials, to cool hot materials well below ambient temperature without external energy supplies or moving parts.

Here, we describe a simple thermal connection containing a novel thermal element, namely, the equivalent of a “thermal inductor” that we had originally designed for precise heat capacity measurements in an actively driven thermal circuit ( 17 ). We show that it can also act in a passive way without any external or internally hidden source of power, and is able to drive the temperature difference between two massive bodies to change sign by imposing a certain thermal inertia on the flow of heat. We demonstrate in an experiment that such a process can occur through a series of quasi-equilibrium states, and show that it still fully complies with the second law of thermodynamics in the sense that the entropy of the whole system monotonically increases over time, albeit heat is temporarily flowing from cold to hot.

Arrows represent the direction of the flow of heat from (light/yellow) or to (dark/purple) the respective warmer object. ( A ) When an initially hot body is thermally connected at time t = 0 to a colder thermal reservoir held at temperature T r , its temperature T b is expected to drop monotonically by the loss of heat Q to the colder reservoir and to approach T r in the limit t → ∞. ( B ) Sketch of a process in which T b undershoots the temperature of the reservoir for t > t 0 , and heat Q is thereafter temporarily transferred from the chilling body to the warmer reservoir. The lowest temperature of the body T b,min < T r is reached at t = t min when the connection can be removed. ( C ) Two similarly connected finite heat capacities are expected to smoothly approach thermodynamic equilibrium at a mean temperature T ¯ , with heat flowing in one direction only and always T b > T r . ( D ) Two bodies showing opposite oscillations in temperature, with an alternating direction of the heat flow and a repeated temporary transfer of heat from cold to hot. The roman numerals (i) to (iv) refer to the four quarters of the period of one full oscillation cycle of T b (t), as elaborated in the text and in the caption of Fig. 2B .

The second law of thermodynamics, on the other hand, imposes strict limits on the efficiency of heat engines, cooling devices, and heat pumps ( 15 ) and suggests a preferred direction of the flow of heat to reach a thermodynamic equilibrium ( 16 ). According to the latter interpretation, a hot body with temperature T b that is connected to a colder object at temperature T r approaches thermodynamic equilibrium with strictly T b > T r , and T b is expected to monotonically fall as a function of time t because heat is not supposed to flow by itself from a cold to a warmer body ( Fig. 1 , A and C). Any undershooting or oscillatory behavior of T b with respect to T r ( Fig. 1 , B and D), with a reversing direction of the heat flow and transferring heat from cold to hot, would normally be ascribed to an active intervention to remove heat with external work to be done, or to a violation of the second law of thermodynamics.

According to the rules of classical thermodynamics, the flow of heat between two thermally connected objects with different temperatures is determined by Fourier’s law of heat conduction, stating that the rate of heat flow between these objects increases with growing temperature difference, and, more generally, by the second law of thermodynamics, which requires that heat can flow by itself only from a warmer to a colder body. The majority of measures for energy-efficient thermal management in everyday infrastructure are based on these laws, be it for the thermal insulation of buildings and heat accumulators, or for harvesting a maximum of mechanical work from a heat engine operating between two different temperatures. While the validity of these fundamental laws is undisputed, there have been exciting technical developments during the past few years, which appear to be at odds with popular interpretations of these laws. In the simplest version of Fourier’s law, for example, the rate of heat ( Q . ) flowing between a body at a temperature T b that is connected to an object with a different temperature T r is given by Q . = k ( T b − T r ) , where the thermal conductance k of the connection is expected to be independent of the sign of the temperature difference T b − T r . Nevertheless, a number of recent experiments demonstrated that a thermal rectification is possible to some extent, thereby opening up the way for a customized thermal management beyond the simple form of Fourier’s law ( 1 ). The operation of such a “thermal diode” (i.e., an analog to a diode rectifying electric current) has been demonstrated, for example, by the use of phononic devices ( 2 – 6 ) and phase change materials ( 7 – 10 ) as well as the application of quantum dots ( 11 ) and engineered metallic hybrid devices at low temperatures ( 12 ), paving the way for even more sophisticated thermal circuits such as thermal transistors and logic gates ( 6 , 13 , 14 ).

RESULTS AND DISCUSSION

Modeling of the oscillating thermal circuit The considered thermal connection consists of an ideal electrical inductor with inductance L and a Peltier element with Peltier coefficient Π = αT, forming a closed electrical circuit (Fig. 2A) (17). Here, α stands for the combined Seebeck coefficient of the used thermoelectric materials and T is the absolute temperature of the considered junction between these materials. When an electric current I is flowing through, heat Q is generated or absorbed at a rate Q . = Π I = α T I , respectively, depending on the direction of the current. A body with heat capacity C and a thermal reservoir (or two finite bodies) are each in thermal contact with the opposite sides of the Peltier element, providing a thermal link by its thermal conductance k. The process is described by Kirchhoff’s voltage law in Eq. 1A containing the generated thermoelectric voltage α(T b − T r ) and by the thermal balance equations (Eqs. 1B and 1C) for the rates of heat removed from (or added to) one body ( Q . b ) and to (from) the other body or the thermal reservoir ( Q . r ) , respectively L I . + R I = α ( T b − T r ) (1A) Q . b = − α T b I + 1 2 R I 2 − k ( T b − T r ) (1B) Q . r = + α T r I + 1 2 R I 2 + k ( T b − T r ) (1C)where R is the internal resistance of the Peltier element and α is taken as a constant for simplicity. We also temporarily ignore parasitic effects due to other sources of electrical resistance or thermal transport through leads or convection, and the heat capacity contribution of the Peltier element is thought to be absorbed in C. We count Q . > 0 for a heat input; however, the choice of the signs of I and α in Eq. 1 turns out to be unimportant. The dissipated joule heating power RI2 is regarded to be equally distributed to both sides of the Peltier element. The individual contributions to the flow of heat in Eqs. 1B and 1C are visualized in Fig. 2B. The set of equations (Eq. 1), but without the inductive term L I . , is standard to describe the flow of heat and charge in a Peltier element (18, 19). Fig. 2 Equivalent electrical network and illustration of the heat flow within the considered thermal connection between a body with heat capacity C at temperature T b and another body or a thermal reservoir at T r . (A) The electrical network consists of a Peltier element (Π) with internal resistance R and thermal conductance k in a closed circuit with an ideal inductance L. The oscillatory current I is ultimately driven by the voltages supplied by the thermoelectric effect due to the temperature difference between the cold and the hot end of the Peltier element, and the induced voltage L I . across L (see Eq. 1A). (B) Sketch of the individual contributions to the flow of heat (open arrowheads, arrow lengths not to scale) in Eqs. 1B and 1C for situations when heat is flowing from (filled light/yellow arrows) or to (filled dark/purple arrows) the warmer end of the Peltier element, drawn for one oscillation cycle of T b (t), as depicted in Fig. 1 (B and D). The thermal oscillator acts during a full period of an oscillation cycle of T b (t) alternately as a thermoelectric generator (i), a cooler (ii), a generator (iii), and a thermoelectric heater (iv). During all these processes, a small amount of electromagnetic power ( L I I . , green double arrows) is exchanged with the inductor, although the total stored magnetic energy 1 2 L I 2 is always less than a fraction Δ 0 /T r of the initially deposited excess heat ~ CΔ 0 (see text and Fig. 5). To consider the situation for the actual experiment to be presented further below, where a finite heat capacity C is connected to an infinite thermal reservoir as shown in Fig. 1 (A and B), we combine Q . b = C T . b and T r = const. with Eqs. 1A and 1B and obtain a nonlinear differential equation for I(t), namely L C I ¨ + ( R C + k L ) I . + ( k R + α 2 T r ) I + 1 2 α R I 2 + α L I I . = 0 (2) This equation can be easily numerically solved with high accuracy. The time evolution T b (t) could then be obtained by inserting the corresponding solution I(t) into Eq. 1A. For a systematic analysis, we restrict ourselves to Δ 0 = T b (0) − T r < < T r with temperature-independent α, k, R, and C, valid within a sufficiently narrow temperature interval ± Δ 0 around T r . Then, the last two terms of the differential equation (Eq. 2) are negligible because with Eq. 1A, 1 2 R I 2 + L I I . < R I 2 + L I I . = α ( T b − T r ) I < < α T r I , and we end up with the equation for a damped harmonic oscillator L C I ¨ + ( R C + k L ) I . + ( k R + α 2 T r ) I ≈ 0 (3)

Discussion of the oscillatory solutions It is very instructive to discuss at first the analytical solutions of this simplified equation. The initial conditions for t = 0, i.e., the time when the virgin thermal connection is inserted, are I(0) = 0 and T b (0) − T r = Δ 0 , thereby fixing I . ( 0 ) = α Δ 0 / L . The corresponding solution may show an overdamped or an oscillatory behavior depending on the combination of the constant factors in the equation. To achieve a possible undershooting of T b (t) below T r , we focus at first on oscillatory solutions of I(t). The condition for their occurrence, 4α2T r > LC(k/C − R/L)2, can always be fulfilled for any value of α, if L is chosen as L* = RC/k. While R and k are given by the characteristics of the Peltier element, C is limited only by the heat capacity of the Peltier element but can otherwise be chosen at will. The solution of interest is I(t) = I 0 exp(− t/τ)sin(ωt), with τ and ω matched to fulfill the differential equation, and I 0 = αΔ 0 /(Lω). The corresponding phase-shifted solution for the temperature is T b (t) − T r = Δ 0 exp(− t/τ)cos(ωt − δ)/cosδ with tanδ = (R − L/τ)/Lω. We now seek the particular solution realizing the weakest possible damping of I(t) within an oscillation cycle. This occurs for a maximum value of ωτ, where L = L*, ω = ω * = α 2 T r k / ( R C 2 ) , and τ = τ* = C/k. Introducing the standard definition of the dimensionless figure of merit for a Peltier element at T = T r (20), ZT r = α2T r /kR = (ω*τ*)2 (hereafter abbreviated as Z T ), the first minimum of T b (t) is attained with these parameters at t min = βτ* with β = ( π / 2 + arctan Z T ) / Z T for T b , min = T r − Δ 0 exp ( − β ) Z T / ( Z T + 1 ) < T r (4) If expressed by the dimensionless quantities (T b (t) − T r )/Δ 0 and t/τ*, the time evolution and (T b,min − T r )/Δ 0 , a measure for the maximum obtained cooling effect, only depend on Z T but are independent of the thermal load C and the other parameters of the system. In Fig. 3A, we show a series of corresponding T b (t) curves that we numerically obtained with Eq. 3 for different values of Z T , expressed in these dimensionless units, with T b (0) − T r = Δ 0 = 80 K and T r = 293 K to mimic a realistic scenario. However, despite the fairly large ratio Δ 0 /T r ≈ 0.27, the difference between the thus obtained values and the results calculated using the explicit solution of Eq. 3 with ω * = Z T / τ * would be barely distinguishable in Fig. 3. Fig. 3 Evolution of the temperature difference between a cooling body and a thermal bath or another finite body, which are connected in an experiment using a thermal inductor. (A) Normalized temperature difference (T b (t) − T r )/Δ 0 between a finite body and a thermal reservoir for L = L* = RC/k and Z T between 0.25 (red) and 5 (blue) in steps of 0.25, obtained from solving Eq. 3. The time is in units of τ* = C/k. The black line represents a corresponding relaxation process with a time constant τ*, which would take place if the Peltier circuit were interrupted from the beginning. If the thermal connection is not removed after reaching the respective T b,min (dashed line), T b (t) approaches thermal equilibrium with eventually T b = T r in all cases. The inset shows the damped oscillations of both T b (t) and I(t). (B) Temperatures T b (t) and T r (t) of two connected finite bodies with equal heat capacities, relative to the mean initial temperature T ¯ = [ T b ( 0 ) + T r ( 0 ) ] / 2 and normalized to the initial temperature difference Δ 0 , for Z T = 5 (time in units of τ*). T av denotes their average value showing local minima around T b ≈ T r (the numbers for T av were calculated for Δ 0 /T r = 0.27). The inset shows the evolution of the total entropy gain as a function of time in corresponding normalized units. According to Eq. 4, the minimum temperature decreases with increasing Z T but is limited to T b,min > T r − Δ 0 for a finite value of Z T so that no catastrophic oscillation can occur. If the thermal connection were removed after the body has reached its minimum temperature, then T b would stay at T b,min < T r under perfectly isolated conditions, as sketched in Fig. 1B. Removing it in a state where I = 0 even leaves the thermal connection in its original virgin state at T b = T r − Δ 0 exp ( − π / Z T ) but still with T b < T r (inset of Fig. 3A). Any external work associated with the act of removing the thermal connection could be made infinitesimally small, for example, by opening a nanometer-sized gap between the body and the thermal connection. If the connection is not removed at all, T b (t) exhibits a damped oscillation around T r , approaching thermal equilibrium with eventually T b = T r . We note that the maximum possible cooling effect is not reached for the above parameters, but for L opt = λ L* with λ(Z T ) > 1 (see section S1). The corresponding solutions for I(t) are overdamped for Z T < 1/3, but the temperature of the body still undershoots T r for all values of Z T > 0. In a closely related scenario, two finite bodies with identical heat capacities 2C and different initial temperatures T b (0) and T r (0) are thermally connected in the same way and observed under completely isolated conditions (Fig. 1D). In the limit Δ 0 = T b ( 0 ) − T r ( 0 ) < < T ¯ with the mean initial temperature T ¯ = [ T b ( 0 ) + T r ( 0 ) ] / 2 , we end up with the same simplified differential equation (Eq. 3) for I(t) but with T r replaced by T ¯ (see section S2). In Fig. 3B, we show the resulting counter-oscillating temperatures T b (t) and T r (t) for Z T = 5, together with the average temperature T av = [ T b ( t ) + T r ( t ) ] / 2 ≤ T ¯ , which is not constant but reaches local minima around the times when the two temperatures are equal.

Oscillatory flow of heat Each time when T b − T r changes its sign (this occurs for the first time as soon as T b drops below T r , for L = L* after t 0 = π/2ω*; Figs. 1 and 3), heat is still continuously flowing from the cold to the warmer object (dark/purple arrows in Figs. 1 and 2B) until |T b − T r | reaches its maximum, where the direction of the heat flow is reversed. The moving charge carriers drive virtually all of this heat directly away from the cold to the warm end without being temporarily stored as energy of the magnetic field residing in the inductor. The maximum amount of magnetic energy, 1 2 L I 2 < 1 2 L I 0 2 = α 2 Δ 0 2 / 2 L * ω * 2 , is less than a fraction Δ 0 /T r < < 1 of the excess heat ~ CΔ 0 that has been initially stored in the warmer body. In this sense, the electrical inductor acts, in interplay with the Peltier element, only as the driver of the temperature oscillation by imposing a certain thermal inertia that temporarily counteracts the flow of heat dictated by Fourier’s law. Thus, we can interpret the role of the circuit as that of a thermal inductor. In analogy to the self-inductance L of an electrical inductor generating a voltage difference ΔV according to L I . = − Δ V , we can even ascribe to the present circuit a thermal self-inductance L th = L/(α2T r ) (17), obeying L th I . th = L th Q ¨ = − Δ T (see section S4). From a refrigeration engineering point of view, we can divide a full period of an oscillation cycle of T b (t) into four stages (see Fig. 2B). In the first quarter of a full period (i), the operation corresponds to that of a thermoelectric generator. In the second quarter (ii), the circuit acts as a thermoelectric cooler even for T b < T r , driven by the electric current that is still flowing in the original direction due to the action of the electric inductor. During the third quarter (iii), the electric current changes sign (thermoelectric generator), while heating persists in the fourth quarter (iv) even for T b > T r , and the device is then operating as a thermoelectric heater. The order of magnitude of the rate of heat flowing between the objects can be chosen arbitrarily low by adjusting the thermal load C. In this way, the electronic oscillator circuit reaches the typically very large time scales encountered in thermal systems (i.e., seconds, minutes, or even longer). This guarantees that the processes, albeit irreversible, can run in a quasistatic way and pass through a series of quasi-equilibrium states with well-defined thermodynamic potentials and state variables of the bodies. This is in marked contrast to nonequilibrium oscillatory processes such as the Belousov-Zhabotinsky chemical reaction (21, 22), other oscillations in complex systems far from thermodynamic equilibrium (23), to thermal inductor type of behaviors associated with the convection of heated fluids (24), or to transient switching operations in light-emitting diodes (25).

Experiments In a real experiment, measurements of sizeable temperature oscillations face certain challenges. At present, the most efficient Peltier elements have a maximum Z T ≈ 2 (26). In a scenario of cooling an amount of boiling water from 100°C down to its freezing temperature at 0°C by connecting it to a heat sink at room temperature (20°C), for example, a Z T ≈ 5 would be required (Fig. 3A), which is out of reach of the present technology. A further challenge is the choice of the inductance L. It should be large enough to allow the cooling of a substantial amount of material while keeping k as small as possible to maximize Z T . With τ* = C/k of the order of several seconds and typical internal resistivities of Peltier elements R ≈ 0.1 ohm and higher, L opt > L* = Rτ* must be of the order of 1 H or larger, although a useful cooling effect could still be achieved for L < L*. Normal conducting inductors with these large inductances suffer from a considerable internal resistance R s , however, thereby reducing the cooling performance well below the theoretical expectations. The incorporation of a corresponding finite resistivity R s switched in series, in addition to the internal resistance R of the Peltier element as it is sketched in the inset of Fig. 4A, leads again to Eq. 3 but with R replaced by R + R s (see section S3). This will reduce the effective dimensionless figure of merit Z T of the Peltier element by a factor R/(R + R s ), which can be substantial as soon as R s becomes of the order of R or even larger, with an associated decrease of the cooling performance, as illustrated in Fig. 3A and in section S1. Fig. 4 Results from experiments with oscillating thermal circuits containing the equivalent of a thermal inductor. (A) Temperature T b (t) data taken for two configurations of superconducting coils with L = 30 and 58.5 H, respectively. In this type of an experiment, the oscillating thermal circuit is entirely passive. The temperature T b (t) of a copper cube that has been thermally connected to a heat reservoir held at T r = 295 K and initially heated by Δ 0 = T b (0) − T r ≈ 82 K substantially undershoots with respect to T r by ≈1.7 K for L = 58.5 H. The inset shows the respective equivalent electrical network, including a parasitic electrical resistance R s in series due to electrical leads and connections. The solid lines are T b (t) data obtained by solving the corresponding relevant differential equations using the parameters from a global fit to the data shown in (B) and with R s = 21 and 43 milliohms for L = 30.0 and 58.5 H, respectively. (B) Experiments using a gyrator-type substitute of an electric inductor with a nominal R s ≈ 0. Main panel: Temperature T b (t) for four different values of nominal inductance L, with a maximum undershoot of T b (t) with respect to T r by ≈2.7 K for L = 90.9 H. The T b (t) data from (A) (green and purple dots) are included for comparison. Inset: Evolution of the electric current flowing through the Peltier element. The solid black lines correspond to a global fit to the four datasets according to the relations given in the main text, with the fitting parameters C, R, k, and Z T . Therefore, we performed an experiment with two configurations of resistanceless superconducting coils (with L = 30 and 58.5 H, respectively) connected to a commercially available Peltier element. With a cube of ≈1 cm3 copper as the thermal load at temperature T b and a massive copper base acting as the thermal reservoir held at T r = 295 K (≈22°C), we are thereby implementing an entirely passive oscillating thermal circuit that should show the predicted temperature oscillations (see Materials and Methods). In Fig. 4A, we present the results of these measurements according to the experimental scheme shown in Fig. 1B and analyzed in Fig. 3A. Initially heated to 377 K (≈104°C), T b dropped for L = 58.5 H by ≈1.7 K below the base temperature T r , verifying that heat has been flowing from the chilling copper cube to the thermal reservoir as soon as T b fell below 295 K around t 0 ≈ 410 s. Using the known values k, R, and α of the Peltier element that we had previously obtained by the procedure described further below, we modeled these experimental T b (t) data by taking into account the parasitic resistance R s in series due to the long electric lines to the superconducting coils (inset of Fig. 4A), according to the corresponding analogon to Eq. 2 (see eq. S5 in section S3). The data are best reproduced with R s = 21 and 43 milliohms for L = 30.0 and 58.5 H, respectively (solid lines in Fig. 4A), in very good agreement with our direct measurements for the total resistivity of the leads and connections, R s ≈ 20 and 45 milliohms (see Materials and Methods). In Fig. 5, we have also plotted the different fractions of the rates of energy flow in this experiment, as we have depicted them in Fig. 2B. The corresponding data were derived from the measured temperature difference ΔT = T b (t) − T r , the electric current I, and the known parameters of the Peltier element. Except for the comparably small electromagnetic power term ( L I I . ) and the parasitic joule heating along R s , all the relevant exchange of energy occurs directly between the Peltier element, the thermal load and the thermal bath, with the thermoelectric contributions αT b I and αT r I dominating the other terms appearing in Eqs. 1B and 1C. Fig. 5 The different fractions of the rates of energy flow in the oscillating thermal circuit for the experiment with L = 58.5 H shown in Fig. 4A The roman numerals (i) and (ii) refer to the definition provided in the text and in the caption of Fig. 2B. The subsequent stages (iii) and (iv) are not discernible in these data because of the still quite low value Z T = 0.432 (see also Fig. 3A). The thermoelectric contributions αT b I and αT r I (right scales) dominate all the other terms in Eqs. 1B and 1C (left scales). To be better able to quantitatively analyze our model and to precisely determine the parameters of the Peltier element, we had previously performed a series of additional experiments using an active gyrator-type substitute of a real inductor (27) that allows simulation of almost arbitrary values of L with negligible effective internal resistivity, R s ≈ 0. Although the thermal connection can then no longer be considered as strictly operating “without external intervention,” no net external work is performed on the system. In Fig. 4B, we present the results of a series of corresponding measurements for four different values of the nominal inductance L of the gyrator (L = 33.2, 53.6, 90.9, and 150 H), with T b reaching a temperature of ≈2.7 K below T r for L = 90.9 H. In the same figure, we also show the results of a global fit to both the measured T b (t) and I(t) data according to the relations given in this work, with only four free parameters: C, R, k, and Z T . We obtain an excellent agreement for an effective Z T = 0.432, C = 4.96 J/K (this value includes a heat capacity contribution of the Peltier element), R = 0.22 ohm, and k = 0.0318 W/K, resulting in a Seebeck coefficient α = Z T k R / T r = 3.21 × 10−3 V/K. The used values L = 33.2 and 90.9 H are very close to the resulting L* = 34.3 H and the optimum inductance L opt = 94.4 H, respectively (see section S1).