Heating and cooling are fundamentally asymmetric and evolve along distinct pathways

The study addresses how systems relax to equilibrium after temperature changes, especially far from equilibrium. Classical non-equilibrium thermodynamics assumes quasi-static relaxation through local equilibrium states for small temperature differences, but this assumption fails for large quenches. Prior anomalous relaxation phenomena (Mpemba and Kovacs effects) illustrate non-trivial far-from-equilibrium kinetics. Recent theory predicted that, for thermodynamically equidistant (TE) temperatures relative to an intermediate temperature, heating from a colder state should be faster than cooling from the hotter state, challenging reciprocity expected from local equilibrium arguments. The authors test and generalize this prediction, asking whether heating and cooling between fixed temperatures relax along the same path and at the same rate. They show that heating and cooling are inherently asymmetric and follow distinct pathways, with heating faster than cooling even between any fixed pair of temperatures.

The paper situates its work within far-from-equilibrium relaxation phenomena, referencing anomalous relaxation such as the Mpemba effect and the Kovacs memory effect as evidence that relaxation paths depend on initial conditions. It cites recent theoretical predictions that heating from a colder state is faster than cooling from a hotter state when the initial states are thermodynamically equidistant from a target temperature. The work connects to stochastic thermodynamics and information geometry, including concepts such as relative entropy, Fisher information, thermodynamic uncertainty relations, and non-equilibrium speed limits, which provide tools to quantify relaxation kinetics and statistical distances between probability distributions.

Experimental: A 1 μm-diameter silica microparticle dispersed in deionized water is trapped in a harmonic (parabolic) optical potential generated by a tightly focused 1,064 nm infrared laser (JPK-Bruker Nano Tracker 2). Particle position is recorded with a quadrant photodiode at 50 kHz. An external noisy electric field (white Gaussian noise from an arbitrary waveform generator, 80 MHz bandwidth) is applied via parallel gold microelectrodes in a custom electrophoretic chamber. The noise mimics an additional white thermal noise source, yielding an effective temperature Teff = T + σ²γ/(2κkB) (reported in Methods as Teff = T + 2κα/γ with notation differences), which increases linearly with the noise variance V²; Teff is calibrated via equipartition ⟨x²⟩ = kBTeff/κ. The noise is modulated by a square signal of 10 ms period to implement temperature quenches; the variance shows a finite ramp (~0.10 ms) between initial and final levels. Experimental datasets comprise M = 24,000 trajectories with N = 250 time points each; histograms P(x,t) are built with 80 bins for each t. Relative entropy D[P||Q] is computed by histogram sums, using the convention 0 log 0 = 0. Fisher information f(t) is estimated from D[P(x,t+Δt)||P(x,t)] with Δt = 0.04 ms, yielding instantaneous statistical velocity v(t) = √f(t). Statistical length L(t) = ∫0^t v(τ)dτ is computed numerically; degree of completion φ(t) = L(t)/L(∞). Reported uncertainties are s.e.m. over ten independent temporal series. Protocols: Two types of TE protocols are implemented. Forward protocol: system is initially at equilibrium at a hotter (Th) or colder (Tc) bath temperature, both TE relative to a warm target Tw, and then relaxes at Tw. Backward protocol: system starts at equilibrium at Tw and relaxes to the respective TE temperatures Th or Tc. A third protocol compares heating and cooling directly between any two fixed temperatures T1 < T2, starting from equilibrium at one and relaxing to the other. Theory: The colloid dynamics are modeled by the overdamped Langevin equation in a harmonic potential U(x)=κx²/2 with friction γ, dx = −(κ/γ)x dt + dξ, with 〈dξi(t)dξj(t′)〉 = 2(kBTi/γ)δijδ(t−t′)dtdt′. For TE quenches, the generalized excess free energy D(t) = kBTw D[P(x,t)||Pw(x)] yields analytic forms: D^W(t) = kBTw[Δ^W(t) − 1 − ln Δ^W(t)] with Δ^W(t) = 1 + (Th/Tw − 1)e^{-2(κ/γ)t} for forward relaxation at Tw, and an analogous expression Δ^B(t) = Tc/Tw + (1 − Tc/Tw)e^{-2(κ/γ)t} for backward. Plotting D/kBTw versus p = Δ collapses data to the master curve f(p) = (p − 1 − ln p)/2. Using information geometry, an infinitesimal line element dl = √f(t) dt defines v(t)=√f(t), the statistical length L(t), and the degree of completion φ(t)=L(t)/L(∞). For the harmonic, overdamped case, L(∞) = |ln(Ti/Tw)|/√2 and φ(t) = 1 − [ln(1 + (Ti/Tw − 1)e^{−2(κ/γ)t})]/[ln(Ti/Tw)]. Theoretical results are compared to experiments without fitting parameters, accounting for finite ramp times by an effective relative temperature τ̂_eff = e^{−2ωτ/κ}(τ̂ − 1)+1 in the variance dynamics. Mathematical proofs (Supplementary Theorems 1–3) establish the observed asymmetries under stated conditions.

- Heating is faster than cooling under TE conditions in both forward and backward protocols: for all t > 0, D^h(t) < D^c(t) in the forward protocol, and D_h(t) > D_c(t) in the backward protocol, confirmed experimentally and proven theoretically for any κ and γ in harmonic landscapes. - Data collapse: D(t)/kBTw versus p = Δ(t) collapses onto the master curve f(p) = (p − 1 − ln p)/2, validating the model (Fig. 2). - Representative parameters: characteristic time τ = γ/κ = 0.1844(3) ms, Tc/Tw = 0.11(1), Th/Tw = 3.56(1) (forward and backward data). Additional TE conditions with τ ≈ 0.099–0.539 ms similarly confirm asymmetry (Extended Data Fig. 3). - Thermal kinematics: statistical length L(∞) = |ln(Ti/Tw)|/√2 is larger for the colder initial state (farther from equilibrium), yet the degree of completion satisfies φ_heat(t) > φ_cool(t) for all 0 < t < ∞ under TE, due to an initial overshoot in statistical velocity v(t) during heating. - Between any pair of fixed temperatures T1 < T2, heating from T1 to T2 is faster than reciprocal cooling from T2 to T1: φ_heat(t) > φ_cool(t) for all 0 < t < ∞, experimentally confirmed for T1 = 302(3) K and T2 = 2753(7) K with τ = 0.1844(3) ms (Fig. 4). - Near-equilibrium limit: For small temperature contrasts (ε ≪ 1), heating and cooling become approximately symmetric in both generalized excess free energy D and degree of completion φ(t), consistent with linear non-equilibrium thermodynamics. - Mechanistic insight: Faster heating arises from more efficient entropy production in the system and an initial free expansion-like regime that accelerates propagation in distribution space; spectral arguments and higher collision rates at higher temperatures support the speed-up.

The findings directly answer whether thermal relaxation is reciprocal between heating and cooling: it is not. Even for simple, single-well (harmonic) systems, heating and cooling follow distinct microscopic pathways and proceed at different rates, contradicting the quasi-static, local-equilibrium picture except very near equilibrium. Under TE conditions, although both initial states are equidistant in free energy from equilibrium, the colder state is statistically farther away yet heats faster, explained by a kinematic framework combining stochastic thermodynamics and information geometry. The intrinsic overshoot in statistical velocity during heating reflects more efficient in-system entropy production compared with heat dissipation in cooling, bounding the lag from equilibrium and accelerating relaxation. The asymmetry persists beyond TE settings to any pair of temperatures, highlighting a fundamental, general feature of far-from-equilibrium relaxation. These insights have implications for the design and operation of microscale energy-conversion devices and Brownian heat engines, where cycle timing and thermal management depend critically on the direction of temperature changes.

This work demonstrates a fundamental asymmetry between heating and cooling in microscale thermal relaxation: heating is faster than cooling under thermodynamically equidistant quenches and even between any fixed temperature pair. Experiments on optically trapped colloids and exact theory for overdamped Langevin dynamics in harmonic potentials establish distinct microscopic pathways and introduce a thermal kinematics framework using Fisher information, statistical length, and degree of completion to quantify relaxation. Near equilibrium, symmetry is recovered, delineating the far-from-equilibrium origin of the effect. The results suggest broad impact on thermal management and the optimization of microscopic engines and devices. The authors note that further examinations of non-instantaneous quench protocols will help establish the limits of validity of the predictions and whether analogous asymmetries extend to more complex energy landscapes and multi-dimensional systems.

- Model and proofs assume overdamped Langevin dynamics in a harmonic (quadratic) potential and effectively one positional degree of freedom; generalization to non-quadratic or multi-well landscapes is not established here. - The backward-process metric D_B(t) used for comparison with forward D_W(t) is not a true excess free energy and relative entropy is not a metric (asymmetry and lack of triangle inequality), motivating the shift to information-geometric measures. - Experimental temperature quenches include a finite ramp (~0.10 ms), sometimes comparable to relaxation times; although accounted for and found not to alter conclusions, this deviates from ideal instantaneous quenches. - Effective heating via external white noise raises the effective temperature without altering dissipation; while standard and calibrated, it is an approximation to a true thermal bath at elevated temperature. - Asymmetry diminishes near equilibrium; within experimental error, small temperature contrasts show near-symmetric behavior.