Anomalous Heat Flow Phenomena

• Anomalous heat flow is the breakdown of classical Fourier diffusion, characterized by superdiffusive transport, size-dependent conductivity, and long-range correlations.
• Theoretical studies use fluctuating hydrodynamics, mode-coupling theory, and nonlocal equations to quantify deviations from normal heat transport.
• Experimental realizations in nanowires, quantum devices, and topological materials reveal unconventional heat currents that prompt new thermal management strategies.

Anomalous heat flow refers to the breakdown of classical, diffusive heat transport as formalized by Fourier’s law. In diverse physical systems—ranging from low-dimensional lattices and nano-materials to open quantum systems—heat may propagate in ways that defy the normal linear relationship between heat current and temperature gradient. Manifestations include superdiffusive or subdiffusive energy spreading, size-dependent thermal conductivities, cold-to-hot heat flow, and topological or quantum-mechanical anomalous currents. The phenomenon fundamentally connects to nontrivial conservation laws, long-range correlations, quantum coherence, and microscopic or topological anomalies.

1. Classical Framework: Diffusive vs. Anomalous Heat Transport

In bulk, three-dimensional solids, the local law $j_h(x) = -\kappa \nabla T(x)$ (Fourier's law) holds, with $\kappa$ a size-independent scalar (thermal conductivity), and energy packets spread diffusively: the mean square displacement (MSD) of an initial excess-energy pulse scales as $\langle \Delta x^2(t) \rangle_E \sim t$. In one-dimensional nonlinear chains, polymer nanofibers, or carbon nanotubes, this paradigm fails. Molecular dynamics simulations and analytic theory consistently find that $\langle \Delta x^2(t) \rangle_E \sim t^\beta$ with $\beta > 1$ (superdiffusion) and that $\kappa(L) \sim L^\alpha$, $\alpha = \beta - 1 > 0$, so the effective conductivity diverges with length, violating Fourier's law (Liu et al., 2011, Ray et al., 2019, Yang et al., 2010, Benenti et al., 2020).

The breakdown is attributed to conserved momentum, dimensional reduction, and the emergence of long-time tails in heat-current autocorrelations. Fluctuating hydrodynamics and mode-coupling theory identify two main universality classes for 1D momentum-conserving systems: one with $\beta = 4/3$ ($\alpha = 1/3$) (KPZ class) and another with $\alpha = 1/2$ (Li et al., 2014, Benenti et al., 2020).

2. Mathematical and Theoretical Formalism

The linear-response theory connects the non-equilibrium spread of energy with equilibrium current fluctuations via the exact formulas: $$\frac{d^2}{dt^2}\langle \Delta x^2(t) \rangle_E = \frac{2 C_{JJ}(t)}{k_B T^2 c}$$

$\kappa(L) \sim \int_0^{L / v_s} C_{JJ}(t) dt$

where $C_{JJ}(t)$ is the total-heat-current autocorrelation function, $c$ is volumetric specific heat, $v_s$ is the maximal signal velocity, and $\kappa(L)$ acquires size dependence under persistent correlations (power-law decay of $C_{JJ}(t)$) (Liu et al., 2011).

In systems possessing additional conservation laws (e.g., stretch, momentum), the nonlinear fluctuating hydrodynamics framework and mode-coupling equations precisely predict dynamical exponents and scaling of both energy and momentum spreads. The conjecture that normal (finite viscosity) momentum diffusion corresponds to normal heat conduction, and superdiffusive (diverging viscosity) momentum spread yields anomalous heat transport, is numerically corroborated for FPU chains, hard-point gases, and models with bounded vs. unbounded interactions (Li et al., 2014).

Anomalous heat diffusion in the strict mathematical sense may also be cast in nonlocal or fractional-derivative equations on finite domains; for example, the steady-state and relaxation of a 1D chain with two conservation laws is governed by a skew-fractional Laplacian of order $3/2$ (Priyanka et al., 2018). Heat pulse propagation in harmonic lattices follows hyperbolic equations with algebraic (not exponential) decay at wavefronts (Sokolov et al., 2017).

3. Experimental Realizations and Signatures

Anomalous heat flow is observed in a wide variety of experimental systems:

• Silicon nanowires and carbon nanotubes: Nonequilibrium MD studies exhibit $\kappa(L) \propto L^\alpha$ with $\alpha \approx 0.15-0.4$ depending on temperature and boundary conditions. Energy-pulse experiments in SiNWs display superdiffusive exponents $\alpha \approx 1.07-1.23$ (Yang et al., 2010). In carbon nanotubes $\kappa(L) \propto L^{1/2}$ and, for graphene, $\kappa(L) \propto \ln L$, with nonlinear steady-state temperature profiles and Lévy-walk scaling (Ray et al., 2019).
• Fractal Metamaterials: Heat conduction in isotropic fractal networks with non-integer dimension exhibits strongly non-exponential (stretched-exponential) decay of temperature and energy profiles in the presence of randomly distributed heat sinks, and optimal absorption is realized at a nontrivial dimension, tunable via sink concentration (Lin et al., 2017).
• NiCu Multilayers Under H$_2$: Photon radiation calorimetry reveals $4$–$6\,\mathrm{W}$ persistent excess heat during hydrogen desorption from NiCu thin films with an integrated energy output per H atom of $410 \pm 108\,\mathrm{keV}$, far exceeding chemical energy scales and signifying non-conventional, possibly nuclear, mechanisms (Kasagi et al., 2023).
• Colloidal Systems: The "inverse Mpemba effect" is observed: a colloidal particle initialized colder can heat up exponentially faster (in $L_1$ distance to equilibrium) than a warm one when the initial state's overlap with the slowest relaxation mode vanishes. The phenomenon is generic for overdamped diffusions with multistable potentials and clean spectral gaps (Kumar et al., 2021).

4. Quantum and Topological Anomalous Heat Flow

In quantum thermodynamic settings, anomalous heat flow encompasses transient or post-selected violations of the classical Clausius law ("heat flows from cold to hot"). Mechanisms include:

• Initial Correlations and Quantum Coherence: Global states with thermal marginals but initial quantum or classical correlations can support spontaneous heat flow from cold to hot (Ma et al., 2022). In finite-dimensional systems, accessible via NMR or gate-based superconducting quantum devices, quantum correlations (e.g., negativity in a Kirkwood–Dirac quasiprobability, quantum discord, or entanglement) are necessary for "quantum anomalous heat flow," as evidenced by direct violation of semi-classical heat flow bounds (Mallik et al., 30 Oct 2024, Comar et al., 14 Jun 2024).
• Indefinite Causal Order (ICO): By coherently superposing the order of thermalizing channels—the quantum SWITCH—a qubit system can conditionally absorb heat from a colder reservoir, counter to classical expectations. This coherence-driven resource enables construction of quantum heat engines that simultaneously refrigerate (transfer heat up the gradient) and produce work, with proof-of-principle photonic experiments confirming regime boundaries and operation (Xue et al., 6 Nov 2025).
• Topological and Gravitational Anomalies: In Weyl semimetals, Berry curvature and chiral anomaly physics manifest as a thermal "gravitational anomaly": applying a temperature gradient parallel to a magnetic field pumps energy between Weyl nodes, generating a $B^2 \nabla T$ anomalous heat current—directly linked to the electronic chiral magnetic effect and governed by the mixed axial-gravitational anomaly (Tanwar et al., 2023). Likewise, in Floquet-engineered 2D materials such as 8-$Pmmn$ borophene, off-resonant circularly-polarized light induces a gap and finite Berry curvature, producing a thermal Hall current whose magnitude and sign are controlled optically (Sengupta et al., 2018).

5. Universal Scaling, Mechanistic Classification, and Physical Consequences

Anomalous heat flow unifies several macroscopic manifestations:

System Class	Scaling Law	Physical Regime
1D nonlinear chain	$\kappa(L) \sim L^\alpha,\, \alpha>0$	Superdiffusive, Fourier breaks down
Carbon nanotube	$\kappa(L) \sim L^{1/2}$	Lévy-walk transport
Quantum system w/ corr.	Heat flows cold$\to$hot (post-selected/early time)	Consumes initial correlations
Topological semimetal	$J_{Q,\mathrm{anom}} \propto B^2 \nabla T$	Chiral–gravitational anomaly
Floquet Dirac material	$\kappa_{xy} \ne 0$, sign via optical control	Berry curvature, photo-induced

In classical systems, the persistence of long-range current correlations, fractional transport equations, and scaling laws are robust under details of microscopic dynamics, provided invariants such as momentum are maintained. In quantum systems, anomalous heat flow is not merely a statistical fluctuation, but can be traced to the operational consumption of nonclassical resources—coherence, mutual information, or causal indefiniteness.

Consequences include:

• Failure of size-independent material constants: $\kappa$ may scale supra-linearly with system length, impacting nanoscale thermal engineering.
• Nonlinear and nonlocal temperature profiles, with direct implications for device heat management.
• Topology-driven and optically switchable transport modes, providing means for active thermal control and thermal logic.
• In the quantum domain, resource-driven heat transport opens routes to new device architectures—combined refrigerators and engines exploiting coherence or entanglement.

6. Limitations, Outlook, and Open Questions

• Linear Response and Ergodicity Assumptions: Most scaling relations hold in or near equilibrium and may fail for strongly driven, aging, or nonergodic states (Liu et al., 2011, Benenti et al., 2020).
• Finite-size/crossover Regimes: There is often a large (but finite) system-size or time transient before anomalous or normal transport emerges; nanodevices may reside in such crossovers.
• Experimental Verification: Measurement of anomalous heat transport requires careful separation of ballistic, superdiffusive, and localized modes—present challenges in both classical MD and quantum hardware fidelity (Mallik et al., 30 Oct 2024).
• Topological Anomaly Robustness: While topological sources of anomalous heat currents are robust to disorder, precise quantification can be masked by competing phonon or magnon contributions (Tanwar et al., 2023).
• Quantum Resource Theory: Classification of quantum sources (coherence, discord, causal order) for anomalous flows remains an active area, along with quantifying operational advantages for quantum thermal machines (Ma et al., 2022, Xue et al., 6 Nov 2025).

A comprehensive understanding of anomalous heat flow links dynamical systems, nonequilibrium statistical mechanics, condensed matter, and quantum information theory, and underpins both fundamental insights and potential technological innovations in energy transport and control.