Heat Transfer Phenomena in Non-Equilibrium Systems

Heat transfer in non-equilibrium systems departs from the classical picture of instantaneous diffusive smoothing of temperature gradients. Instead, it embraces finite speeds of thermal propagation, memory effects and coupling between heat flux and additional state variables. These phenomena become prominent at micro- and nano-scales, in heterogeneous media, under rapid thermal loading or within strongly anisotropic materials, where the assumptions of local equilibrium and Fourier’s law break down. Theoretical frameworks such as extended irreversible thermodynamics and internal-variable formalisms introduce higher-order fluxes or relaxation times to capture wave-like heat pulses, size-dependent conductivity and dynamic coupling with mechanical deformation.

Experimental observations of non-Fourier behaviour include thermal waves in cryogenic crystals, ballistic phonon transport in thin films, and directional heat flow controlled by parity-time symmetries. Mathematical models range from hyperbolic heat equations to dual-phase-lag formulations and two-temperature descriptions separating electron and phonon subsystems. Computational schemes that respect causality and stability are crucial for reliable predictions in applications from ultrafast laser processing to thermal management in microelectronics.

The global significance of these advances spans energy conversion, where non-equilibrium transport can boost thermoelectric efficiency; materials science, where tailored thermal responses enable adaptive insulation; and environmental engineering, where predictive models of heat waves improve climate-control designs. Ongoing research continues to refine constitutive laws, validate them against ultra-resolved measurements and integrate multi-physics couplings for real-world systems.

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Heat Transfer Phenomena in Non-Equilibrium Systems publication trend

The graph below shows the total number of articles in heat transfer phenomena in non-equilibrium systems across all publications each year (not limited to Nature Index journals).

Technical terms

Non-equilibrium system: A physical system in which thermodynamic variables vary in time or space, preventing the establishment of thermodynamic equilibrium.

Fourier’s law: A constitutive relation stating that heat flux is proportional to the negative temperature gradient, describing diffusive heat conduction.

Maxwell–Cattaneo–Vernotte (M–C–V) equation: A modification of Fourier’s law introducing finite thermal propagation speed through a relaxation-time term.

Hyperbolic heat equation: A form of heat conduction equation incorporating time derivatives of heat flux, allowing wave-like propagation of thermal signals.

Two-temperature model: A framework treating electrons and phonons as separate subsystems, each with its own temperature, to capture non-equilibrium thermal transport.

References

1. Parity-time and anti-parity-time symmetries in heat transfer. National Science Review (2024).
2. Derivations of the stress-strain relations for viscoanelastic media and the heat equation in irreversible thermodynamic with internal variables. International Journal of Mathematics and Computer in Engineering (2024).
3. Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model. Zeitschrift für angewandte Mathematik und Physik (2020).