Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons (lattice vibration waves), electrons, fluid particles, and photons. Heat is defined as thermal energy in transit resulting from a spatial temperature distribution. Thermal energy is stored in the temperature-dependent motion of particles, including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical, quantum, and statistical mechanics. The energy is converted through interaction processes among various carriers. When these principal energy carriers are not in equilibrium, they are transported or converted toward an equilibrium distribution identified by carrier properties and statistical mechanics as (i = p, e, f, ph corresponding to phonon, electron, fluid particle, and photon, respectively). The heat transfer processes are governed by the rates (or kinetics) at which various related physical phenomena occur, such as (for example) the frequency of particle collisions in classical mechanics. These states and kinetics together determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level (atom or molecule length scale) to the macroscale are the laws of thermodynamics, including conservation of energy.
Principal energy carriers
Phonon A phonon is a quantized mode of vibration occurring in rigid crystal lattices. In classical mechanics, any lattice vibration can be decomposed into a superposition of non-localized normal modes. When these modes are analyzed using quantum mechanics, they are found to possess particle-like properties and are classified as quasiparticles. The energy (Ep) and momentum (Pp) of a phonon are defined as Ep = ħωp and Pp = ħ κp, where ħ is the reduced Planck constant, ωp is the angular frequency and κp is the wavevector; the relationship between these two variables is known as the dispersion relation. Because phonons are indistinguishable quasiparticles with zero spin, they are considered bosons, and their equilibrium distribution follows Bose–Einstein distribution. Phonons are primary thermal energy carriers that contribute to heat capacity (sensible heat storage) and conductive heat transfer in the condensed phase. Furthermore, they play a critical role in thermal energy conversion through interactions with electrons and fluid particles.
Electron An electron is a subatomic elementary particle with a negative elementary electric charge. Beyond their role as charge carriers, electrons are primary carriers of thermal energy transport in condensed matter. In particular, they are the dominant thermal transport carriers in metallic materials. However, at room temperature, the electronic contribution to thermal energy storage (heat capacity) is significantly lower than that of phonons. Due to their extremely small mass, electrons exhibit prominent wave-like characteristics. As particles with half-integer spin, they obey the Pauli exclusion principle and are classified as fermions; consequently, their equilibrium occupancy follows the Fermi-Dirac distribution. The state of electrons within a crystal is described by their band structure, which defines the relationship between energy (Ee) and wavevector (κe). Electrons engage in various interaction kinetics with other principal energy carriers, including photons (electromagnetic waves), phonons (lattice vibrations), and fluid particles, all of which are central to thermal energy conversion processes.
Fluid particle A fluid particle is the smallest unit (atom or molecule) in a fluid phase (gas, liquid, or plasma) that can be defined without breaking chemical bonds. The total energy of a fluid particle is categorized into potential, electronic, translational, vibrational, and rotational components. Thermal energy storage within a fluid particle occurs through temperature-dependent particle motions, specifically translational, vibrational, and rotational energies. Electronic energy is typically only considered if the temperature is high enough to cause ionization, dissociation, or other electronic transitions. According to the Hamiltonian, the energy state (Ef) of a fluid particle is quantized. However, fluid particles are generally treated as classical particles due to their large momentum and extremely short de Broglie wavelengths; consequently, their equilibrium distribution follows Maxwell–Boltzmann (MB) statistics. Fluid particles interact with other principal energy carriers. Vibrational or rotational modes, which possess relatively high energy, can be excited or decay through interactions with photons. Gas lasers utilize the interaction kinetics between fluid particles and photons, and laser cooling has been applied in CO2 gas laser systems. Furthermore, fluid particles can be adsorbed onto solid surfaces via physisorption or chemisorption. In these cases, the frustrated vibrational modes of the adsorbates decay by creating electron-hole (e−-h+) pairs or phonons. These interaction rates are determined through ab initio calculations and the application of Fermi's Golden Rule. Photon A photon is the quantum of electromagnetic (EM) radiation and serves as the primary energy carrier for radiative heat transfer. While classical EM waves are governed by Maxwell's equations, the quantization of the EM field is necessary to explain phenomena such as blackbody radiation and to resolve the ultraviolet catastrophe. The energy of a photon with angular frequency ωph is defined as Eph = ħωph. Photons exhibit a linear dispersion relation, where frequency is directly proportional to the wavevector (κph); consequently, the phase and group speeds are equal to the photon speed (uph), i.e., uph = dωph/dκph = ωph/κph. The propagation speed of a photon in a vacuum is a universal constant. As bosons, photons follow the Bose–Einstein distribution function (fph). Photons possess the widest range of energy among the principal carriers and are central to various energy conversion processes. They interact with electric and magnetic entities, such as electric dipoles (which may be excited by optical phonons or fluid particle vibrations) or transition dipole moments during electronic transitions. In heat transfer physics, these interaction kinetics are typically treated using an interaction Hamiltonian and perturbation theory, specifically Fermi's Golden Rule. Length and time scales
Thermophysical properties of matter and the kinetics of interaction and energy exchange among the principal carriers are fundamentally determined by their atomic-level configuration and interaction. Transport properties, such as thermal conductivity, are calculated from these atomic-level attributes using classical and quantum physics. Quantum states of principal carriers (e.g., momentum and energy) are derived from the Schrödinger equation (called first-principles or ab initio) and their interaction rates (for kinetics) are calculated using the quantum states and the quantum perturbation theory (formulated as the Fermi golden rule) . A variety of ab initio (Latin for from the beginning) solvers (software) exist to perform these calculations (e.g., ABINIT, CASTEP, Gaussian, Q-Chem, Quantum ESPRESSO, SIESTA, VASP, WIEN2k). Because electrons in the inner shells (core) are not involved in heat transfer, computational costs can be significantly reduced by employing proper approximations about the inner-shells electrons.
Due to intensive computational demands, fully quantum-mechanical treatments—including equilibrium and nonequilibrium ab initio molecular dynamics (AIMD)—are strictly limited to small length and time scales. Consequently, alternative frameworks utilizing simplifying assumptions are required to capture larger-scale kinetics. In classical (Newtonian) MD, the trajectories of atoms or molecules (particles) are governed by the empirical or effective interaction potentials. These potentials are traditionally parameterized by fitting to ab initio calculations or experimental thermophysical properties, or they are trained using machine learning frameworks. From the ensembles of simulated particles, static or dynamic thermal properties or scattering rates are derived. At yet larger length scales (mesoscale, involving many mean free paths), the Boltzmann transport equation (BTE) which is based on the classical Hamiltonian-statistical mechanics is applied. BTE considers particle states in terms of position and momentum vectors (x,p) and this is represented as the state occupation probability. The occupation has equilibrium distributions (the known boson, fermion, and Maxwell–Boltzmann particles) and transport of energy (heat) is due to nonequilibrium (cause by a driving force or potential). Central to the transport is the role of scattering which turns the distribution toward equilibrium. The scattering is presented by the relations time or the mean free path. The relaxation time (or its inverse which is the interaction rate) is found from other calculations (ab initio or MD) or empirically. BTE can be numerically solved with Monte Carlo method, etc.
The appropriate modeling framework (ab initio, MD, or BTE) is selected based on the target length and time scales of the system. Heat transfer physics analyses can couple these frameworks—for example, by feeding scattering rates calculated from ab initio or classical MD directly into the BTE—to systematically evaluate the states and kinetics governing thermal energy storage, transport, and transformation. Ultimately, heat transfer physics bridges classical and quantum mechanical perspectives to characterize the kinetics of the four principal energy carriers. This foundation enables robust multiscale workflows (from ab initio to MD, BTE, and macroscale) capable of capturing low-dimensionality and size effects.
Storage (Heat Capacity) From a fundamental perspective, thermal energy is stored within principal energy carriers, such as phonons, electrons, fluid particles, and photons. Consequently, the stored thermal energy is determined by the energy levels and population distributions of these carriers. The population at a specific energy level is given by the product of the density of states [the number of available quantum or classical states at a given energy level, D(E)] and the occupancy function [the probability that a state is occupied, f(E,T)]. Depending on the nature of the carrier, this distribution typically follows Bose-Einstein statistics (for phonons and photons), Fermi-Dirac statistics (for electrons), or Maxwell-Boltzmann statistics (for classical particles). The energy storage (sensible heat) associated with each carrier is quantified by the specific heat capacity at either constant volume or constant pressure (cv or cp). Characterizing these heat capacities requires identifying the energy distribution among the principal carriers.
Phonon The phonon dispersion relation describes all possible phonon modes within the Brillouin_zone (the primitive cell in reciprocal space) and provides the phonon density of states, Dp. Because phonons are bosonic particles, their occupancy follows the Bose–Einstein distribution {fpo = [exp(ħωp/kBT)-1]−1, kB: Boltzmann constant}. Using the phonon density of states and this occupancy distribution, the phonon energy can be expressed as Ep(T) = ∫Dp(ωp)fp(ωp,T)ħωpdωp, and the phonon number density is given by np(T) = ∫Dp(ωp)fp(ωp,T)dωp. The phonon specific heat capacity is then derived from the temperature sensitivity of the internal energy distribution:
Electron