# Heat current

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A heat current is a kinetic exchange rate between molecules, relative to the material in which the kinesis occurs. It is defined as ${\displaystyle {\frac {dQ}{dt}}}$, where ${\displaystyle Q}$ is heat and ${\displaystyle t}$ is time.

For conduction, heat current is defined by Fourier's law as

${\displaystyle {\frac {\partial Q}{\partial t}}=-k\oint _{S}{{\overrightarrow {\nabla }}T\cdot \,{\overrightarrow {dS}}}}$

where

${\displaystyle {\big .}{\frac {\partial Q}{\partial t}}{\big .}}$ is the amount of heat transferred per unit time [W] and
${\displaystyle {\overrightarrow {dS}}}$ is an oriented surface area element [m2]

The above differential equation, when integrated for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as:

${\displaystyle {\big .}{\frac {\Delta Q}{\Delta t}}=-kA{\frac {\Delta T}{\Delta x}}}$

where

A is the cross-sectional surface area,
${\displaystyle \Delta T}$ is the temperature difference between the ends,
${\displaystyle \Delta x}$ is the distance between the ends.

For thermal radiation, heat current is defined as

${\displaystyle W=\sigma \cdot A\cdot T^{4}}$

where the constant of proportionality ${\displaystyle \sigma }$ is the Stefan–Boltzmann constant, ${\displaystyle A}$ is the radiating surface area, and ${\displaystyle T}$ is temperature.

Heat current can also be thought of as the total phonon distribution multiplied by the energy of one phonon, times the group velocity of the phonons. The phonon distribution of a particular phonon mode is given by the Bose-Einstein factor, which is dependent on temperature and phonon energy.