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The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan , who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.
The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation. This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known.
The law was formulated by Josef Stefan in 1879 and later derived by Ludwig Boltzmann. The formula E = σT 4 is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and σ = 5.670 367 × 10 −8 W·m −2 ⋅K −4 is the Stefan–Boltzmann constant. [28]
The solution of the above integral yields a remarkably elegant equation for the total emissive power of a blackbody, the Stefan-Boltzmann law, which is given as, = where is the Steffan-Boltzmann constant.
Boltzmann tried for many years to "prove" the second law of thermodynamics using his gas-dynamical equation – his famous H-theorem. However the key assumption he made in formulating the collision term was " molecular chaos ", an assumption which breaks time-reversal symmetry as is necessary for anything which could imply the second law.
The surface emits a radiative flux density F according to the Stefan–Boltzmann law: = where σ is the Stefan–Boltzmann constant. A key to understanding the greenhouse effect is Kirchhoff's law of thermal radiation. At any given wavelength the absorptivity of the atmosphere will be equal to the emissivity. Radiation from the surface could be ...
Combining the formulas for the Schwarzschild radius of the black hole, the Stefan–Boltzmann law of blackbody radiation, the above formula for the temperature of the radiation, and the formula for the surface area of a sphere (the black hole's event horizon), several equations can be derived. The Hawking radiation temperature is: [2] [20] [21]
Boltzmann's equation—carved on his gravestone. [1]In statistical mechanics, Boltzmann's equation (also known as the Boltzmann–Planck equation) is a probability equation relating the entropy, also written as , of an ideal gas to the multiplicity (commonly denoted as or ), the number of real microstates corresponding to the gas's macrostate: