nLab inverse temperature

Idea

In formulas in thermodynamics it is often the (multiplicative) inverse of the temperature $T$ that is the natural variable, then traditionally abbreviated

$$\beta \;\coloneqq\; \frac{1}{k T} \,,$$

where $k$ denotes the Boltzmann constant, a physical unit which for purposes of pure mathematics may be identified with unity and hence omitted from the formula.

This inverse temperature appears, in particular, in the formula for the Boltzmann distribution, which is the probability distribution such that the probability density for a physical system to be in a state with energy $E$ is proportional to the exponential of $E$ weighted by minus the inverse temperature:

$$d p \;=\; e^{- \beta E} \, d E \,.$$

Essentially a special case of this is the formula for the heat kernel on a Riemannian manifold, which is an integral kernel proportial to the exponential of the Riemannian distance squared and again weighted by minus the inverse temperature.

$$K_{Heat}(x,y) \;\propto\; e^{ - \beta \cdot d(x,y)^2 } \,.$$

Extrapolating from this case, the scale of the exponent in various kernels may often be understood as akin to inverse temperature.

Finally, when understanding thermal/Euclidean field theory as the result of Wick rotation of relativistic field theory (when possible), the inverse temperature $\beta$ is proportional to the radius of the “thermal circle” on which the Euclidean field theory must be understood to be KK-compactified, see there for more.

Related concepts

• temperature, absolute zero

• thermal field theory, infinite-temperature thermal field theory

• kernel methods