Local Thermal Equilibrium and Thermal Observables

Local thermal equilibrium compares a possibly nonstationary state with equilibrium reference states through a deliberately chosen family of pointlike observables. It is weaker than the KMS condition and more precise than assigning a temperature from a single detector reading. The comparison can determine local mean temperature, energy density, or a rest frame, but only to the resolution carried by the chosen observables.

Required background. C*-dynamical systems and the KMS condition defines the equilibrium reference states; relativistic KMS analyticity and spectrum supplies their covariant temperature-vector interpretation; nonequilibrium steady states and entropy production provides a contrasting nonequilibrium notion based on stationarity and currents. Helpful background. Local equilibrium and hydrostatics gives the hydrodynamic closure problem; Tolman–KMS local temperature treats stationary redshift.

Thermal reference states and local probes

In Minkowski space, write $V_+$ for the open future timelike cone and let $\omega_\beta$ be a KMS state with inverse-temperature four-vector $\beta\in V_+$. To admit unresolved temperature fluctuations or phase mixtures, use the convex reference states

$$\omega_\rho(A)=\int_B\omega_\beta(A)\,d\rho(\beta),$$

where $B\subset V_+$ is compact and $\rho$ is a probability measure. A local thermal observable at $x$ is an idealized point field whose equilibrium expectation defines a thermal function

$$\Phi(\beta)=\omega_\beta(\phi(x)).$$

Balanced derivatives of Wick powers are useful because they probe short-distance correlations without introducing a preferred global time. Their construction requires the usual energy bounds and a specified renormalization prescription; the pointlike notation abbreviates a controlled limit of smeared fields. The passage from local fields to thermal functions is developed in Buchholz, Ojima, and Roos 2002, §3, pp. 224–229.

Choose a finite-dimensional space $S_x$ of such observables. A state $\omega$ is $S_x$-thermal if some reference state $\omega_{\rho_x}$ satisfies

$$\omega(\phi(x))=\omega_{\rho_x}(\phi(x)) =\int_B\Phi(\beta)\,d\rho_x(\beta) \qquad\text{for every }\phi(x)\in S_x.$$

The definition is explicitly resolution-dependent. Enlarging $S_x$ imposes more moment constraints and may destroy compatibility. Requiring the condition at every $x$ in a region, with smoothly varying thermal functions, yields a local-equilibrium field of expectations rather than a globally stationary state. The exact compatibility criterion and its propagation constraints appear in Buchholz, Ojima, and Roos 2002, §4, pp. 230–235.

Wick square as a thermometer

For the free massless scalar in four-dimensional Minkowski space, vacuum normal ordering gives

$$\omega_\beta\!\left({:}\phi^2{:}(x)\right)=\frac1{12\,\beta^2}, \qquad \beta^2=\beta_\mu\beta^\mu>0.$$

Thus, for a sharp rest-frame temperature, one may define $T^2=12\,\omega({:}\phi^2{:})$. Balanced second derivatives give a thermal energy tensor whose equilibrium function is

$$E^{\mu\nu}(\beta)=\frac{\pi^2}{90} \left(4\beta^\mu\beta^\nu-\beta^2\eta^{\mu\nu}\right)(\beta^2)^{-3}.$$

Together, the Wick square and enough components of $E^{\mu\nu}$ can distinguish a sharp inverse-temperature vector from nearby alternatives in this model. These formulae and their domain are established in Buchholz, Ojima, and Roos 2002, §5.3, pp. 237–239.

For a mixture, however, the Wick square measures $\int T^2d\rho$, not a unique microscopic temperature. A pure reference at $T=1$ and the equal mixture of $T=1/\sqrt2$ and $T=\sqrt{3/2}$ have the same second moment,

$$\int T^2d\rho=1,$$

but fourth moments $1$ and $5/4$. A stress-energy observable, which scales as $T^4$, separates them. This is a concrete reason to speak of thermal observables and their resolution rather than a universal local thermometer.

Curvature and renormalization

On curved spacetime, the locally covariant Wick square admits a finite curvature renormalization, schematically

$${:}\phi^2{:}_{c}={:}\phi^2{:}_{0}+cR.$$

Consequently, $12\,\omega({:}\phi^2{:})$ cannot be called a geometry-independent temperature squared until $c$ has been fixed by a renormalization condition. A stationary KMS state also carries a Tolman redshift relation tied to a Killing flow, whereas the $S_x$-thermal criterion can be applied to nonstationary states and may return a probability distribution rather than a scalar. The first curved-spacetime application makes this distinction explicit in Tolman–KMS local temperature.

Failure boundaries and nonconverses

Agreement on one observable never proves local KMS behavior. Even agreement on a finite $S_x$ fixes only finitely many moments of $\rho_x$; distinct mixtures may remain indistinguishable. Conversely, failure for an unnecessarily large set can coexist with an accurate hydrodynamic description at coarser resolution. Positivity is another constraint: arbitrary prescribed numbers for the thermal functions need not lie in the convex range of equilibrium values, so no positive reference measure need exist.

The definition also does not identify a unique velocity field unless $S_x$ contains observables sensitive to the direction of $\beta^\mu$. A scalar Wick-square thermometer determines at most a temperature moment. Finally, local thermality neither implies stationarity nor rules out heat flow: neighboring points may be matched by different reference measures.

Exercises

1. Moment ambiguity. Verify the two-temperature example above and compute its fourth moment.

Solution

The equal mixture has second moment $\tfrac12(1/2+3/2)=1$, the same as the sharp state at $T=1$. Its fourth moment is $\tfrac12(1/4+9/4)=5/4$, whereas the sharp state’s fourth moment is $1$.

2. Covariant Stefan–Boltzmann form. Put $\beta^\mu=T^{-1}u^\mu$ with $u^2=1$ into $E^{\mu\nu}(\beta)$.

Solution

Since $\beta^2=T^{-2}$, one obtains

$$E^{\mu\nu}=\frac{\pi^2T^4}{90}(4u^\mu u^\nu-\eta^{\mu\nu}),$$

the perfect-fluid tensor with energy density $\pi^2T^4/30$ and pressure one third of that density.

3. Renormalization shift. How does the Wick-square temperature proxy change under ${:}\phi^2{:}\mapsto{:}\phi^2{:}+cR$?

Solution

The proxy changes by $12cR$. It is therefore invariant in Minkowski space but curvature-dependent in general; a renormalization condition is part of the thermometer’s definition.

References

• Buchholz, D., Ojima, I., and Roos, H. (2002). “Thermodynamic properties of non-equilibrium states in quantum field theory.” Annals of Physics 297, 219–242. DOI. Open PDF.
• Solveen, C. (2012). “Local thermal equilibrium in quantum field theory on flat and curved spacetimes.” Classical and Quantum Gravity 29, 245015. DOI. Open PDF.