Thermodynamic uncertainty relations

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Bohr and Heisenberg suggested that the thermodynamical quantities of temperature and energy are complementary in the same way as position and momentum in quantum mechanics. Roughly speaking, their idea was that a definite temperature can be attributed to a system only if it is submerged in a heat bath, in which case energy fluctuations are unavoidable. On the other hand, a definite energy can only be assigned to systems in thermal isolation, thus excluding the simultaneous determination of its temperature.

Rosenfeld extended this analogy with quantum mechanics and obtained a quantitative uncertainty relation in the form Delta U Delta (1/T) >= k where k is Boltzmann's constant. The two `extreme' cases of this relation would then characterize this complementarity between isolation (U definite) and contact with a heat bath (T definite). Other formulations of the thermodynamical uncertainty relations were proposed by Mandelbrot (1956, 1989), Lindhard (1986) and Lavenda (1987, 1991). This work, however, has not led to a consensus in the literature.

It is shown here that the uncertainty relation for temperature and energy in the version of Mandelbrot is indeed exactly analogous to modern formulations of the quantum mechanical uncertainty relations. However, his relation holds only for the canonical distribution, describing a system in contact with a heat bath. There is, therefore, no complementarity between this situation and a thermally isolated system.