Isometric Uncertainty Relations

We generalize the link between fluctuation theorems and thermodynamic uncertainty relations by deriving a bound on the variance of fluxes that satisfy an isometric fluctuation theorem. The resulting bound, which depends on the system's dimension d, naturally interpolates between two known bounds. The bound derived from the entropy production fluctuation theorem is recovered for $d=1$, and the original entropy production thermodynamic uncertainty relation is obtained in the $d\to \infty$ limit. We show that our result can be generalized to order parameters in equilibrium systems, and we illustrate the results on a Heisenberg spin chain.