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In this paper we consider the energy and momentum transport in(1+1)-dimension conformal field theories (CFTs) that are deformed by anirrelevant operator $T\bar{T}$, using the integrability based generalizedhydrodynamics, and holography. The two complementary methods allow us to studythe energy and momentum transport after the in-homogeneous quench, derive theexact non-equilibrium steady states (NESS) and calculate the Drude weights andthe diffusion constants. Our analysis reveals that all of these quantitiessatisfy universal formulae regardless of the underlying CFT, therebygeneralizing the universal formulae for these quantities in pure CFTs. As asanity check, we also confirm that the exact momentum diffusion constant agreeswith the conformal perturbation. These fundamental physical insights haveimportant consequences for our understanding of the $T\bar{T}$-deformed CFTs.First of all, they provide the first check of the $T\bar{T}$-deformed$\mathrm{AdS}_3$/$\mathrm{CFT}_2$ correspondence from the dynamical standpoint.And secondly, we are able to identify a remarkable connection between the$T\bar{T}$-deformed CFTs and reversible cellular automata.

Thermal transport in $T\bar{T}$-deformed conformal field theories: from integrability to holography