Thermodynamic evolution of the $z=1.75$ galaxy cluster IDCS J1426.5+3508

We present resolved thermodynamic profiles out to 500 kpc, about $r_{500}$, of the $z=1.75$ galaxy cluster IDCS J1426.5+3508 with 40 kpc resolution. Thanks to the combination of Sunyaev-Zeldovich and X-ray datasets, IDCS J1426.5+3508 becomes the most distant cluster with resolved thermodynamic profiles. These are derived assuming a non-parametric pressure profile and a very flexible model for the electron density profile. The shape of the pressure profile is flatter than the universal pressure profile. The IDCS J1426.5+3508 temperature profile is increasing radially out to 500 kpc. To identify the possible future evolution of IDCS J1426.5+3508 , we compared it with its local descendants that numerical simulations show to be $0.65pm0.12$ dex more massive. We found no evolution at 30 kpc, indicating a fine tuning between cooling and heating at small radii. At $30<r<300$ kpc, our observations show that entropy and heat must be deposited with little net gas transfer, while at 500 kpc the gas need to be replaced by a large amount of cold, lower entropy gas, consistent with theoretical expectation of a filamentary gas stream, which brings low entropy gas to 500 kpc and energy at even smaller radii. At $r gtrsim 400$ kpc the polytropic index takes a low value, which indicates the presence of a large amount of non-thermal pressure. Our work also introduces a new definition of the evolutionary rate, which uses unscaled radii, unscaled thermodynamic quantities, and different masses at different redshifts to compare ancestors and descendants. It has the advantage of separating cluster evolution, dependence on mass, pseudo-evolution and returns a number with unique interpretation, unlike other definitions used in literature.