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We study the entropic fluctuations of a general XY spin chain where initiallythe left(x<0)/right(x>0) part of the chain is in thermal equilibrium at inversetemperature Tl/Tr. The temperature differential results in a non-trivialenergy/entropy flux across the chain. The Evans-Searles (ES) entropicfunctional describes fluctuations of the flux observable with respect to theinitial state while the Gallavotti-Cohen (GC) functional describes thesefluctuations with respect to the steady state (NESS) the chain reaches in thelarge time limit. We also consider the full counting statistics (FCS) of theenergy/entropy flux associated to a repeated measurement protocol, thevariational entropic functional (VAR) that arises as the quantization of thevariational characterization of the classical Evans-Searles functional and anatural class of entropic functionals that interpolate between FCS and VAR. Wecompute these functionals in closed form in terms of the scattering data of theJacobi matrix h canonically associated to the XY chain. We show that all thesefunctionals are identical if and only if h is reflectionless (we call thisphenomenon entropic identity). If h is not reflectionless, then the ES and GCfunctionals remain equal but differ from the FCS, VAR and interpolatingfunctionals. Furthermore, in the non-reflectionless case, the ES/GC functionaldoes not vanish at 1 (i.e., the Kawasaki identity fails) and does not have thecelebrated ES/GC symmetry. The FCS, VAR and interpolating functionals alwayshave this symmetry. In the cases where h is a Schr\"odinger operator, theentropic identity leads to some unexpected open problems in the spectral theoryof one-dimensional discrete Schr\"odinger operators.

Entropic fluctuations in XY chains and reflectionless Jacobi matrices