V. Jaksic, B. Landon, C.-A. Pillet
Entropic fluctuations in XY chains and reflectionless Jacobi matrices
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ABSTRACT. We study the entropic fluctuations of a general XY spin chain where
initially the left(x<0)/right(x>0) part of the chain is in thermal
equilibrium at inverse temperature Tl/Tr. The temperature differential
results in a non-trivial energy/entropy flux across the chain. The
Evans-Searles (ES) entropic functional describes fluctuations of the
flux observable with respect to the initial state while the
Gallavotti-Cohen (GC) functional describes these fluctuations with
respect to the steady state (NESS) the chain reaches in the large time
limit. We also consider the full counting statistics (FCS) of the
energy/entropy flux associated to a repeated measurement protocol, the
variational entropic functional (VAR) that arises as the quantization
of the variational characterization of the classical Evans-Searles
functional and a natural class of entropic functionals that
interpolate between FCS and VAR. We compute these functionals in
closed form in terms of the scattering data of the Jacobi matrix h
canonically associated to the XY chain. We show that all these
functionals are identical if and only if h is reflectionless (we call
this phenomenon entropic identity). If h is not reflectionless, then
the ES and GC functionals remain equal but differ from the FCS, VAR
and interpolating functionals. Furthermore, in the non-reflectionless
case, the ES/GC functional does not vanish at 1 (i.e., the Kawasaki
identity fails) and does not have the celebrated ES/GC symmetry. The
FCS, VAR and interpolating functionals always have this symmetry. In
the cases where h is a Schr dinger operator, the entropic identity
leads to some unexpected open problems in the spectral theory of
one-dimensional discrete Schr dinger operators.