Differentiating the absolutely continuous invariant measure of an interval map f with respect to f

Let the map $f:[-1,1]\to[-1,1]$ have a.c.i.m. $\rho$ (absolutely continuous $f$-invariant measure with respect to Lebesgue). Let $\delta\rho$ be the change of $\rho$ corresponding to a perturbation $X=\delta f\circ f^{-1}$ of $f$. Formally we have, for differentiable $A$, $$\delta\rho(A)=\sum_{n=0}^\infty\int\rho(dx) X(x){d\over dx}A(f^nx)$$ but this expression does not converge in general. For $f$ real-analytic and Markovian in the sense of covering $(-1,1)$ $m$ times, and assuming an {\it analytic expanding} condition, we show that $$\lambda\mapsto\Psi(\lambda)=\sum_{n=0}^\infty\lambda^n \int\rho(dx) X(x){d\over dx}A(f^nx)$$ is meromorphic in ${\bf C}$, and has no pole at $\lambda=1$. We can thus formally write $\delta\rho(A)=\Psi(1)$.Comment: 10 pages, plain Te

Entropy production in quantum spin systems

We consider a quantum spin system consisting of a finite subsystem connected to infinite reservoirs at different temperatures. In this setup we define nonequilibrium steady states and prove that the rate of entropy production in such states is nonnegative.Comment: In honor of J.L. Lebowit

How should one define entropy production for nonequilibrium quantum spin systems?

This paper discusses entropy production in nonequilibrium steady states for infinite quantum spin systems. Rigorous results have been obtained recently in this area, but a physical discussion shows that some questions of principle remain to be clarified.Comment: 8 page