Generalized Boltzmann distributions for systems strongly coupled to large finite bath -- a microcanonical approach

Abstract

The theory of probability shows that, as the fraction $X_n/Y\to 0$, the conditional probability for $X_n$, given $X_n+Y\in h_{\delta }:=[h,h+\delta ]$, has a limit law $f_{X_n}(x)e^{-{\psi }_n(h_{\delta })x}$, where ${\psi }_n(h_{\delta })$ equals to $[\partial \ln P(Y\in y_{\delta })/\partial y{]}_{y=h}$ plus an additional term, contributed from the correlation between $X_n$ and bath $Y$. By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form of a redefined Hamiltonian and corrected Boltzmann temperature that reflects the modification due to strong system-bath coupling and the large but finite bath.

arXiv commentAbstract revised; Minor revisions to the main text