Thermodynamics from relative entropy

Thermodynamics can be developed from a microscopic starting point in terms of entropy and the maximum entropy principle. We investigate here to what extent one can replace entropy with relative entropy which has several advantages, for example, in the context of local quantum field theory. We find that the principle of maximum entropy can be replaced by a principle of minimum expected relative entropy. Various ensembles and their thermodynamic potentials can be defined through relative entropy. We also show that thermal fluctuations are in fact governed by a relative entropy. Furthermore, we reformulate the third law of thermodynamics using relative entropy only.

Figure 1

Left panel: Curves of relative entropies $S(p\parallel q)$ for a two-state system with probabilities $p_1=p, p_2=1-p$ with respect to different model distributions $q_1=q, q_2=1-q$ as parametrized by the value of $q$. Right panel: Entropy $S\left(q\right)$ and area function $B\left(q\right)$ of the model $q$. The maximum of entropy and the minimum of $B\left(q\right)$ are both at the uniform distribution, $q=1/2$.