Shannon meets Carnot: Generalized second thermodynamic law

The classical thermodynamic laws fail to capture the behavior of systems with the energy Hamiltonian being an explicit function of the temperature. Such Hamiltonian arises, for example, in modeling information processing systems, like communication channels, as thermal systems. Here we generalize the second thermodynamic law to encompass systems with temperature-dependent energy levels, , where 〈〉 denotes averaging over the Boltzmann distribution and reveal a new definition to the basic notion of temperature. Furthermore, it is shown that the principles of the traditional thermodynamic framework are maintained, e.g., two systems in equilibrium have the same generalized temperature and heat always flows from high to low generalized temperature. This generalization enables to express, for instance, the mutual information of the Gaussian channel as a consequence of the laws of nature - the laws of thermodynamics.