Functional thermo-dynamics: a generalization of dynamic density functional theory to non-isothermal situations

The following article appeared in Journal of Chemical Physics 139.3 (2013): 034106 and may be found at http://scitation.aip.org/content/aip/journal/jcp/139/3/10.1063/1.4811655We present a generalization of Density Functional Theory (DFT) to non-equilibrium non-isothermal situations. By using the original approach set forth by Gibbs in his consideration of Macroscopic Thermodynamics (MT), we consider a Functional Thermo-Dynamics (FTD) description based on the density field and the energy density field. A crucial ingredient of the theory is an entropy functional, which is a concave functional. Therefore, there is a one to one connection between the density and energy fields with the conjugate thermodynamic fields. The connection between the three levels of description (MT, DFT, FTD) is clarified through a bridge theorem that relates the entropy of different levels of description and that constitutes a generalization of Mermin's theorem to arbitrary levels of description whose relevant variables are connected linearly. Although the FTD level of description does not provide any new information about averages and correlations at equilibrium, it is a crucial ingredient for the dynamics in non-equilibrium states. We obtain with the technique of projection operators the set of dynamic equations that describe the evolution of the density and energy density fields from an initial non-equilibrium state towards equilibrium. These equations generalize time dependent density functional theory to non-isothermal situations. We also present an explicit model for the entropy functional for hard spheresP.E. acknowledges the hospitality and support of the Freiburg Institute for Advanced Studies and BIFI. Financial support under MICINN projects FIS2010-22047-C05-01,FIS2010-22047-C05-03, and MODELICO-S2009/ESP-1691 (Comunidad de Madrid) are greatly acknowledge