On a microscopic equation of state for Lennard-Jones chains

The local thermodynamic equilibrium hypothesis is fundamental to the use of thermodynamic notions and the relations linking them, in equilibrium as well as nonequilibrium states. Its domain of applicability can vary widely across different systems. Monodimensional particle systems often violate the locality condition required by that hypothesis, because correlations can persist over long distances and times. Using a microscopic analogue of the van der Waals equation, which involves the microscopic mechanical equivalent of pressure, temperature and density, we investigate the validity of one of the nonequilibrium relations recently proposed as an equation of state concerning the microscopic mechanical equivalent of temperature and density, for monodimensional chains of particles. We do this without invoking the local equilibrium hypothesis: as we deal with mechanical, not thermodynamic, quantities, we do not need the local equilibrium hypothesis. The origin and validity of such a relation are described and indications are given as to why it works in various regimes of interest. Interestingly, that relation appears to be better satisfied, in our Lennard-Jones systems, when the microscopic “temperature” grows locally or the microscopic “density” decreases. These conditions are respectively favoured by high “temperature” differences at the boundaries, and by increasing the length of the chains. At those state points, that linear relation can be seen as a microscopic analogue of the van der Waals equation we propose. In some cases, a simpler approximation based on that link also exists. These types of relations work quite well for a surprisingly large and physically significant range of nonequilibrium states.