Microscopic state equation for oscillator chains

Systems allowing anomalous transport of mass, momentum energy, etc., such as low-dimensional particles systems or highly confining media, are hard to characterize thermodynamically. Indeed, local thermodynamic equilibrium may not be established and their behaviour often strongly depends on many microscopic parameters, including the symmetry of the interaction potentials. Thermodynamic state equations, on the other hand, involve a small set of observables, which are obtained averaging in time and over the large number of particles that populate mesoscopic cells in which local equilibrium can be realized. In this work we show that a linear relation discovered earlier, that connects the average distance between pairs of consecutive particles with their kinetic energy, applies to quite a large set of 1-dimensional particle systems known to produce anomalous transport. This relation is microscopic in nature, since the quantities involved are neither averaged over many particles, neither over very large times. Nevertheless, its robustness is under variations of the external parameters, and the limited set of quantities it involves qualify it as a state equation, analogously to thermodynamic relations. We provide conditions for which the relation can be violated within a limited range of parameters values, and we find that it can be extended to two-dimensional networks of coupled oscillators. The validity of this relation further shows that the states of aggregation of matter in low-dimensional systems are often different from standard macroscopic ones.