Thermodynamic Signalling in Underdamped Networks

Thermodynamic communication (movement of energy) between underdamped degrees of freedom or "nodes" is mediated by the momentum-coordinate correlation tensor Ω :=〈 pq^T〉. We may formally regard the components Ω_ij as a field on the discrete space of node pairs (i, j). The fluctuation -dissipation relation dictates a discrete field equation for the Ω_ij. In this linear, imhomogeneous equation, the "source" on the space of node pairs is generated by the thermally active nodes, subject to fluctuation and dissipation. If the network has D-dimensional topology, we may say that "thermodynamic communication lives on a discrete space with D + D dimensional topology." This is more than the trivial observation, that the domain of the field is the cartesian product of the node space with itself. For a network with nearest neighbor coupling, a continuum limit yields a boundary value problem for an ultra hyperbolic equation on D + D dimensional space. Think of it as a "wave equation on D + D dimensional Minkowski space." Domains of dependence and geometric attenuation shall behave accordingly. In this sense, the world of thermodynamic communication between pairs of nodes in D dimensions is truely D + D dimensional.