On the weak thermodynamic knot in vacuum (2+1)-dimensional space-time far from a source

We propose a novel idea of the existence of a topological object, a weak thermodynamics knot, in the framework of an Abelian Chern-Simons action in (2+1)-dimensional Minkowski space-time. This proposal is based on the idea that thermodynamics is treated as $U(1)$ local gauge theory where the thermodynamic force i.e. the gradient of entropy acts as gauge potential, and the field strength tensor could consist of a set of subset fields that satisfy the non-trivial Hopf maps, leading to topological structures. The corresponding field equations are derived. The field equations show that the phase has the topological structure that remains invariant under small variation. This result suggests that the weak thermodynamic knot may be a novel topological feature in low-dimensional thermodynamics.