Thermodynamics with non-integer dimensional space of states: Fractality of state set

A generalization of the standard equilibrium thermodynamics for the non-integer dimensional space (NIDS) of thermodynamic parameters is proposed by using the calculus of differential forms in NIDS. Spaces with non-integer dimensions are actively used in statistical physics, quantum field theory and models of fractal media. An axiomatic basis of integration in spaces with non-integer dimensions was first proposed by Wilson in 1973. The formulation of the self-consistent calculus of derivatives and integrals in NID space, including the calculus of differential forms in NID space, are proposed in a large multi-page paper [Fractal and Fractional. 9 (2025) 714] of about 200 pages. The NIDS generalizations of thermodynamic work, amount of heat and amount of mass transfer are proposed. The fundamental thermodynamic relations for TD systems with NIDS are obtained. Various examples of calculations of thermodynamic work for TD processes in NIDS are given. The proposed NIDS thermodynamics can be interpreted as thermodynamic with fractal distribution of the states on the set of independent thermodynamics parameters. The paper proposes methods for the experimental derivation (measurement) of non-integer dimensions of the space of thermodynamic states of the systems and media.