Thermomechanics of odd matter: from deterministic to stochastic media

Abstract Odd matter exhibits non-reciprocal constitutive response while remaining compatible with balance laws and the second law of thermodynamics. We develop a unified continuum framework based on Edelen’s primitive thermodynamics, which decomposes generalized forces into gradient (hyperdissipative) and orthogonal (powerless) components. We show that this convex decomposition in velocity space admits an elastic analogue: hyperelastic response corresponds to integrable stresses derived from a stored-energy potential, while odd elasticity arises as the non-integrable, yet not derivable from a stored-energy potential branch of admissible response. Within the odd elasticity, we distinguish two cases: an instantaneous power orthogonality (the stronger one) or a configurational orthogonality (the weaker one). Working within the case of configurational orthogonality we construct the (linear) thermo-poromechanical theory, derive a finite strain odd neo-Hookean solid, and establish preservation of coercivity and exponential energy decay in random fluid and solid odd continua. Scale-dependent homogenization preserves coercivity of the even part and orthogonality of the odd part. In driven granular Couette flow, odd stresses modify fluctuation statistics without producing entropy, consistent with the fluctuation theorem. The results provide a systematic thermodynamic foundation for odd matter in deterministic and stochastic settings.