Non-commutative connections of the second type or hom-connections and associated integral forms are studied as generalisations of right connections of Manin. First, it is proven that the existence of hom-connections with respect to the universal differential graded algebra is tantamount to the injectivity, and that every injective module admits a hom-connection with respect to any differential graded algebra. The bulk of the article is devoted to describing a method of constructing hom-connections from twisted multi-derivations . The notion of a free twisted multi-derivation is introduced and the induced first order differential calculus is described. It is shown that any free twisted multi-derivation on an algebra A induces a unique hom-connection on A (with respect to the induced differential calculus Ω^1(A) ) that vanishes on the dual basis of Ω^1(A) . To any flat hom-connection ∇ on A one associates a chain complex, termed a complex of integral forms on A . The canonical cokernel morphism to the zeroth homology space is called a ∇ - integral . Examples of free twisted multi-derivations, hom-connections and corresponding integral forms are provided by covariant calculi on Hopf algebras (quantum groups). The example of a flat hom-connection within the 3D left-covariant differential calculus on the quantum group \mathcal{O}_q(\mathrm{SL}(2)) is described in full detail. A descent of hom-connections to the base algebra of a faithfully flat Hopf–Galois extension or a principal comodule algebra is studied. As an example, a hom-connection on the standard quantum Podle’s sphere \mathcal{O}_q(\mathrm{SL}(2)) is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the ∇ -integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.

An intriguing phenomenon in nonequilibrium quantum thermodynamics is the asymmetry of thermal processes. Relaxation to thermal equilibrium is the most important dissipative process, being a key concept for the design of heat engines and refrigerators, contributing to the study of foundational questions of thermodynamics, and being relevant for quantum computing through the process of algorithmic cooling. Despite their importance, the dynamics of these processes are far from being understood. We show that the free relaxation to thermal equilibrium follows intrinsically different paths depending on whether it involves the temperature of the system to increase or to decrease. Our theory is exemplified using the recently developed thermal kinematics based on information geometry theory, utilizing three prototypical examples: a two-level system, the quantum harmonic oscillator, and a trapped quantum Brownian particle, in all cases showing faster heating than cooling under the appropriate conditions. A general understanding is obtained based on the spectral decomposition of the Liouvillian and the spectral gap of reciprocal processes.