The German physicist Rudolf Clausius, in the 1850s, was the first to mathematically quantify the discovery of irreversibility in nature through his introduction of the concept of entropy. In his 1854 memoir "On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat," Clausius states:
Simply, Clausius states that it is impossible for a system to transfer heat from a cooler body to a hotter body. For example, a cup of hot coffee placed in an area of room temperature  will transfer heat to its surroundings and thereby cool down with the temperature of the room slightly increasing (to ). However, that same initial cup of coffee will never absorb heat from its surroundings, causing it to grow even hotter, with the temperature of the room decreasing (to ). Therefore, the process of the coffee cooling down is irreversible unless extra energy is added to the system.
However, a paradox arose when attempting to reconcile microanalysis of a system with observations of its macrostate. Many processes are mathematically reversible in their microstate when analyzed using classical Newtonian mechanics. This paradox clearly taints microscopic explanations of macroscopic tendency towards equilibrium, such as James Clerk Maxwells 1860 argument that molecular collisions entail an equalization of temperatures of mixed gases. From 1872 to 1875, Ludwig Boltzmann reinforced the statistical explanation of this paradox in the form of Boltzmanns entropy formula, stating that an increase of the number of possible microstates a system might be in, will increase the entropy of the system, making it less likely that the system will return to an earlier state. His formulas quantified the analysis done by William Thomson, 1st Baron Kelvin, who had argued that:
Another explanation of irreversible systems was presented by French mathematician Henri Poincaré. In 1890, he published his first explanation of nonlinear dynamics, also called chaos theory. Applying chaos theory to the second law of thermodynamics, the paradox of irreversibility can be explained in the errors associated with scaling from microstates to macrostates and the degrees of freedom used when making experimental observations. Sensitivity to initial conditions relating to the system and its environment at the microstate compounds into an exhibition of irreversible characteristics within the observable, physical realm.