Editing Loschmidt's paradox (section)

== Dynamical systems ==
{{main|Entropy as an arrow of time}}

Current {{As of?|date=August 2023}} research in dynamical systems offers one possible mechanism for obtaining irreversibility from reversible systems. The central argument is based on the claim that the correct way to study the dynamics of macroscopic systems is to study the [[transfer operator]] corresponding to the microscopic equations of motion. It is then argued {{by whom|date=August 2023}} that the transfer operator is not [[Unitary (physics)|unitary]] (''i.e.'' is not reversible) but has [[Eigenvalues and eigenvectors|eigenvalues]] whose magnitude is strictly less than one; these eigenvalues corresponding to decaying physical states. This approach is fraught with various difficulties; it works well for only a handful of exactly solvable models.<ref>Dean J. Driebe, ''Fully Chaotic Maps and Broken Time Symmetry'', (1999) Kluwer Academic {{ISBN|0-7923-5564-4}}.</ref>

Abstract mathematical tools used in the study of [[dissipative system]]s include definitions of [[mixing (mathematics)|mixing]], [[wandering set]]s, and [[ergodic theory]] in general.