  In accordance with the entropy principle of thermodynamics, under spontaneous
evolutions, physical systems always evolve towards states with equal or greater
randomness. But, where does this randomness originate? Renowned Birkhoff-von
Neumann theorem, often referred to as Birkhoff theorem, identifies source of
this randomness to be the stochastic application of reversible operations on
the system under study, thereby ensuring its epistemic origin. Analogue of this
theorem is known to fail in the quantum case. Here, we extend this
investigation beyond quantum mechanics to a broader class of operational
theories described within the framework of general probabilistic theories
(GPTs). In this generalized framework, we establish Birkhoff-violation as the
prevalent trait; in fact the asymptotic variant of the theorem gets violated.
We then demonstrate that Birkhoff-violation in GPTs can lead to consequences
that are atypical to quantum theory. For instance, we report manifestation of
Birkhoff-violation in a communication task, which otherwise is not observed in
quantum world. We also show that, unlike the quantum case, in other operational
theories the state transformation criteria can be distinct under mixtures of
reversible transformations and doubly stochastic evolutions, leading to
different resource theories of purity. Despite these exotic implications, we
analyze how to define a coherent notion of entropy in this generalized
framework, while upholding alignment with von Neumann's thought experiment.
