Depending on context, the term entropy is used for a thermodynamic quantity, a measure of available choice, a quantity to measure information, or in a statistical inference context as a maximum configuration predictor. For systems in equilibrium or processes without memory, the mathematical expression for these different entropy concepts appears to be the so-called Boltzmann–Gibbs–Shannon entropy, H. For processes with memory, such as driven- or self-reinforcing-processes, this is no-longer true: the different entropy concepts lead to distinct functionals that generally differ from H.

Here we focus on the maximum configuration entropy (that predicts empirical distribution functions) in the context of driven dissipative systems. We develop the corresponding framework and derive the entropy functional that describes the distribution of observable states as a function of the details of the driving process. We do this for sample space reducing (SSR) processes, which are an analytically tractable model of driven dissipative systems with controllable driving.

The fact that a consistent framework for a maximum configuration entropy exists for arbitrarily driven non-equilibrium systems demonstrates the possibility to derive a full statistical theory of driven dissipative systems. It might equip us with the technical means to derive a thermodynamic theory of driven processes based on the statistical theory. We briefly discuss the loss of Legendre structure for driven systems.

R. Hanel, S. Thurner, Maximum configuration principles for driven systems with arbitrary driving, Entropy 20 (2018) 838