This is followed by the strongly non-local GENERIC-based model formulation in the "Strongly non-local model formulation" section.
In this context, the resulting models are inherently spatially strongly non-local (i.e., functional) and non-isothermal in character.
Analogous to the case of classical density functional theory (CDFT), consider now the approximation of the above strongly non-local formulation by a weakly non-local one.
In this context, the resulting models for conservative and non-conservative dynamics are inherently spatially strongly non-local (i.e., functional) and non-isothermal in nature.
This could be incorporated by including in the friction matrix element (M_{varepsilon varepsilon }^) a strongly non-local contribution that satisfies the condition (29 1 for the conservation of energy, with (M_{varepsilon varepsilon }^) being strongly non-local in the sense of not making use of any spatial derivative operator.
The current approach is based in particular on periodic microelasticity (Wang and Jin, 2001; Bulatov and Cai, 2006; Wang and Li, 2010) to model the strongly non-local elastic interaction of dislocation lines via their (residual) strain fields.
This specific example, (84), is relevant for the reduction of the strongly non-local model formulation to the weakly non-local case in the "Special case: weakly non-local model formulation" section.
In the last part of the work, the strongly non-local model formulation is reduced to weakly non-local form with the help of generalized gradient approximation of the energy and entropy functionals.
The strongly non-local form of the GENERIC formulated in the last section is now applied to the formulation of a model for a non-isothermal, heat-conducting mixture of displacively transforming thermoelastic solid phases and diffusing chemical constituents.
By analogy with CDFT, one can pursue weakly non-local approximations of the energy and entropy functionals of the strongly non-local formulation such as local density or generalized gradient approximations.
As shown in the last part of the work, the current strongly non-local model formulation reduces consistently to the weakly non-local one of Gladkov et al. (2016).
