Memory integral formulation

Here the strain y f) is taken relative to the current state, so y(t) = 0. This memory integral formulation is a special case of Boltzmann s superposition integral,... 

One may also use a memory function m(t) within the integral formulation of linear and nonlinear viscoelasticity this memory fimction m(t), which is the derivative of the relaxation function GKt), is not a measurable function. 

Due to complexity of the real world, all QDT descriptions involve practically certain approximations or models. As theoretical construction is concerned, the infiuence functional path integral formulation of QDT may by far be the best [4]. The main obstacle of path integral formulation is however its formidable numerical implementation to multilevel systems. Alternative approach to QDT formulation is the reduced Liouville equation for p t). The formally exact reduced Liouville equation can in principle be constructed via Nakajima-Zwanzig-Mori projection operator techniques [5-14], resulting in general two prescriptions. One is the so-called chronological ordering prescription (COP), characterized by a time-ordered memory dissipation superoperator 7(t, r) and read as... 

The integrals, Eqs. (B.3), exist quite generally by closing the contours C in the lower and upper complex planes, respectively, for extensions to the nonself adjoint case, see Figures 2.4, 2.5 and Appendix D. The retarded-advanced formulation including memory terms becomes... 

The equivalence between the differential and integral equation formulations exists only when the memory function m(f) is expressed as a sum of one or more exponentials, but the exponential representation is invariably used for the memory function. 

---