A question about memristors: These semiconductor entities have been defined in terms of magnetic flux leakage, and a non-linear relationship to the electric charge that has flowed. In essence, the devices' electrical resistance changes as the amount of electric charge that has flown through it. So, they can readily be seen as adaptable to transistors. The question, is what physics laws govern the operation of these devices. Thermodynamic laws seem to restrict their operation, but they are seen as having applications in quantum computing, or even in the general case what computing applications could they have?
As you mentioned memristance governs nonlinear behavior of electric or magnetic circuit based on the amount of electric charge which has passed through it. In this paper Strukov et al. from HP labs described properties of memristors and provided fundamental mathematical model.
As a physical model they employed metal/oxide/metal circuit where metal is Pt and oxide is thin - tens of nanometers TiO2 film. This oxide film consists of two layers: a layer of pure TiO2 and another layer of oxygen poor TiO2. In oxygen poor TiO2 oxygen vacancies serve as mobile 2+ charges which can diffuse in the direction of external electric field. Controlling the thickness of the oxygen poor TiO2 layer will change overall resistance of the circuit and produce hysteretic pattern upon sweep.
Thus memristivity naturally occurs as a result of diffusion of cations in wide bandgap oxide semiconductors. Memristivity can be intrinsic or extrinsic, i.e. in the example above TiO2 is an intrinsic memristor since no external dopants were introduced into the system and only oxygen vacancy served as a conductive agent. Obviously extrinsic memristor is doped with ions.