A question about Memristors

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.

Answering to: "The question, is what physics laws govern the operation of these devices." and to "what computing applications could they have", let me quote a paragraph from a recent work I coauthored: "Towards peptide-based tunable multistate memristive materials", Cardona Serra et al, Phys Chem Chem Phys 2021, DOI 10.1039/D0CP05236A

The first memristive material was proposed as late as in 2008 in HP laboratories, and was based on an oxygen migration mechanism at the interface between TiO2/TiO2−x. In this specific case, the interface between both inorganic phases acts as a mobile barrier where oxo anions move as a function of the current passing through the device. The relative position of the barrier between TiO2/TiO2−x results in different states of resistance. This mechanism gives rise to the so-called ‘memristive switching’ effect. A more recent type of memristive material uses nanoscale spintronic oscillators, a totally different approach, where magnetism and electronics interplay for building the neuronal units. This route is based on the concept of spin-valve magnetoresistance, where the total resistance of the magnetic tunnel junction (MTJ) depends on the relative orientation of a soft-variable ferromagnet with respect to a hard-fixed ferromagnet. Thus, the current passing through the junction generates a torque on the magnetization of the soft-ferromagnet which leads to a spin precession with frequencies varying from 100 MHz to tens of GHz. These spin precessions can be converted to voltage oscillations through magnetoresistance. Due to their response to thermal noise, MTJs have been presented as non-volatile magnetic memories that can contribute with a certain stochasticity in the resistance transition. This property has been recently exploited for random number generation and other noise-based computing applications. While this spintronic-based mechanism is physically very different from the oxygen ion migration described above, substantially the result is the same: a process whereby the electric current through a component results in a controllable change of electrical resistance. Indeed, a variety of physical processes can give rise to analogous memristive behaviors, and different materials have been found to be more or less adequate for different memristive applications.

While this does not even aim to be exhaustive, it provides context for the answer which in short is: a variety of physical mechanisms can give rise in practice to memristive behavior, always based in materials or devices that are out of thermodynamic equilibrium in the relevant operating time scale. And a variety of computing applications can benefit from this, perhaps most notably among those hardware-based neuromorphic computing.