A question about Memristors 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? 
 A: 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.
A: 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.
