Josephson equations

Two equations derived from first principles by Brian Josephson that characterize the device, somewhat like an I-V curve:where:

Note how these equations are not a typical I-V curve, as they are not an instantaneous dependency between voltage and current: the history of the voltage matters! Or in other words, the system has an internal state, represented by the Josephson phase at a given point in time.

To understand them better, it is important to look at some important cases separately:

Maximum current that can flow across a Josephson junction, as can be directly seen from the Josephson equations.

Is a fixed characteristic value of the physical construction of the junction.

A function $\mathbb{R}\to \mathbb{R}$ defined by the second of the Josephson equations plus initial conditions.

It represents an internal state of the junction.