How the negative resistance feature of tunnel diodes used in oscillators?
One way to think of this is to look at the tank circuit with parallel resistor as a damped second-order system. The damping factor is \$ \zeta = \$\$ 1\over 2R \$\$\sqrt{L \over C}  \$, where R could be considered to include losses in the real L and C parts (so the model is three ideal passive components in parallel, and in parallel with the Esaki diode.  

If the tunnel diode is in a negative resistance region of operation, it is in parallel with R (also series with R1 || R2, but let's say that resistance is much smaller than R or the Esaki diode negative resistance). The total parallel resistance is the parallel combination of R and the tunnel diode negative resistance, and if the negative resistance of the tunnel diode is lower than the positive resistance, the result will be an overall negative resistance, negative damping, and continuous oscillation. 

If the damping factor is 0 then the oscillation continues at a constant amplitude. A damping factor that is negative implies oscillation of increasing amplitude. What happens is that the oscillation increases from a small value (started by noise) until the amplitude increases to the point where the voltage swing is such that the tunnel diode no longer has enough negative resistance to allow the amplitude to increase, and stabilizes there.