Hidden free energy released by explicit parity-time-symmetry breaking

Abstract

It is shown that the familiar two-stream instability is the result of spontaneous parity-time (PT)-symmetry breaking in a conservative system, and more importantly, explicit PT-symmetry breaking by viscosity can destabilize the system in certain parameter regimes that are stable when viscosity vanishes. This reveals that complex systems may possess hidden free energies protected by PT-symmetry and viscosity, albeit dissipative, can expose the systems to these freed energies by breaking PT-symmetry explicitly. Such a process is accompanied by instability and total variation growth.

Figures (3)

• Figure 1: Spontaneous (a) and explicit (b) PT-symmetry breaking for a 4D system. Red and blue circles indicate positive- and negative-action modes, respectively.The route of destabilization illustrated in (b) is forbidden by PT symmetry.
• Figure 2: Two-stream instabilities for $\omega_{2}/\omega_{1}=100.$ Plotted are the four eigen-frequencies $\omega_{R}+i\omega_{I}$ of the system as functions of the differential velocity $\Delta v$. Absolute log scale was used for (b) and (d). The large empty space between $-1.48<\omega_{I}<-0.10$ is omitted in (c) for better resolution. The system parameters for (a) and (b) are $(\nu_{1},\nu_{2},v_{t1},v_{t2},v_{10},v_{20},\omega_{1},\omega_{2})=(0,0,0,6,0,\Delta v,0.04,4).$ There are two Krein collisions in (a) at $\Delta v=2.44$ and $\Delta v=4.96$. The system parameters for (c) and (d) are $(\nu_{1},\nu_{2},v_{t1},v_{t2},v_{10},v_{20},\omega_{1},\omega_{2})=(0,1,0,6,0,\Delta v,0.04,4).$
• Figure 3: Two-stream instabilities for $\omega_{2}/\omega_{1}=1.$ Plotted are the four eigen-frequencies $\omega_{R}+i\omega_{I}$ of the system as functions of the differential velocity $\Delta v$. The system parameters for (a) and (b) are $(\nu_{1},\nu_{2},v_{t1},v_{t2},v_{10},v_{20},\omega_{1},\omega_{2})=(0,0,1,1,0,\Delta v,1,1).$ There are two Krein collisions in (a) at $\Delta v=2$ and $\Delta v=3.46$. The system parameters for (c) and (d) are $(\nu_{1},\nu_{2},v_{t1},v_{t2},v_{10},v_{20},\omega_{1},\omega_{2})=(0,0.3,1,1,0,\Delta v,1,1).$