Statistical analysis of complex systems with nonclassical invariant measures

I investigate the problem of finding a statistical description of a complex many-body system whose invariant measure cannot be constructed stemming from classical thermodynamics ensembles. By taking solitons as a reference system and by employing a general formalism based on the Ablowitz-Kaup-Newell-Segur scheme, I demonstrate how to build an invariant measure and, within a one-dimensional phase space, how to develop a suitable thermodynamics. A detailed example is provided with a universal model of wave propagation, with reference to a transparent potential sustaining gray solitons. The system shows a rich thermodynamic scenario, with a free-energy landscape supporting phase transitions and controllable emergent properties. I finally discuss the origin of such behavior, trying to identify common denominators in the area of complex dynamics.

Figure 1

(a) Theoretical (solid line) and numerical (red circles) soliton invariant measure versus $\lambda /2\rho$ calculated for grayness $w=0.9$. (b) Theoretical (solid line) and numerical (red circles) free energy $\mathcal{X}$ versus inverse temperature $\beta$ for $w=0.9$.

Figure 2

Free-energy profile for varying temperature $1/\beta$ and grayness $w$.

Figure 3

Time-dependent dynamics of the many-body soliton ensemble for $w=1$ (a), $w=0.95$ (b), $w=0.9$ (c), $w=0.85$ (d). In all simulations the amplitude of the input ${\psi }_0$ is set to $\rho =30$.

Figure 4

Time position of the wave breaking point (dashed line) versus grayness $w$, with insets showing the corresponding field intensity profiles (solid lines).