Incoherent Fermi-Pasta-Ulam Recurrences and Unconstrained Thermalization Mediated by Strong Phase Correlations

The long-standing and controversial Fermi-Pasta-Ulam problem addresses fundamental issues of statistical physics, and the attempt to resolve the mystery of the recurrences has led to many great discoveries, such as chaos, integrable systems, and soliton theory. From a general perspective, the recurrence is commonly considered as a coherent phase-sensitive effect that originates in the property of integrability of the system. In contrast to this interpretation, we show that convection among a pair of waves is responsible for a new recurrence phenomenon that takes place for strongly incoherent waves far from integrability. We explain the incoherent recurrence by developing a nonequilibrium spatiotemporal kinetic formulation that accounts for the existence of phase correlations among incoherent waves. The theory reveals that the recurrence originates in a novel form of modulational instability, which shows that strongly correlated fluctuations are spontaneously created among the random waves. Contrary to conventional incoherent modulational instabilities, we find that Landau damping can be completely suppressed, which unexpectedly removes the threshold of the instability. Consequently, the recurrence can take place for strongly incoherent waves and is thus characterized by a reduction of nonequilibrium entropy that violates the $H$ theorem of entropy growth. In its long-term evolution, the system enters a secondary turbulent regime characterized by an irreversible process of relaxation to equilibrium. At variance with the expected thermalization described by standard Gibbsian statistical mechanics, our thermalization process is not dictated by the usual constraints of energy and momentum conservation: The inverse temperatures associated with energy and momentum are zero. This unveils a previously unrecognized scenario of unconstrained thermalization, which is relevant to a variety of weakly dispersive wave systems. Our work should stimulate the development of new experiments aimed at observing recurrence behaviors with random waves. From a broader perspective, the spatiotemporal kinetic formulation we develop here paves the way to the study of novel forms of global incoherent collective behaviors in wave turbulence, such as the formation of incoherent breather structures.

Popular Summary

Disorder tends to increase in nature. A drop of ink in water, for example, will spread until the ink particles are uniformly dispersed. But sometimes a system can revert to an initial, more ordered state. Computer experiments performed by Enrico Fermi, John Pasta, and Stanislaw Ulam in 1953 showed that oscillations among atoms in a crystal, rather than monotonously increasing in disorder (or entropy), can produce recurring orderly patterns. It is generally believed that, in wave systems, these momentary decreases in entropy require a correlation among the initial points known as phase coherence. In contrast to this common belief, we show that turbulent waves—more like waves at the bottom of a waterfall than on the surface of a calm lake—can also exhibit recurrence behaviors.

We develop a theory of wave interactions that reveals that some recurring “hidden order” can be created even among waves that are initially disordered. These recurrences show a reduction of (nonequilibrium) entropy that is in contrast to our everyday experience—a violation of what is known as the Boltzmann $H$ theorem of entropy growth. Phase coherence and integrability (the property that systems exhibit essentially regular periodic evolution) are not prerequisites for this behavior.

Recurrence behavior among coherent waves is well studied in experiments with optical fibers and wave tanks. These new findings will hopefully motivate experiments with incoherent waves as well. Our work paves the way for studying collective behaviors of such systems and suggests the existence of a novel form of large, unexpected waves (known as rogue waves) that are inherently random structures.

Figure 1

Incoherent recurrences. Evolutions of the (a) power ratio $N_{u_p}(t)/N_u$ and (b) total entropy $\tilde{\mathcal{S}}(t)$, for $w=5$ (black line), $w=9$ (blue line), and $w=25$ (red line) from NLSE simulations: The recurrences get more pronounced as $w$ increases, which violates the $H$ theorem of entropy growth ($\sigma =0.47$, $\eta =1$). (c) Probability density function (PDF) in logarithmic scale of the intensity $|u{|}^2$ of the wave at $t=50$ (blue circles) and $t=60$ (dashed red line) corresponding to $w=25$ [red lines in (a) and (b)]: Gaussian statistics is preserved throughout incoherent recurrences, as confirmed by the evolution of the kurtosis (d). Note that the dashed dark line in (c) shows the purely exponential law inherent to Gaussian statistics of the wave amplitudes.

Figure 2

Microscopic vs macroscopic fluctuations. NLSE simulation showing the spatiotemporal evolution of $|u_s{|}^2(x,t)$ (a) and corresponding spatial profile of the intensity at $t=35$ (b) [see the dashed line in (a)]. Panel (c) shows a zoom on a particular small region within the blue rectangle in (b): Aside from the microscopic fluctuations with correlation length ${\lambda }_c\sim 1$ (c), the waves exhibit macroscopic fluctuations with the large scale ${\ell }_c\sim 4\pi w/\kappa \simeq 10^3$ (b). The envelope of the macroscopic fluctuations is reported by the dark bold line in (b), which denotes the spatial profile of the correlator $n_{u_s}(x,t=35)$, whose dynamics is coupled to the other correlators $(n_{{\phi }_j},m_{\mu })$ by the phase-correlation kinetic equations; see Eqs. (11) and (12).

Figure 3

Creation of local phase correlations. (a) Spatiotemporal evolution of the product $|u_p(x,t)v_s^{*}(x,t)|$ obtained from NLSE simulations, showing that phase correlations grow and exhibit a nonhomogeneous statistics. (b),(c) Spatiotemporal evolutions of the phases of the random waves in local spatial regions of size ${\ell }_c$ [white rectangles in (a)]: The similarity of the phases of the random waves $u_p$ and $v_s$ reflects their strong correlations [note that the space-time windows in the phase plots (b),(c) correspond to those in the white rectangles in (a)]. (d),(e) Evolutions of the normalized phase correlation coefficient ${\varrho }_{uv}(t)$ given in Eq. (8) (blue line), and power ratio $N_{u_p}(t)/N_u$ (red line), in local spatial regions [white rectangles in (a)]. Note that there are no correlations between the initial random waves, ${\varrho }_{uv}(t=0)\simeq 0$: As a result of the phase-correlation modulational instability [Eq. (16)], the correlation grows up to almost unity, ${\varrho }_{uv}\simeq 1$. The dashed dark lines in (d) and (e) show the correlation coefficient ${\varrho }_{uv}(t)$ computed over the whole numerical spatial window, showing that there is no global phase correlation, ${\varrho }_{uv}(t)\lesssim 0.1$: A complete phase correlation (${\varrho }_{uv}\simeq 1$) solely emerges in local spatial regions of scale ${\ell }_c$, which in turn leads to almost complete local recurrences, $N_{u_p}/N_u\sim 1$ [red lines in (d) and (e)]. Parameters are $\sigma =7.5$, $\eta =1$, $\kappa =2/3$, $w=25$.

Figure 4

Phase-correlation modulational instability. (a) Growth rate of phase-correlation instability $\mathrm{Re}[\lambda (p)]$ from Eq. (15) for $\eta =1$ (blue line), $\eta =1.01$, $\sigma =7.5$ (black line), $\eta =1.1$, $\sigma =2$ (red line): Landau damping is suppressed for $\eta =1$, thus leading to an efficient phase-correlation growth. The length scale of macroscopic (nonhomogeneous) fluctuations is ${\ell }_c\simeq 2\pi /p_{\mathrm{MI}}$; see Fig. 2 and Eq. (17). (b) Temporal evolution of the energy in the sideband ($u_s$) in the presence of coherent (dashed black line) and incoherent (solid blue line) pump waves. When the pump waves $(u_p,v_p)$ are incoherent, the phase correlations are unstable, while the sideband components are stable, so that the growth of $u_s$ is delayed by a significant time, ${\tau }_d\simeq 13$ (solid blue line). In contrast, when the pump waves are coherent, the standard modulational instability leads to an instantaneous growth of the sideband $u_s$ (dashed black line).

Figure 5

Long-term evolution: Secondary turbulent regime and unconstrained thermalization. (a),(b) Simulations of the phase-correlation kinetic equations (20) and (21) showing the evolutions of the normal correlator $n_{u_p}^0$ (a) and anomalous phase correlator $|m_{uv}^0|$ (b): After a few incoherent recurrences $(t\gtrsim 80)$, the correlators enter a secondary turbulent regime. (d),(e) Corresponding evolutions of the spatial averages of the correlators $⟨n_{u_p}^0⟩(t)$ (c) and $⟨|m_{uv}^0|⟩(t)$ (d), obtained from the simulations of the phase-correlation kinetic equations (20) and (21) (dashed red line) and NLSE (blue line): For large times ($t\gtrsim 150$), the fields relax to the equilibrium state predicted by the theory (horizontal black dashed lines); see Eqs. (35) and (36). Parameters are $N_u=0.6$, $N_v=0.4$, $\kappa =2/3$, $w=54$. Unexpectedly, this equilibrium state is not constrained by Hamiltonian and momentum conservation (see Sec. 4).

Figure 6

Property of equilibrium thermodynamics. Schematic illustration of the collision of two incoherent waves with two different velocities. After the collision, the two waves propagate with the same average velocity, so as to prevent a relative internal motion among the two waves: An isolated system at equilibrium can only exhibit a uniform motion of translation as a whole, while any macroscopic internal motion is not possible at equilibrium [73]. This equilibrium thermodynamic property is verified for a system of strongly dispersive (weakly nonlinear) waves, as described by the wave turbulence theory [46]. However, this property is not verified for weakly dispersive (strongly nonlinear) waves ruled by the phase-correlation kinetic equations: A maximum entropy state is achieved for $\nu =\beta =0$; see Eq. (30).

Figure 7

Unconstrained thermalization: PDFs of the correlators. Equilibrium PDFs of the normal correlator $n_{u_p}^0$ (a) and anomalous phase correlator $|m_{uv}^0|$ (b), obtained from simulations of the phase correlation kinetic equations (20) and (21). A good agreement is obtained with the theory; see the equilibrium PDFs [Eqs. (33) and (34)] in red lines. The equilibrium state is not constrained by Hamiltonian and momentum conservation ($N_u=0.6$, $N_v=0.4$, $\kappa =2/3$, $w=54$).