Ultraviolet cascade in the thermalization of the classical ${\varphi }^4$ theory in $3+1$ dimensions

We investigate the dynamics of thermalization and the approach to equilibrium in the classical ${\varphi }^4$ theory in $3+1$ spacetime dimensions. The nonequilibrium dynamics is studied by numerically solving the equations of motion in a light-cone-like discretization of the model for a broad range of initial conditions and energy densities. A smooth cascade of energy towards the ultraviolet is found to be the basic mechanism of thermalization. After an initial transient stage, at a time scale of several hundred inverse masses, the squared magnitude of the field spatial gradient becomes larger than the nonlinear term and there emerges a stage of universal cascade, independent of the details of the initial conditions. As the cascade progresses, the modes with higher wave numbers, but well behind the forefront of the cascade, exhibit weaker and weaker nonlinearities well described by the Hartree approximation, while the infrared modes retain strong self-interactions. As a consequence, two time scales for equilibration appear as characteristic of two time-dependent wave number regions. For $k^2\gtrsim \overline{{\varphi }^2}(t)$, we observe an effective equilibration to a time-dependent powerlike spectrum with a time scale in the hundreds of inverse masses; cutoff effects are absent and the Hartree approximation holds for $k^2\gg \overline{{\varphi }^2}(t)$. On the other hand, infrared modes with $k^2\lesssim \overline{{\varphi }^2}(t)$ equilibrate only by time scales in the millions of inverse masses when the cutoff effects become dominant and complete thermalization is setting in. Accordingly, we observe in the field correlator a relatively large and long-lived deviation from the Hartree behavior of a nonperturbative character. There corresponds an effective mass governing the long distance behavior of the correlator which turns out to be significantly smaller than the Hartree mass which is exhibited by the modes with $k^2\gtrsim \overline{{\varphi }^2}(t)$. Virialization and the equation of state start to set in much earlier than thermalization. The applicability of these results in quantum field theory for large occupation numbers and small coupling is analyzed.

Figure 1

$⟨{\varphi }^2⟩$, $⟨(\nabla \varphi {)}^2⟩$, and the ratio $⟨{\varphi }^4⟩/⟨{\varphi }^2{⟩}^2$ as a function of $T/a$ in thermal equilibrium from Monte Carlo simulations.

Figure 2

$1/⟨|{\tilde{\varphi }}_{\mathit{k}}{|}^2⟩$, the inverse of the direction-averaged $\varphi$ power spectrum, as a function of $a^{-2}[1-[B(⟨{\varphi }^2⟩)C(\mathit{k}a){]}^2]$, which reproduces $k^2+1+3⟨{\varphi }^2⟩$ for small $ka$ while including the lattice artifacts when $ka$ becomes of order 1. In the present case the UV cutoff is $\Lambda =125.6637$. The almost linear behavior of $1/⟨|{\tilde{\varphi }}_{\mathit{k}}{|}^2⟩$ down to relatively small $ka$ (see the inset) supports the accuracy of the Hartree approximation.

Figure 3

$Z_k$ vs the radial wave number (in the upper plot) and the power spectrum $⟨|{\tilde{\pi }}_{\mathit{k}}{|}^2⟩$ (lower plot), which is indeed almost flat and equal to the temperature $T$.

Figure 4

Log-log plot of the time evolution of local observables with sliding time average. Initial conditions are localized wave packets of Lorentzian shape without any zero-mode condensate. No average is performed over initial packet amplitudes or positions.

Figure 5

Log-log plot of the time evolution of local observables. The time averaging interval is quite small, $\tau =2$, and the initial conditions are plane waves with a large zero-mode condensate. No average is performed over initial amplitudes or phases.

Figure 6

Log-log plot of the time evolution of local observables. The time averaging interval is $\tau =40$ and the initial conditions are plane waves with a large zero-mode condensate. No average is performed over initial amplitudes or phases.

Figure 7

Log-log plot of the time evolution of local observables. Initial conditions are plane waves without a zero-mode condensate. The time averaging interval is $\tau =200$ and an extra average is performed over 40 random choices of initial amplitudes and phases. This explains why the curves in this picture are smoother than those in Figs. 5, 6.

Figure 8

Log-log plot of the time evolution of local observables with sliding time average and very small energy density. Initial conditions are of the same type as in Fig. 4. No average is performed over initial packet amplitudes or positions. Time averaging is performed here as by Eqs. (2.17, 2.18).

Figure 9

$\log |\overline{\varphi }(t)|$ as a function of the logarithm of the time $t$ for $E/V=1000$ and $E/V=10$ with $L=6.4$ and $a=0.1$. No average is performed over initial packet amplitudes or positions. Time averaging is performed here as by Eqs. (2.17, 2.18).

Figure 10

$⟨{\varphi }_{+}^4⟩$ vs $⟨{\varphi }^2⟩$ in thermal equilibrium (obtained from Monte Carlo simulations) and $\overline{{\varphi }_{+}^4}(t)$ vs $\overline{{\varphi }^2}(t)$ from time averages. Both $\overline{{\varphi }_{+}^4}(t)$ and $\overline{{\varphi }^2}(t)$ decrease for increasing time.

Figure 11

The normalized left-hand side of the virial theorem $\Delta (t)$ vs the logarithm of the time $t$ for $E/V=\rho =1$, 10, 100, and 1000 for $a=0.1$ and $L=6.4$ [see Eq. (4.3)]. No average is performed over initial packet amplitudes or positions. Time averaging is performed here as by Eqs. (2.17, 2.18).

Figure 12

The equation of state $\overline{p}/\rho$ vs the logarithm of the time $t$ for $\rho =100$ and 10 with $a=0.05$ and for $\rho =5$ and 0.1 with $a=0.1$. We have in all four cases $L=6.4$. No average is performed over initial packet amplitudes or positions. Time averaging is performed here as by Eqs. (2.17, 2.18).

Figure 13

The power $4\pi k^2\overline{|{\tilde{\pi }}_k{|}^2}(t)$ vs $k=|\mathit{k}|$ at 20 different times ranging, in an approximately exponential way, from $t=0$ to $t=12.75$ (left panel) and from $t=12.75$ to $t=2948$ (right panel). The time averaging interval is $\tau =2$ and the initial conditions are infrared random plane waves, as apparent from the IR peaks at early times. The front of the UV cascade arrives close to the cutoff $\Lambda$ for the latest times.

Figure 14

The power $4\pi k^2\overline{|{\tilde{\pi }}_k{|}^2}(t)$ vs $k$ as in the right of Fig. 13 but at values of the UV cutoff $\Lambda$ scaled by $1/4$, $1/2$, 2, and 4. When $\Lambda =251.328$ the last time $t=2978$ is missing. The cascades shown in the lower panels are cutoff independent since they are evolving well below the cutoff $\Lambda$.

Figure 15

Log-log plot of $\overline{|{\tilde{\varphi }}_k{|}^2}(t)$ vs time. Initial conditions are as in Fig. 13. The four thicker lines correspond to the modes initially filled (after averaging over directions). The zero-mode (dashed line) was not filled at $t=0$. Notice the peculiar behavior of the lowest $k$ modes compared with the rest of the $k$ modes.

Figure 16

As in Fig. 15, but for $\overline{|{\tilde{\pi }}_k{|}^2}(t)$ vs time. Notice the peculiar behavior of the lowest $k$ modes compared with the rest of the $k$ modes.

Figure 17

Log-log plot of the time evolution of average wave number $\overline{k}(t)$, for five values of the lattice spacing $2a$, with initial conditions as in Figs. 13, 14.

Figure 18

$\log [\overline{|{\tilde{\pi }}_k{|}^2}(t)]$ vs $k$ for several values of time. All relevant parameters are at the indicated values. Initial conditions were random plane waves. $\overline{|{\tilde{\pi }}_k{|}^2}(t)$ starts decreasing with $k$ as $\sim k^{-\alpha }$ and later it dies exponentially for $\Lambda >k>\overline{k}(t)$.

Figure 19

$\log [4\pi k^2\overline{|{\tilde{\pi }}_k{|}^2}(t)]$ vs $\log k$ in a fixed wave number window for several values of time. All relevant parameters are at the indicated values.

Figure 20

The power $4\pi k^2\overline{|{\tilde{\pi }}_k{|}^2}(t)$ vs $k=|\mathit{k}|$ at several times, $E/V=89.5$ and two lattice spacings. The initial conditions are infrared random plane waves with a zero-mode condensate dominating the energy. Notice the peaks due to parametric resonance.

Figure 21

Log-log plot of $\overline{|{\tilde{\varphi }}_k{|}^2}(t)$ vs time. Initial conditions are as in Fig. 20. Only the zero-mode (thick line) was macroscopically filled at $t=0$.

Figure 22

Log-log plot of the time evolution of space-averaged local observables and of the average wave number. The time averaging interval is $\tau =200$ and the initial conditions are infrared random plane waves. This plot is analogous to Fig. 5 but with the time evolution lasting for significantly later times.

Figure 23

$4\pi k^2\overline{|{\tilde{\pi }}_k{|}^2}(t)$ vs $k$ at the times indicated. All parameters are as in Fig. 22.

Figure 24

Log-log plot of $\overline{|{\tilde{\pi }}_k{|}^2}(t)$ and $\overline{|{\tilde{\varphi }}_k{|}^2}(t)$ vs $k$. Data points are explicitly indicated. All parameters are as in Fig. 22. Comparison with Fig. 22 shows that modes with $k^2<\overline{{\varphi }^2}(t)$ thermalize much slower than those with $k^2>\overline{{\varphi }^2}(t)$.

Figure 25

Log-log plot $\overline{|{\tilde{\pi }}_k{|}^2}(t)$ and $\overline{|{\tilde{\varphi }}_k{|}^2}(t)$ vs $\ln t$. Only modes up to $k\le \Lambda$ are shown. All parameters are as in Fig. 22. We see that modes with $k^2<\overline{{\varphi }^2}(t)$ thermalize here the last. [Compare with Fig. 22 to see $\log \overline{{\varphi }^2}(t)$.]

Figure 26

Comparison of $\log \overline{|{\tilde{\pi }}_k{|}^2}(t)$ vs $Z_k(t)$. Diamonds in the bottom panel represent equilibrium values. All parameters are as in Fig. 22. We see that $Z_k(t)$ equilibrates much earlier than $\overline{|{\tilde{\pi }}_k{|}^2}(t)$.

Figure 27

Scaling properties of the $Z_k(t)$ for two low-lying ranges of the radial wave number $k$.

Figure 28

The number of modes over spherical shells, $4\pi k^2{\overline{n}}_k(t)$, vs $k=|\mathit{k}|$ at the different times indicated. The initial conditions are infrared random plane waves. The UV cascade is clearly seen as $k^2{\overline{n}}_k(t)$ is depleted for low $k$ while it grows for large $k$ as time grows.

Figure 29

$M_{\mathrm{eff}}^2(t)-1$ as a function of the logarithm of the time $t$ for $E/V=0.1$, 10, 100, and 1000 with $L=6.4$ and $a=0.1$. No average is performed over initial packet amplitudes or positions. Time averaging is performed here as by Eqs. (2.17, 2.18).

Figure 30

$R\equiv [M_{\mathrm{eff}}^2(t)-1]/\overline{{\varphi }^2}(t)$ as a function of the logarithm of time $t$ for $E/V=0.1$, 10, 100, and $E/V=1000$ with $L=6.4$ and $a=0.1$. Notice that $R$ varies in a much narrower interval than $M_{\mathrm{eff}}^2(t)$ depicted in Fig. 29. No average is performed over initial packet amplitudes or positions. Time averaging is performed here as by Eqs. (2.17, 2.18).

Figure 31

The space average of $\varphi (\mathit{x},t)$ as a function of time $t$ for $E/V=100$ with $L=6.4$ and $a=0.1$ over a sample time interval. Notice the fast oscillations displayed here which are erased by the time averaging. Thanks to time averaging the slow dynamics becomes visible in Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25.

Figure 32

The inverse of the ratio $Z_k^{\xi }$ [defined by Eq. (C2)] vs the radial wave number for the values of time and of $\xi$ indicated. All parameters are as in Fig. 22.

Figure 33

$R({\varphi }_0)=[M_{\mathrm{eff}}^2-1]/<{\varphi }^2>$ vs $\rho$ for the cnoidal solution Eq. (D1). Here $⟨\cdots ⟩$ stands for the time average over one period.