Chapters 2,3 and 4: Dissipation rate and deformations of chaotic billiards

The dynamics of a particle inside a cavity (billiard) in $d=2$ or 3 dimensions is a major theme in studies of classical and quantum chaos [154,91,25]. Whereas the physics of time-independent chaotic systems is extensively explored, less is known about the physics when such a system is `driven' (time-dependent chaotic Hamiltonian). The main exceptions are the studies of the kicked rotator and related systems [75]. However, the rotator (with no kicks) is a $d=1$ integrable system, whereas we are interested in chaotic ($d\ge2$) cavities.

Driven cavities have been of special interest since the 1970s in studies of the so-called `one-body' dissipation rate in vibrating nuclei [29,120,118,106,107]. A renewed interest in this problem is anticipated in the field of mesoscopic physics. Quantum dots [20,65] can be regarded as small 2D cavities whose shape is controlled by electrical gates. Quasiparticle motion inside the dot can have long coherence (dephasing) times, and enable the semiclassical regime to be approached (many wavelengths across the system).

In Chapter 2 I give tutorial review of the theory of dissipation in general driven ergodic systems, which is quite a young field. The Hamiltonian is controlled by a single parameter $x$, whose time-dependence will be $x(t)=A\sin(\omega t)$ where $A$ is the amplitude and $\omega$ is the driving frequency. In both the classical (Section 2.1) and quantum-mechanical (Section 2.2) pictures, dissipation is a result of stochastic energy spreading. Once this spreading is established, the pictures can be unified [46]. Irreversible growth of energy (heating) is then a result of biased diffusion (a random walk) in energy. I will confine myself to a regime where linear response theory (LRT) is valid. In the quantum case this is known as the Kubo formalism, although the language of the energy spreading picture appears different (I connect the two pictures in Section 2.2.5). The heating rate is given by

$$\frac{d}{dt}\langle {\mathcal{H}} \rangle \ = \ \mu(\omega) \cdot \mbox{\small$\frac{1}{2}$}(A\omega)^2,$$ (1.1)

where the `friciton coefficient' $\mu(\omega)$ is related linearly<sup>1.1</sup>to $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$, the correlation spectrum of the time-dependent fluctuating quantity $-\partial {\mathcal{H}}/\partial x$. The latter (a generalized `force' on the parameter $x$) specifies the random `kicks' up or down in energy that particles receive. In the quantum case $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ is the `band profile' (average off-diagonal shape) of the matrix $\partial {\mathcal{H}}/\partial x$ in the energy representation. In Section 2.3 I demonstrate theoretically and numerically the semiclassical equivalence of the classical and quantum versions of $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$, and examine quantum effects beyond the band profile.

In Chapters 3 and 4 we specialize to a system of non-interacting particles inside a billiard whose walls are deformed by the parameter $x$. One can specify any deformation by a function $D({\mathbf s})$, where ${\mathbf s}$ specifies location of a wall element on the boundary (surface) of the billiard, and $D({\mathbf s})x$ is the resulting normal displacement of this wall element. We will be interested in low-frequency driving, meaning $\omega \ll 1/\tau_{{\mbox{\tiny bl}}}$ the mean collision frequency. In the $\omega\rightarrow0$ limit (uniform parameter velocity $\dot{x}$), the heating rate (1.1) is given by the dc friction $\mu(0)$ proportional to $\nu_{{\mbox{\tiny E}}}= \tilde{C}_{{\mbox{\tiny E}}}(\omega\rightarrow0)$. An assumption of uncorrelated collisions (the white noise approximation or WNA) gives an estimate for $\nu_{{\mbox{\tiny E}}}$, which in turn in $d=3$ leads to the well-known `wall formula' [29] (from the nuclear application),

$$\mu_{{\mbox{\tiny E}}} \ = \ \frac{N}{\mathsf{V}} m v_{{\mbox{\tiny E}}} \oint D({\mathbf s})^2 d{\mathbf s}.$$ (1.2)

This ($\omega$-independent) estimate of the friction applies to a microcanonical ensemble of $N$ particles with speed $v_{{\mbox{\tiny E}}}$ in a billiard volume ${\mathsf{V}}$.

We analyze $\mu(\omega)$ numerically in 2D billiard shapes (generalized Sinai, and Bunimovich stadium). We believe this is the first study of frequency-dependent heating rate in billiard systems. The chief discovery (Section 3.3) is a class of deformations whose heating rate vanishes in the $\omega\rightarrow0$ limit, like a power-law $\tilde{C}_{{\mbox{\tiny E}}}(\omega)\sim \omega^\gamma$. This holds even for billiards with strong chaos, and goes completely against the WNA prediction. The class of `special' deformations turns out to be just the class which preserves the billiard shape. For translations and dilations $\gamma = 4$ and for rotations $\gamma = 2$. This is to be compared to the case of a generic deformation, for which $\gamma = 0$ as the WNA prediction would predict. We give classical explanations for the power-laws (which rely on correlation on short timescales $\sim \tau_{{\mbox{\tiny bl}}}$). Importantly, the special class is manifested in the quantum band profile too. Thus the special nature of dilation, believed to be new in the literature, corresponds to a quasi-orthogonality relation between eigenstates on the boundary, which in turn will be the key to the powerful numerical method of Chapter 6. We also discuss (Section 3.4) non-generic shape-dependent effects (such as marginally-stable orbits) which may alter the power-laws given above.

The goal of Chapter 4 is as follows: given a general deformation $D{({\mathbf s})}$, in a given billiard shape, we seek an analytical estimate of $\nu_{{\mbox{\tiny E}}}$ (and hence $\mu(0)$). It is an exact result that $\nu_{{\mbox{\tiny E}}}$ is a quadratic form in the function space of $D{({\mathbf s})}$. The WNA fails to take into account that $\nu_{{\mbox{\tiny E}}}$ vanishes for special deformations (which form a linear subspace in $D{({\mathbf s})}$). We show that how it is possible to systematically subtract (project out) the `special' components of a general $D{({\mathbf s})}$. Applying the WNA only to the remaining (`normal') component gives an improved estimate of $\nu_{{\mbox{\tiny E}}}$. We analytically and numerically justify this projection procedure, and test the quality of the improved formula. The quality is limited by that of the WNA estimate of the `normal' component, which relies on the assumption of strongly chaotic motion. However, in the generalized Sinai billiard the improved formula is found to perform much better than the original WNA.

Our work replaces all ad hoc corrections which had been introduced [29] to account for the intuitive result that translations and rotations cause no heating at $\omega=0$. We thereby clear up some inconsistent habits in the nuclear community (Section 4.3.2). More significantly, the incorporation of the special nature of dilation is entirely new.

Note that the effect of interaction between the particles is not considered. If the mean free path for inter-particle collisions is large compared with the size of the cavity, then we expect that our analysis still applies. (However if the mean free path is much smaller, then we get into the hydrodynamic regime, where viscosity becomes the dominant dissipative effect).

This work was performed in collaboration with Doron Cohen. I also benefitted greatly from use of a classical billiard trajectory code written by Michael Haggerty.