Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Abstract

We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for N=1, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol’d-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

Quantum approach to the thermalization of the toppling pencil interacting with a finite bath

“…Our model system is a double well, which is bilinearly coupled to a finite number N of harmonic oscillators [11]. A quartic double-well is a bistable system with two symmetry related minima.…”

“…Although both, initial position as well as momentum expectations are zero, due to its finite width, there is a finite energy content in the wavepacket. This initial state is the motivation for the naming "toppling pencil"; a pencil balanced tip down on a flat surface, prone to fall over [11].…”

“…), an effect which is not intended by coupling to the environment [11]. This driving apart can, however, be compensated by including the so-called counter term proportional to the square of the system coordinate in the potential of the total Hamiltonian (system plus bath) to complete the squares with respect to the dependence on the oscillator coordinates.…”

“…Ever since the pioneering works of Fermi, Pasta, Ulam and Tsingou (FPUT) [1,2], the puzzle of energy exchange between subsystems, eventually leading to equilibration and thermalization in closed systems with a finite number of degrees of freedom has intrigued researchers in the fields of classical [3][4][5][6][7][8][9][10][11], as well as semiclassical [12,13] and quantum mechanics [3,[14][15][16][17][18][19][20]. In the quantum context, an important concept is the eigenstate thermalization hypothesis (ETH) [21,22].…”

“…In the present contribution, we will investigate the toppling pencil model, studied recently by Dittrich and Pena Martínez (DPM) [11]. In their classical mechanics study, they focused on a particle on top of the barrier in a quartic double well that is coupled to a small (from just one single up to the order of ten and higher) number of harmonic oscillators.…”