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 , 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.
Highlights
For not few of its pioneers, the theory of deterministic chaos came with the hope for a deterministic description of an important part of physical phenomena, which till had been relegated to the realm of randomness
By contrast to the theory of deterministic chaos, we here study the origin of randomness in discrete time series, such as those generated by games of luck, in a deterministic dynamics
We substantiate our approach by constructing a detailed model of a bistable system interacting with a many-body environment, a quartic double well coupled to a bath comprising only a finite number of harmonic oscillators, which evolves in time as a closed Hamiltonian system, conserving information and energy
Summary
For not few of its pioneers, the theory of deterministic chaos came with the hope for a deterministic description of an important part of physical phenomena, which till had been relegated to the realm of randomness. Under certain conditions on the frequency dependence of coupling and spectral density of the oscillators, the bath becomes an irreversible sink of information and energy, inducing relaxation to a stationary state and dissipation in the central system [7] In this point, we adopt a more recent development in statistical mechanics, in that we keep the number of oscillators large, N 1, but finite [8,9,10,11,12], so that the dynamics of the total system can be treated in the framework of the time-reversal invariant Hamiltonian mechanics of closed systems.
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