Abstract :
[en] For classical systems, statistical description is usually required when the dimension of the system phase space is important. This description is used when considering many body problems such as the thermodynamical description of fluids, granular media or the diffusion of Brownian particles in the presence of a thermal reservoir. Can this limit be reached at the single particle level? We address this question by considering a single particle coupled to its own past, via a wave-mediated size-controllable memory, which stores the past trajectory of the particle. Such an object has been turn into reality by Couder et al [1,2] by using liquid bouncing droplets that interact with the waves they generate by impacting a liquid surface. In those experiments, the memory is made by stationary waves with a tunable damping time. The wave memory triggers various regime from individual self-propelled particle motion to structured chaos [3] or self-organization processes between a particle and its waves [4].
In the limit of a large amount of “souvenirs” stored in the wave field, we will show that the dynamics can be well described by the tools of non-equilibrium statistical mechanics. Both the dynamics of the particle and the wave-mediated memory are investigated. We show that, from the particle-point of view, the memory acts as a random force. The particle dynamics is adequately described by a Langevin equation for self-propelled systems. The corresponding Fokker-Planck equation shows that the memory can be seen as a thermal reservoir whose temperature is tuned via the amount of “souvenirs” in the dynamics. The features of this unusual wave-like thermal reservoir are investigated from an energetic and entropic perspective. We show that the global wave field is on average weaker that one would expect from superposition of random wave fields. This depletion originates from destructive interferences that the particle generates along its trajectory, corresponding to a kind-of minimization principle.
[1] Y. Couder, S. Protière, E. Fort & A. Boudaoud, “Walking and orbiting droplets”, Nature, 437, 208 (2005).
[2] A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi & Y. Couder, “Information stored in Faraday waves: the origin of a path memory”, J. Fluid Mech., 674, 433- 463 (2011).
[3] S. Perrard, M. Labousse, E. Fort & Y. Couder, “Chaos driven by interfering memory”, Phys. Rev. Lett, 113, 104101 (2014).
[4] S. Perrard, M. Labousse, M. Miskin, E. Fort & Y. Couder, “Self-organization into quantized eigenstates of a classical wave-driven particle”, Nat. Comm., 5, 3219 (2014).
