Integrability vs. statistical behavior in the FPU problem: an endless story

Slides

The talk is devoted to the celebratd FPU problem, namely the interplay between dynamics and statistics in a chain of weakly nonlinear oscillators. The perspective we shall follow is that FPU should be regarded as a perturbation of the (completely integrable) Toda model. Normal statistical behavior requires rather large time scales, scaling as 𝜀^(-5/2) where 𝜀=E/N is the energy per particle. On substantially shorter times, FPU is practically indistinguishable from Toda: nevertheless its statistical behavior is not well understood, since Toda itself, although integrable, in the thermodynamic limit is not understood. A crucial question for statistical mechanics is the relation between Toda actions and (linear) normal modes, which looks quite puzzling for large N.

Path large deviations for kinetic theories: beyond the Boltzmann, the Landau, the Balescu—Lenard—Guernsey, and the weak turbulence kinetic equations

Joint works with Gregory Eyink, Ouassim Feliachi and Jules Guioth

Slides

In many physical systems one seeks to describe effectively mesoscopic or macroscopic variables. Kinetic theories and kinetic equations are examples where the average mesoscopic dynamics is obtained through very clear theoretical procedures and can possibly lead to mathematical proofs, for instance the Boltzmann equation for dilute gases, the Landau or the Balescu—Guernsey—Lenard equations in plasma physics, or the wave kinetic equation for weak turbulence theory. A few works go beyond the average evolution and describe, for instance, Gaussian fluctuations. However, for many physical systems, rare events can be of importance, and Gaussian fluctuations are not relevant. This is the case for instance if one wants to understand the irreversibility paradox associated to the kinetic equations, or to understand the dynamics that leads to rare events with big impact.

The aim of this presentation is to describe recent results where we derived explicitly the functional that describes the path large deviations for the empirical measure of dilute gases, plasma, systems of particles with long range interactions, and waves with weak interactions. The associated kinetic equations (the average evolution) are then either the Boltzmann, the Landau, the Balescu--Lenard—Guernsey, or the weak turbulence kinetic equations. After making the classic assumptions in theoretical physics textbooks for deriving the kinetic equation, our derivation of the large deviation functional is exact.

These path large deviation principles give a very nice and transparent new interpretation of the classical irreversibility paradox. This new explanation is fully compatible with the classical one, but it gives a deeper insight.

References

For the large deviations associated to the Boltzmann equation (dilute gazes), and a general introduction (published in J. Stat. Phys. in 2020): F. Bouchet, 2020, Is the Boltzmann equation reversible? A large deviation perspective on the irreversibility paradox and the Boltzmann equation, Journal of Statistical Physics, 181, 515–550, https://link.springer.com/article/10.1007/s10955-020-02588-y, https://arxiv.org/abs/2002.10398

For the large deviations associated to the Landau equation (plasma below the Debye length, accepted for publication in J. Stat. Phys. in March 2021): O. Feliachi and F. Bouchet, 2021, Dynamical large deviations for plasma below the Debye length and the Landau equation, Journal of Statistical Physics, 183, 42, https://link.springer.com/article/10.1007/s10955-021-02771-9, https://arxiv.org/abs/2101.04455.

For the large deviations associated with the Balescu—Guernsey--Lenard equation (plasma and systems with long range interactions): O. Feliachi and F. Bouchet, 2022, Dynamical Large Deviations for Homogeneous Systems with Long Range Interactions and the Balescu–Guernsey–Lenard Equation, Journal of Statistical Physics 186, 22, and https://arxiv.org/abs/2105.05644

For the large deviations associated with the weak turbulence kinetic equation that describe weakly interacting waves: J. Guioth, F. Bouchet and G. Eyink, 2022, Path large deviations for the kinetic theory of weak turbulence, https://arxiv.org/abs/2203.11737

Information and thermodynamics: optimizing information processing using underdamped systems

Slides

The energetic cost to perform basic operations on a 1-bit logic gate is bounded by a fundamental theoretical limit: the Landauer principle states that at least k_B T ln 2 of energy is required to erase a 1-bit memory ([RESET] operation), with k_B T the thermal energy of the system. Practical erasures implementations require an overhead to the Landauer’s bound, observed to scale as k_B T  B/tau, with tau the protocol duration and B close to the system position response time. Most experiments use over-damped systems, for which minimizing the overhead means minimizing the dissipation. Underdamped systems thus sounds appealing to reduce this energetic cost. That is why we use an underdamped system to build an optimized logic-gate in terms of processing speed and energetic cost. The one-bit memory consists in an underdamped micro-mechanical oscillator confined in a double-well potential created by a feedback loop [3].

The resulting virtual potential can be shaped within the few kBT range with high precision and can follow elaborate procedures. We measure, within the stochastic thermodynamic framework, the work and the heat of any operation.

We demonstrate that, using this underdamped system, the Landauer bound is reached with a 1% uncertainty, with protocols as short as 100 ms [2], several of magnitude faster than the state-of-the-art using over-damped memories. Nevertheless, we show experimentally and theoretically that in the underdamped regime, fast erasures induce a heating of the memory: the work influx is not instantaneously compensated by the inefficient heat transfert to the thermostat. This temperature rise results in a kinetic and potential energy contribution superseding the viscous dissipation term. Our model covering all damping regimes allows new optimisation strategies in information processing, based on the thorough understanding of the energy exchanges [1]. We illustrate such perspectives with applications to several logical operations of the 1-bit logic gate: repeated [RESET] operations and [NOT] operations. Besides, we elaborate optimal procedures to lower even further the information processing cost. We finally pave the way to shortcut procedures to create faster protocols designed for a continuous use of the memory.

References

[1] Dago, S., Bellon, L.: Dynamics of information erasure and extension of landauer’s bound to fast processes. Phys. Rev. Lett. 128, 070604 (Feb 2022), https://doi.org/10.1103/PhysRevLett.128.070604

[2] Dago, S., Pereda, J., Barros, N., Ciliberto, S., Bellon, L.: Information and thermodynamics: Fast and precise approach to landauer’s bound in an underdamped micromechanical oscillator. Phys. Rev. Lett. 126, 170601 (2021), https://doi.org/10.1103/PhysRevLett. 126.170601

[3] Dago, S., Pereda, J., Ciliberto, S., Bellon, L.: Virtual double well potential for an underdamped oscillator created by a feedback loop (2022), arXiv.2201.09870, to be published in JSTAT.

Frozen dynamics in the discrete nonlinear Schroedinger (DNLS) equation

Slides

Tall breathers turn out to be nearly decoupled from the surrounding background, even when the latter operates in a chaotic regime of nonnegligible amplitude.

This phenomenon, which almost blocks relaxation to equilibrium, is discussed first with reference to the full DNLS model and then studying a simplified master-slave setup, where the breather is affected by the background dynamics, but not vice-versa.

This way I can attribute the exponentially small effective coupling to the existence of an adiabatic invariant, which is determined via two different perturbative techniques.

An unexpected similarity with Levy processes is uncovered, although the sporadic mechanisms leading to the breaking of the adiabatic invariant still need to be accurately identified.