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Register a new userThermal bath Engineering for Swift Equilibration
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Authors
Marie Chupeau
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We propose a new protocol that ensures the fast equilibration of an overdamped harmonic oscillator by a joint time-engineering of the confinement strength and of the effective temperature of the thermal bath. We demonstrate experimentally the effectiveness of our protocol with an optically trapped Brownian particle and report an equilibrium recovering time reduced by about two orders of magnitude compared to the natural relaxation time. Our scheme paves the way towards reservoir engineering in nano-systems.
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Engineered swift equilibration (ESE) is a class of driving protocols that enforce an equilibrium distribution with respect to external control parameters at the beginning and end of rapid state transformations of open, classical non-equilibrium systems. ESE protocols have previously been derived and experimentally realized for Brownian particles in simple, one-dimensional, time-varying trapping potentials; one recent study considered ESE in two-dimensional Euclidean configuration space. Here we extend the ESE framework to generic, overdamped Brownian systems in arbitrary curved configuration space and illustrate our results with specific examples not amenable to previous techniques. Our approach may be used to impose the necessary dynamics to control the full temporal configurational distribution in a wide variety of experimentally realizable settings.
In the context of stochastic thermodynamics, a minimal model for non equilibrium steady states has been recently proposed: the Brownian Gyrator (BG). It describes the stochastic overdamped motion of a particle in a two dimensional harmonic potential, as in the classic Ornstein-Uhlenbeck process, but considering the simultaneous presence of two independent thermal baths. When the two baths have different temperatures, the steady BG exhibits a rotating current, a clear signature of non equilibrium dynamics. Here, we consider a time-dependent potential, and we apply a reverse-engineering approach to derive exactly the required protocol to switch from an initial steady state to a final steady state in a finite time $tau$. The protocol can be built by first choosing an arbitrary quasi-static counterpart - with few constraints - and then adding a finite-time contribution which only depends upon the chosen quasi-static form and which is of order $1/tau$. We also get a condition for transformations which - in finite time - conserve internal energy, useful for applications such as the design of microscopic thermal engines. Our study extends finite-time stochastic thermodynamics to transformations connecting non-equilibrium steady states.
We study the stationary state of a rough granular sphere immersed in a thermal bath composed of point particles. When the center of mass of the sphere is fixed the stationary angular velocity distribution is shown to be Gaussian with an effective temperature lower than that of the bath. For a freely moving rough sphere coupled to the thermostat via inelastic collisions we find a condition under which the joint distribution of the translational and rotational velocities is a product of Gaussian distributions with the same effective temperature. In this rather unexpected case we derive a formula for the stationary energy flow from the thermostat to the sphere in accordance with Fourier law.
We consider a particle in a one-dimensional box of length $L$ with a Maxwell bath at one end and a reflecting wall at the other end. Using a renewal approach, as well as directly solving the master equation, we show that the system exhibits a slow power law relaxation with a logarithmic correction towards the final equilibrium state. We extend the renewal approach to a class of confining potentials of the form $U(x) propto x^alpha$, $x>0$, where we find that the relaxation is $sim t^{-(alpha+2)/(alpha-2)}$ for $alpha >2$, with a logarithmic correction when $(alpha+2)/(alpha-2)$ is an integer. For $alpha <2$ the relaxation is exponential. Interestingly for $alpha=2$ (harmonic potential) the localised bath can not equilibrate the particle.
The time evolution of a finite fermion system towards statistical equilibrium is investigated using analytical solutions of a nonlinear partial differential equation that had been derived earlier from the Boltzmann collision term. The solutions of this fermionic diffusion equation are rederived in closed form, evaluated exactly for simplified initial conditions, and applied to hadron systems at low energies in the MeV-range, as well as to quark systems at relativistic energies in the TeV-range where antiparticle production is abundant. Conservation laws for particle number including created antiparticles, and for the energy are discussed.
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