Brownian oscillators driven by correlated noise in a moving trap
Lukas Glod, Gabriela Vasziova, Jana Tothova, Vladimir Lisy
备注：
Presented at the 17th International Conference Applied Physics of Condensed Matter (APCOM 2011), Novy Smokovec, High Tatras, Slovakia, June 22 - 24, 2011
Brownian oscillator, i.e. a micron-sized or smaller particle trapped in a thermally fluctuating environment is studied. The confining harmonic potential can move with a constant velocity. As distinct from the standard Langevin theory, the chaotic force driving the particle is correlated in time. The dynamics of the particle is described by the generalized Langevin equation with the inertial term, a coloured noise force, and a memory integral. We consider two kinds of the memory in the system. The first one corresponds to the exponentially correlated noise in a weakly viscoelastic fluid and in the second case the memory naturally arises within the Navier-Stokes hydrodynamics. Exact analytical solutions are obtained in both the cases using a simple and effective method not applied so far in this kind of problems.
We use a first-passage time approach to study the statistics of the trapping times induced by persistent motion of active particles colliding with flat boundaries. The angular first-passage time distribution and mean first-passage time is calculated exactly for active Brownian and run-and-tumble particles and the results are compared. Theoretical predictions are in excellent agreement with Langevin simulations. Our results shed further light onto how active particles with different dynamics may be equivalent in the bulk, yet behave differently near boundaries or obstacles.
Exact dynamics and dimensional effects in torque-driven trapped active Brownian particles
Indian Institute of Science Education and Research Mohali（印度科学教育研究学院莫哈利分校）；Tata Institute of Fundamental Research（塔塔基础研究所）；University of Vermont（佛蒙特大学）；Institute of Physics（物理研究所）；Homi Bhabha National Institute（霍米·巴巴国民研究所）
We find an exact series solution for the steady-state probability distribution of a harmonically trapped active Brownian particle in two dimensions, in the presence of translational diffusion. This series solution allows us to efficiently explore the behavior of the system in different parameter regimes. Identifying "active" and "passive" regimes, we predict a surprising re-entrant active-to-passive transition with increasing trap stiffness. Our numerical simulations validate this finding. We discuss various interesting limiting cases wherein closed form expressions for the distributions can b…
We consider an active Brownian particle in a $d$-dimensional harmonic trap, in the presence of translational diffusion. While the Fokker-Planck equation can not in general be solved to obtain a closed form solution of the joint distribution of positions and orientations, as we show, it can be utilized to evaluate the exact time dependence of all moments, using a Laplace transform approach. We present explicit calculation of several such moments at arbitrary times and their evolution to the steady state. In particular we compute the kurtosis of the displacement, a quantity which clearly shows t…
We study the Quantum Brownian motion of a charged particle moving in a harmonic potential in the presence of an uniform external magnetic field and linearly coupled to an Ohmic bath through momentum variables. We analyse the growth of the mean square displacement of the particle in the classical high temperature domain and in the quantum low temperature domain dominated by zero point fluctuations. We also analyse the Position Response Function and the long time tails of various correlation functions. We notice some distinctive features, different from the usual case of a charged quantum Browni…
We provide insights into energetics of a Brownian oscillator in contact with a heat bath and driven by an external unbiased time-periodic force that takes the system out of thermodynamic equilibrium. Solving the corresponding Langevin equation, we compute average kinetic and potential energies in the long-time stationary state. We also derive the energy balance equation and study the energy flow in the system. In particular, we identify the energy delivered by the external force, the energy dissipated by a thermal bath and the energy provided by thermal equilibrium fluctuations. Next, we illus…
We apply the concept of a frequency-dependent effective temperature based on the fluctuation-dissipation ratio to a driven Brownian particle in a nonequilibrium steady state. Using this system as a thermostat for a weakly coupled harmonic oscillator, the oscillator thermalizes according to a canonical distribution at the respective effective temperature across the entire frequency spectrum. By turning the oscillator from a passive "thermometer" into a heat engine, we realize the cyclic extraction of work from a single thermal reservoir, which is feasible only due to its nonequilibrium nature.
We study analytically the probability distribution of the heat released by an ensemble of harmonic oscillators to the thermal bath, in the nonequilibrium relaxation process following a temperature quench. We focus on the asymmetry properties of the heat distribution in the nonstationary dynamics, in order to study the forms taken by the Fluctuation Theorem as the number of degrees of freedom is varied. After analysing in great detail the cases of one and two oscillators, we consider the limit of a large number of oscillators, where the behavior of fluctuations is enriched by a condensation tra…
Recently, the quantum counterpart of energy equipartition theorem has drawn considerable attention. Motivated by this, we formulate and investigate an analogous statement for the free energy of a quantum oscillator linearly coupled to a passive heat bath consisting of an infinite number of independent harmonic oscillators. We explicitly demonstrate that the free energy of the Brownian oscillator can be expressed in the form $F(T) = \langle f(ω,T) \rangle $ where $f(ω,T)$ is the free energy of an individual bath oscillator. The overall averaging process involves two distinct averages: the first…
We study the dynamics of the center of mass of a Brownian particle levitated in a Paul trap. We focus on the overdamped regime in the context of levitodynamics, comparing theory with our numerical simulations and experimental data from a nanoparticle in a Paul trap. We provide an exact analytical solution to the stochastic equation of motion, expressions for the standard deviation of the motion, and thermalization times by using the WKB method under two different limits. Finally, we prove the power spectral density of the motion can be approximated by that of an Ornstein-Uhlenbeck process and …
We consider a model of a particle trapped in a harmonic optical trap but with the addition of a non-conservative radiation induced force. This model is known to correctly describe experimentally observed trapped particle statistics for a wide range of physical parameters such as temperature and pressure. We theoretically analyse the effect of non-conservative force on the underlying steady state distribution as well as the power spectrum for the particle position. We compute perturbatively the probability distribution of the resulting non-equilibrium steady states for all dynamical regimes, un…
We propose a reduction scheme for a system constituted by two coupled harmonically-bound Brownian oscillators. We reduce the description by constructing a lower dimensional model which inherits some of the basic features of the original dynamics and is written in terms of suitable transport coefficients. The proposed procedure is twofold: while the deterministic component of the dynamics is obtained by a direct application of the invariant manifold method, the diffusion terms are determined via the Fluctuation-Dissipation Theorem. We highlight the behavior of the coefficients up to a critical …
We investigate the influence of external forces on the collective dynamics of interacting active Brownian particles in two as well as three spatial dimensions. Via explicit coarse graining, we derive predictive models that are applicable for space- and time-dependent external force fields. We study these models for the cases of gravity and harmonic traps. In particular, we derive a generalized barometric formula for interacting active Brownian particles under gravity that is valid for low to high concentrations and activities of the particles. Furthermore, we show that one can use an external …
The overdamped Brownian dynamics of a harmonic oscillator is a paradigmatic system in non-equilibrium statistical mechanics, which reliably models relevant stochastic systems such as colloidal particles submitted to optical confinement. In this work, optimal thermal protocols are tailored to minimise the connection time between equilibrium states of overdamped $d$-dimensional oscillators. Application of control theory reveals that these optimal protocols are of bang-bang type, that is, the temperature of the bath has to take alternatively the minimum and maximum values allowed. Minimum connect…
A Brownian harmonic oscillator, which dissipates energy either by friction or via emission of electromagnetic radiation, is considered. This Brownian emitter is driven by the surrounding thermo-quantum fluctuations, which are theoretically described by the fluctuation-dissipation theorem. It is shown how the AbrahamLorentz force leads to dependence of the half-width on the peak frequency of the oscillator amplitude spectral density. It is also found that for the case of a charged particle, moving in vacuum at zero temperature, its root-mean-square velocity fluctuation is a universal constant, …
The cost of speed: Time-optimal thermal control of trapped Brownian particles
Universidad de Granada（格拉纳达大学）；Simon Fraser University（西蒙菲莎大学）；Facultad de Educación, Economía y Tecnología de Ceuta（休达教育、经济与技术学院）；Instituto Carlos I de Física Teórica y Computacional（卡洛斯一世理论与计算物理研究所）；Universidad de Sevilla（塞维利亚大学）
We characterize throughout the spectral range of an optical trap the nature of the noise at play and the ergodic properties of the corresponding Brownian motion of an overdamped trapped single microsphere, comparing experimental, analytical and simulated data. We carefully analyze noise and ergodic properties $(i)$ using the Allan variance for characterizing the noise and $(ii)$ exploiting a test of ergodicity tailored for experiments done over finite times. We derive these two observables in the low-frequency Ornstein-Uhlenbeck trapped-diffusion regime and study analytically their evolution t…
In this paper we study the nonequilibrium evolution of a quantum Brownian oscillator, modeling the internal degree of freedom of a harmonic atom or an Unruh-DeWitt detector, coupled to a nonequilibrium, nonstationary quantum field and inquire whether a fluctuation-dissipation relation can exist after/if it approaches equilibration. This is a nontrivial issue since a squeezed bath field cannot reach equilibration and yet, as this work shows, the system oscillator indeed can, which is a necessary condition for FDRs. We discuss three different settings: A) The bath field essentially remains in a …
We study the quantum counterpart of the theorem on energy equipartition for classical systems. We consider a free quantum Brownian particle modelled in terms of the Caldeira-Leggett framework: a system plus thermostat consisting of an infinite number of harmonic oscillators. By virtue of the theorem on the averaged kinetic energy $E_k$ of the quantum particle, it is expressed as $E_k = \langle \mathcal E_k \rangle$, where $\mathcal E_k$ is thermal kinetic energy of the thermostat per one degree of freedom and $\langle ...\rangle$ denotes averaging over frequencies $ω$ of thermostat oscillators…
We study quantum non-Markovian dynamics of the Caldeira-Leggett model, a prototypical model for quantum Brownian motion describing a harmonic oscillator linearly coupled to a reservoir of harmonic oscillators. Employing the exact analytical solution of this model one can determine the size of memory effects for arbitrary couplings, temperatures and frequency cutoffs. Here, quantum non-Markovianity is defined in terms of the flow of information between the open system and its environment, which is quantified through the Bures metric as distance measure for quantum states. This approach allows u…
