We consider a quantum linear oscillator coupled to a bath in equilibrium at an arbitrary temperature and then exposed to an external field arbitrary in form and strength. We then derive the reduced density operator in closed form of the coupled oscillator in a non-equilibrium state at an arbitrary time.
We investigate the Brownian motion of a charged particle in a magnetic field. We study this in the high temperature classical and low temperature quantum domains. In both domains, we observe a transition of the mean square displacement from a monotonic behaviour to a damped oscillatory behaviour as one increases the strength of the magnetic field. When the strength of the magnetic field is negligible, the mean square displacement grows linearly with time in the classical domain and logarithmically with time in the quantum domain. We notice that these features of the mean square displacement ar…
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…
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, …
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 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 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…
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…
This paper reviews the formulation of the Feynman-Vernon model of linear dissipative systems for a standard Brownian particle moving in an external potential $V(x,t)$ and introduces the formulation of a generalized oscillator model of a Brownian particle coupled to a thermal environment moving with a given velocity $v_{env}$. Diffusion processes in a moving environment are of interest e.g. in the study of the motion of vortices in superfluids. The starting point of the paper is the formulation of the oscillator model that takes into account space and time invariance of a thermal environment [M…
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…
This article discusses the numerical result predicted by the quantum Langevin equation of the generalized diffusion function of a Brownian particle immersed in an Ohmic quantum bath of harmonic oscillators. The time dependence of the standard deviation of the reduced Wiener function of the system, obtained by integrating the whole function in the momentum space, is determined as well. The complexity of the equations leads to resort to a much simpler calculation based in the position correlation function. They are done for the three possible regimes of the system, namely, periodic, aperiodic an…
The problem of the initial conditions for the oscillator model of quantum dissipative systems is studied. It is argued that, even in the classical case, the hypothesis that the environment is in thermal equilibrium implies a statistical correlation between environment oscillators and central system. A simple form of initial conditions for the quantum problem, taking into account such a correlation in analogy with the classical ones, is derived on the base of symmetry considerations. The same symmetries also determine unambiguously the form of the Lagrangian. As a check of the new form of corre…
The Brownian motion of a single particle is a paradigmatic model of the nonequilibrium dynamics of dissipative systems. In the system-plus-reservoir approach, one can derive the particle's equations of motion from the reversible dynamics of the system coupled to a bath of oscillators representing its thermal environment. However, extending the system-plus-reservoir approach to multiple particles in a collective environment is not straightforward, and conflicting models have been proposed to that end. Here, we set out to reconcile some aspects of the nonlinear and the bilinear models of two Bro…
The motion of a free quantum particle in a thermal environment is usually described by the quantum Langevin equation, where the effect of the bath is encoded through a dissipative and a noise term, related to each other via the fluctuation dissipation theorem. The quantum Langevin equation can be derived starting from a microscopic model of the thermal bath as an infinite collection of harmonic oscillators prepared in an initial equilibrium state. The spectral properties of the bath oscillators and their coupling to the particle determine the specific form of the dissipation and noise. Here we…
We consider a chain of weakly harmonic coupled oscillators perturbed by a conservative noise. We show that by tuning accordingly the coupling constant energy can diffuse like a Brownian motion or superdiffuse like a maximally 3/2-stable asymmetric L{é}vy process. For a critical value of the coupling, the energy diffusion is described by a family of L{é}vy processes which interpolates between these two processes.
We propose a simulation method for Brownian dynamics of hard rods in one dimension for arbitrary continuous external force fields. It is an event-driven procedure based on the fragmentation and mergers of clusters formed by particles in contact. It allows one to treat particle interactions in addition to the hard-sphere exclusion as long as the corresponding interaction forces are continuous functions of the particle coordinates. We furthermore develop a treatment of sticky hard spheres as described by Baxter's contact interaction potential.
In this work, we study a system of passive Brownian (non-self-propelled) particles in two dimensions, interacting only through a social-like force (velocity alignment in this case) that resembles Kuramoto's coupling among phase oscillators. We show that the kinematical stationary states of the system go from a phase in thermal equilibrium with no net flux of particles, to far-from-equilibrium phases exhibiting collective motion by increasing the coupling among particles. The mechanism that leads to the instability of the equilibrium phase relies on the competition between two time scales, name…
We study parametrically driven quantum oscillators and show that, even for weak coupling between the oscillators, they can exhibit various many-body states with broken time-translation symmetry. In the quantum-coherent regime, the symmetry breaking occurs via a nonequilibrium quantum phase transition. For dissipative oscillators, the main effect of the weak coupling is to make the switching rate of an oscillator between its period-2 states dependent on the states of other oscillators. This allows mapping the oscillators onto a system of coupled spins. Away from the bifurcation parameter values…
Physical scenarios that require a relativistic treatment are ubiquitous in nature, ranging from cosmological objects to charge carriers in Dirac materials. Interestingly all of these situations have in common that the systems typically operate very far from thermal equilibrium. Therefore, if and how the framework of stochastic thermodynamics applies at relativistic energies is a salient question. In the present work we generalize the notions of stochastic heat and work for the relativistic Langevin equation and derive the fluctuation theorems without and with feedback. For processes with feedb…
