Steady-state Correlation Function Research Articles

Steady state correlations and induced trapping of an inertial AOUP particle

We study the dynamics of an inertial active Ornstein–Uhnlenbeck particle self-propelling in a confined harmonic well. The transport behavior of the particle is investigated by analyzing the particle trajectories, steady state correlations and mean square displacement (MSD). The steady state correlation functions for the position as well as velocity are exactly calculated using different methods. We explore how the inertia affects the dynamical behavior, when the particle is confined in a harmonic trap as well as when it is set free. From the exact calculation of MSD, it is observed that the initial time regimes are ballistic for both harmonically confined particle and free particle, whereas the long time regimes are diffusive for a free particle and nondiffusive for a harmonically confined particle. One of our interesting observations is that the harmonically confined particle gets more and more confined with increase in the self-propulsion time or activity time of the dynamics and finally it gets trapped for very large value of the self-propulsion time. For a free particle, the velocity correlation decays by the complex interplay between the inertial time scale and the self-propulsion time scale of the dynamics. Moreover, decorrelation in velocity happens only when these two time scales are of equal order.

A statistical quasi-particles thermofield theory with Gaussian environments: System-bath entanglement theorem for nonequilibrium correlation functions.

For open quantum systems, the Gaussian environmental dissipative effect can be represented by statistical quasi-particles, namely, dissipatons. We exploit this fact to establish the dissipaton thermofield theory. The resulting generalized Langevin dynamics of absorptive and emissive thermofield operators are effectively noise-resolved. The system-bath entanglement theorem is then readily followed between an important class of nonequilibrium steady-state correlation functions. All these relations are validated numerically. A simple corollary is the transport current expression, which exactly recovers the result obtained from the nonequilibrium Green's function formalism.

Nonequilibrium Green-Kubo relations for hydrodynamic transport from an equilibrium-like fluctuation-response equality

Near equilibrium, Green-Kubo relations provide microscopic expressions for macroscopic transport coefficients in terms of equilibrium correlation functions. At their core, they are based on the intimate relationship between response and fluctuations as embodied by the equilibrium fluctuation-dissipation theorem, a connection generically broken far from equilibrium. In this work, we identify a class of perturbations whose response around far-from-equilibrium steady states is linked to steady-state correlation functions via an equilibrium-like fluctuation-response equality. We then utilize this prediction to substantiate linearized hydrodynamic transport equations that describe how spatial inhomogeneities in macroscopic nonequilibrium systems relax. As a consequence, we derive nonequilibrium Green-Kubo relations for the transport coefficients of two types of hydrodynamic variables: local conserved densities and broken-symmetry modes. A byproduct of this work is to provide a theoretical foundation for the validity of Onsager's regression hypothesis around nonequilibrium steady states. Our predictions are analytically and numerically corroborated for two model systems: density diffusion in a fluid of soft, spherical active Brownian particles and phase diffusion in the noisy Kuramoto model on a square lattice.

Neutrino Oscillations in the Presence of Matter and Continuous Non-Selective Measurement

We investigate three-flavor neutrino oscillation affected by an environment mimicking a continuous non-selective measurement. We show that such a coupling that is given by a measured observable affects probability of inter-flavor neutrino transition and a steady-state correlation function of the neutrino’s flavor. We juxtapose and compare our predictions influenced by matter’s scattering and CP-violation.

Linear response theory for quantum Gaussian processes

Fluctuation dissipation theorems (FDTs) connect the linear response of a physical system to a perturbation to the steady-state correlation functions. Until now, most of these theorems have been derived for finite-dimensional systems. However, many relevant physical processes are described by systems of infinite dimension in the Gaussian regime. In this work, we find a linear response theory for quantum Gaussian systems subject to time dependent Gaussian channels. In particular, we establish a FDT for the covariance matrix that connects its linear response at any time to the steady state two-time correlations. The theorem covers non-equilibrium scenarios as it does not require the steady state to be at thermal equilibrium. We further show how our results simplify the study of Gaussian systems subject to a time dependent Lindbladian master equation. Finally, we illustrate the usage of our new scheme through some examples. Due to broad generality of the Gaussian formalism, we expect our results to find an application in many physical platforms, such as opto-mechanical systems in the presence of external noise or driven quantum heat devices.

Mode-coupling theory for the steady-state dynamics of active Brownian particles.

We present a theory for the steady-state dynamics of a two-dimensional system of spherically symmetric active Brownian particles. The derivation of the theory consists of two steps. First, we integrate out the self-propulsions and obtain a many-particle evolution equation for the probability distribution of the particles' positions. Second, we use the projection operator technique and a mode-coupling-like factorization approximation to derive an equation of motion for the density correlation function. The nonequilibrium character of the active system manifests itself through the presence of a steady-state correlation function that quantifies spatial correlations of microscopic steady-state currents of the particles. This function determines the dependence of the short-time dynamics on the activity. It also enters into the expression for the memory matrix and thus influences the long-time glassy dynamics.

Coexisting spin and Rabi oscillations at intermediate time regimes in electron transport through a photon cavity.

In this work, we theoretically model the time-dependent transport through an asymmetric double quantum dot etched in a two-dimensional wire embedded in a far-infrared (FIR) photon cavity. For the transient and the intermediate time regimes, the current and the average photon number are calculated by solving a Markovian master equation in the dressed-states picture, with the Coulomb interaction also taken into account. We predict that in the presence of a transverse magnetic field the interdot Rabi oscillations appearing in the intermediate and transient regime coexist with slower non-equilibrium fluctuations in the occupation of states for opposite spin orientation. The interdot Rabi oscillation induces charge oscillations across the system and a phase difference between the transient source and drain currents. We point out a difference between the steady-state correlation functions in the Coulomb blocking and the photon-assisted transport regimes.

Two coupled nonlinear cavities in a driven-dissipative environment

We investigate two coupled nonlinear cavities that are coherently driven in a dissipative environment. We perform semiclassical, numerical and analytical quantum studies of this dimer model when both cavities are symmetrically driven. In the semiclassical analysis, we find steady-state solutions with different photon occupations in two cavities. Such states can be considered analogs of the closed system double well symmetry breaking states. We analyze the occurrence and properties of these localized states in the system parameter space and examine how the symmetry breaking states, in form of a bistable pair, are associated to the single cavity bistable behavior. In a full quantum calculation of the master equation dynamics that includes quantum fluctuations, the symmetry breaking states and bistability disappear due to the quantum fluctuations. In quantum trajectory picture, we observe enhanced quantum jumps and switching which indicate the presence of the underlying semiclassical symmetry breaking states. Finally, we present a set of analytical solutions for the steady state correlation functions using the complex P-representation and discuss its regime of validity.

Spatial correlations, additivity, and fluctuations in conserved-mass transport processes.

We exactly calculate two-point spatial correlation functions in steady state in a broad class of conserved-mass transport processes, which are governed by chipping, diffusion, and coalescence of masses. We find that the spatial correlations are in general short-ranged and, consequently, on a large scale, these transport processes possess a remarkable thermodynamic structure in the steady state. That is, the processes have an equilibrium-like additivity property and, consequently, a fluctuation-response relation, which help us to obtain subsystem mass distributions in the limit of subsystem size large.

Theory for the dynamics of dense systems of athermal self-propelled particles.

We present a derivation of a recently proposed theory for the time dependence of density fluctuations in stationary states of strongly interacting, athermal, self-propelled particles. The derivation consists of two steps. First, we start from the equation of motion for the joint distribution of particles' positions and self-propulsions and we integrate out the self-propulsions. In this way we derive an approximate, many-particle evolution equation for the probability distribution of the particles' positions. Second, we use this evolution equation to describe the time dependence of steady-state density correlations. We derive a memory function representation of the density correlation function and then we use a factorization approximation to obtain an approximate expression for the memory function. In the final equation of motion for the density correlation function the nonequilibrium character of the active system manifests itself through the presence of a new steady-state correlation function that quantifies spatial correlations of the velocities of the particles. This correlation function enters into the frequency term, and thus it describes the dependence of the short-time dynamics on the properties of the self-propulsions. More importantly, the correlation function of particles' velocities enters into the vertex of the memory function and through the vertex it modifies the long-time glassy dynamics.

On the Steady State Correlation Functions of Open Interacting Systems

We address the existence of steady state Green-Keldysh correlation functions of interacting fermions in mesoscopic systems for both the partitioning and partition-free scenarios. Under some spectral assumptions on the non-interacting model and for sufficiently small interaction strength, we show that the system evolves to a NESS which does not depend on the profile of the time-dependent coupling strength/bias. For the partitioned setting we also show that the steady state is independent of the initial state of the inner sample. Closed formulae for the NESS two-point correlation functions (Green-Keldysh functions), in the form of a convergent expansion, are derived. In the partitioning approach, we show that the 0th order term in the interaction strength of the charge current leads to the Landauer-Buettiker formula, while the 1st order correction contains the mean-field (Hartree-Fock) results.

The fluctuation-dissipation theorem for stochastic kinetics—Implications on genetic regulations

The Fluctuation-Dissipation theorem (FDT) connects the "memory" in the fluctuation in equilibrium to the response of a system after a perturbation, which has been a fundamental ground in many branches of physics. When viewing a cell as a stochastic biochemical system, the cell's response under a perturbation is related to its intrinsic steady-state correlation functions via the FDT, a theorem we derived and present in this work. FDT allows us to use the noise to derive dynamic response and infer dynamic properties in the system. We tested FDT's validity with gene regulation models and found that it is limited to the linear response. For an indirect regulation pathway where unknown components may exist, FDT still works within the linear response region. Thus, FDT may be used for systems with partial knowledge, and it is potentially possible to identify the existence of unobserved components. With FDT, the dynamic response can be composed of steady-state measurements without the complete detailed knowledge for the regulation or kinetics. The response function derived can give important insights into the dynamics and time scales of the system.

The relaxation time of processes in a FitzHugh–Nagumo neural system with time delay

In this paper, we study the relaxation time (RT) of the steady-state correlation function in a FitzHugh–Nagumo neural system under the presence of multiplicative and additive white noises and time delay. The noise correlation parameter λ can produce a critical behavior in the RT as functions of the multiplicative noise intensity D, the additive noise intensity Q and the time delay τ. That is, the RT decreases as the noise intensities D and Q increase, and increases as the time delay τ increases below the critical value of λ. However, above the critical value, the RT first increases, reaches a maximum, and then decreases as D, Q and τ increase, i.e. a noise intensity D or Q and a time delay τ exist, at which the time scales of the relaxation process are at their largest. In addition, the additive noise intensity Q can also produce a critical behavior in the RT as a function of λ. The noise correlation parameter λ first increases the RT of processes, then decreases it below the critical value of Q. Above the critical value, λ increases it.

Approximative analytical method for some Langevin dynamical systems

We develop an approach which leads to an analytically treatable integral representation for the steady state correlation functions of some Langevin dynamical systems, which are related to the microscopic Hamiltonian description of the heat mechanism. For the simpler case of harmonic interactions (where rigorous results exist and a comparison with our method is possible), we perform the computations for the thermal conductivity (given by two-point functions) and show that our approximative scheme works perfectly well.

Dynamics of fluctuation-dominated phase ordering: Hard-core passive sliders on a fluctuating surface

We study the dynamics of a system of hard-core particles sliding downwards on a one-dimensional fluctuating interface, which in a special case can be mapped to the problem of a passive scalar advected by a Burgers fluid. Driven by the surface fluctuations, the particles show a tendency to cluster, but the hard-core interaction prevents collapse. We use numerical simulations to measure the autocorrelation function in steady state and in the aging regime, and space-time correlation functions in steady state. We have also calculated these quantities analytically in a related surface model. The steady-state autocorrelation is a scaling function of t/L(z), where L is the system size and z is the dynamic exponent. Starting from a finite intercept, the scaling function decays with a cusp, in the small argument limit. The finite value of the intercept indicates the existence of long-range order in the system. The space-time correlation, which is a function of r/L and t/L(z), is nonmonotonic in t for fixed r. The aging autocorrelation is a scaling function of t(1) and t(2) where t(1) is the waiting time and t(2) is the time difference. This scaling function decays as a power law for t(2)>>t(1); for t(1)>>t(2), it decays with a cusp as in steady state. To reconcile the occurrence of strong fluctuations in the steady state with the fact of an ordered state, we measured the distribution function of the length of the largest cluster. This shows that fluctuations never destroy ordering, but rather the system meanders from one ordered configuration to another on a relatively rapid time scale.

Resonance fluorescence in a band-gap material: Direct numerical simulation of non-Markovian evolution

A numerical method of calculating the non-Markovian evolution of a driven atom radiating into a structured continuum is developed. The formal solution for the atomic reduced density matrix is written as a Markovian algorithm by introducing a set of additional, virtual density matrices which follow, to the level of approximation of the algorithm, all the possible trajectories of the photons in the electromagnetic field. The technique is perturbative in the sense that more virtual density matrices are required as the product of the effective memory time and the effective coupling strength become larger. The number of density matrices required is given by $3^{M}$ where $M$ is the number of timesteps per memory time. The technique is applied to the problem of a driven two-level atom radiating close to a photonic band gap and the steady-state correlation function of the atom is calculated.

The far-infrared molecular dynamics of circular dichroism and optical rotatory dispersion

The nature of the far-infrared molecular dynamics of circular dichroism (CD) and optical rotatory dispersion (ORD) is investigated with field applied molecular dynamics simulation (FMD) in chiral (S)-CHBrCIF and achiral water ensembles. Rosenfeld tensor symmetries show that femtosecond transients and steady state correlation functions are different in the chiral ensemble but identical in the achiral ensemble for right and left circularly polarised components of a laser.

Dynamics of Multidimensional Barrier Crossing in the Overdamped Limit

AbstractTwo methods for numerical solution of multidimensional diffusion problems are presented and applied to the two dimensional barrier crossing problem in the overdamped limit. One of these methods is based on evaluating the smallest non‐vanishing eigenvalue of the Smoluchowski equation, and the other is based on an adaption of Chandler's steady state correlation function approach. Both methods make use of the fast Fourier transform algorithm for solving a transformed version of the Smoluchowski equation. The numerical solutions are compared to results based on the Kramers theory and some observations concerning effects of the dynamics of barrier crossing problems are made.

Theory for correlation functions of processes driven by external colored noise.

We study steady-state correlation functions of nonlinear stochastic processes driven by external colored noise. We present a methodology that provides explicit expressions of correlation functions approximating simultaneously short- and long-time regimes. The non-Markov nature is reduced to an effective Markovian formulation, and the nonlinearities are treated systematically by means of double expansions in high and low frequencies. We also derive some exact expressions for the coefficients of these expansions for arbitrary noise by means of a generalization of projection-operator techniques.

Linear relaxation times of stochastic processes driven by non-Gaussian noises

The linear relaxation time (LRT) associated with steady-state correlation functions is studied for Langevin equations with non-Gaussian noises: dichotomous Markov noise and Poissonian white shot noise. Exact results for arbitrary models are obtained and compared with results for Gaussian noises. Some general features of LRTs are discussed. The concept of dynamic effective diffusion is introduced and the existence of an optimal effective Fokker-Planck approximation is discussed. Explicit examples for prototype models are presented and briefly compared with the analogs for Gaussian noises.