non-Markovian Langevin Equation Research Articles

Brownian motion with time-dependent friction and single-particle dynamics in liquids.

A microscopic theory of molecular motion in classical monatomic liquids, proposed by Glass and Rice [Phys. Rev. 176, 239 (1968)10.1103/PhysRev.176.239], is revisited and extended to incorporate the dynamic friction in the Brownian description of the atomic diffusion in a mean-time-dependent harmonic force field. A modified, non-Markovian Langevin equation is utilized to derive an equation of motion for the velocity autocorrelation function with time-dependent friction coefficient. Numerical solution of the equation gives an excellent account of the velocity autocorrelation function in Lennard-Jones liquids, liquid alkali, and transition metals over a broad range of density and temperature. Derivation of the equation of motion leads to a self-consistent expression for the time dependence of friction coefficient. Our results demonstrate that the nature of time dependence of the friction coefficient changes dramatically with the liquid density. At low and moderate densities, the dynamic friction decays exponentially whereas it increases exponentially at high liquid densities. Our findings provide an opportunity for a different outlook of the Brownian description of atomic dynamics in liquids.

NonMarkovianity in cosmology: Memories kept in a quantum field

In this work we ask how an Unruh–DeWitt (UD) detector with harmonic oscillator internal degrees of freedom Q measuring an evolving quantum matter field Φ(x,t) in an expanding universe with scale factor a(t) responds. We investigate the detector’s response which contains non-Markovian information about the quantum field squeezed by the dynamical spacetime. The challenge is in the memory effects accumulated over the evolutionary history. We first consider a detector W, the ‘Witness’, which co-existed and evolved with the quantum field from the beginning. We derive a nonMarkovian quantum Langevin equation for the detector’s Q by integrating over the squeezed quantum field. The solution of this integro-differential equation would answer our question, in principle, but very challenging, in practice. Striking a compromise, we then ask, to what extent can a detector D introduced at late times, called the ‘Detective’, decipher past memories. This situation corresponds to many cosmological experiments today probing specific stages in the past, such as COBE targeting activities at the surface of last scattering. Somewhat surprisingly we show that it is possible to retrieve to some degree certain global physical quantities, such as the resultant squeezing, particles created, quantum coherence and correlations. The reason is because the quantum field has all the fine-grained information from the beginning in how it was driven by the cosmic dynamics a(t). How long the details of past history can persist in the quantum field depends on the memory time. The fact that a squeezed field cannot come to complete equilibrium under continuous driving, as in an evolving spacetime, actually helps to retain the memory. We discuss interesting features and potentials of this ‘archeological’ perspective toward cosmological issues.

Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions

We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise (FGN) in a superharmonic external potential of the form U(x) ∝ x 2n (). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential U(x). We analyse in detail the temporal approach to stationarity as well as the shape of the non-Boltzmann stationary PDF. A typical characteristic is that subdiffusive, antipersistent (with negative autocorrelation) motion tends to effect an accumulation of probability close to the origin as compared to the corresponding Boltzmann distribution while the opposite trend occurs for superdiffusive (persistent) motion. For this latter case this leads to distinct bimodal shapes of the PDF. This property is compared to a similar phenomenon observed for Markovian Lévy flights in superharmonic potentials. We also demonstrate that the motion encoded in the fractional Langevin equation driven by FGN always relaxes to the Boltzmann distribution, as in this case the fluctuation-dissipation theorem is fulfilled.

Correlation functions of non-Markovian systems out of equilibrium: analytical expressions beyond single-exponential memory

This paper is concerned with correlation functions of stochastic systems with memory, a prominent example being a molecule or colloid moving through a complex (e.g. viscoelastic) fluid environment. Analytical investigations of such systems based on non-Markovian stochastic equations are notoriously difficult. A common approximation is that of a single-exponential memory, corresponding to the introduction of one auxiliary variable coupled to the Markovian dynamics of the main variable. As a generalization, we here investigate a class of ‘toy’ models with altogether three degrees of freedom, giving rise to more complex forms of memory. Specifically, we consider, mainly on an analytical basis, the under- and overdamped motion of a colloidal particle coupled linearly to two auxiliary variables, where the coupling between variables can be either reciprocal or non-reciprocal. Projecting out the auxiliary variables, we obtain non-Markovian Langevin equations with friction kernels and colored noise, whose structure is similar to that of a generalized Langevin equation. For the present systems, however, the non-Markovian equations may violate the fluctuation–dissipation relation as well as detailed balance, indicating that the systems are out of equilibrium. We then study systematically the connection between the coupling topology of the underlying Markovian system and various autocorrelation functions. We demonstrate that already two auxiliary variables can generate surprisingly complex (e.g. non-monotonic or oscillatory) memory and correlation functions. Finally, we show that a minimal overdamped model with two auxiliary variables and suitable non-reciprocal coupling yields correlation functions resembling those describing hydrodynamic backflow in an optical trap.

Microrheological approach for the viscoelastic response of gels

In this paper I present a simple and self-consistent framework based on microrheology that allows one to obtain the mechanical response of viscoelastic fluids and gels from the motion of probe particles immersed on it. By considering a non-markovian Langevin equation, I obtain general expressions for the mean-squared displacement and the time-dependent diffusion coefficient that are directly related to the memory kernels and the response function, and which allow one to obtain estimates for the complex shear modulus and the complex viscosity of the material. The usefulness of such approach is demonstrated by applying it to describe experimental data on chemically cross-linked polyacrylamide through its sol-gel transition.

Equations of motion for position-dependent coarse-grain mappings obtained with Mori-Zwanzig theory.

A position-dependent transformation is introduced for mapping a system of atomistic particles to a system of coarse-grained (CG) variables, which under some circumstances might be considered particles. This CG mapping allows atomistic particles to simultaneously contribute to more than a single CG particle and to change in time the CG particle they are associated with. That is, the CG mapping is dynamic. Mori-Zwanzig theory is then used to obtain the equations of motion for this CG mapping, resulting in conservative, dissipative, and random force terms in generalized, non-Markovian Langevin equations. In addition to the usual forces arising from the effective CG potential derived from atomistic interactions, new forces arise from the dynamic changes in the CG mapping itself. These new forces effectively account for changes arising from fluxes of atomistic particles into and out of CG ones as time progresses. Several examples are given showing the range of problems that can be addressed with this new CG mapping. These range from the usual case where atomistic particles are grouped into large molecular-like chunks, with mappings that remain fixed in time and for which an atomistic particle is part of only a single CG one, to the case where CG particles resemble fluid elements, containing many hundreds of independent atomistic particles. The new CG mapping also allows for hybrid descriptions, in which a part of the system remains atomistic or molecular-like and a part is highly coarse-grained to mesoscopic fluid element-like particles, for example. In the latter case, the equations of motion then provide the correct formalism for determining the forces, beyond the usual conservative ones. This provides a theoretical foundation upon which approximate equations of motion can be formulated to thus build numerical algorithms for expanded applications of accurate CG molecular dynamics.

Capture cross section with quantum diffusion approach

A C++ code for calculating the capture of a projectile by target nucleus is described. The code is based on the quantum diffusion model developed for considering collisions of nuclei at energies below and above the Coulomb barrier. The code provides the capture cross sections and other characteristics of reaction as functions of Ec.m.. The formalism of the model is briefly described. The code contains the Fortran subroutine to calculate the nucleus–nucleus potential. Program summaryProgram title: NuclearCaptureProgram files doi: http://dx.doi.org/10.17632/n66yg8mdzs.1Licensing provisions: GNU General Public License v2Programming language: C++, FortranExternal routines/libraries: GSL-2.2.1 GNU Scientific Library in C++ is required. It is available from https://www.gnu.org/software/gsl .Nature of problem: Capture of colliding nuclei at energies below and above the Coulomb barrierSolution method: Non-Markovian quantum Langevin equations

Thermodynamic work statistics for Ornstein–Uhlenbeck-type heat baths

In this work we explore the validity of the transient work fluctuation theorem as well as the Jarzynski equality. In the case of a Brownian particle dragged through a fluid by an optical trap, the fluid plays the role of a non-Markovian heat bath characterized by an specific Ornstein–Uhlenbeck friction memory kernel. To achieve our goal, we used an alternative method which transformed the generalized Langevin equation into an equivalent third-order non-Markovian Langevin equation. By means of the exact analytical solution of this Langevin equation, we calculated the statistics of thermodynamic work which established the Crooks relation, and then the Jarzynski equality came immediately. The fluctuation theorems are applicable to small systems where energy fluctuations with respect to the average behavior are big enough to be measured. The examples of such a class of systems are found inside living organisms.

Non-Markovian coherent feedback control of quantum dot systems

In this paper we present a non-Markovian coherent feedback scheme for decoherence suppression in single quantum dot systems. The feedback loop is closed via a quantum tunnelling junction between the natural source and drain baths of the quantum dot. The exact feedback-controlled non-Markovian Langevin equation is derived for describing the dynamics of the quantum dot. To deal with the nonlinear memory function in the Langevin equation, we analyse the Green’s function-based root locus, from which we show that the decoherence of the quantum dot can be suppressed via increasing the feedback coupling strength. The effectiveness of decoherence suppression induced by non-Markovian coherent feedback is demonstrated by a single quantum dot example bathed with Lorentzian noises.

Application of the theory of open quantum systems to nuclear physics problems

Quantum diffusion equations with transport coefficients explicitly depending on time are derived from the generalized non-Markovian Langevin equations. The asymptotic behavior of the friction and diffusion coefficients is investigated in the case of the FC and RWA couplings between the collective and internal subsystems. An asymptotic expression is obtained for the propagator of the density matrix of the open quantum system with the general quadratic Hamiltonian, linearly coupled (in coordinate and momentum) to internal degrees of freedom. The effect of different sets of transport coefficients on the decoherence and decay rate of the metastable state is investigated using the master equation for the reduced density matrix of open quantum systems. The developed approach is used to study the capture of the projectile nucleus by the target nucleus at energies near the Coulomb barrier. Capture cross sections in asymmetric reactions are well described with allowance for the calculated capture probabilities. Particular cases where dissipation favors penetration through the potential barrier are found. The generalized Kramers formula for the quasi-stationary decay rate of the quantum metastable systems is analytically derived.

Quantum entanglement between amide-I and amide-site in Davydov-Scott model

In this paper, we firstly derive non-Markovian operator Langevin equations of the Davydov monomer in its environment. Next, we replace the equations with the c-number quantum general Langevin equations (QGLEs) by calculating statistical and quantum averages of the operator Langevin equations. Then, by using the c-number QGLEs we investigate the evolutions of the subsystems amide-I and amide-site. The evolution of a parameter θ describing quantum entanglement of the coupling subsystems with continuous variable Hamiltonian has also been investigated. It is shown that there is certain entanglement between the amide-I and amide-site in the Davydov-Scott monomer.

On a Class of Non-Markovian Langevin Equations

This paper proposes a generalized Langevin's equation for a small classical mechanical system embedded in a reservoir. The interaction of the main system with the reservoir is given by a Gaussian transform as introduced in our previous paper [8]. Thus, a first result proves the existence of a strong solution to this equation in the space where the Gaussian transform (or non-Markovian noise) is defined. The interpretation of the noise is obtained by considering a finite number n of oscillating particles with discrete frequencies in the reservoir. The action of this discrete reservoir on the small system is described by a memory kernel and a sequence of zero-mean Gaussian processes. So, an integro-differential equation for the evolution of a generic particle in the main system arises for each n. This equation has a unique solution Xn which converges in distribution towards the solution of the initial non-Markovian Langevin's equation.

Stochastic resonance in bistable systems with nonlinear dissipation and multiplicative noise: A microscopic approach

We have studied the stochastic resonance (SR) of bistable systems coupled to a bath with a nonlinear system–bath interaction, by using the microscopic, generalized Caldeira–Leggett (CL) model. The adopted CL model yields the non-Markovian Langevin equation with nonlinear dissipation and state-dependent (multiplicative) diffusion which preserve the fluctuation–dissipation relation (FDR). Results of our simulations are given as follows: (1) the spectral power amplification (SPA) exhibits SR not only for a and b but also for τ while the stationary probability distribution function is independent of them where a and b denote magnitudes of multiplicative and additive noises, respectively, and τ expresses the relaxation time of Ornstein–Uhlenbeck (OU) colored noise; (2) the SPA for coexisting additive and multiplicative noises has a single-peak but two-peak structure as functions of a, b and/or τ. Results (1) and (2) are qualitatively different from previous ones obtained by phenomenological Langevin models where the FDR is not held or indefinite. These show an importance of the FDR in a study on SR of open bistable systems.

Rocking Subdiffusive Ratchets: Origin, Optimization and Efficiency

We study origin, parameter optimization, and thermodynamic efficiency of isothermal rocking ratchets based on fractional subdiffusion within a generalized non-Markovian Langevin equation approach. A corresponding multi-dimensional Markovian embedding dynamics is realized using a set of auxiliary Brownian particles elastically coupled to the central Brownian particle (see video on the journal web site). We show that anomalous subdiffusive transport emerges due to an interplay of nonlinear response and viscoelastic effects for fractional Brownian motion in periodic potentials with broken space-inversion symmetry and driven by a time-periodic field. The anomalous transport becomes optimal for a subthreshold driving when the driving period matches a characteristic time scale of interwell transitions. It can also be optimized by varying temperature, amplitude of periodic potential and driving strength. The useful work done against a load shows a parabolic dependence on the load strength. It grows sublinearly with time and the corresponding thermodynamic efficiency decays algebraically in time because the energy supplied by the driving field scales with time linearly. However, it compares well with the efficiency of normal diffusion rocking ratchets on an appreciably long time scale.

Decoherence suppression via non-Markovian coherent feedback control

In this paper, we present a coherent feedback control scheme for non-Markovian bosonic systems, in which an engineered quantum control field is introduced to couple both the system and the noise bath. The closed-loop dynamics of the system is described by an exact non-Markovian quantum Langevin equation, where the spectral density functions of the noise and the quantum control field, as well as their coupling, are combined into a single memory kernel function. We show that the coupling between the quantum control field with the noise bath can be used as a feedback control to modulate the memory kernel function. As a result, the noise bath can be driven out of resonance with the system and the decoherence can be efficiently suppressed. The effectiveness of the controllability is demonstrated with a photonic circuit in photonic crystals.

ANISOTROPIC BEHAVIOR OF A POINT CHARGE MOVING IN THE PRESENCE OF THE CONDUCTING PLATE UNDER THE INFLUENCE OF THE STOCHASTIC LORENTZ FORCES

The influence of quantized electromagnetic fields on a nonrelativistic charged particle near a conducting plate is investigated by studying the the non-Markovian Langevin equation of the particle. This stochastic approach incorporates not only the stochastic noise manifested from electromagnetic vacuum fluctuations, but also dissipation backreaction on a charge in the form of the retarded Lorentz forces. Since the imposition of the boundary is expected to anisotropically modify the effects of the fields on the evolution of the particle, we consider the motion of a charge undergoing small-amplitude oscillations in the direction either parallel or normal to the plane boundary. Laplace transform technique is applied to find the solutions to these two configurations.

Classical open systems with nonlinear nonlocal dissipation and state-dependent diffusion: Dynamical responses and the Jarzynski equality

We have studied dynamical responses and the Jarzynski equality (JE) of classical open systems described by the generalized Caldeira-Leggett model with the nonlocal system-bath coupling. In the derived non-Markovian Langevin equation, the nonlinear nonlocal dissipative term and state-dependent diffusion term yielding multiplicative colored noise satisfy the fluctuation-dissipation relation. Simulation results for harmonic oscillator systems have shown the following: (a) averaged responses of the system (x) to applied sinusoidal and step forces significantly depend on model parameters of magnitudes of additive and multiplicative noises and the relaxation time of colored noise, although stationary marginal probability distribution functions are independent of them, (b) a combined effect of nonlinear dissipation and multiplicative colored noise induces enhanced fluctuations ([x-(x)](2)) for an applied sinusoidal force, and (c) the JE holds for an applied ramp force independently of the model parameters with a work distribution function which is (symmetric) Gaussian and (asymmetric) non-Gaussian for additive and multiplicative noises, respectively. It has been shown that the non-Markovian Langevin equation in the local and overdamped limits is quite different from the widely adopted, phenomenological Markovian Langevin equation subjected to additive and multiplicative noises.

Quantum statistical effects in nuclear reactions, fission, and open quantum systems

Quantum diffusion equations with time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytical formulas for calculating friction and diffusion coefficients in nuclear processes are obtained. The asymptotics of the transport coefficients and of the correlation functions are investigated. The problem of correlation decay in quantum dissipative systems is studied. A comparative analysis of diffusion coefficients for the harmonic and inverted oscillators is performed. The role of quantum statistical effects during passage through a parabolic potential barrier is investigated. Sets of diffusion coefficient assuring the purity of states at any time instant are found in cases of non-Markovian dynamics. The influence of different sets of transport coefficients on the rate of decay from a metastable state is studied in the framework of the master equation for reduced density matrices describing open quantum systems. The approach developed is applied to investigation of fission processes and the processes of projectile-nuclei capture by target nuclei for bombarding energies in the vicinity of the Coulomb barrier. The influence of dissipation and fluctuation on these processes is taken into account in a self-consistent way. The evaporation residue cross sections for asymmetric fusion reactions are calculated from the derived capture probabilities averaged over all orientations of the deformed projectile and target nuclei.

Nuclear Quantum Effects in Solids Using a Colored-Noise Thermostat

We present a method, based on a non-Markovian Langevin equation, to include quantum corrections to the classical dynamics of ions in a quasiharmonic system. By properly fitting the correlation function of the noise, one can vary the fluctuations in positions and momenta as a function of the vibrational frequency, and fit them so as to reproduce the quantum-mechanical behavior, with minimal a priori knowledge of the details of the system. We discuss the application of the thermostat to diamond and to ice Ih. We find that results in agreement with path-integral methods can be obtained using only a fraction of the computational effort.

Stochastic quantization for complex actions

We use the stochastic quantization method to study systems with complex valued path integral weights. We assume a Langevin equation with a memory kernel and Einstein’s relations with colored noise. The equilibrium solution of this non-Markovian Langevin equation is analyzed. We show that for a large class of elliptic non-Hermitian operators acting on scalar functions on Euclidean space, which define different models in quantum field theory, converge to an equilibrium state in the asymptotic limit of the Markov parameter τ→∞. Moreover, as we expected, we obtain the Schwinger functions of the theory.