Nonlinear Langevin Equation Research Articles

Activated dynamics in dense fluids of attractive nonspherical particles. II. Elasticity, barriers, relaxation, fragility, and self-diffusion

In paper II of this series we apply the center-of-mass version of Nonlinear Langevin Equation theory to study how short-range attractive interactions influence the elastic shear modulus, transient localization length, activated dynamics, and kinetic arrest of a variety of nonspherical particle dense fluids (and the spherical analog) as a function of volume fraction and attraction strength. The activation barrier (roughly the natural logarithm of the dimensionless relaxation time) is predicted to be a rich function of particle shape, volume fraction, and attraction strength, and the dynamic fragility varies significantly with particle shape. At fixed volume fraction, the barrier grows in a parabolic manner with inverse temperature nondimensionalized by an onset value, analogous to what has been established for thermal glass-forming liquids. Kinetic arrest boundaries lie at significantly higher volume fractions and attraction strengths relative to their dynamic crossover analogs, but their particle shape dependence remains the same. A limited universality of barrier heights is found based on the concept of an effective mean-square confining force. The mean hopping time and self-diffusion constant in the attractive glass region of the nonequilibrium phase diagram is predicted to vary nonmonotonically with attraction strength or inverse temperature, qualitatively consistent with recent computer simulations and colloid experiments.

Theory of correlated two-particle activated glassy dynamics: General formulation and heterogeneous structural relaxation in hard sphere fluids

We generalize the nonlinear Langevin equation theory of activated single particle dynamics to describe the correlated motion of two tagged spherical particles in a glass- or gel-forming fluid as a function of their initial separation. The theory is built on the concept of a two-dimensional dynamic free energy surface which quantifies the forces on two particles moving in a cooperative manner. For the hard sphere fluid, above a threshold volume fraction we generically find two relaxation channels corresponding largely, but not exclusively, to a center-of-mass-like displacement and a radial separation of the two tagged particles. The entropic barriers and mean first passage times are computed and found to systematically vary with volume fraction and initial particle separation; both oscillate as a function of the latter in a manner related to the equilibrium pair correlation function. A dynamic correlation length is estimated as the length scale beyond which the two-particle activated dynamics becomes uncorrelated in space and time, and is found to modestly grow with increasing mean relaxation time. The theory is also applied to a simplified model of cage escape, the elementary step of structural relaxation. Predictions for characteristic relaxation times, translation-relaxation decoupling, and stretched-exponential decay of time correlation functions are obtained. A novel mechanism for understanding why strong decoupling emerges in the activated regime, but stretched nonexponential time correlation functions do not change shape as the mean relaxation time grows, is presented and favorably compared with experiment. The theory may serve as a starting point for constructing a predictive model of multiple correlated caging and hopping (forward and backward) events of a pair of tagged particles.

Critical dynamics near QCD critical point based on nonlinear Langevin equation

Critical dynamics near QCD critical point based on nonlinear Langevin equation

Existence of Solutions to Nonlinear Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions

We study a boundary value problem to Langevin equation involving two fractional orders. The Banach fixed point theorem and Krasnoselskii's fixed point theorem are applied to establish the existence results.

Phase Coupling Estimation from Multivariate Phase Statistics

Coupled oscillators are prevalent throughout the physical world. Dynamical system formulations of weakly coupled oscillator systems have proven effective at capturing the properties of real-world systems and are compelling models of neural systems. However, these formulations usually deal with the forward problem: simulating a system from known coupling parameters. Here we provide a solution to the inverse problem: determining the coupling parameters from measurements. Starting from the dynamic equations of a system of symmetrically coupled phase oscillators, given by a nonlinear Langevin equation, we derive the corresponding equilibrium distribution. This formulation leads us to the maximum entropy distribution that captures pairwise phase relationships. To solve the inverse problem for this distribution, we derive a closed-form solution for estimating the phase coupling parameters from observed phase statistics. Through simulations, we show that the algorithm performs well in high dimensions (d = 100) and in cases with limited data (as few as 100 samples per dimension). In addition, we derive a regularized solution to the estimation and show that the resulting procedure improves performance when only a limited amount of data is available. Because the distribution serves as the unique maximum entropy solution for pairwise phase statistics, phase coupling estimation can be broadly applied in any situation where phase measurements are made. Under the physical interpretation, the model may be used for inferring coupling relationships within cortical networks.

Continuous Markovian model for Lévy random walks with superdiffusive and superballistic regimes

We consider a previously devised model describing Levy random walks (Phys. Rev E 79, 011110; 80, 031148, (2009)). It is demonstrated numerically that the given model describes Levy random walks with superdiffusive, ballistic, as well as superballistic dynamics. Previously only the superdiffusive regime has been analyzed. In this model the walker velocity is governed by a nonlinear Langevin equation. Analyzing the crossover from small to large time scales we find the time scales on which the velocity correlations decay and the walker motion essentially exhibits Levy statistics. Our analysis is based on the analysis of the geometric means of walker displacements and allows us to tackle probability density functions with power-law tails and, correspondingly, divergent moments.

Theory of aging, rejuvenation, and the nonequilibrium steady state in deformed polymer glasses

The nonlinear Langevin equation theory of segmental relaxation, elasticity, and mechanical response of polymer glasses is extended to describe the coupled effects of physical aging, mechanical rejuvenation, and thermal history. The key structural variable is the amplitude of density fluctuations, and segmental dynamics proceeds via stress-modified activated barrier hopping on a dynamic free-energy profile. Mechanically generated disorder (rejuvenation) is quantified by a dissipative work argument and increases the amplitude of density fluctuations, thereby speeding up relaxation beyond that induced by the landscape tilting mechanism. The theory makes testable predictions for the time evolution and nonequilibrium steady state of the alpha relaxation time, density fluctuation amplitude, elastic modulus, and other properties. Model calculations reveal a rich dependence of these quantities on preaging time, applied stress, and temperature that reflects the highly nonlinear competition between physical aging and mechanical disordering. Thermal history is "erased" in the long-time limit, although the nonequilibrium steady state is not the literal "fully rejuvenated" freshly quenched glass. The present work provides the conceptual foundation for a quantitative treatment of the nonlinear mechanical response of polymer glasses under a variety of deformation protocols.

Statistical properties of directed avalanches

A two-dimensional directed stochastic sandpile model is studied both numerically and analytically. One of the known analytical approaches is extended by considering general stochastic toppling rules. The probability density distribution for the first-passage time of stochastic process described by a nonlinear Langevin equation with power-law dependence of the diffusion coefficient is obtained. Large-scale Monte Carlo simulations are performed with the aim to analyze statistical properties of the avalanches, such as the asymmetry between the initial and final stages, scaling of voids and the width of the thickest branch. Comparison with random walks description is drawn and different plausible scenarios for the avalanche evolution and the scaling exponents are suggested.

Dynamic free energies, cage escape trajectories, and glassy relaxation in dense fluids of uniaxial hard particles

We extend the naïve mode coupling theory and nonlinear Langevin equation theory of coupled translational-rotational activated dynamics in dense fluids of uniaxial hard particles to more anisotropic rods, and mechanistically analyze in depth the dynamic free-energy surface, hopping process, kinetic vitrification, and fragility. Universal behavior is predicted for the transient center-of-mass (CM) localization length and angle based on a differential volume fraction that quantifies the distance from the dynamic crossover and proper geometric nondimensionalization of the localization quantities. The thermally activated real space cage escape process is increasingly controlled by the CM translation relative to the rotational motion as the particle aspect ratio grows. The mean first passage or structural relaxation time grows faster than exponentially with volume fraction, and is a nonmonotonic function of aspect ratio. The latter results in a kinetic vitrification volume fraction and dynamic fragility that vary nonmonotonically with shape anisotropy. The barrier hopping time based on the simplified CM theory where particle rotation is dynamically frozen is massively reduced by ∼2-3.5 orders of magnitude if the cooperative rotation-translation paths are exploited to escape local cage constraints.

Barnett kinetic coefficients in dense charged and neutral media

An approach based on the nonlinear response theory is proposed for determining nonlinear transport properties of nonideal multielement charged and neutral media. A version of this theory developed for describing these properties involves the comparison of phenomenological conservation equations for dense media in the Barnett approximation and microscopic equations for operators of dynamic variables. The Mori algorithm used for formulating the microscopic equations makes it possible to obtain these equations for non-ideal media in the form of generalized nonlinear Langevin equations. As a result, macroscopic expressions are derived for nonlinear kinetic coefficients in the second order in thermal perturbations (of temperature, mass velocity, etc.). The expressions for nonlinear and linearized Barnett kinetic coefficients are compared.

Why does the rouse model fairly describe the dynamic characteristics of polymer melts at molecular masses below critical mass?

Generalization of the Rouse model without any use of the postulates concerning the Gaussian distribution of the vector connecting the ends of segments is advanced. In the initial (in general, nonlinear) Langevin equations, self-averaging over continuous fragments of a macromolecule naturally defines a linear term for the tagged chain, and this term differs from the entropy term of the classical Rouse model only by the numerical coefficient. According to the inertia-free approximation, the initial decay rates of correlation functions for the normal modes are described by the Rouse model independently of the character of fluctuations of the vector connecting the ends of the Kuhn segment. This statement is valid for any moment if the initial Langevin equations are treated in terms of the approximation of dynamic self-consistency. Simulation of the Fraenkel chains by the method of Brownian dynamics shows that decay of autocorrelation functions of shortwave normal modes is fairly described by the linearized equations for a given model of a chain and that the Rouse equation can be used for the long-wave modes. The results of this study make it possible to explain a marked difference between the lengths of the Kuhn and Rouse segments that is estimated from static and dynamic experiments.

A path integral approach to random motion with nonlinear friction

Using a path integral approach, we derive an analytical solution of a nonlinear and singular Langevin equation, which has been introduced previously by P-G de Gennes as a simple phenomenological model for the stick-slip motion of a solid object on a vibrating horizontal surface. We show that the optimal (or most probable) paths of this model can be divided into two classes of paths, which correspond physically to a sliding or slip motion, where the object moves with a non-zero velocity over the underlying surface, and a stick-slip motion, where the object is stuck to the surface for a finite time. These two kinds of basic motions underlie the behavior of many more complicated systems with solid/solid friction and appear naturally in de Gennes' model in the path-integral framework.

Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions

We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii′s fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.

Anomalous thermostat and intraband discrete breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels Γ ( t ) and noise term ζ ( t ) for the anharmonic coordinates q ( t ) = ( q α ( t ) ) . For zero temperature, i.e. for ζ ( t ) ≡ 0 , we prove that the support of the Fourier transform of Γ ( t ) and of the time averaged velocity–velocity correlation functions K ( t ) of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory q ( t ) with frequency Ω we find that the energy E T transferred to the harmonic system up to time T is proportional to T α . If Ω equals one of the phonon frequencies ω ν , it is α = 2 . We prove that there is a zero measure set L such that for Ω in its full measure complement R ∖ L , it is α = 0 , i.e. there is no energy dissipation. Under certain conditions L contains a subset L ′ such that for Ω ∈ L ′ the dissipation rate is nonzero and may be subdissipative ( 0 ≤ α < 1 ) or superdissipative ( 1 < α ≤ 2 ) , compared to ordinary dissipation ( α = 1 ) . Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency Ω exist for all Ω in a Cantor set C ( k ) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for G ( t ; Ω ) , related to Γ ( t ) . For Ω ∈ C ( k ) the small denominators do not lead to divergencies such that G ( t ; Ω ) is a smooth and bounded function in t .

Theory of coupled translational-rotational glassy dynamics in dense fluids of uniaxial particles

The naïve mode coupling theory (NMCT) for ideal kinetic arrest and the nonlinear Langevin equation theory of activated single-particle barrier hopping dynamics are generalized to treat the coupled center-of-mass (CM) translational and rotational motions of uniaxial hard objects in the glassy fluid regime. The key dynamical variables are the time-dependent displacements of the particle center-of-mass and orientational angle. The NMCT predicts a kinetic arrest diagram with three dynamical states: ergodic fluid, plastic glass, and fully nonergodic double glass, the boundaries of which meet at a "triple point" corresponding to a most difficult to vitrify diatomic of aspect ratio approximately 1.43. The relative roles of rotation and translation in determining ideal kinetic arrest are explored by examining three limits of the theory corresponding to nonrotating, pure rotation, and rotationally ergodic models. The ideal kinetic arrest boundaries represent a crossover to activated dynamics described by two coupled stochastic nonlinear Langevin equations for translational and rotational motions. The fundamental quantity is a dynamic free-energy surface, which for small aspect ratios in the high-volume fraction regime exhibits two saddle points reflecting a two-step activated dynamics where relatively rapid rotational dynamics coexists with slower CM translational motions. For large-enough aspect ratios, the dynamic free-energy surface has one saddle point which corresponds to a system-specific coordinated translation-rotation motion. The entropic barriers as a function of the relative amount of rotation versus translation are determined.

Influence of the noise spectrum on the anomalous diffusion in a stochastic system

We use an effective Markovian description to study the long-time behavior of a nonlinear second-order Langevin equation with a Gaussian noise. When dissipation is neglected, the energy of the system grows as with time a power law with an anomalous scaling exponent that depends both on the confining potential and on the high-frequency distribution of the noise. The asymptotic expression of the probability distribution function in phase space is calculated analytically. The results are extended to the case where small dissipative effects are taken into account.

The influence of shape on the glassy dynamics of hard nonspherical particle fluids. II. Barriers, relaxation, fragility, kinetic vitrification, and universality

We extend and apply the nonlinear Langevin equation theory of activated barrier hopping dynamics in glassy fluids and colloidal suspensions to study broad families of one-, two-, and three-dimensional hard nonspherical particles. Beyond the ideal kinetic arrest volume fraction, entropic barriers emerge with heights (alpha relaxation times, inverse diffusion constants) that increase nonlinearly (nonexponentially) with volume fraction and in a manner that becomes stronger with particle dimensionality. Partial collapse of the volume fraction dependence of barrier heights and reduced relaxation times of different particle shapes within a fixed dimensionality class are achieved based on a difference volume fraction variable that quantifies the distance from the ideal mode coupling theory dynamic crossover. However, the barrier, alpha relaxation time, and self-diffusion constant results of all shapes are remarkably well collapsed onto a single universal master curve based on a theoretically motivated coupling constant which quantifies the renormalized mean square force on a tagged particle. The latter is determined mainly by the square of the intermolecular site-site contact value of the pair correlation function, thereby providing an explicit microscopic connection between local packing, binary collisions, and slow dynamics. A large variation of the dynamic fragility with particle shape is found with compact cluster particles being the most fragile. A kinetic glass transition map is constructed that is relevant to vitrification of laboratory colloidal suspensions. The possible relevance of the hard particle results for understanding the dynamic fragility of thermal van der Waals liquids is discussed.

Nonlinear transport properties of non-ideal systems

The theory of nonlinear transport is elaborated to determine the Burnett transport properties of non-ideal multi-element plasma and neutral systems. The procedure for the comparison of the phenomenological conservation equations of a continuous dense medium and the microscopic equations for dynamical variable operators is used for the definition of these properties. The Mori algorithm is developed to derive the equations of motion of dynamical value operators of a non-ideal system in the form of the generalized nonlinear Langevin equations. In consequence, the microscopic expressions of transport coefficients corresponding to second-order thermal disturbances (temperature, mass velocity, etc) have been found in the long wavelength and low frequency limits.

Pitchfork And Hopf Bifurcations In Stochastic Regulatory Networks With Delayed Feedback

The bifurcation diagram of a model nonlinear Langevin equation appropriate for delayed feedback loops in regulatory networks is obtained. We show that under parametric fluctuation the bifurcation remains sharp, both in the ranges of direct and oscillatory bifurcation. At threshold, the stationary distribution function becomes a power law in the dynamical variable. The combined effect of delay and stochasticity leads to a shift in the location of the bifurcation threshold relative to the deterministic limit of no fluctuation. The shift scales linearly with the noise intensity.

Cooperative activated dynamics in dense mixtures of hard and sticky spheres

The coupled activated dynamics in dense mixtures of repulsive and sticky hard spheres is studied using stochastic nonlinear Langevin equation theory. The effective free energy surface, barriers, saddle point trajectories, and mean first passage times depend in a rich manner on mixture composition, (high) total volume fraction, and attractive interaction strength. In general, there are three types of saddle point trajectories or relaxation pathways: a pure sticky or pure repulsive particle displacement keeping the other species localized, and a cooperative motion involving repulsive and attractive particle displacements. The barrier for activated hopping usually increases with the ratio of sticky to repulsive particle displacement. However, at intermediate values of the displacement ratio it can attain a broad plateau value, and can even exhibit a local maximum, and hence nonmonotonic behavior, at high sticky particle mixture compositions if the attraction strength is modest. The mean first passage, or hopping, times are computed using multidimensional Kramers theory. In most cases the hopping time trends reflect the behavior of the barrier height, especially as the sticky particle attraction strengths become large. However, there are dramatic exceptions associated with cooperative repulsive and attractive particle trajectories where the barriers are high but a greatly enhanced number of such trajectories exist near the saddle point.