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Non-Local Kinetics: Revisiting and Updates Emphasizing Fractional Calculus Applications. Symmetry (Basel) 2023. [DOI: 10.3390/sym15030632] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/06/2023] Open
Abstract
Non-local kinetic problems spanning a wide area of problems where fractional calculus is applicable have been analyzed. Classical fractional kinetics based on the Continuum Time Random Walk diffusion model with the absence of stationary states, real-world problems from pharmacokinetics, and modern material processing have been reviewed. Fractional allometry has been considered a potential area of application. The main focus in the analysis has been paid to the memory functions in the convolution formulation, crossing from the classical power law to versions of the Mittag-Leffler function. The main idea is to revisit the non-local kinetic problems with an update updating on new issues relevant to new trends in fractional calculus.
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Sprenger AR, Bair C, Löwen H. Active Brownian motion with memory delay induced by a viscoelastic medium. Phys Rev E 2022; 105:044610. [PMID: 35590653 DOI: 10.1103/physreve.105.044610] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2022] [Accepted: 04/11/2022] [Indexed: 01/17/2023]
Abstract
By now active Brownian motion is a well-established model to describe the motion of mesoscopic self-propelled particles in a Newtonian fluid. On the basis of the generalized Langevin equation, we present an analytic framework for active Brownian motion with memory delay assuming time-dependent friction kernels for both translational and orientational degrees of freedom to account for the time-delayed response of a viscoelastic medium. Analytical results are obtained for the orientational correlation function, mean displacement, and mean-square displacement which we evaluate in particular for a Maxwell fluid characterized by a kernel which decays exponentially in time. Further, we identify a memory-induced delay between the effective self-propulsion force and the particle orientation which we quantify in terms of a special dynamical correlation function. In principle, our predictions can be verified for an active colloidal particle in various viscoelastic environments such as a polymer solution.
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Affiliation(s)
- Alexander R Sprenger
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
| | - Christian Bair
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
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Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus. FRACTAL AND FRACTIONAL 2020. [DOI: 10.3390/fractalfract4030045] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper.
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Singh S, Singh RK, Kumar S. Transport and tumbling of polymers in viscoelastic shear flow. Phys Rev E 2020; 102:012605. [PMID: 32794989 DOI: 10.1103/physreve.102.012605] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/29/2020] [Accepted: 06/19/2020] [Indexed: 11/07/2022]
Abstract
Polymers in shear flow are ubiquitous and we study their motion in a viscoelastic fluid under shear. Employing Hookean dumbbells as representative, we find that the center-of-mass motion follows: 〈x_{c}^{2}(t)〉∼γ[over ̇]^{2}t^{α+2}, generalizing the earlier result: 〈x_{c}^{2}(t)〉∼γ[over ̇]^{2}t^{3}(α=1). Here 0<α<1 is the coefficient defining the power-law decay of noise correlations in the viscoelastic media. Motion of the relative coordinate, on the other hand, is quite intriguing in that 〈x_{r}^{2}(t)〉∼t^{β} with β=2(1-α), for small α. This implies nonexistence of the steady state, making it inappropriate for addressing tumbling dynamics. We remedy this pathology by introducing a nonlinear spring with FENE-LJ interaction and study tumbling dynamics of the dumbbell. We find that the tumbling frequency exhibits a nonmonotonic behavior as a function of shear rate for various degrees of subdiffusion. We also find that this result is robust against variations in the extension of the spring. We briefly discuss the case of polymers.
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Affiliation(s)
- Sadhana Singh
- Department of Physics, Banaras Hindu University, Varanasi 221005, India
| | - R K Singh
- Department of Physics, Banaras Hindu University, Varanasi 221005, India.,Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India
| | - Sanjay Kumar
- Department of Physics, Banaras Hindu University, Varanasi 221005, India
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Sevilla FJ, Rodríguez RF, Gomez-Solano JR. Generalized Ornstein-Uhlenbeck model for active motion. Phys Rev E 2019; 100:032123. [PMID: 31640041 DOI: 10.1103/physreve.100.032123] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2019] [Indexed: 06/10/2023]
Abstract
We investigate a one-dimensional model of active motion, which takes into account the effects of persistent self-propulsion through a memory function in a dissipative-like term of the generalized Langevin equation for particle swimming velocity. The proposed model is a generalization of the active Ornstein-Uhlenbeck model introduced by G. Szamel [Phys. Rev. E 90, 012111 (2014)10.1103/PhysRevE.90.012111]. We focus on two different kinds of memory which arise in many natural systems: an exponential decay and a power law, supplemented with additive colored noise. We provide analytical expressions for the velocity autocorrelation function and the mean-squared displacement, which are in excellent agreement with numerical simulations. For both models, damped oscillatory solutions emerge due to the competition between the memory of the system and the persistence of velocity fluctuations. In particular, for a power-law model with fractional Brownian noise, we show that long-time active subdiffusion occurs with increasing long-term memory.
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Affiliation(s)
- Francisco J Sevilla
- Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000, Ciudad de México, México
| | - Rosalío F Rodríguez
- Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000, Ciudad de México, México
- FENOMEC, Universidad Nacional Autónoma de México, Apdo. Postal 20-726, 01000, Ciudad de México, México
| | - Juan Ruben Gomez-Solano
- Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000, Ciudad de México, México
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Mankin R, Laas K, Sauga A. Generalized Langevin equation with multiplicative noise: temporal behavior of the autocorrelation functions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061131. [PMID: 21797326 DOI: 10.1103/physreve.83.061131] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2011] [Revised: 05/20/2011] [Indexed: 05/31/2023]
Abstract
The temporal behavior of the mean-square displacement and the velocity autocorrelation function of a particle subjected to a periodic force in a harmonic potential well is investigated for viscoelastic media using the generalized Langevin equation. The interaction with fluctuations of environmental parameters is modeled by a multiplicative white noise, by an internal Mittag-Leffler noise with a finite memory time, and by an additive external noise. It is shown that the presence of a multiplicative noise has a profound effect on the behavior of the autocorrelation functions. Particularly, for correlation functions the model predicts a crossover between two different asymptotic power-law regimes. Moreover, a dependence of the correlation function on the frequency of the external periodic forcing occurs that gives a simple criterion to discern the multiplicative noise in future experiments. It is established that additive external and internal noises cause qualitatively different dependences of the autocorrelation functions on the external forcing and also on the time lag. The influence of the memory time of the internal noise on the dynamics of the system is also discussed.
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Affiliation(s)
- R Mankin
- Institute of Mathematics and Natural Sciences, Tallinn University, Tallinn, Estonia
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Suciu N. Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:056301. [PMID: 20866316 DOI: 10.1103/physreve.81.056301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2009] [Revised: 03/03/2010] [Indexed: 05/29/2023]
Abstract
Whenever one uses translation invariant mean Green's functions to describe the behavior in the mean and to estimate dispersion coefficients for diffusion in random velocity fields, the spatial homogeneity of the transition probability of the transport process is implicitly assumed. This property can be proved for deterministic initial conditions if, in addition to the statistical homogeneity of the space-random velocity field, the existence of unique classical solutions of the transport equations is ensured. When uniqueness condition fails and translation invariance of the mean Green's function cannot be assumed, as in the case of nonsmooth samples of random velocity fields with exponential correlations, asymptotic dispersion coefficients can still be estimated within an alternative approach using the Itô equation. Numerical simulations confirm the predicted asymptotic behavior of the coefficients, but they also show their dependence on initial conditions at early times, a signature of inhomogeneous transition probabilities. Such memory effects are even more relevant for random initial conditions, which are a result of the past evolution of the process of diffusion in correlated velocity fields, and they persist indefinitely in case of power law correlations. It was found that the transition probabilities for successive times can be spatially homogeneous only if a long-time normal diffusion limit exits. Moreover, when transition probabilities, for either deterministic or random initial states, are spatially homogeneous, they can be explicitly written as Gaussian distributions.
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Affiliation(s)
- Nicolae Suciu
- Friedrich-Alexander University, Erlangen-Nuremberg, Germany and Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj Napoca, Romania.
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Suciu N, Vamoş C, Radu FA, Vereecken H, Knabner P. Persistent memory of diffusing particles. Phys Rev E 2010; 80:061134. [PMID: 20365145 DOI: 10.1103/physreve.80.061134] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2009] [Indexed: 11/07/2022]
Abstract
The variance of the advection-diffusion processes with variable coefficients is exactly decomposed as a sum of dispersion terms and memory terms consisting of correlations between velocity and initial positions. For random initial conditions, the memory terms quantify the departure of the preasymptotic variance from the time-linear diffusive behavior. For deterministic initial conditions, the memory terms account for the memory of the initial positions of the diffusing particles. Numerical simulations based on a global random walk algorithm show that the influence of the initial distribution of the cloud of particles is felt over hundreds of dimensionless times. In case of diffusion in random velocity fields with finite correlation range the particles forget the initial positions in the long-time limit and the variance is self-averaging, with clear tendency toward normal diffusion.
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Affiliation(s)
- N Suciu
- Chair for Applied Mathematics I, Friedrich-Alexander University Erlangen-Nuremberg, Erlangen, Germany.
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Mankin R, Rekker A. Memory-enhanced energetic stability for a fractional oscillator with fluctuating frequency. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041122. [PMID: 20481692 DOI: 10.1103/physreve.81.041122] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2010] [Revised: 03/18/2010] [Indexed: 05/29/2023]
Abstract
The long-time limit behavior of the variance and the correlation function for the output signal of a fractional oscillator with fluctuating eigenfrequency subjected to a periodic force is considered. The influence of a fluctuating environment is modeled by a multiplicative white noise and by an additive noise with a zero mean. The viscoelastic-type friction kernel with memory is assumed as a power-law function of time. The exact expressions of stochastic resonance (SR) characteristics such as variance and signal-to-noise ratio (SNR) have been calculated. It is shown that at intermediate values of the memory exponent the energetic stability of the oscillator is significantly enhanced in comparison with the cases of strong and low memory. A multiresonancelike behavior of the variance and SNR as functions of the memory exponent is observed and a connection between this effect and the memory-induced enhancement of energetic stability is established. The effect of memory-induced energetic stability encountered in case the harmonic potential is absent, is also discussed.
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Affiliation(s)
- Romi Mankin
- Institute of Mathematics and Natural Sciences, Tallinn University, 25 Narva Road, 10120 Tallinn, Estonia
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Soika E, Mankin R, Ainsaar A. Resonant behavior of a fractional oscillator with fluctuating frequency. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011141. [PMID: 20365357 DOI: 10.1103/physreve.81.011141] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2009] [Revised: 09/25/2009] [Indexed: 05/29/2023]
Abstract
The long-time behavior of the first moment for the output signal of a fractional oscillator with fluctuating frequency subjected to an external periodic force is considered. Colored fluctuations of the oscillator eigenfrequency are modeled as a dichotomous noise. The viscoelastic type friction kernel with memory is assumed as a power-law function of time. Using the Shapiro-Loginov formula, exact expressions for the response to an external periodic field and for the complex susceptibility are presented. On the basis of the exact formulas it is demonstrated that interplay of colored noise and memory can generate a variety of cooperation effects, such as multiresonances versus the driving frequency and the friction coefficient as well as stochastic resonance versus noise parameters. The necessary and sufficient conditions for the cooperation effects are also discussed. Particularly, two different critical memory exponents have been found, which mark dynamical transitions in the behavior of the system.
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Affiliation(s)
- Erkki Soika
- Institute of Mathematics and Natural Sciences, Tallinn University, 10120 Tallinn, Estonia.
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Chakrabarti R, Sebastian KL. A lower bound to the survival probability and an approximate first passage time distribution for Markovian and non-Markovian dynamics in phase space. J Chem Phys 2009; 131:224504. [PMID: 20001054 DOI: 10.1063/1.3269613] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We derive a very general expression of the survival probability and the first passage time distribution for a particle executing Brownian motion in full phase space with an absorbing boundary condition at a point in the position space, which is valid irrespective of the statistical nature of the dynamics. The expression, together with the Jensen's inequality, naturally leads to a lower bound to the actual survival probability and an approximate first passage time distribution. These are expressed in terms of the position-position, velocity-velocity, and position-velocity variances. Knowledge of these variances enables one to compute a lower bound to the survival probability and consequently the first passage distribution function. As examples, we compute these for a Gaussian Markovian process and, in the case of non-Markovian process, with an exponentially decaying friction kernel and also with a power law friction kernel. Our analysis shows that the survival probability decays exponentially at the long time irrespective of the nature of the dynamics with an exponent equal to the transition state rate constant.
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Affiliation(s)
- Rajarshi Chakrabarti
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India.
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Despósito MA, Viñales AD. Subdiffusive behavior in a trapping potential: mean square displacement and velocity autocorrelation function. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:021111. [PMID: 19792081 DOI: 10.1103/physreve.80.021111] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2009] [Indexed: 05/28/2023]
Abstract
A theoretical framework for analyzing stochastic data from single-particle tracking in viscoelastic materials and under the influence of a trapping potential is presented. Starting from a generalized Langevin equation, we found analytical expressions for the two-time dynamics of a particle subjected to a harmonic potential. The mean-square displacement and the velocity autocorrelation function of the diffusing particle are given in terms of the time lag. In particular, we investigate the subdiffusive case. Using a power-law memory kernel, exact expressions for the mean-square displacement and the velocity autocorrelation function are obtained in terms of Mittag-Leffler functions and their derivatives. The behaviors for short-, intermediate-, and long-time lags are investigated in terms of the involved parameters. Finally, the validity of usual approximations is examined.
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Affiliation(s)
- M A Despósito
- Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina.
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Chatterjee D, Cherayil BJ. Brownian particles in stationary and moving traps: the mean and variance of the heat distribution function. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:011118. [PMID: 19658664 DOI: 10.1103/physreve.80.011118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2009] [Revised: 06/02/2009] [Indexed: 05/28/2023]
Abstract
A recent theoretical model developed by Imparato [Phys. Rev. E 76, 050101(R) (2007)] of the experimentally measured heat and work effects produced by the thermal fluctuations of single micron-sized polystyrene beads in stationary and moving optical traps has proved to be quite successful in rationalizing the observed experimental data. The model, based on the overdamped Brownian dynamics of a particle in a harmonic potential that moves at a constant speed under a time-dependent force, is used to obtain an approximate expression for the distribution of the heat dissipated by the particle at long times. In this paper, we generalize the above model to consider particle dynamics in the presence of colored noise, without passing to the overdamped limit, as a way of modeling experimental situations in which the fluctuations of the medium exhibit long-lived temporal correlations, of the kind characteristic of polymeric solutions, for instance, or of similar viscoelastic fluids. Although we have not been able to find an expression for the heat distribution itself, we do obtain exact expressions for its mean and variance, both for the static and for the moving trap cases. These moments are valid for arbitrary times and they also hold in the inertial regime, but they reduce exactly to the results of Imparato in appropriate limits.
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Affiliation(s)
- Debarati Chatterjee
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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Viñales AD, Wang KG, Despósito MA. Anomalous diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:011101. [PMID: 19658647 DOI: 10.1103/physreve.80.011101] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2009] [Revised: 05/13/2009] [Indexed: 05/28/2023]
Abstract
The diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise is studied. Using the Laplace analysis we derive exact expressions for the relaxation functions of the particle in terms of generalized Mittag-Leffler functions and its derivatives from a generalized Langevin equation. Our results show that the oscillator displays an anomalous diffusive behavior. In the strictly asymptotic limit, the dynamics of the harmonic oscillator corresponds to an oscillator driven by a noise with a pure power-law autocorrelation function. However, at short and intermediate times the dynamics has qualitative difference due to the presence of the characteristic time of the noise.
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Affiliation(s)
- A D Viñales
- Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
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Chaudhury S, Chatterjee D, Cherayil BJ. The dynamics of single enzyme reactions: A reconsideration of Kramers' model for colored noise processes. J Chem Phys 2008; 129:075104. [DOI: 10.1063/1.2969767] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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