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# Nonlinear drag forces and the thermostatistics of overdamped motion

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## Abstract

Diverse processes in statistical physics are usually analyzed on the assumption that the drag force acting on a test particle moving in a resisting medium is linear on the velocity of the particle. However, nonlinear drag forces do appear in relevant situations that are currently the focus of experimental and theoretical work. Motivated by these developments, we explore the consequences of nonlinear drag forces for the thermostatistics of systems of interacting particles performing overdamped motion. We derive a family of nonlinear Fokker-Planck equations for these systems, taking into account the effects of nonlinear drag forces. We investigate the main properties of these evolution equations, including an H-theorem, and obtain exact solutions of the stretched q-exponential form.

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... Moreover, such equations have properties that are relevant and/or interesting from both a physical and mathematical point of view. For instance, they admit exact analytical solutions of the q-Gaussian form that can be interpreted as maximum entropy densities obtainable from the optimization (under appropriate constraints) of S q ; they have been studied in the context of entropy production [13,14]; with different drift forces [15,16]; they obey an H-theorem formulated in terms of a free-energy-like quantity [17,18], and so on. In particular, we highlight an experimental work on granular media [5], that verified within great precision (2% error) the Tsallis and Bukman's [10] prediction 19 years after the original proposal. ...
... (k) on the right-hand side of (A4) is aq-Gaussian distribution (up to normalization), given by (16). Hence, F q [S N ](k) isq-Gaussian of the form: ...
... with p = 2 + d(1 − q). Comparing the above equation with (16), the variance of S N reads ...
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... Among uncountable other illustrations, we may mention lateral force microscope experiments in silicon, Si(100) surfaces [33], and similar phenomena in bio-inspired, asymmetrically-structured surfaces [34]. In the case of over-damped systems described by NLFPEs, isotropic, non-linear drag forces were considered in [35], and non-isotropic, linear drag forces in [36]. In the present work, we investigate a system with drag forces that are both non-linear and non-isotropic. ...
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We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those moving to the left. The dynamics of the system, effectively described by a non-linear, Fokker–Planck equation, exhibits peculiar features related to the way in which the drag force depends on velocity. The evolution equation satisfies an H-theorem involving the Sq nonadditive entropy, and admits particular, exact, time-dependent solutions closely related, but not identical, to the q-Gaussian densities. The departure from the canonical, q-Gaussian shape is related to the fact that in one spatial dimension, in contrast to what occurs in two or more spatial dimensions, the drag’s dependence on direction entails that its dependence on velocity is necessarily (and severely) non-linear. The results reported here provide further evidence of the deep connections between overdamped, many-body systems, non-linear Fokker–Planck equations, and the Sq-thermostatistics.
... The standard dissipative acceleration α disss , equation (36), is a parabolic function of angular velocity ω [93]. Some authors have used power-laws [30,64,[94][95][96][97][98][99]. In order to test if such models provide better fits we considered the following power-law ...
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... Therein, the first correction in ζ 0 introduces a quadratic dependence on v. [34][35][36] In some situations, nonlinearities in the drag coefficient have quite strong physical implications. [36][37][38][39][40] The main goal of this paper is to study the Mpemba effect in the kinetic theory framework we have just described, i.e., the Enskog-Fokker-Planck equation with nonlinear drag. To meet this end, we work using the first Sonine approximation, in which the time evolution of temperature is coupled to that of excess kurtosis. ...
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... The standard dissipative acceleration α disss , equation (35), is a parabolic function of angular velocity ω [91]. Some authors have used power-laws [30,61,[92][93][94][95][96][97]. In order to test if such models provide better fits we considered the following power-law ...
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... Therein, the first correction in ζ 0 introduces a quadratic dependence on v. [31][32][33] In some situations, nonlinearities in the drag coefficient have quite strong physical implications. [33][34][35][36][37] The main goal of this paper is to study the Mpemba effect in the kinetic theory framework we have just described, i.e., the Enskog-Fokker-Planck equation with nonlinear drag. To meet this end, we work in the first Sonine approximation, in which the time evolution of the temperature is coupled to that of the excess kurtosis. ...
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The problem of the limits of validity of the Langevin equation is considered in detail in the case of (microscopic) test-particles in very dilute gases. It is shown that, in this case, the current Langevin equation follows from the Newton’s law in an exact way only in the Maxwell test-particle–gas-particle interaction model, and in an approximate way only in the Rayleigh-gas limit and in the low-velocity limit, while in any other interaction model, or limit, only a Langevin-like equation with speed-dependent friction coefficient and speed-dependent fluctuating force can be written. Such a circumstance, although probably limited to the particular physical situation considered in this paper, suggests that, in general, some preliminary, specific check of the validity of the Langevin equation should be performed before using the said equation to interpret laboratory experiments.
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We consider nonextensive systems that are related to the nonextensive entropy proposed by Tsallis and can be described by means of the nonlinear porous medium equation and the nonlinear Fokker–Planck equation proposed by Plastino and Plastino. We show how to determine the degree of nonextensivity of these systems from experimental data. Both transient and stationary cases are addressed.
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A model is developed of the interstellar diffusion of galactic civilizations which takes into account the population dynamics of such civilizations. The problem is formulated in terms of potential theory, with a family of nonlinear partial differential and difference equations specifying population growth and diffusion for an organism with advantageous genes that undergoes random dispersal while increasing in population locally, and a population at zero population growth. In the case of nonlinear diffusion with growth and saturation, it is found that the colonization wavefront from the nearest independently arisen galactic civilization can have reached the earth only if its lifetime exceeds 2.6 million years, or 20 million years if discretization can be neglected. For zero population growth, the corresponding lifetime is 13 billion years. It is concluded that the earth is uncolonized not because interstellar spacefaring civilizations are rare, but because there are too many worlds to be colonized in the plausible colonization lifetime of nearby civilizations, and that there exist no very old galactic civilizations with a consistent policy of the conquest of inhabited worlds.
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The formalism of statistical mechanics can be generalised by starting from more general measures of information than the Shannon entropy and maximising those subject to suitable constraints. We discuss some of the most important examples of information measures that are useful for the description of complex systems. Examples treated are the Rényi entropy, Tsallis entropy, Abe entropy, Kaniadakis entropy, Sharma?Mittal entropies, and a few more. Important concepts such as the axiomatic foundations, composability and Lesche stability of information measures are briefly discussed. Potential applications in physics include complex systems with long-range interactions and metastable states, scattering processes in particle physics, hydrodynamic turbulence, defect turbulence, optical lattices, and quite generally driven nonequilibrium systems with fluctuations of temperature.
Article
We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2008
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A system of interacting vortices under overdamped motion, which has been commonly used in the literature to model flux-front penetration in disordered type-II superconductors, was recently related to a nonlinear Fokker-Planck equation, characteristic of nonextensive statistical mechanics, through an analysis of its stationary state. Herein, this connection is extended by means of a thorough analysis of the time evolution of this system. Numerical data from molecular-dynamics simulations are presented for both position and velocity probability distributions P(x,t) and P(v(x),t), respectively; both distributions are well fitted by similar q-Gaussian distributions, with the same index q=0, for all times considered. Particularly, the evolution of the system occurs in such a way that P(x,t) presents a time behavior for its width, normalization, and second moment, in full agreement with the analytic solution of the nonlinear Fokker-Planck equation. The present results provide further evidence that this system is deeply associated with nonextensive statistical mechanics.
Article
We briefly discuss the state of the art on the anomalous dynamics of the Hamiltonian mean field (HMF) model. We stress the important role of the initial conditions for understanding the microscopic nature of the intriguing metastable quasi-stationary states (QSS) observed in the model and the connections to Tsallis statistics and glassy dynamics. We also present new results on the existence of metastable states in the Kuramoto model and discuss the similarities with those found in the HMF model. The existence of metastability seems to be quite a common phenomenon in fully coupled systems, whose origin could be also interpreted as a dynamical mechanism preventing or hindering synchronization.
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Tsallis’ q-triplet [C. Tsallis, Dynamical scenario for nonextensive statistical mechanics, Physica A 340 (2004) 1–10] is the best empirical quantifier of nonextensivity. Here we study it with reference to an experimental time-series related to the daily depth-values of the stratospheric ozone layer. Pertinent data are expressed in Dobson units and range from 1978 to 2005. After the evaluation of the three associated Tsallis’ indices one concludes that nonextensivity is clearly a characteristic of the system under scrutiny.
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Tsallis [Physica A 340 (2004) 1] identified a set of numbers, the “q-triplet” ≡{qstat, qsen, qrel}, for a system described by nonextensive statistical mechanics. The deviation of the q's from unity is a measure of the departure from thermodynamic equilibrium. We present observations of the q-triplets derived from two sets of daily averages of the magnetic field strength B observed by Voyager 1 in the solar wind near 40 AU during 1989 and near 85 AU during 2002, respectively. The results for 1989 do not differ significantly from those for 2002. We find qstat=1.75±0.06, qsen=-0.6±0.2, and qrel=3.8±0.3.
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Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.
Article
We show through a nonlinear Fokker-Planck formalism, and confirm by molecular dynamics simulations, that the overdamped motion of interacting particles at T=0, where T is the temperature of a thermal bath connected to the system, can be directly associated with Tsallis thermostatistics. For sufficiently high values of T, the distribution of particles becomes Gaussian, so that the classical Boltzmann-Gibbs behavior is recovered. For intermediate temperatures of the thermal bath, the system displays a mixed behavior that follows a novel type of thermostatistics, where the entropy is given by a linear combination of Tsallis and Boltzmann-Gibbs entropies.
Article
Driven anomalous diffusions (such as those occurring in some surface growths) are currently described through the nonlinear Fokker-Planck-like equation (∂/∂t)pmu=-(∂/∂x)[F(x)pmu]+D(∂2/∂x2 )pnu [(mu,nu)∈R2 F(x)=k1-k2x is the external force; k2>=0]. We exhibit here the (physically relevant) exact solution for all (x,t). This solution was found through an ansatz based on the generalized entropic form Sq[p]=\{1-∫du[p(u)]q\}/(q-1) (with q∈R), in a completely analogous manner through which the usual entropy S1[p]=-∫dup(u)lnp(u) is known to provide the correct ansatz for exactly solving the standard Fokker-Planck equation (mu=nu=1). This remarkably simple unification of normal diffusion (q=1), superdiffusion (q>1) and subdiffusion (q
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We investigate probability density functions of velocity differences at different distances r measured in a Couette-Taylor flow for a range of Reynolds numbers Re. There is good agreement with the predictions of a theoretical model based on nonextensive statistical mechanics (where the entropies are nonadditive for independent subsystems). We extract the scale-dependent nonextensitivity parameter q(r,Re) from the laboratory data.
Article
Multidimensional nonlinear Fokker-Planck equations of mean-field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions. Two types of eigenvalue equations are studied. The nonlinear Fokker-Planck equations are shown to exhibit an H theorem with a Liapunov functional that takes the form of a free energy involving generalized entropies of Tsallis. The second-order variation of the Liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean-field coupling potential. Assuming quasiequilibrium for the velocity distribution, the reduced eigenvalue equation with space coordinates alone is also obtained. The alternative type of eigenvalue equation based on the linearization of the nonlinear Fokker-Planck equations is presented. Taking the mean-field coupling potential to be the gravitational one, the nonlinear Fokker-Planck equation in terms of three-dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean-field model of self-gravitating system. By solving the eigenvalue equation for the eigenfunction with 0 eigenvalue, the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed.
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We introduce a class of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy functional until a maximum entropy state is reached. Nonlinear Fokker-Planck equations associated with Tsallis entropies are a special case of these equations. Applications of these results to stellar dynamics and vortex dynamics are proposed. Our prime result is a relaxation equation that should offer an easily implementable parametrization of two-dimensional turbulence. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations can have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in "classes of equivalence" and provide an aesthetic connection between topics (vortices, stars, bacteria, em leader ) which were previously disconnected.
Article
A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard--linear Fokker-Planck equation--may be also related to a family of nonlinear Fokker-Planck equations.