Non-linear kinetics underlying generalized statistics

Kaniadakis, Giorgio

doi:10.1016/s0378-4371(01)00184-4

Cited by 530 publications

(631 citation statements)

References 41 publications

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“…As shown in [49], Eq. (195) can be derived from the generalized Boltzmann equation introduced by Kaniadakis [46] in a weak deflexion limit. This equation conserves mass and energy and increases the generalized entropy S = − C(f )d D rd D v. This corresponds to a microcanonical structure while the Fokker-Planck equation has a canonical structure as it decreases the free energy at fixed mass and temperature.…”

Section: Generalized Landau Equationmentioning

confidence: 99%

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

204

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We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the FokkerPlanck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N → +∞, a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of the Vlasov equation. To order 1/N , the collision term of a homogeneous system has the form of the Lenard-Balescu operator. It reduces to the Landau operator when collective effects are neglected. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard-Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N → +∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ → +∞, or for large times t ≫ ξ −1 , it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, two-dimensional vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker-Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.

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Section: Generalized Landau Equationmentioning

confidence: 99%

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

204

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“…Recently, similar motivations underlying generalized statistics also lead to a new example, namely, Kaniadakis entropies [7], such as…”

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confidence: 99%

“…As one may check, we can have and the above functions could reduce to the standard exponential and logarithm if we let . In this framework, the generalized -velocity distribution function can be obtained either by maximizing the [7] or by the -H theorem in the kinetic theory [8]. It can be expressed as…”

mentioning

confidence: 99%

The κ parameter and κ-distribution in κ-deformed statistics for the systems in an external field

Guo

2007

Physics Letters A

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The -deformed statistics has been studied in many papers. It is naturally important question for us to ask what should the parameter stand for and under what physical situation should the -deformed statistics be suitable for the statistical description of a system. In this paper, we have derived a formula expression of parameter based on the -H theorem, the -velocity distribution and the generalized Boltzmann equation in the framework of -deformed statistics. We thus obtain a physical interpretation for the parameter with regard to the temperature gradient and the external force field. We show that, as the q-statistics based on Tsallis entropy, the -deformed statistics may also be the candidate one suitable for the statistical description of the systems in external fields when being in the nonequilibrium stationary state, but has different physical characteristics. Namely, the -distribution is found to describe the nonequilibrium stationary state of the system where the external force should be vertical to the temperature gradient.

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“…This issue has been discussed in literature by several authors, both in a very general nonlinear kinetic framework [22,23,24] and, more specifically, in the Tsallis entropy framework [21,25,26]. Quite generally, the NFPE associated with a given entropy S[p] can be obtained from the relation…”

Section: Nonlinear Fokker-planck Equation In the (− Q)-formalismmentioning

confidence: 99%

Equivalence among different formalisms in the Tsallis entropy framework

Scarfone

Wada

2007

Physica A: Statistical Mechanics and its Applications

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In a recent paper [Phys. Lett. A 335, 351 (2005)] the authors discussed the equivalence among the various probability distribution functions of a system in equilibrium in the Tsallis entropy framework. In the present letter we extend these results to a system which is out of equilibrium and evolves to a stationary state according to a nonlinear Fokker-Planck equation. By means of time-scale conversion, it is shown that there exists a "correspondence" among the self-similar solutions of the nonlinear Fokker-Planck equations associated with the different Tsallis formalisms. The time-scale conversion is related to the corresponding Lyapunov functions of the respective nonlinear Fokker-Planck equations.

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Non-linear kinetics underlying generalized statistics

Cited by 530 publications

References 41 publications

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

The κ parameter and κ-distribution in κ-deformed statistics for the systems in an external field

Equivalence among different formalisms in the Tsallis entropy framework

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