Kinetic equation for classical particles obeying an exclusion principle

Kaniadakis, Giorgio; Quarati, Piero

doi:10.1103/physreve.48.4263

Cited by 83 publications

(65 citation statements)

References 33 publications

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“…(iii) For γ = 1 and σ 0 < +∞, we get the fermionic (K = +1) or bosonic (K = −1) Smoluchowski equation, which is a GFP equation with a normal diffusion and a variable mobility taking into account exclusion or inclusion constraints in position space [20,[27][28][29]32,36,37,39,42,43,46,53]. It is associated with the Fermi-Dirac (K = +1) or Bose-Einstein (K = −1) entropy in position space…”

Section: A New Form Of Generalized Entropymentioning

confidence: 99%

“…In that case, the motion of the particles is biased, resulting in anomalous diffusion or anomalous mobility. The corresponding generalized FP equations are associated with generalized free energies and have equilibrium states different from the Boltzmann distribution [24,[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] (see [45,46] for reviews). In the situations described above, generalized entropies arise because the system experiences small-scale (hidden) constraints so that the a priori accessible microstates are not equiprobable.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Stochastic Fokker-Planck Equations

Chavanis

2015

Entropy

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We consider a system of Brownian particles with long-range interactions. We go beyond the mean field approximation and take fluctuations into account. We introduce a new class of stochastic Fokker-Planck equations associated with a generalized thermodynamical formalism. Generalized thermodynamics arises in the case of complex systems experiencing small-scale constraints. In the limit of short-range interactions, we obtain a generalized class of stochastic Cahn-Hilliard equations. Our formalism has application for several systems of physical interest including self-gravitating Brownian particles, colloid particles at a fluid interface, superconductors of type II, nucleation, the chemotaxis of bacterial populations, and two-dimensional turbulence. We also introduce a new type of generalized entropy taking into account anomalous diffusion and exclusion or inclusion constraints.

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Section: A New Form Of Generalized Entropymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized Stochastic Fokker-Planck Equations

Chavanis

2015

Entropy

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“…[33], on the basis of the generalized exclusion-inclusion principle the authors introduced a family of NFPEs describing the evolution of a classical system of particles whose statistical behavior interpolates between bosonic and fermionic particles. The equilibrium distribution governed by the EIP can be obtained by maximizing the following entropy 18) with −1 ≤ κ ≤ 1.…”

Section: Interpolating Bosons-fermions-entropymentioning

confidence: 99%

“…As working examples we present the quantization of some classical systems described by entropies already known in the literature: BG-entropy, Tsallis-entropy [32], Kaniadakis-entropy [26] and the interpolating quantum statistics entropy [33].…”

Section: Introductionmentioning

confidence: 99%

Canonical quantization of nonlinear many-body systems

Scarfone

2005

Phys. Rev. E

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We study the quantization of a classical system of interacting particles obeying a recently proposed kinetic interaction principle (KIP) [G. Kaniadakis, Physica A 296, 405 (2001)]. The KIP fixes the expression of the Fokker-Planck equation describing the kinetic evolution of the system and imposes the form of its entropy. In the framework of canonical quantization, we introduce a class of nonlinear Schrödinger equations (NSEs) with complex nonlinearities, describing, in the mean field approximation, a system of collectively interacting particles whose underlying kinetics is governed by the KIP. We derive the Ehrenfest relations and discuss the main constants of motion arising in this model. By means of a nonlinear gauge transformation of third kind it is shown that in the case of constant diffusion and linear drift the class of NSEs obeying the KIP is gauge-equivalent to another class of NSEs containing purely real nonlinearities depending only on the field ρ = |ψ| 2 .

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“…This equation is also used to study the dynamics of stochastic differential equations with Gaussian noise. Considerable interest to a description of a wide class of stochastic processes on the basis of the Fokker-Planck-Kolmogorov equation with various nonlinearity types has aroused recently (see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]). …”

Section: Introductionmentioning

confidence: 99%

Nonlinear Fokker-Planck-Kolmogorov Equation in the Semiclassical Coherent Trajectory Approximation

et al. 2005

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A semiclassical asymptotic of the Cauchy problem is constructed for the two-dimensional Fokker-Planck equation with nonlocal nonlinearity in the class of trajectory concentrated functions based on the complex WKB-Maslov method. The system of Einstein-Ehrenfest equations describing the dynamics of average values of the coordinate operator and centered moments is derived. The results obtained are illustrated by a number of examples.

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Kinetic equation for classical particles obeying an exclusion principle

Cited by 83 publications

References 33 publications

Generalized Stochastic Fokker-Planck Equations

Generalized Stochastic Fokker-Planck Equations

Canonical quantization of nonlinear many-body systems

Nonlinear Fokker-Planck-Kolmogorov Equation in the Semiclassical Coherent Trajectory Approximation

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