Classical model of bosons and fermions

Kaniadakis, Giorgio; Quarati, Piero

doi:10.1103/physreve.49.5103

Cited by 95 publications

(96 citation statements)

References 8 publications

(2 reference statements)

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“…Equation (29) determines the equilibrium state from the knowledge of the generalized entropy. Inversely, in certain cases, we know the equilibrium state and we want to determine the corresponding entropy.…”

Section: Equilibrium Statesmentioning

confidence: 99%

“…Therefore, a steady state is determined by the condition (26), where µ is a constant of integration which plays the role of a chemical potential. This condition is equivalent to Equations (27)- (29). Therefore, a steady state of the GFP equation (50) is a critical point of free energy F at fixed mass.…”

Section: Generalized H-theoremmentioning

confidence: 99%

“…(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: Equilibrium Statesmentioning

confidence: 99%

Section: Generalized H-theoremmentioning

confidence: 99%

Section: A New Form Of Generalized Entropymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Stochastic Fokker-Planck Equations

Chavanis

2015

Entropy

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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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Asymptotic properties of nonlinear diffusion, nonlinear drift-diffusion, and nonlinear reaction-diffusion equations

Frank

2004

Ann. Phys.

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We review a Fokker-Planck approach to nonlinear evolution equations such as nonlinear diffusion equations and nonlinear drift-diffusion equations and extend this approach to nonlinear reaction-diffusion equations. Using this Fokker-Planck approach along with appropriately defined entropy and free energy measures, we show that transient solutions converge to stationary ones in the long time limit. Implications for bifurcation theory are also addressed. Asymptotic propertiesLet ρ(x, t) denote the density of the subsystems of a many-body system at time t, where x ∈ Ω denotes the state variable of a single subsystem and Ω describes the phase space of the subsystem. We consider *

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Classical model of bosons and fermions

Cited by 95 publications

References 8 publications

Generalized Stochastic Fokker-Planck Equations

Generalized Stochastic Fokker-Planck Equations

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

Asymptotic properties of nonlinear diffusion, nonlinear drift-diffusion, and nonlinear reaction-diffusion equations

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