Generalized Stochastic Fokker-Planck Equations

Chavanis, Pierre‐Henri

doi:10.3390/e17053205

Cited by 34 publications

(80 citation statements)

References 91 publications

(208 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equation (1.1), where u is a probability density, is known in the literature as the nonlinear Fokker-Planck equation (NFPE) and it is relevant in the kinetic theory of statistical mechanics as a generalized mean field Smoluchowski equation for the case where the diffusion and transport coefficients depend on the density u. (See [13], [18]- [20] [26].) The case of the classical Smoluchowski equation is recovered for b ≡ 1 and β(r) ≡ r. The existence and uniqueness of a kinetic, respectively generalized entropic, solution to (1.1) in L 1 (R d ) was firstly proved by G.Q.…”

Section: Introductionmentioning

confidence: 99%

Generalized solutions to nonlinear Fokker–Planck equations

Barbu

2016

Journal of Differential Equations

View full text Add to dashboard Cite

One proves the H-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equationand under appropriate hypotheses on β, D and b the convergence inrespectively, for some t n → ∞ of the solution u(t n ) to an equilibrium state of the equation for a large set of nonnegative initial data in L 1 . These results are new in the literature on nonlinear Fokker-Planck equations. Furthermore, the solution to the McKean-Vlasov stochastic differential equation corresponding to (1), which is a nonlinear distorted Brownian motion, is shown to have this equilibrium state as its unique invariant measure.

show abstract

Section: Introductionmentioning

confidence: 99%

Generalized solutions to nonlinear Fokker–Planck equations

Barbu

2016

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…C ′′ (f ) > 0). This is what we call a generalized entropy [214,221,224]. These functionals appeared in relation to the notion of "generalized thermodynamics" pioneered by Tsallis [225] who introduced a particular form of non-Boltzmannian entropy (of a power-law type) called the Tsallis entropy.…”

Section: Ultrarelativistic Limitmentioning

confidence: 99%

“…This also suggests that what we call "generalized thermodynamics" is just "standard thermodynamics" with a generalized form of entropy taking into account microscopic constraints[221,224].…”

mentioning

confidence: 99%

Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

Chavanis

2020

Eur. Phys. J. Plus

Self Cite

View full text Add to dashboard Cite

We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi-Dirac entropy for fermions. The maximization of entropy at fixed mass-energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium and the Tolman-Klein relations. Our paper provides all the necessary equations that are needed to construct the caloric curves of selfgravitating fermions in general relativity as done in recent works. We consider the nonrelativistic limit c → +∞ and recover the equations obtained within the framework of Newtonian gravity. We also discuss the inequivalence of statistical ensembles as well as the relation between the dynamical and thermodynamical stability of self-gravitating systems in Newtonian gravity and general relativity.PACS numbers: 04.40. Dg, 05.70.Fh, 95.30.Sf, 95.35.+d I. INTRODUCTIONSelf-gravitating fermions play an important role in different areas of astrophysics. They appeared in the context of white dwarfs, neutron stars and dark matter halos where the fermions are electrons, neutrons and massive neutrinos respectively. We start by a brief history of the subject giving an exhaustive list of references. 1 Soon after the discovery of the quantum statistics by Fermi [3,4] and Dirac [5] in 1926, Fowler [6] used this "new thermodynamics" to solve the puzzle of the extreme high density of white dwarfs, which could not be explained by classical physics [7]. He understood that white dwarfs owe their stability to the quantum pressure of the degenerate electron gas resulting from the Pauli exclusion principle [8]. He considered a completely degenerate electron gas at T = 0 based on the fact that the temperature in white dwarfs is much smaller than the Fermi temperature (T ≪ T F ). He also used Newtonian gravity which is a very good approximation to describe white dwarfs in general. The first models [9-11] of white dwarfs were based on the nonrelativistic equation of state of a Fermi gas and provided the corresponding mass-radius relation. Stoner [9] developed an analytical approach based on a uniform density approximation for the star. Chandrasekhar [11] derived the exact mass-radius relation of nonrelativistic white dwarfs by applying the theory of polytropes of index n = 3/2 [12]. It was then realized that special relativity must be taken into account at high densities. When the relativistic equation of state is employed it was found that white dwarfs can exist only below a maximum mass M max = 1.42 M ⊙ [13-25], now known as the Chandrasekhar limiting mass. Frenkel [13] was the first to mention that relativistic effects become important when the mass of white dwarfs becomes larger than the solar mass but he did not envision the existence of an upper mass limit. The maxi...

show abstract

“…In the laste decades, several authors have tried to generalize standard thermodynamics and Brownian theory (see, e.g., [42,51,53,54] for reviews). For example, one can encounter situations in which the diffusion coefficient of the particles depends on their density as a power-law.…”

Section: Introductionmentioning

confidence: 99%

Generalized Euler, Smoluchowski and Schrödinger equations admitting self-similar solutions with a Tsallis invariant profile

Chavanis

2019

Eur. Phys. J. Plus

Self Cite

View full text Add to dashboard Cite

The damped isothermal Euler equations, the Smoluchowski equation and the damped logarithmic Schrödinger equation with a harmonic potential admit stationary and self-similar solutions with a Gaussian profile. They satisfy an H-theorem for a free energy functional involving the von Weizsäcker functional and the Boltzmann functional. We derive generalized forms of these equations in order to obtain stationary and self-similar solutions with a Tsallis profile. In particular, we introduce a nonlinear Schrödinger equation involving a generalized kinetic term characterized by an index q and a power-law nonlinearity characterized by an index γ. We derive an H-theorem satisfied by a generalized free energy functional involving a generalized von Weizsäcker functional (associated with q) and a Tsallis functional (associated with γ). This leads to a notion of generalized quantum mechanics and generalized thermodynamics. When q = 2γ − 1, our nonlinear Schrödinger equation admits an exact self-similar solution with a Tsallis invariant profile. Standard quantum mechanics (Schrödinger) and standard thermodynamics (Boltzmann) are recovered for q = γ = 1.2 In the absence of friction (ξ = 0) we get the quantum isothermal Euler equations and in the strong friction limit (ξ → +∞) we get the quantum Smoluchowski equation [58]. This provides an interesting formal connection between quantum mechanics and Brownian theory [55].

show abstract

Generalized Stochastic Fokker-Planck Equations

Cited by 34 publications

References 91 publications

Generalized solutions to nonlinear Fokker–Planck equations

Generalized solutions to nonlinear Fokker–Planck equations

Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

Generalized Euler, Smoluchowski and Schrödinger equations admitting self-similar solutions with a Tsallis invariant profile

Contact Info

Product

Resources

About