Non-linear relativistic diffusions

Haba, Z.

doi:10.1016/j.physa.2011.03.025

Cited by 7 publications

(8 citation statements)

References 78 publications

(65 reference statements)

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“…The classical (Boltzmann) statistics can be described by ν = 0. The relation to Kompaneets equation [26][27]is discussed in [25].…”

Section: Relativistic Diffusionmentioning

confidence: 99%

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Einstein gravity of a diffusing fluid

Haba

2014

Class. Quantum Grav.

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We discuss Einstein gravity for a fluid consisting of particles interacting with an environment of some other particles. The environment is described by a time-dependent cosmological term which is compensating the lack of the conservation law of the energy-momentum of the dissipative fluid. The dissipation is approximated by a relativistic diffusion in the phase space. We are interested in a homogeneous isotropic flat expanding Universe described by a scale factor a. At an early stage the particles are massless. We obtain explicit solutions of the diffusion equation for a fluid of massless particles at finite temperature. The solution is of the form of a modified Jüttner distribution with a time-dependent temperature. At later time Universe evolution is described as a diffusion at zero temperature with no equilibration. We find solutions of the diffusion equation at zero temperature which can be treated as a continuation to a later time of the finite temperature solutions describing an early stage of the Universe. The energy-momentum of the diffusing particles is defined by their phase space distribution. A conservation of the total energy-momentum determines the cosmological term up to a constant. The resulting energymomentum inserted into Einstein equations gives a modified Friedmann equation. Solutions of the Friedmann equation depend on the initial value of the cosmological term. The large value of the cosmological constant implies an exponential expansion. If the initial conditions allow a power-like solution for a large time then it must be of the form a ≃ τ (no deceleration, τ is the cosmic time) . The final stage of the Universe evolution is described by a non-relativistic diffusion of a cold dust.

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“…The classical (Boltzmann) statistics can be described by ν = 0. The relation to Kompaneets equation [26][27]is discussed in [25].…”

Section: Relativistic Diffusionmentioning

confidence: 99%

“…A model with friction leading to the Jüttner equilibrium distribution [20] is discussed in [21][22][23][24]. An equilibration to a quantum distribution requires a nonlinear diffusion [25] which describes the bunching of Bose particles and repulsion of Fermi particles. We treat the linear diffusion as an approximation to the nonlinear one.…”

Section: Introductionmentioning

confidence: 99%

Einstein gravity of a diffusing fluid

Haba

2014

Class. Quantum Grav.

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“…△ m H is the generator of the diffusion defined first by Schay [10] and Dudley [11]. The drift (77) can also be restricted to the mass-shell [22]- [23] (it determines the exponential speed of the decay to the equilibrium, see [34]).…”

Section: Random Dynamicsmentioning

confidence: 99%

“…Using the vector w µ we can define a drift by means of an expectation value of the electromagnetic field F µν . We have introduced such a drift in our earlier papers [22] [23] as a friction which brings the diffusion to the Jüttner equilibrium on the basis of the detailed balance condition (a generalization of the Ornstein-Uhlenbeck process). In sec.5 we show that the diffusion process with a continuous mass spectrum can be decomposed into the processes on the mass-shell introduced first by Schay [10] and Dudley [11].…”

Section: Introductionmentioning

confidence: 99%

Relativistic diffusion of particles with a continuous mass spectrum

Haba

2012

Preprint

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We discuss general positivity conditions necessary for a definition of a relativistic diffusion on the phase space. We show that Lorentz covariant random vector fields on the forward cone p 2 ≥ 0 lead to a definition of a generator of Lorentz covariant diffusions. We discuss in more detail diffusions arising from particle dynamics in a random electromagnetic field approximating the quantum field at finite temperature. We develop statistical mechanics of a gas of diffusing particles. We discuss viscosity of such a gas in an expansion of the energy momentum tensor in gradients of the fluid velocity.

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“…Thus, the notion of distance remains well-defined also in the spatially inhomogeneous case. Recall that the Kullback-Leibler distance, which has many applications (see, e.g., [5,18,21] and references therein), is defined only in the spatially homogeneous case.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Fokker–Planck Equation: Stability, Distance and the Corresponding Extremal Problem in the Spatially Inhomogeneous Case

Sakhnovich

2015

Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes

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We start with a global Maxwellian M k , which is a stationary solution, with the constant total density (ρ(t) ≡ ρ), of the Fokker-Planck equation. The notion of distance between the function M k and an arbitrary solution f (with the same total density ρ at the fixed moment t) of the Fokker-Planck equation is introduced. In this way, we essentially generalize the important Kullback-Leibler distance, which was studied before. Using this generalization, we show local stability of the global Maxwellians in the spatially inhomogeneous case. We compare also the energy and entropy in the classical and quantum cases.

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Non-linear relativistic diffusions

Cited by 7 publications

References 78 publications

Einstein gravity of a diffusing fluid

Einstein gravity of a diffusing fluid

Relativistic diffusion of particles with a continuous mass spectrum

Nonlinear Fokker–Planck Equation: Stability, Distance and the Corresponding Extremal Problem in the Spatially Inhomogeneous Case

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