64 research outputs found
Statistical mechanics of the shallow-water system with a prior potential vorticity distribution
We adapt the statistical mechanics of the shallow-water equations to the case
where the flow is forced at small scales. We assume that the statistics of
forcing is encoded in a prior potential vorticity distribution which replaces
the specification of the Casimir constraints in the case of freely evolving
flows. This determines a generalized entropy functional which is maximized by
the coarse-grained PV field at statistical equilibrium. Relaxation equations
towards equilibrium are derived which conserve the robust constraints (energy,
mass and circulation) and increase the generalized entropy
Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
We study the growth of correlations in systems with weak long-range
interactions. Starting from the BBGKY hierarchy, we determine the evolution of
the two-body correlation function by using an expansion of the solutions of the
hierarchy in powers of 1/N in a proper thermodynamic limit .
These correlations are responsible for the ``collisional'' evolution of the
system beyond the Vlasov regime due to finite effects. We obtain a general
kinetic equation that can be applied to spatially inhomogeneous systems and
that takes into account memory effects. These peculiarities are specific to
systems with unshielded long-range interactions. For spatially homogeneous
systems with short memory time like plasmas, we recover the classical Landau
(or Lenard-Balescu) equations. An interest of our approach is to develop a
formalism that remains in physical space (instead of Fourier space) and that
can deal with spatially inhomogeneous systems. This enlightens the basic
physics and provides novel kinetic equations with a clear physical
interpretation. However, unless we restrict ourselves to spatially homogeneous
systems, closed kinetic equations can be obtained only if we ignore some
collective effects between particles. General exact coupled equations taking
into account collective effects are also given. We use this kinetic theory to
discuss the processes of violent collisionless relaxation and slow collisional
relaxation in systems with weak long-range interactions. In particular, we
investigate the dependence of the relaxation time with the system size and
provide a coherent discussion of all the numerical results obtained for these
systems
Improvement of leucocytic Na+K+ pump activity in uremic patients on low protein diet
Improvement of leucocytic Na+ K+ pump activity in uremic patients on low protein diet. Leucocytic Na+K+ pump activity was assessed in 20 patients with advanced renal failure. Na+K+-ATPase activity was reduced when compared with the values obtained from normal subjects (101.8 ± 48.6 versus 165.13 ± 8.9 µM of Pi hr-1 · g-1 P < 0.001) and the mean 86Rb uptake by U 937 cells was depressed by 38% after the addition of patients' sera. Subsequently, patients were put on a diet providing 0.3g protein/kg body weight daily and supplemented with ketoacids. After three months of dietary treatment Na+K+-ATPase activity increased to 142 ± 48.3 (P < 0.01) and reached normal values at the sixth month (162.8 ± 54.70 µM of Pi hr-1 · g-1; P < 0.001) whereas 86Rb uptake increased by 23 percent when compared to initial values. These data suggest that among the different mechanisms which have been advanced to explain the defects in the Na+ pump observed in uremic patients, circulating inhibitors deriving from alimentary protein intake may affect cation transport
Brownian theory of 2D turbulence and generalized thermodynamics
We propose a new parametrization of 2D turbulence based on generalized
thermodynamics and Brownian theory. Explicit relaxation equations are obtained
that should be easily implementable in numerical simulations for three typical
types of turbulent flows. Our parametrization is related to previous ones but
it removes their defects and offers attractive new perspectives.Comment: Submitted to Phys. Rev. Let
Hamiltonian and Brownian systems with long-range interactions
We discuss the dynamics and thermodynamics of systems with long-range
interactions. We contrast the microcanonical description of an isolated
Hamiltonian system to the canonical description of a stochastically forced
Brownian system. We show that the mean-field approximation is exact in a proper
thermodynamic limit. The equilibrium distribution function is solution of an
integrodifferential equation obtained from a static BBGKY-like hierarchy. It
also optimizes a thermodynamical potential (entropy or free energy) under
appropriate constraints. We discuss the kinetic theory of these systems. In the
limit, a Hamiltonian system is described by 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. We also consider the motion of a test particle in a bath
of field particles and derive the general form of the Fokker-Planck equation.
The diffusion coefficient is anisotropic and depends on the velocity of the
test particle. This can lead to anomalous diffusion. For Brownian systems, in
the limit, the kinetic equation is a non-local Kramers equation.
In the strong friction limit , or for large times , 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.Comment: The original paper has been split in two parts with some new material
and correction
Kinetic theory of Onsager's vortices in two-dimensional hydrodynamics
Starting from the Liouville equation, and using a BBGKY-like hierarchy, we
derive a kinetic equation for the point vortex gas in two-dimensional (2D)
hydrodynamics, taking two-body correlations and collective effects into
account. This equation is valid at the order 1/N where N>>1 is the number of
point vortices in the system (we assume that their individual circulation
scales like \gamma ~ 1/N). It gives the first correction, due to graininess and
correlation effects, to the 2D Euler equation that is obtained for
. For axisymmetric distributions, this kinetic equation
does not relax towards the Boltzmann distribution of statistical equilibrium.
This implies either that (i) the "collisional" (correlational) relaxation time
is larger than Nt_D, where t_D is the dynamical time, so that three-body,
four-body... correlations must be taken into account in the kinetic theory, or
(ii) that the point vortex gas is non-ergodic (or does not mix well) and will
never attain statistical equilibrium. Non-axisymmetric distributions may relax
towards the Boltzmann distribution on a timescale of the order Nt_D due to the
existence of additional resonances, but this is hard to prove from the kinetic
theory. On the other hand, 2D Euler unstable vortex distributions can
experience a process of "collisionless" (correlationless) violent relaxation
towards a non-Boltzmannian quasistationary state (QSS) on a very short
timescale of the order of a few dynamical times. This QSS is possibly described
by the Miller-Robert-Sommeria (MRS) statistical theory which is the
counterpart, in the context of two-dimensional hydrodynamics, of the
Lynden-Bell statistical theory of violent relaxation in stellar dynamics
Brownian particles with long and short range interactions
We develop a kinetic theory of Brownian particles with long and short range
interactions. We consider both overdamped and inertial models. In the
overdamped limit, the evolution of the spatial density is governed by the
generalized mean field Smoluchowski equation including a mean field potential
due to long-range interactions and a generically nonlinear barotropic pressure
due to short-range interactions. This equation describes various physical
systems such as self-gravitating Brownian particles (Smoluchowski-Poisson
system), bacterial populations experiencing chemotaxis (Keller-Segel model) and
colloidal particles with capillary interactions. We also take into account the
inertia of the particles and derive corresponding kinetic and hydrodynamic
equations generalizing the usual Kramers, Jeans, Euler and Cattaneo equations.
For each model, we provide the corresponding form of free energy and establish
the H-theorem and the virial theorem. Finally, we show that the same
hydrodynamic equations are obtained in the context of nonlinear mean field
Fokker-Planck equations associated with generalized thermodynamics. However, in
that case, the nonlinear pressure is due to the bias in the transition
probabilities from one state to the other leading to non-Boltzmannian
distributions while in the former case the distribution is Boltzmannian but the
nonlinear pressure arises from the two-body correlation function induced by the
short-range potential of interaction. As a whole, our paper develops
connections between the topics of long-range interactions, short-range
interactions, nonlinear mean field Fokker-Planck equations and generalized
thermodynamics. It also justifies from a kinetic theory based on microscopic
processes, the basic equations that were introduced phenomenologically in
gravitational Brownian dynamics, chemotaxis and colloidal suspensions with
attractive interactions
Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
We introduce a new set of generalized Fokker-Planck equations that conserve
energy and mass and increase a generalized entropy until a maximum entropy
state is reached. The concept of generalized entropies is rigorously justified
for continuous Hamiltonian systems undergoing violent relaxation. Tsallis
entropies are just a special case of this generalized thermodynamics.
Application of these results to stellar dynamics, vortex dynamics and Jupiter's
great red spot are proposed. Our prime result is a novel relaxation equation
that should offer an easily implementable parametrization of geophysical
turbulence. This relaxation equation depends on a single key parameter related
to the skewness of the fine-grained vorticity distribution. Usual
parametrizations (including a single turbulent viscosity) correspond to the
infinite temperature limit of our model. They forget a fundamental systematic
drift that acts against diffusion as in Brownian theory. Our generalized
Fokker-Planck equations may have applications in other fields of physics such
as chemotaxis for bacterial populations. We propose the idea of a
classification of generalized entropies in classes of equivalence and provide
an aesthetic connexion between topics (vortices, stars, bacteries,...) which
were previously disconnected.Comment: Submitted to Phys. Rev.
Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations
We develop a theory of fluctuations for Brownian systems with weak long-range
interactions. For these systems, there exists a critical point separating a
homogeneous phase from an inhomogeneous phase. Starting from the stochastic
Smoluchowski equation governing the evolution of the fluctuating density field,
we determine the expression of the correlation function of the density
fluctuations around a spatially homogeneous equilibrium distribution. In the
stable regime, we find that the temporal correlation function of the Fourier
components of the density fluctuations decays exponentially rapidly with the
same rate as the one characterizing the damping of a perturbation governed by
the mean field Smoluchowski equation (without noise). On the other hand, the
amplitude of the spatial correlation function in Fourier space diverges at the
critical point (or at the instability threshold ) implying
that the mean field approximation breaks down close to the critical point and
that the phase transition from the homogeneous phase to the inhomogeneous phase
occurs sooner. By contrast, the correlations of the velocity fluctuations
remain finite at the critical point (or at the instability threshold). We give
explicit examples for the Brownian Mean Field (BMF) model and for Brownian
particles interacting via the gravitational potential and via the attractive
Yukawa potential. We also introduce a stochastic model of chemotaxis for
bacterial populations generalizing the mean field Keller-Segel model by taking
into account fluctuations
Low incidence of SARS-CoV-2, risk factors of mortality and the course of illness in the French national cohort of dialysis patients
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