1,324 research outputs found
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.
Heat and work distributions for mixed Gauss-Cauchy process
We analyze energetics of a non-Gaussian process described by a stochastic
differential equation of the Langevin type. The process represents a
paradigmatic model of a nonequilibrium system subject to thermal fluctuations
and additional external noise, with both sources of perturbations considered as
additive and statistically independent forcings. We define thermodynamic
quantities for trajectories of the process and analyze contributions to
mechanical work and heat. As a working example we consider a particle subjected
to a drag force and two independent Levy white noises with stability indices
and . The fluctuations of dissipated energy (heat) and
distribution of work performed by the force acting on the system are addressed
by examining contributions of Cauchy fluctuations to either bath or external
force acting on the system
Spin diffusion from an inhomogeneous quench in an integrable system
Generalised hydrodynamics predicts universal ballistic transport in
integrable lattice systems when prepared in generic inhomogeneous initial
states. However, the ballistic contribution to transport can vanish in systems
with additional discrete symmetries. Here we perform large scale numerical
simulations of spin dynamics in the anisotropic Heisenberg spin
chain starting from an inhomogeneous mixed initial state which is symmetric
with respect to a combination of spin-reversal and spatial reflection. In the
isotropic and easy-axis regimes we find non-ballistic spin transport which we
analyse in detail in terms of scaling exponents of the transported
magnetisation and scaling profiles of the spin density. While in the easy-axis
regime we find accurate evidence of normal diffusion, the spin transport in the
isotropic case is clearly super-diffusive, with the scaling exponent very close
to , but with universal scaling dynamics which obeys the diffusion
equation in nonlinearly scaled time.Comment: 8 pages, 7 figures, version as accepted by Nature Communication
Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
We study a general class of nonlinear mean field Fokker-Planck equations in
relation with an effective generalized thermodynamical formalism. We show that
these equations describe several physical systems such as: chemotaxis of
bacterial populations, Bose-Einstein condensation in the canonical ensemble,
porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model,
Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian
particles, Debye-Huckel theory of electrolytes, two-dimensional turbulence...
In particular, we show that nonlinear mean field Fokker-Planck equations can
provide generalized Keller-Segel models describing the chemotaxis of biological
populations. As an example, we introduce a new model of chemotaxis
incorporating both effects of anomalous diffusion and exclusion principle
(volume filling). Therefore, the notion of generalized thermodynamics can have
applications for concrete physical systems. We also consider nonlinear mean
field Fokker-Planck equations in phase space and show the passage from the
generalized Kramers equation to the generalized Smoluchowski equation in a
strong friction limit. Our formalism is simple and illustrated by several
explicit examples corresponding to Boltzmann, Tsallis and Fermi-Dirac entropies
among others
Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology
We discuss a general class of nonlinear mean-field Fokker-Planck equations
[P.H. Chavanis, Phys. Rev. E, 68, 036108 (2003)] and show their applications in
different domains of physics, astrophysics and biology. These equations are
associated with generalized entropic functionals and non-Boltzmannian
distributions (Fermi-Dirac, Bose-Einstein, Tsallis,...). They furthermore
involve an arbitrary binary potential of interaction. We emphasize analogies
between different topics (two-dimensional turbulence, self-gravitating systems,
Debye-H\"uckel theory of electrolytes, porous media, chemotaxis of bacterial
populations, Bose-Einstein condensation, BMF model, Cahn-Hilliard
equations,...) which were previously disconnected. All these examples (and
probably many others) are particular cases of this general class of nonlinear
mean-field Fokker-Planck equations
Entropy production in the cyclic lattice Lotka-Volterra model
The cyclic Lotka-Volterra model in a -dimensional regular lattice is
considered. Its ``nucleus growth'' mode is analyzed under the scope of Tsallis'
entropies , . It is shown both
numerically and by means of analytical considerations that a linear increase of
entropy with time, meaning finite asymptotic entropy rate, is achieved for the
entropic index . Although the lattice exhibits fractal patterns
along its evolution, the characteristic value of can be interpreted in
terms of very simple features of the dynamics.Comment: 7 pages, 6 figure
Heavy-tailed targets and (ab)normal asymptotics in diffusive motion
We investigate temporal behavior of probability density functions (pdfs) of
paradigmatic jump-type and continuous processes that, under confining regimes,
share common heavy-tailed asymptotic (target) pdfs. Namely, we have shown that
under suitable confinement conditions, the ordinary Fokker-Planck equation may
generate non-Gaussian heavy-tailed pdfs (like e.g. Cauchy or more general
L\'evy stable distribution) in its long time asymptotics. For diffusion-type
processes, our main focus is on their transient regimes and specifically the
crossover features, when initially infinite number of the pdf moments drops
down to a few or none at all. The time-dependence of the variance (if in
existence), with , in principle may be
interpreted as a signature of sub-, normal or super-diffusive behavior under
confining conditions; the exponent is generically well defined in
substantial periods of time. However, there is no indication of any universal
time rate hierarchy, due to a proper choice of the driver and/or external
potential.Comment: Major revisio
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