7,601 research outputs found
A parametrization of two-dimensional turbulence based on a maximum entropy production principle with a local conservation of energy
In the context of two-dimensional (2D) turbulence, we apply the maximum
entropy production principle (MEPP) by enforcing a local conservation of
energy. This leads to an equation for the vorticity distribution that conserves
all the Casimirs, the energy, and that increases monotonically the mixing
entropy (-theorem). Furthermore, the equation for the coarse-grained
vorticity dissipates monotonically all the generalized enstrophies. These
equations may provide a parametrization of 2D turbulence. They do not generally
relax towards the maximum entropy state. The vorticity current vanishes for any
steady state of the 2D Euler equation. Interestingly, the equation for the
coarse-grained vorticity obtained from the MEPP turns out to coincide, after
some algebraic manipulations, with the one obtained with the anticipated
vorticity method. This shows a connection between these two approaches when the
conservation of energy is treated locally. Furthermore, the newly derived
equation, which incorporates a diffusion term and a drift term, has a nice
physical interpretation in terms of a selective decay principle. This gives a
new light to both the MEPP and the anticipated vorticity method.Comment: To appear in the special IUTAM issue of Fluid Dynamics Research on
Vortex Dynamic
Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution
Using a Maximum Entropy Production Principle (MEPP), we derive a new type of
relaxation equations for two-dimensional turbulent flows in the case where a
prior vorticity distribution is prescribed instead of the Casimir constraints
[Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a
Gaussian prior is specifically treated in connection to minimum enstrophy
states and Fofonoff flows. These relaxation equations are compared with other
relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776
(1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a
small-scale parametrization of 2D turbulence or serve as numerical algorithms
to compute maximum entropy states with appropriate constraints. We perform
numerical simulations of these relaxation equations in order to illustrate
geometry induced phase transitions in geophysical flows.Comment: 21 pages, 9 figure
Moment free energies for polydisperse systems
A polydisperse system contains particles with at least one attribute
(such as particle size in colloids or chain length in polymers) which takes
values in a continuous range. It therefore has an infinite number of conserved
densities, described by a density {\em distribution} . The free
energy depends on all details of , making the analysis of phase
equilibria in such systems intractable. However, in many (especially
mean-field) models the {\em excess} free energy only depends on a finite number
of (generalized) moments of ; we call these models truncatable.
We show, for these models, how to derive approximate expressions for the {\em
total} free energy which only depend on such moment densities. Our treatment
unifies and explores in detail two recent separate proposals by the authors for
the construction of such moment free energies. We show that even though the
moment free energy only depends on a finite number of density variables, it
gives the same spinodals and critical points as the original free energy and
also correctly locates the onset of phase coexistence. Results from the moment
free energy for the coexistence of two or more phases occupying comparable
volumes are only approximate, but can be refined arbitrarily by retaining
additional moment densities. Applications to Flory-Huggins theory for
length-polydisperse homopolymers, and for chemically polydisperse copolymers,
show that the moment free energy approach is computationally robust and gives
new geometrical insights into the thermodynamics of polydispersity.Comment: RevTeX, 43 pages including figure
Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit
We complete the literature on the statistical mechanics of point vortices in
two-dimensional hydrodynamics. Using a maximum entropy principle, we determine
the multi-species Boltzmann-Poisson equation and establish a form of virial
theorem. Using a maximum entropy production principle (MEPP), we derive a set
of relaxation equations towards statistical equilibrium. These relaxation
equations can be used as a numerical algorithm to compute the maximum entropy
state. We mention the analogies with the Fokker-Planck equations derived by
Debye and H\"uckel for electrolytes. We then consider the limit of strong
mixing (or low energy). To leading order, the relationship between the
vorticity and the stream function at equilibrium is linear and the maximization
of the entropy becomes equivalent to the minimization of the enstrophy. This
expansion is similar to the Debye-H\"uckel approximation for electrolytes,
except that the temperature is negative instead of positive so that the
effective interaction between like-sign vortices is attractive instead of
repulsive. This leads to an organization at large scales presenting
geometry-induced phase transitions, instead of Debye shielding. We compare the
results obtained with point vortices to those obtained in the context of the
statistical mechanics of continuous vorticity fields described by the
Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results
but differences appear at the next order. In particular, the MRS theory
predicts a transition between sinh and tanh-like \omega-\psi relationships
depending on the sign of Ku-3 (where Ku is the Kurtosis) while there is no such
transition for point vortices which always show a sinh-like \omega-\psi
relationship. We derive the form of the relaxation equations in the strong
mixing limit and show that the enstrophy plays the role of a Lyapunov
functional
- âŠ