Landau damping in the multiscale Vlasov theory

Vlasov kinetic theory is extended by adopting an extra one particle distribution function as an additional state variable characterizing the micro-turbulence internal structure. The extended Vlasov equation keeps the reversibility, the Hamiltonian structure, and the entropy conservation of the original Vlasov equation. In the setting of the extended Vlasov theory we then argue that the Fokker-Planck type damping in the velocity dependence of the extra distribution function induces the Landau damping. The same type of extension is made also in the setting of fluid mechanics

Multiscale Thermodynamics

Multiscale thermodynamics is a theory of relations among levels of investigation of complex systems. It includes the classical equilibrium thermodynamics as a special case but it is applicable to both static and time evolving processes in externally and internally driven macroscopic systems that are far from equilibrium and are investigated on microscopic, mesoscopic, and macroscopic levels. In this paper we formulate the multiscale thermodynamics, explain its origin, and illustrate it in mesoscopic dynamics that combines levels.Comment: 53 pages no figure

Thermodynamics and Rate Thermodynamics

Approach of mesoscopic state variables to time independent equilibrium sates (zero law of thermodynamics) gives birth to the classical equilibrium thermodynamics. Approach of fluxes and forces to fixed points (equilibrium fluxes and forces) that drive reduced mesoscopic dynamics gives birth to the rate thermodynamics that is applicable to driven systems. We formulate the rate thermodynamics and dynamics, investigate its relation to the classical thermodynamics, to extensions involving more details, to the hierarchy reformulations of dynamical theories, and to the Onsager variational principle. We also compare thermodynamic and dynamic critical behavior observed in closed and open systems. Dynamics and thermodynamics of the van der Waals gas provides an illustration.Comment: 35 page

One and two-fiber orientation kinetic theories of fiber suspensions

http://dx.doi.org/10.1016/j.jnnfm.2012.10.009The morphology influencing rheological properties of suspensions of rigid spheres constitutes the flow induced collective ordering of the spheres characterized by two or more sphere distribution functions. When the rigid spheres are replaced by rigid fibers, the collective order in the position of the spheres is replaced by the flow induced orientation of the fibers that suffices to be characterized by one-fiber orientation distribution function. A flow induced collective ordering of fibers (both in position and orientation), that can only be characterized by two or more fiber distribution functions, can still however constitute an important part of the morphology. We show that two types of interaction among fibers, one being the Onsager-type topological interaction entering the free energy and the other the hydrodynamics interaction entering the dissipative part of the time evolution, give indeed rise to a collective order in the orientation influencing the rheology of fiber suspensions

Multiscale Theory

Boltzmann kinetic equation is put into the form of an abstract time evolution equation representing links connecting autonomous mesoscopic dynamical theories involving varying amount of details. In the chronological order we present results that led to the abstract time equation evolution in both state space and the space of vector fields. In the final section we list some open problems.Comment: Accepted in Journal of Nonequilibrium Thermodynamic

Solid-fluid dynamics of yield-stress fluids

On the example of two-phase continua experiencing stress induced solid-fluid phase transitions we explore the use of the Euler structure in the formulation of the governing equations. The Euler structure guarantees that solutions of the time evolution equations possessing it are compatible with mechanics and with thermodynamics. The former compatibility means that the equations are local conservation laws of the Godunov type and the latter compatibility means that the entropy does not decrease during the time evolution. In numerical illustrations, in which the one-dimensional Riemann problem is explored, we require that the Euler structure is also preserved in the discretization.Comment: 51 pages, 7 figure

Role of thermodynamics in extensions of mesoscopic dynamical theories

Complex macroscopic systems (like for instance those encountered in nanotechnology and biology) need to be investigated in a family of mesoscopic theories involving varying amount of details. In this paper we formulate a general thermodynamics providing a uni- versal framework for such multiscale viewpoint of mesoscopic dynamics. We then discuss its role in making extensions (i.e. in lifting a mesoscopic theory to a more microscopic level that involves more details)

Extra mass flux in fluid mechanics

The conditions of existence of extra mass flux in single component dissipative non-relativistic fluids are clarified. By considering Galilean invariance we show that if total mass flux is equal to total momentum density, then mass, momentum, angular momentum and booster (center-of-mass) are conserved. However, these conservation laws may be fulfilled also by other means. We show an example of weakly non-local hydrodynamics where the conservation laws are satisfied as well although the total mass flux is different from momentum density

Hamiltonian Coupling of Electromagnetic Field and Matter

Reversible part of evolution equations of physical systems is often generated by a Poisson bracket. We discuss geometric means of construction of Poisson brackets and their mutual coupling (direct, semidirect and matched-pair products) as well as projections of Poisson brackets to less detailed Poisson brackets. This way the Hamiltonian coupling between transport of mixtures and electrodynamics is elucidated