Deriving a GENERIC system from a Hamiltonian system

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain’s PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain’s works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation.

Kinetic theory describes a dilute monatomic gas using a distribution function f(q,p,t), the expected phase-space density of particles at a given position q with a given momentum p. The distribution function evolves according to the collisionless Boltzmann equation in the high Knudsen number limit. Fluid dynamics provides an alternative description of the gas using macroscopic hydrodynamic variables that are functions of position and time only. The mass, momentum and entropy densities of the gas evolve according to the compressible Euler equations in the limit of vanishing viscosity and thermal diffusivity. Both systems can be formulated as noncanonical Hamiltonian systems. Each configuration space is an infinite-dimensional Poisson manifold, and the dynamics is the flow generated by a Hamiltonian functional via a Poisson bracket. We construct a map J1 from the space of distribution functions to the space of hydrodynamic variables that respects the Poisson brackets on the two spaces. This map is therefore a Poisson map. It maps the p-integral of the Boltzmann entropy flogf to the hydrodynamic entropy density. This map belongs to a family of Poisson maps to spaces that include generalised entropy densities as additional hydrodynamic variables. The whole family can be generated from the Taylor expansion of a further Poisson map that depends on a formal parameter. If the kinetic-theory Hamiltonian factors through the Poisson map J1, an exact reduction of kinetic theory to fluid dynamics is possible. However, this is not the case. Nonetheless, by ignoring the contribution to the Hamiltonian from the entropy of the distribution function relative to its local Maxwellian, a distribution function defined by the p-moments ∫dnp(1,p,|p|2)f, we construct an approximate Hamiltonian that factors through the map. The resulting reduced Hamiltonian, which depends on the hydrodynamic variables only, generates the compressible Euler equations. We can thus derive the compressible Euler equations as a Hamiltonian approximation to kinetic theory. We also give an analogous Hamiltonian derivation of the compressible Euler–Poisson equations with non-constant entropy, starting from the Vlasov–Poisson equation.

For the Landau-Lifshitz equation a Lie type linear system representation in the Minkowski space M(superscript 3+1) has been derived previously [25]. The internal symmetry group is a proper orthochronous Lorentz group SO0(3,1), and the numerical method based on the internal symmetry was developed in [29]. This paper derives another four new representations of the Landau-Lifshitz equation. We prove that this equation admits two generators: one conservative and one dissipative, as well as two brackets: Poisson bracket and dissipative bracket. Upon embedding the Landau-Lifshitz equation into a skew-symmetric matrix space, we can develop a double-bracket flow representation. The conserved magnetization' magnitude is just the result of the isospectrality for an iso-spectral flow equation. Finally, on the cotangent bundle of an invariant manifold of the constant magnetization magnitude, we introduce the Lie-Poisson bracket to construct an evolutional differential equations system. The magnetization trajectory traces a coadjoint orbit in the Poisson manifold under a coadjoint action of the rotation group SO(3). The six different representations including the one by Bloch et al. [3] are compared.

Optimal reduction of controlled Hamiltonian system with Poisson structure and symmetry

Thermodynamics is a powerful theory that has been succesfully applied to describe the properties and behavior of macroscopic systems under a very wide range of physical conditions (Callen, 1985). Thermal and caloric information is obtained by performing experiments that agree with the constraints imposed (Callen, 1985; Kondepudi & Prigogine, 1999; Ragone, 1995). However, a considerable number of experimental, practical problems and systems of interest are often found in nonequilibrium or metastable states in which static thermodynamic relations are only valid locally, and frequently are insufficient to describe complicated time dependent situations (Demirel, 2007). Macroscopic systems in flow conditions are good examples of this peculiar behavior through non-Newtonian rheological effects, phase transitions and generalized statistics of turbulent motion. In fact, although there is a large amount of theoretical work proposing generalizations of the thermodynamic theory to nonequilibrium or quasiequilibrium situations (Beck & Cohen, 2003; Beris & Edwards, 1994; C. Beck & Swinney, 2005; de Groot & Mazur, 1984; Demirel, 2007; Kondepudi & Prigogine, 1999; Onuki, 2004; Rodriguez & Santamaria Holek, 2007), some of them very succesful (de Groot & Mazur, 1984; Demirel, 2007; Kondepudi & Prigogine, 1999), they are restricted due to the assumption and validity of local thermodynamic equilibrium. Recently, some interesting generalizations of thermodynamics to describe macroscopic systems in the presence of flow have been systematically developed in Refs. (Beris & Edwards, 1994; Onuki, 2004). Nevertheless, there are still a good deal of open questions, for instance those concerning the validity of the usual relations for the thermal and caloric equations of state, arising further research on this subject. Related to these general questions, there are several particular manifestations of the effects of flow on the thermodynamic behavior of systems. For example, diffusion of suspended particles in a heat bath in equilibrium, may strongly differ from that when the particles diffuse in a heat bath under the presence of shear flow. These effects have been analyzed along the years in studies ranging from experiments (Breedveld, 1998; Guasto & Gollub, 2007; Pine, 2005; Taylor & Friedman, 1996) and computer simulations (Sarman, 1992) to kinetic 5

We define a Poisson structure on the Nualart-Pardoux test algebra associated to the path space of a finite dimensional Lie algebra.

The recently developed quantum surface of section method is applied to a search for extremely high-lying energy levels in a simple but generic Hamiltonian system between integrability and chaos, namely the semiseparable 2-dim oscillator. Using the stretch of 13,445 consecutive levels with the sequential number around $1.8\cdot 10^7$ (eighteen million) we have clearly demonstrated the validity of the semiclassical Berry-Robnik level spacing distribution while at 1000 times smaller sequential quantum numbers we find the very persistent quasi universal phenomenon of power-law level repulsion which is globally very well described by the Brody distribution.

The authors introduce a generalized bracket which is capable of generating the dynamical equations governing the flow of both elastic and viscoelastic media. This generalized bracket is divided into two parts: a noncanonical Poisson bracket and a new dissipation bracket. The non-canonical Poisson bracket is the Eulerian equivalent of the canonical Lagrangian Poisson bracket corresponding to an ideal (non-dissipative) continuum. It is derived for a nonlinear elastic medium, and then it is used to obtain the Eulerian equations of motion in nonlinear elasticity valid for large deformations. It is shown that the proposed non-canonical bracket naturally leads to a materially objective relation involving the upper-convected time derivative of a strain tensor, as suggested by Oldroyd 40 years ago. The dissipation bracket for linear irreversible thermodynamics is next proposed in a general, phenomenological circumstance, so that the dissipation processes occurring in real systems (viscous and relaxation phenomena) can be incorporated into the Hamiltonian formalism. This bracket diverges from previously proposed dissipation brackets in that it uses the same generating functional (i.e. the Hamiltonian) as the Poisson bracket, rather than an entropy functional or a dissipative potential. It is shown that, in combination with the choice of an appropriate Hamiltonian functional, the generalized bracket proposed here can generate the governing equations for many viscoelastic media, including the Voigt solid and the Maxwell viscoelastic fluid.

This paper aims to construct the bracket formalism of mixture continua by using the method of Lagrangian- to-Eulerian (LE) transformation. The LE approach first builds up the transformation relations between the Eulerian state variables and the Lagrangian canonical variables, and then transforms the bracket in Lagrangian form to the bracket in Eulerian form. For the conservative part of the bracket formalism, this study systematically generates the noncanonical Poisson brackets of a two-component mixture. For the dissipative part, we deduce the Eulerian-variable-based dissipative brackets for viscous and diffusive mechanisms from their Lagrangian-variable-based counterparts. Finally, the evolution equations of a micromorphic fluid, which can be treated as a multi-component mixture, are derived by constructing its Poisson and dissipative brackets.

Interactions of quantum systems with their environment play a crucial role in resource-theoretic approaches to thermodynamics in the microscopic regime. Here, we analyze the possible state transitions in the presence of "small" heat baths of bounded dimension and energy. We show that for operations on quantum systems with fully degenerate Hamiltonian (noisy operations), all possible state transitions can be realized exactly with a bath that is of the same size as the system or smaller, which proves a quantum version of Horn's lemma as conjectured by Bengtsson and Zyczkowski. On the other hand, if the system's Hamiltonian is not fully degenerate (thermal operations), we show that some possible transitions can only be performed with a heat bath that is unbounded in size and energy, which is an instance of the third law of thermodynamics. In both cases, we prove that quantum operations yield an advantage over classical ones for any given finite heat bath, by allowing a larger and more physically realistic set of state transitions.

The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.

We considered a Hamiltonian system that can be described by two generalized variables. One of them relaxes quickly when the system is in contact with a heat bath at fixed temperature, while the second one, the slow variable, mimics the interaction of the system with another heat bath at a lower temperature. The coupling between these variables leads to an energy flow between the heat baths. Allahverdyan and Nieuwenhuizen [Phys. Rev. E 62, 845 (2000)] proposed a formalism to deal with such problem and calculated the steady states of the system and some related properties as entropy production, energy dissipation, etc. In this work we applied the formalism to a coupled system of ideal gases and also to an ideal gas interacting with a harmonic oscillator. If the temperatures of the heat baths are not too close, the Onsager relations do not apply.

ON THE LIE ALGEBRA OF MOTION INTEGRALS FOR TWO-DIMENSIONAL HYDRODYNAMIC EQUATIONS IN CLEBSH VARIABLES

We employ the generalized bracket formalism of nonequilibrium thermodynamics by Beris and Edwards to derive Lorentz-covariant time-evolution equations for an imperfect fluid with viscosity, dilatational viscosity, and thermal conductivity. Following closely the analysis presented by Öttinger (Physica A, 259, 1998, 24–42; Physica A, 254, 1998, 433–450) to the same problem but for the GENERIC formalism, we include in the set of hydrodynamic variables a covariant vector playing the role of a generalized thermal force and a covariant tensor closely related to the velocity gradient tensor. In our work here, we derive first the nonrelativistic equations and then we proceed to obtain the relativistic ones by elevating the thermal variable to a four-vector, the mechanical force variable to a four-by-four tensor, and by representing the Hamiltonian of the system with the time component of the energy-momentum tensor. For the Poisson and dissipation brackets we assume the same general structure as in the nonrelativistic case, but with the phenomenological coefficients in the dissipation bracket describing friction to heat and viscous effects being properly constrained for the resulting dynamic equations to be manifest Lorentz-covariant. The final relativistic equations are identical to those derived by Öttinger but the present approach seems to be more general in the sense that one could think of alternative forms of the phenomenological coefficients describing friction that could ensure Lorentz-covariance.

Poisson schemes for Hamiltonian systems on Poisson manifolds