Showing papers on "Dissipative system published in 1981"

Nonlinear differential equations in abstract spaces

Abstract: Preface Directional derivatives Mean value theorems The Cauchy problem Successive approximations Types of approximate solutions Dissipative type conditions Dissipative operators The exponential formula Difference approximations Convergence of difference approximations Global existence Fundamental properties Differential inequalities in cones Existence of solutions in weak topology Equations with delay Boundary value problems Monotone iterative methods

Nonlinear Evolution Equations - Global Behavior of Solutions

Abstract: Generalities and local theory.- The global existence problem.- Theory of monotone operators and applications.- Smoothing effect for some nonlinear evolution equations.- Schrodinger and wave equations with a logarithmic nonlinearity.- The linear case: Hilbertian theory and applications.- Some nonlinear monotone cases.- Some nonlinear, non monotone cases.- Autonomous dissipative systems.- General results for quasi-autonomous periodic systems.- More on asymptotic behavior for solutions of the nonlinear dissipative forced wave equation.- Boundedness of trajectories for quasi-autonomous dissipative systems.- Almost-periodic quasi-autonomous dissipative systems in a Hilbert space.

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem

Abstract: The paper is devoted to the theory of thermal fluctuations in nonlinear macroscopic systems and to the derivation of variational principles of nonlinear nonequilibrium thermodynamics. In the first part of the paper rigorous universal fluctuation-dissipation relations for nonlinear classical and quantum systems, subjected to dynamic as well as thermodynamic perturbations, are derived and analyzed. General expressions for dissipative fluxes and nonlinear transfer coefficients with the help of fluctuation cumulants are found. The canonical structure of nonlinear evolution equations of macrovariables is derived and the rule of introducing langevinian random forces into these equations, in accordance with fluctuation-dissipation relations. A Markovian theory of fluctuations in a stationary nonequilibrium state is constructed.

Urban Evolution, Self-Organization, and Decisionmaking

Abstract: A dynamical model of a central place system is described which, derived from the concepts underlying dissipative structures, takes into account the self-organizing aspects of urban evolution, and s...

A simple nonlinear dissipative quantum evolution equation

Abstract: The author has considered a nonlinear dissipative evolution equation that generalises the Schrodinger equation. In the corresponding evolution all the stationary states of the usual Schrodinger equation have a behaviour of semistable limit cycles, except the ground state which is stable. The model is applied to the spin-1/2 and to the damped harmonic oscillator. For the latter it is shown that the coherent states remain coherent and evolve as in the corresponding classical problem.

Dissipative contributions of internal multiplicative noise: I. mechanical oscillator

Abstract: Langevin equations for closed systems with multiplicative fluctuations must also include appropriate dissipative terms that ensure eventual equilibration of the system. We consider an oscillator coupled to a heat bath and show that a particular nonlinear coupling to a harmonic heat bath leads to a fluctuating frequency and to nonlinear dissipative terms . We also analyze the effects of the multiplicative fluctuations and of the corresponding nonlinear dissipation on the temporal evolution of the average oscillator energy. We find that the rate of equilibration of this system can be significantly different from that of an oscillator with only additive fluctuations and linear dissipation.

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: II. Kinetic potential and variational principles for nonlinear irreversible processes

Abstract: On the basis of a complete system of fluctuation-dissipation relations, considered in the first part of this series, a variational principle for nonlinear irreversible processes is derived. According to this principle the virtual entropy production functional (analogous to the action in mechanics) has an absolute minimum meaning on the real trajectory of a system. The universal structure of the “kinetic potential” and the “lagrangian” of a system, each contain complete information about fluctuations of macrovariables. The connection of the lagrangian with the markovian kinetic operator of macrovariables is stated. Fundamental properties of dissipative potentials, reflecting microscopic reversibility, are considered. The derived variational principle can be applied to closed systems (the steady state of which is equilibrium) as well as to open ones (when external dynamic forces cause entropy flux through the system and put it into a steady non-equilibrium state). Canonical transformations of macrovariables are considered.

Evolution problems for a class of dissipative materials

Abstract: This work examines the problems of dynamic and quasi-static evolution for a large class of dissipative materials, including viscoplastic, viscoelastic, and elastic perfectly plastic materials. We show that when the potential of dissipation is regular, the displacement solution is regular; however, in the case of perfect plasticity, where the potential is irregular, the solution can be discontinuous. A suitable framework is used in order to account for these discontinuities. Existence theorems are stated, and the boundary conditions are discussed. The evolution equations encountered in this work are strongly nonlinear but with a monotone time-dependent nonlinearity. A direct method of resolution is proposed, since the known results do not apply in this case.

The Thermodynamic Origin of Ecosystems

Abstract: A general theory of the origin of ecosystems in thermodynamic terms is developed. Regular fluctuations in the availablility of energy give rise to dissipative structures in systems where the bounda...

General properties of gausson-conserving descriptions of quantal damped motion

Abstract: We investigate the time-evolution of correlated critical states (gaussons) for a time-dependent harmonic oscillator coupled to a loss mechanism. A general dissipativity condition can be formulated and a derivation of the structure of gausson-conserving, dissipative as well as nondissipative, Hamiltonians, is presented. The already known frictional Hamiltonians are found to be particular cases of this description of damped motion.

Some vorticity theorems and conservation laws for non-barotropic fluids

Abstract: Some theorems concerning the vorticity in barotropic flows of perfect fluids are generalized for non-barotropic flows. The generalization involves replacing the velocity in certain parts of the equations by a time-dependent quantity which is a function of the velocity and thermodynamic properties of the fluid. Results which are generalized include Kelvin's circulation theorem and conservation laws for potential vorticity and helicity. It is shown how the results can be further generalized to include dissipative effects. The possibility of using some of the results in deriving a complete set of Lagrangian conservation laws for perfect fluids is discussed.

Self-Synchronization of Nonlinear Oscillations in the Presence of Fluctuations

Abstract: A statistical mechanical theory is presented for the self-organization of a macroscopic oscillation with the presence of external fluctuations in a system of Van der Pol oscillators coupled through dissipative interactions. Starting from Langevin equations for the Van der Pol oscillators, the static and dynamic characteristics are studied. The threshold condition is given by the relative size between the fluctuation and the interaction. The transitions between synchronous and asynchronous phases are well discussed by a Landau-type equation. The steady state value of the order parameter and the onset time are compared between the theory and the computer experiments and a good agreement is obtained.

Irreversible thermodynamics of overdriven shocks in solids

Abstract: An isotropic solid capable of transporting heat and of undergoing dissipative plastic flow, is treated. The shock is assumed to be a steady wave, and any phase changes or macroscopic inhomogeneities which might be induced by the shock are neglected. Under these conditions it is established that for an overdriven shock, no solution is possible without heat transport, and when the heat transport is governed by the steady conduction equation, no solution is possible without plastic dissipation as well. Upper and lower bounds are established for the thermodynamic variables, namely the shear stress, temperature, entropy, plastic strain, and heat flux, as functions of compression through the shock.

Physical consequences of the choice of the Lagrangian

Abstract: The physical consequences of the existence of inequivalent Lagrangians associated with a given equation of motion are examined in quantum mechanics. It is shown that in the case of conservative systems one additional condition is sufficient to select the physically correct Lagrangian. It is remarked that in the case of dissipative systems the situation is quite different, and no conditions are known to single out a unique Lagrangian.

Experimental and theoretical study of drift-waves in a quadrupole

Abstract: Spontaneous oscillations observed in the UMIST steady-state quadrupole are identified as drift waves with one full wave-length around a closed line of force, and with the potential perturbation anti-symmetric about the maximum field point. Existing theories do not properly describe this mode, for which trapped particles are important. The authors develop the theory of the mode and solve the resulting eigenvalue equation to obtain the dispersion curve; the result agrees with experimental evidence within experimental error. The excitation is identified as due to the 'dissipative trapped particle' mechanism, and the corresponding theoretical growth rate also agrees closely with experiment. In their conditions the excitation is due to electron-neutral rather than electron-ion collisions; as a result the temperature gradient is destabilising rather than stabilising, and they show that this also agrees with experiment.

Calculation of the Statistical Properties of Strange Attractors

Abstract: A path integral method is developed for the calculation of the statistical properties of a class of discrete, dissipative mappings which exhibit strange attractors. Exact analytic results are derived for the low order statistical moments. These non-trivial results are verified numerically.

Resonant oscillations of a gas in an open-ended tube

Abstract: An investigation is made of the oscillations near resonance induced in a gas contained in a tube that is open at one end and has an oscillating piston at the other. When all dissipative processes are neglected it is shown that there are discontinuities in the oscillations within a certain frequency band near resonance. The modifying effects of the following dissipative mechanisms are also investigated: (i) compressive viscosity; (ii) damping arising from radiation at the open end; (iii) the effect of the boundary layer at the wall of the tube; (iv) nonlinear dissipation such as might arise from eddy formation when separation occurs at the sharp lip of the open end. The main role of the first two mechanisms is to replace the discontinuities by sharp transitions and, more importantly, to determine their position. Under typical conditions it is possible for the boundary-layer effect to dominate the oscillations, which then differ substantially from those predicted by inviscid theory. In particular the oscillations are smooth throughout the whole range of frequencies. The oscillations are also considerably modified if a substantial proportion of the energy ejected at the open end is dissipated in eddy formation. A related problem is that of a plane wave approaching the open end from an arbitrary direction outside the tube. If the tube is closed at the other end, the boundary condition induced at the closed end by the disturbance transmitted into the tube simulates the effect of the piston. The two problems lend themselves to a similar analysis.

Non-Equilibrium, Chance and Change in Family Therapy.

Abstract: The model of the systemic approach that we commonly use in family therapy is mainly an homeostatic model insisting essentially on negative feedback and allowing little room for amplification phenomena and for abrupt changes that may occur. Our approach, called “systemic,” is related principally to models applying in cases of processes occurring at thermodynamic equilibrium or at stationary states close to equilibrium. In those instances, stability is maintained no matter what external or internal perturbations are exerted on the system. But our practice shows that a modification limited to one part of the family system extends quickly to the whole system. How are those changes brought about? How is an open system, such as the family in which feedback loops exist, pulled away from a stationary state? It was partly to answer such questions that we started to give attention to organizational forms likely to appear away from thermodynamic equilibrium and particularly to the work done in this field by the team ofllya Prigogine. Starting with a few examples of “dissipative structures” some concepts will be introduced which seem likely to be fruitful in our field.