H. Dekker

On the phase space quantization of the linearly damped harmonic oscillator

The quantal form of the classical Liouville equation will be investigated on the basis of a recently developed generalized Hamiltonian theory. The essential novelty inthat theory comprises the use of complex classical coordinates and momenta. We first show how for the nondissipative harmonic oscillator driven by an external classical force, the theory leads to the correct well-known quantum analogue of the classical Liouville equation. We then generalize this procedure to include frictional phenomena for which the novel theory has been observed to be particularly suited. The resulting quantal master equation for the simple linearly damped harmonic oscillator demonstrates that one cannot expect to find a proper quantum mechanical description of dissipative systems in terms of a single Schrodinger wave function. The master equation will then be transformed into its Wigner representation, providing a convinient form for discussion. The diffusion coefficients occuring in the resultant Fokker-Planck equation will be seen to be intimately connected with the survival of Heisenbergs uncertainty principle for dissipative systems. Apart from conceptual elegance, the present approach has superiority to a previous one in at least three aspects: i) there is no need to introduce ad-hoc quantal noise operators, ii) the above mentioned diffusion coefficients are specified and emerge in a natural way, and iii) the present approach has the important advantage of easy extension to more general systems.

Stationary momentum space solution of the Fokker-Planck equation for a simple model of a laser oscillator exhibiting spatial dispersion

Abstract Starting from a multimode hamiltonian for a system of radiating oscillators coupled with atomic reservoirs, the secular master equation for the radiation-density operator is calculated in the interaction picture after elimination of the atomic variables. Using the differential operator representation for coherent states this equation is transcribed into a multimode Fokker-Planck equation. The stationary solution in momentum space is given for the threshold region. Fourier transformation to configuration space results in a quasi-free energy formula for a laser oscillator exhibiting spatial dispersion.

Eigenvalues of a diffusion process with a critical point

The eigenvalues of a Fokker-Planck equation involving a critical point have been computed by means of a simple discretization technique. The results smoothly connect the monostable case above the critical point with the bistable case below it.

Time-local gaussian processes, path integrals and nonequilibrium nonlinear diffusion

In this paper we discuss the concept of time-local gaussian processes. These are processes for which the state variable at time t + τ is gaussian distributed around its most probable value at that time, for a specified realization a small time interval τ earlier. On one hand it will be shown that these processes are related to a very simple path sum. On the other hand the associated stochastic differential equation is derived by means of the Kramers-Moyal method, and will be seen to be the most general nonlinear Fokker-Planck equation. The significance of the present formulation for nonequilibrium processes and the comprehension of critical phenomena will be evaluated.

On the quantization of dissipative systems in the lagrange-Hamilton formalism

Suppose one knows the classical equation of motion for a certain dissipating system. Of course, the dissipation arises from the systems quantum mechanical interaction with its damping reservoirs. However, the question can be raised how much information one is able to obtain about the quantum mechanical features of the system without having any detailed knowledge concerning the quantum nature of these underlying interactions. In this paper the quantization of dissipative systems is considered in the Lagrange-Hamilton formalism. Making use of complex variables it is possible to formulate a classical Lagrange-Hamilton theory where the Hamiltonian is non-Hermitian. The real part of the Hamiltonian can be quantized in the usual manner. For the imaginary part a procedure is presented, which results in a self-consistent quantum mechanical formulation. It will be seen that the quantum information that can be obtained about the classical dissipation may be expressed in terms of an anti-commutator. In the appendix a dissipation-fluctuation theorem is given which in the case of a simple damped harmonic oscillator yields a well-known expression.

A fundamental constraint on quantum mechanical diffusion coefficients

Abstract A general quantum mechanical master equation for the damped oscillator, which can be represented as a phase space Fokker-Planck equation for the Wigner function, will be investigated with respect to the uncertainty principle. This leads to a condition to be imposed on the diffusion coefficients, which (i) is not fulfilled by an often quoted quasi-classical choice, and (ii) provides a uniqueness criterium in Dekkers theory.

Simulation and Control of a HD Diesel Engine Equipped with New EGR Technology

A dynamic model of a Heavy Duty (HD) turbocharged and aftercooled diesel engine was developed. The engine was equipped with high pressure diesel injection, a Variable Geometry Turbine (VGT) and an Exhaust Gas Recirculation (EGR) system. This engine was targeted at meeting EURO4 emission requirements. The final emission results were 2.4 g/kWh NOx and 0.107 g/kWh particulates for the European 13 mode test. Better than 3.0 g/kWh NOx and 0.10 g/kWh particulates are expected to be characteristic EURO4 emission requirements (approximate year of implementation is 2004). In the design of the EGR system the model provided initial asessments of the properties of this system. Associated engine and turbocharger behaviour as well as optimal control strategies were predicted. A transient engine control algorithm was developed using the dynamic engine model. The VGT is closed loop controlled and EGR is shut off during a short time after a load increase. The simulation results were confirmed by actual measurements, demonstrating acceptable transient behaviour.

Correlation time expansion for multidimensional weakly non-markovian gaussian processes

Abstract The stochastic Liouville equation is evaluated explicitly through first order in an expansion in terms of correlation times of multiplicative gaussian coloured noise for a general multidimensional weakly non-markovian process by means of Novikovs theorem. It is explained why the present result is new.

On the path integral for diffusion in curved spaces

The Onsager-Machlup Lagrangian for diffusion processes in curved spaces is determined by evaluating the covariant path integral by means of a spectral analysis of smooth trajectories in Riemannian normal coordinates. The Lagrangian involves a novel curvature scalar potential term v=−(18)R. The present treatment replaces an earlier one.

Critical dynamics the expansion of the master equation including a critical point

The master equation for a diffusion process that takes place in an external potential will be evaluated systematically in terms of a small parameter, namely the diffusion coefficient. Contrary to the known expansion the present solution is not only uniformly valid in the normal monostable and bistable cases, but also applies at the critical point. This has been achieved by using in zeroth order approximation the complete set of eigenfunctions belonging to the appropriate irreducible description of the process. Successive higher order corrections are evaluated explicitly.