Fluctuations and stability of stationary non-equilibrium systems in detailed balance

A generalized thermodynamic potential for Markoffian systems with detailed balance and far from thermal equilibrium has been derived in a previous paper. It was shown that the principle of detailed balance is equivalent to a set of conditions fulfilled by this potential (“potential conditions”). The properties of this potential allow us to extend the validity of a number of thermodynamic concepts well known for systems in or near thermal equilibrium to stationary states far from thermal equilibrium. The concept of symmetry breaking phase transitions for these systems is introduced in analogy to thermal equilibrium systems by considering the dependence of the stationary probability density of the system on a set of externally controlled parameters {λ}. A functional of the time dependent probability density of the system is defined in close analogy to the Gibb's definition of entropy. This functional has the properties of a Ljapunov functional of the governing Fokker-Planck equation showing the stability of the stationary probability density. The Langevin equations connected with the Fokker-Planck equation are considered. It is shown that, by means of the potential conditions, generalized “thermodynamic” fluxes and forces may be defined in such a way that the smoothly varying part of the Langevin equations (kinetic equations) constitutes a linear relation between fluxes and forces. The matrix of coefficients is given by the diffusion matrix of the Fokker-Planck equation. The symmetry relations which hold for this matrix due to the potential conditions then lead to the Onsager-Casimir symmetry relations extended to systems with detailed balance near stationary states far from thermal equilibrium. Finally it is shown that under certain additional assumptions the generalized thermodynamic potential may be used as a Ljapunov function of the kinetic equations.     

Frequency shifts of laser modes in solid state and gaseous systems

The present paper calculates the frequency shifts of laser modes which are brought about by the nonlinear response of the active laser material. The active material is supposed to consist of a set of N atoms each with three or four levels either at random lattice sites or moving with a Gaussian velocity distribution. The formalism developed in our previous work in which a homogeneously broadened line was considered is now applied to a line which is in addition inhomogeneously broadened. The physically most important case of an inhomogeneous Gaussian is treated in detail up to orderγ/α, whereγ is the natural linewidth and α that of the superimposed inhomogeneous broadening. Starting from first principles our present microscopic theory represents especially a foundation of Bennetts concept of hole burning effects. Furthermore it contains the effects of time-dependent nonlinearities as well as that of polarization. Besides the well known mode pulling effect of an inhomogeneous Gaussian several new effects show up. If only one mode lasers an additional repulsive term occurs in both solid state and gaseous systems, which stems from the time independent response of the atomic system. It is small in the solid state case but can be pronounced in the gas laser if the mode frequency is close to the center of the broadened line. If two modes oscillate simultaneously, the mode interaction leads to a repulsive term which can be appreciable both for solid state and gaseous lasers. The results for a solid state laser are in qualitative agreement with line pushing as observed bySnitzer. For comparison with gas laser experiments the He-Ne-Laser is considered in detail. If wide angle scattering of Ne-atoms is slow, the frequency pushing is governed by the time independent response. Our theoretical results are in favorable agreement with various experimental findings byBennett. Finally, the stability of simultaneous laser modes is treated in detail both for the solid state and gaseous case.     

Attractors of convex maps with positive Schwarzian derivative in the presence of noise

One-dimensional maps have proved to be useful models for understanding the transition to turbulence. We investigate a smooth perturbation of the well-known logistic system in order to examine numerically the change in the bifurcation behavior which is observed, when the Schwarzian derivative is allowed to become positive. We find coexistence of a fixed point attractor and a periodic or chaotic two-band-attractor. The chaotic two-band attractor can disappear by yielding a preturbulent state which will asymptotically settle down to a fixed-point. The chaotic behavior of some systems can be destroyed by arbitrarily small amounts of external noise. The concept of (ε, δ)-diffusions is used to describe the sensitivity of attractors against external noise. We also observe a direct transition from a fixed-point to a chaotic one-band attractor. This can be interpreted as type-III-intermittency of Pomeau and Manneville but with an almost linear scaling behavior of the Lyapunov exponent.     

A new access to path integrals and Fokker Planck equations via the maximum calibre principle

Under the assumption that the underlying process is continuous Markovian and using simple short time correlation functions as constraints in the maximum calibre principle of Jaynes we derive the explicit path integrals from which then the corresponding Fokker-Planck equation may be deduced. Our approach is valid for all systems irrespective of whether they are close to or far away from thermal equilibrium and it applies even to nonphysical systems.     

Information and information gain close to nonequilibrium phase transitions. Numerical results

Using a general result on the behavior of information and information gain close to instability points of self-organizing systems we calculate explicitly the information of a single order parameter close to a nonequilibrium phase transition. We also discuss by means of the result why the relevant quantities are interpreted as information rather than as entropy.