Form Of Fluctuation-dissipation Theorem Research Articles

Derivation of a generalized Schrödinger equation from the theory of scale relativity

Using Nottale’s theory of scale relativity relying on a fractal space-time, we derive a generalized Schrodinger equation taking into account the interaction of the system with the external environment. This equation describes the irreversible evolution of the system towards a static quantum state. We first interpret the scale-covariant equation of dynamics stemming from Nottale’s theory as a hydrodynamic viscous Burgers equation for a potential flow involving a complex velocity field and an imaginary viscosity. We show that the Schrodinger equation can be directly obtained from this equation by performing a Cole-Hopf transformation equivalent to the WKB transformation. We then introduce a friction force proportional and opposite to the complex velocity in the scale-covariant equation of dynamics in a way that preserves the local conservation of the normalization condition. We find that the resulting generalized Schrodinger equation, or the corresponding fluid equations obtained from the Madelung transformation, involve not only a damping term but also an effective thermal term. The friction coefficient and the temperature are related to the real and imaginary parts of the complex friction coefficient in the scale-covariant equation of dynamics. This may be viewed as a form of fluctuation-dissipation theorem. We show that our generalized Schrodinger equation satisfies an H-theorem for the quantum Boltzmann free energy. As a result, the probability distribution relaxes towards an equilibrium state which can be viewed as a Boltzmann distribution including a quantum potential. We propose to apply this generalized Schrodinger equation to dark matter halos in the Universe, possibly made of self-gravitating Bose-Einstein condensates.

Dynamic linear response theory for a nonextensive system based on the Tsallis prescription.

We develop here a dynamic linear response theory of many-body nonextensive systems based on the maximization of the Tsallis entropy associated with the density matrix and the concomitant suitably defined mean total energy, number, etc., where the averaging is over the qth power of the density matrix, q being a parameter characterizing the nonextensivity. This formulation is shown to preserve causality (Kramers-Kronig), time reversal, and Onsager reciprocity, while a different form of fluctuation-dissipation theorem is obtained. The traditional theory for extensive systems is obtained in the special case where $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.

Quantum theory of a one-dimensional optical cavity with output coupling. II. Thermal radiation field and the fluctuation-dissipation theorem

The normally ordered correlation function and the power spectrum (cross spectral tensor) of a thermal radiation field existing inside and outside a one-dimensional optical cavity which transmits radiation at one of its end surfaces are calculated on the basis of a multimode formulation of the field developed in a previous paper. The correlation function is found to consist of a series of trigamma functions resulting from retarded correlations due to reflections at the cavity end surfaces. The power spectrum is a product of the Planck distribution and the mode functions of the universe which have a periodic factor with the period being equal to the cavity mode separation. For the purpose of obtaining a description of the cavity which allows one to deduce its thermal noise property, a form of fluctuation-dissipation theorem is derived which connects, in a one-dimensional space with arbitrary structure, the normally ordered correlation function with the admittance for the electric field probed by a current. Application of this theorem to the model cavity leads to an explicit and simultaneous description of the fluctuation and dissipation associated with all the cavity modes. The dissipation is shown to be due to radiation of electromagnetic energy into the free space. The above relation is reduced, under the assumption of high cavity quality factor and of single-cavity-mode selection, to that of a Markovian noise associated with a single decay constant. The consistency of these results with the field commutation relation is examined.