Functional Fokker-Planck treatment of electromagnetic field propagation in a thermal medium

Abstract

The quantum statistics of continuous space time dependent electromagnetic fields is analyzed by means of functionats. The case of a field propagating in a thermal reservoir serves as a simple example to illustrate the succeeding steps: a masterequation is derived for the density operator which is a functional of the field operators. By means of the coherent state representation for continuous fields the masterequation is transformed into a functional differential equation in the function space, spanned by the coherent state amplitudes. This equation is of the Fokker-Planck type and determines a Gaussian process for a continuum of variables or a field. It is solved by determining the characteristics in function space of the associated equation of motion for the characteristic functional and subsequent functional integration. The solution is used to calculate some correlation functions and the spectral function of the field. w 1. Introduction Both in classical physics and in quantum theory the statistics of space time dependent fields has been in the focus of interest for a long time. It found widespread applications 1 in the theories of fluid dynamics, wave propagation, phase transitions, superconductivity, semiconductors etc.It was only recently, however, that powerful methods were developed to treat the quantum statistical problem of the electromagnetic field coupled to a system of atoms, both coupled to thermal reservoirs 2. The application of these methods offered a broad insight into the quantum statistical behaviour of the modes of the electromagnetic field in damped oscillators or self-sustained oscillators as the laser 2 or the parametric oscillator 3. In these applications the point of interest was not the electro-1 A good survey of the general theory and some applications is given by Beran, M. J. :Statistical continuum theories.

Potential and flux field landscape theory. II. Non-equilibrium thermodynamics of spatially inhomogeneous stochastic dynamical systems

“…The first expression is a result of Eqs. (9), (36), and (43). The third expression is derived from the first one using the property V s · D −1 · V r = 0, which is equivalent to V s · ∇A = 0 (already proven in the supplementary material 118 ) considering the relative dynamical constraint equation (Eq.…”

“…This non-equilibrium thermodynamic entropy flow decomposition equation (as well as the heat flow decomposition equation) is a direct reflection of the stationary dynamical decomposition equation F = −D · ∇U + V s , as seen from the expressions in Eqs. (37), (43) and (44). SinceṠ ad ≥ 0, Eq.…”

“…(36) and the adiabatic entropy production rate in Eq. (43) indicates there is another part of entropy production, termed nonadiabatic entropy production. [72][73][74] This is expressed as an entropy production decomposition equation:…”

“…28 Large classes of spatially inhomogeneous systems with stochastic fluctuations can be described by functional Langevin equations (including stochastic partial differential equations), 1,17,[21][22][23][24][29][30][31][32] functional Fokker-Planck equations, 1,17,[21][22][23][24][32][33][34][35] and (spatial) master equations. 1,21,[35][36][37] They have been used to study various problems in physics, chemistry, and biology, [21][22][23][24][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] such as surface growth, 42 electromagnetic field propagation, 43 reaction-diffusion of proteins in Drosophila embryo development, 35...…”

Quantum atom optics with trapped Bose-Einstein condensates

“…The positive-P field equations may be obtained by introducing the functional P -distribution [16,17] …”