Linear systems with adiabatic fluctuations

Abstract

We consider a dynamical system subjected to weak but adiabatically slow fluctuations of external origin. Based on the "adiabatic following" approximation we carry out an expansion in α|µ| −1 , where α is the strength of fluctuations and |µ| −1 refers to the time scale of evolution of the unperturbed system to obtain a linear differential equation for the average solution. The theory is applied to the problems of a damped harmonic oscillator and diffusion in a turbulent fluid. The result is the realization of 'renormalised' diffusion constant or damping constant for the respective problems. The applicability of the method has been critically analyzed.PACS NO. 05.20 Dd , 05.40 +j 1 Dedicated to Prof. Mihir Chowdhury on his 60th birthday 1

Advantage of Quantum Theory over Nonclassical Models of Communication

“…A more general mathematical modeling of an operational theory is possible under the framework of generalized probability theories (GPTs) which incorporates several nonclassical features of quantum theory and thus manifests many advantageous protocols. [ 9–13 ] For example, in the distributed computing setting, where several spatially separated computing devices are allowed to exchange limited communications in order to perform some computational task, quantum nonlocal correlations can provide surprising advantages. [ 14,15 ] Interestingly, in such cases, one can come up with more dramatic correlations that satisfy the relativistic causality or more broadly no‐signaling (NS) principle but at the same time exhibit advantage over the quantum correlations—Popescu–Rohrlich (PR) correlation is one such celebrated example in the bipartite setting.…”

Theory of adiabatic fluctuations: third-order noise

“…The equation is second order in α, i.e., of the order of α 2 |µ| −1 , where |µ| refers to the eigenvalue of A 0 . In our earlier communication [5] we have shown the convergence of the series in α|µ| −1 , pertaining to the separation of the time-scales implied in II in Sec.I. We also remark that it is possible to extend the treatment to higher order, in general.…”

“…Our strategy here is to follow a perturbative approach, pertaining to the separation of the timescale (II) without keeping any above-mentioned restriction on the type of stochastic behavior. Based on the 'adiabatic following approximation' [4] we have recently [5] carried out an expansion in α|µ| −1 to obtain a linear differential equation for the average solution. In this paper we extend this analysis to treat nonlinear stochastic differential equations for construction of appropriate master equations.…”

“…The outlay of the paper is as follows : In the next section we review the basic aspects of adiabatic fluctuations in linear processes as dealt in our earlier paper [5].…”

“…The two basic assumptions, e.g., the adiabatic following approximation and the decoupling approximation as well as validity and convergence of perturbative expansion were discussed in detail in the earlier paper [5]. To make this paper self-contained and readable we review its salient features.…”