Quasiprobability density diffusion equations for the quantum Brownian motion in a potential 
Mulligan, Bernard Patrick James



THESIS 8416 Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] representation of the density operator as a cnumber quasiprobablity distribution in phase space allowing quantum mechanical averages involving the density matrix to be calculated as phase space averages just as classical averages originally used by him to obtain quantum mechanical corrections to classical thermodynamic equilibrium i.e. to the MaxwellBoltzmann distribution so applying to closed quantum systems is extended to open quantum systems comprising a canonical ensemble of Brownian particles in a potential. This is accomplished via an idea of Gross and Lebowitz [E. P. Gross and J. L. Lebowitz, Phys. Rev. 1956, 104, 1528]. They suggested that using Wigner's representation the connection between classical and quantum collision kernels, (i.e. in classical mechanics the Stosszahlanzatz describing the bathparticle interaction in the open system in the Boltzmann equation for the single particle distribution function) is much more transparent than in the density operator formalism. Moreover the quantum kernel should closely correspond to the classical one. Hence the idea developed in this Thesis that in the quantum Brownian motion the collision term in a quantum master equation in Wigner's representation should be described by a KramersMoyal like expansion truncated at the second term (leading of course in the classical limit to the FokkerPlanck equation) as in the classical Brownian motion. Imposition of the Wigner equilibrium distribution as the stationary solution of this equation (which is akin to the FokkerPlanck equation) in the manner used by Einstein [A. Einstein, in R. H. F?rth, Ed., Investigations on the Theory of the Brownian Movement, Methuen, London, 1926; reprinted Dover, New York, 1954] used to calculate diffusion coefficients in the FokkerPlanck equation by imposing the MaxwellBoltzmann distribution as the stationary distribution then leads to the most important results of this Thesis. Namely the diffusion coefficients in the master equation become functions of the quantum parameter and the derivatives of the potential. Moreover all the solution techniques (matrix continued fractions etc) developed for the FokkerPlanck equation carry over to the quantum case as is illustrated by calculating the reaction rate and dynamical structure factor of a particle in a periodic cosine potential where the results are in agreement with those predicted by quantum reaction rate theory.

Keyword(s):

Electronic Engineering, Ph.D.; Ph.D. Trinity College Dublin 
Publication Date:

2007 
Type:

Doctoral thesis 
PeerReviewed:

Unknown 
Language(s):

English 
Institution:

Trinity College Dublin 
Citation(s):

Bernard Patrick James Mulligan, 'Quasiprobability density diffusion equations for the quantum Brownian motion in a potential', [thesis], Trinity College (Dublin, Ireland). Department of Electronic & Electrical Engineering, 2007, pp 211 
Publisher(s):

Trinity College (Dublin, Ireland). Department of Electronic & Electrical Engineering 
Supervisor(s):

Coffey, W.T. 
First Indexed:
20161102 05:57:58 Last Updated:
20170706 05:31:30 