38 research outputs found
Phase space reduction of the one-dimensional Fokker-Planck (Kramers) equation
A pointlike particle of finite mass m, moving in a one-dimensional viscous
environment and biased by a spatially dependent force, is considered. We
present a rigorous mapping of the Fokker-Planck equation, which determines
evolution of the particle density in phase space, onto the spatial coordinate
x. The result is the Smoluchowski equation, valid in the overdamped limit,
m->0, with a series of corrections expanded in powers of m. They are determined
unambiguously within the recurrence mapping procedure. The method and the
results are interpreted on the simplest model with no field and on the damped
harmonic oscillator.Comment: 13 pages, 1 figur
Another Derivation of a Sum Rule for the Two-Dimensional Two-Component Plasma
In a two-dimensional two-component plasma, the second moment of the number
density correlation function has the simple value , where is the dimensionless coupling
constant. This result is derived directly by using diagrammatic methods.Comment: 10 pages, uses axodraw.sty, elsart.sty, elsart12.sty, subeq.sty;
accepted for publication in Physica A, May 200
Driven diffusion in a periodically compartmentalized tube: homogeneity versus intermittency of particle motion
We study the effect of a driving force F on drift and diffusion of a point Brownian particle in a tube formed by identical ylindrical compartments, which create periodic entropy barriers for the particle motion along the tube axis. The particle transport exhibits striking features: the effective mobility monotonically decreases with increasing F, and the effective diffusivity diverges as F â â, which indicates that the entropic effects in diffusive transport are enhanced by the driving force. Our consideration is based on two different scenarios of the particle motion at small and large F, homogeneous and intermittent, respectively. The scenarios are deduced from the careful analysis of statistics of the particle transition times between neighboring openings. From this qualitative picture, the limiting small-F and large-F behaviors of the effective mobility and diffusivity are derived analytically. Brownian dynamics simulations are used to find these quantities at intermediate values of the driving force for various compartment lengths and opening radii. This work shows that the driving force may lead to qualitatively different anomalous transport features, depending on the geometry design
Inequivalent representations of commutator or anticommutator rings of field operators and their applications
Hamiltonian of a system in quantum field theory can give rise to infinitely
many partition functions which correspond to infinitely many inequivalent
representations of the canonical commutator or anticommutator rings of field
operators. This implies that the system can theoretically exist in infinitely
many Gibbs states. The system resides in the Gibbs state which corresponds to
its minimal Helmholtz free energy at a given range of the thermodynamic
variables. Individual inequivalent representations are associated with
different thermodynamic phases of the system. The BCS Hamiltonian of
superconductivity is chosen to be an explicit example for the demonstration of
the important role of inequivalent representations in practical applications.
Its analysis from the inequivalent representations' point of view has led to a
recognition of a novel type of the superconducting phase transition.Comment: 25 pages, 6 figure
Universal behavior of quantum Green's functions
We consider a general one-particle Hamiltonian H = - \Delta_r + u(r) defined
in a d-dimensional domain. The object of interest is the time-independent Green
function G_z(r,r') = . Recently, in one dimension (1D),
the Green's function problem was solved explicitly in inverse form, with
diagonal elements of Green's function as prescribed variables. The first aim of
this paper is to extract from the 1D inverse solution such information about
Green's function which cannot be deduced directly from its definition. Among
others, this information involves universal, i.e. u(r)-independent, behavior of
Green's function close to the domain boundary. The second aim is to extend the
inverse formalism to higher dimensions, especially to 3D, and to derive the
universal form of Green's function for various shapes of the confining domain
boundary.Comment: 46 pages, the shortened version submitted to J. Math. Phy
A Generalization of the Stillinger-Lovett Sum Rules for the Two-Dimensional Jellium
In the equilibrium statistical mechanics of classical Coulomb fluids, the
long-range tail of the Coulomb potential gives rise to the Stillinger-Lovett
sum rules for the charge correlation functions. For the jellium model of mobile
particles of charge immersed in a neutralizing background, the fixing of
one of the -charges induces a screening cloud of the charge density whose
zeroth and second moments are determined just by the Stillinger-Lovett sum
rules. In this paper, we generalize these sum rules to the screening cloud
induced around a pointlike guest charge immersed in the bulk interior of
the 2D jellium with the coupling constant ( is the
inverse temperature), in the whole region of the thermodynamic stability of the
guest charge . The derivation is based on a mapping technique of
the 2D jellium at the coupling = (even positive integer) onto a
discrete 1D anticommuting-field theory; we assume that the final results remain
valid for all real values of corresponding to the fluid regime. The
generalized sum rules reproduce for arbitrary coupling the standard
Z=1 and the trivial Z=0 results. They are also checked in the Debye-H\"uckel
limit and at the free-fermion point . The generalized
second-moment sum rule provides some exact information about possible sign
oscillations of the induced charge density in space.Comment: 16 page
Equation of state in the fugacity format for the two-dimensional Coulomb gas
We derive the exact general form of the equation of state, in the fugacity
format, for the two-dimensional Coulomb gas. Our results are valid in the
conducting phase of the Coulomb gas, for temperatures above the
Kosterlitz-Thouless transition. The derivation of the equation of state is
based on the knowledge of the general form of the short-distance expansion of
the correlation functions of the Coulomb gas. We explicitly compute the
expansion up to order in the activity . Our results are in
very good agreement with Monte Carlo simulations at very low density
Sine-Gordon/Coulomb Gas Soliton Correlation Functions and an Exact Evaluation of the Kosterlitz-Thouless Critical Exponent
We present an exact derivation for the asymptotic large distance behavior of
the spin two-point correlation function in the XY-model. This allows for the
exact obtainment of the critical exponent at the Kosterlitz-Thouless
transition that occurs in this model and in the 2D neutral Coulomb gas and
which has been previously obtained by scaling arguments. In order to do that,
we use the language of sine-Gordon theory to obtain a Coulomb Gas description
of the XY-model spin correlation function, which becomes identified with the
soliton correlator of that theory. Using a representation in terms of bipolar
coordinates we obtain an exact expression for the asymptotic large distance
behavior of the relevant correlator at , which corresponds to the
Kosterlitz-Thouless transition. The result is obtained by approaching this
point from the plasma (high-temperature) phase of the gas. The vortex
correlator of the XY-model is also obtained using the same procedure.Comment: To appear in J. Stat. Phys., 11 page