24 research outputs found
Plane polymer configurations enclosing a fixed area in an electric field: generating functional and statistical mechanical properties
The statistical mechanical properties of plane polymer configurations which enclose a fixed area and are subject to an external electric field are investigated. For this purpose an exact expression for the generating functional is obtained and subsequently used to derive: (a) the distribution function for the enclosed area; (b) the mean square distance of a given repeating unit from the origin; (c) the entropic force on a repeating unit
Statistical mechanical properties of polymer configurations which enclose a constant area
The statistical mechanical properties of plane polymer loops enclosing a constant area are investigated, using a continuous model from the start. For this purpose an analytic expression for the generating functional is obtained, which in turn is used to derive (1) the distribution function for the enclosed area, (2) the average squared distance of a given repeating unit from the origin, and (3) the entropic force on a repeating unit
Stochastic area distributions: optimal trajectories, Maslov indices and asymptotic results
In this paper we study the semi-classical approximation for the distribution of area associated with (i) planar polymer rings constrained to enclose a fixed algebraic area and (ii) planar rings subject to an external electric field and constrained to enclose a fixed algebraic area. We demonstrate that the results are accurate in the asymptotic regime. Moreover, we also show that in case (i) it is possible to reconstruct the exact expression for the distribution, provided the contributions from all optimal trajectories are taken into account, as well as the proper Maslov indices
On a path integral with a topological constraint
We discuss a new method to evaluate a path integral with a topological constraint involving a point singularity in a plane. The path integration is performed explicitly in the universal covering space. Our method is an alternative to an earlier method of Inomata
Area distribution of two-dimensional random walks on a square lattice
The algebraic area probability distribution of closed planar random walks of
length N on a square lattice is considered. The generating function for the
distribution satisfies a recurrence relation in which the combinatorics is
encoded. A particular case generalizes the q-binomial theorem to the case of
three addends. The distribution fits the L\'evy probability distribution for
Brownian curves with its first-order 1/N correction quite well, even for N
rather small.Comment: 8 pages, LaTeX 2e. Reformulated in terms of q-commutator
Path integral solution for an angle-dependent anharmonic oscillator
We have given a straightforward method to solve the problem of noncentral
anharmonic oscillator in three dimensions. The relative propagator is presented
by means of path integrals in spherical coordinates. By making an adequate
change of time we were able to separate the angular motion from the radial one.
The relative propagator is then exactly calculated. The energy spectrum and the
corresponding wave functions are obtained.Comment: Corrected typos and mistakes, To appear in Communications in
Theoretical Physic
Field theoretic description of charge regulation interaction
In order to find the exact form of the electrostatic interaction between two
proteins with dissociable charge groups in aqueous solution, we have studied a
model system composed of two macroscopic surfaces with charge dissociation
sites immersed in a counterion-only ionic solution. Field-theoretic
representation of the grand canonical partition function is derived and
evaluated within the mean-field approximation, giving the Poisson-Boltzmann
theory with the Ninham-Parsegian boundary condition. Gaussian fluctuations
around the mean-field are then analyzed in the lowest order correction that we
calculate analytically and exactly, using the path integral representation for
the partition function of a harmonic oscillator with time-dependent frequency.
The first order (one loop) free energy correction gives the interaction free
energy that reduces to the zero-frequency van der Waals form in the appropriate
limit but in general gives rise to a mono-polar fluctuation term due to charge
fluctuation at the dissociation sites. Our formulation opens up the possibility
to investigate the Kirkwood-Shumaker interaction in more general contexts where
their original derivation fails.Comment: 12 pages, 9 figures, submitted to EPJ
Exact quantum states of a general time-dependent quadratic system from classical action
A generalization of driven harmonic oscillator with time-dependent mass and
frequency, by adding total time-derivative terms to the Lagrangian, is
considered. The generalization which gives a general quadratic Hamiltonian
system does not change the classical equation of motion. Based on the
observation by Feynman and Hibbs, the propagators (kernels) of the systems are
calculated from the classical action, in terms of solutions of the classical
equation of motion: two homogeneous and one particular solutions. The kernels
are then used to find wave functions which satisfy the Schr\"{o}dinger
equation. One of the wave functions is shown to be that of a Gaussian pure
state. In every case considered, we prove that the kernel does not depend on
the way of choosing the classical solutions, while the wave functions depend on
the choice. The generalization which gives a rather complicated quadratic
Hamiltonian is simply interpreted as acting an unitary transformation to the
driven harmonic oscillator system in the Hamiltonian formulation.Comment: Submitted to Phys. Rev.
Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential
The unitary operator which transforms a harmonic oscillator system of
time-dependent frequency into that of a simple harmonic oscillator of different
time-scale is found, with and without an inverse-square potential. It is shown
that for both cases, this operator can be used in finding complete sets of wave
functions of a generalized harmonic oscillator system from the well-known sets
of the simple harmonic oscillator. Exact invariants of the time-dependent
systems can also be obtained from the constant Hamiltonians of unit mass and
frequency by making use of this unitary transformation. The geometric phases
for the wave functions of a generalized harmonic oscillator with an
inverse-square potential are given.Comment: Phys. Rev. A (Brief Report), in pres