Pavol Kalinay

Effective one-dimensional description of confined diffusion biased by a transverse gravitational force.

Diffusion of pointlike noninteracting particles in a two-dimensional channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. A recurrence mapping procedure is applied, which enables the derivation of an effective one-dimensional (1D) evolution equation that governs the 1D density of the particles in the channel. In the limit of stationary flow, an extended Fick-Jacobs equation is reached, which is corrected by an effective diffusion coefficient D(x) that depends on the longitudinal coordinate x. The result is an approximate formula for D(x) that also involves the influence of the transverse force. The calculations are verified by the stationary diffusion in a linear cone, which is exactly solvable.

The Sixth-Moment Sum Rule for the Pair Correlations of the Two-Dimensional One-Component Plasma: Exact Result

The system under consideration is a two-dimensional one-component plasma in the fluid regime, at density n and arbitrary coupling Γ=βe2 (e=unit charge, β=inverse temperature). The Helmholtz free energy of the model, as the generating functional for the direct pair correlation c, is treated in terms of a convergent renormalized Mayer diagrammatic expansion in density. Using specific topological transformations within the bond-renormalized Mayer expansion, we prove that the nonzero contributions to the regular part of the Fourier component of c up to the k2-term originate exclusively from the ring diagrams (unable to undertake the bond-renormalization procedure) of the Helmholtz free energy. In particular, ĉ(k)=−Γ/k2+Γ/(8πn)−k2/[96(πn)2]+O(k4). This result fixes via the Ornstein–Zernike relation, besides the well-known zeroth-, second-, and fourth-moment sum rules, the new sixth-moment condition for the truncated pair correlation h, n(πΓn/2)3 ∫ r6h(r) dr=3(Γ−6)(8−3Γ)/4.

Translation Symmetry Breaking in the One-Component Plasma on the Cylinder

The two-dimensional one-component plasma, i.e. the system of point-like charged particles embedded in a homogeneous neutralizing background, is studied on the surface of a cylinder of finite circumference, or equivalently in a semiperiodic strip of finite width. The model has been solved exactly by Choquardet al. at the free-fermion couplingΓ = 2: in the thermodynamic limit of an infinitely long strip, the particle density turns out to be a nonconstant periodic function in space and the system exhibits long-range order of the Wigner-crystal type. The aim of this paper is to describe, qualitatively as well as quantitatively, the crystalline state for a larger set of couplingsΓ = 2γ (γ = 1,2,..., a positive integer) when the plasma is mappable onto a one-dimensional fermionic theory. The fermionic formalism, supplemented by some periodicity assumptions, reveals that the density profile results from a hierarchy of Gaussians with a uniform variance but with different amplitudes. The number and spatial positions of these Gaussians within an elementary cell depend on the particular value of γ. Analytic results are supported by the exact solution at γ = 1 (Γ = 2) and by exact finite-size calculations at γ = 2,3.

Thermodynamic properties of the two-dimensional Coulomb gas in the low-density limit

The model under consideration is the two-dimensional Coulomb gas of ± charged hard disks with diameter σ. For the case of pointlike charges (σ=0), the system is stable against collapse of positive-negative pairs of charges in the range of inverse temperatures 0≤β<2, where its full exact thermodynamics was obtained recently. In the present work, we derive the leading correction to the exact thermodynamics of pointlike charges due to presence of the hard core σ which enables us to extend the treatment beyond the collapse point β=2. Our results, which are conjectured to be exact in the low-density limit in the interval 0≤β<3, reproduce correctly the singularities of thermodynamic quantities at the collapse point and agree well with Monte-Carlo simulations. The “subtraction” mechanism within the ansatz proposed by M. E. Fisher et al. [J. Stat. Phys.79:1 (1995)], which excludes the existence of intermediate phases between the collapse point β=2 and the Kosterlitz–Thouless transition point βKT=4, is confirmed, however, a different analytic structure of this ansatz is suggested.

Effective diffusion coefficient in 2D periodic channels

Calculation of the effective diffusion coefficient D(x), depending on the longitudinal coordinate x in 2D channels with periodically corrugated walls, is revisited. Instead of scaling the transverse lengths and applying the standard homogenization techniques, we propose an algorithm based on formulation of the problem in the complex plane. A simple model is solved to explain the behavior of D(x) in the channels with short periods L, observed by Brownian simulations of Dagdug et al. [J. Chem. Phys. 133, 034707 (2010)].

Two definitions of the hopping time in a confined fluid of finite particles

We consider a fluid of hard disks diffusing in a flat long narrow channel of width approaching from above the doubled diameter of the disks. In this limit, the disks can pass their neighbors only rarely, in a mean hopping time growing to infinity, so the disks start by diffusing anomalously. We study the hopping time, which is the crucial parameter of the theory describing the subsequent transition to normal diffusion. We show that two different definitions of this quantity, based either on the mean first passage time calculated from solution of the Fick-Jacobs equation, or coming from transition state theory, are incompatible. They have different physical interpretation and also, they give different dependencies of the hopping time on the width of the channel.

When is the next extending of Fick-Jacobs equation necessary?

Applicability of the effective one-dimensional equations, such as Fick-Jacobs equation and its extensions, describing diffusion of particles in 2D or 3D channels with varying cross section A(x) along the longitudinal coordinate x, is studied. The leading nonstationary correction to Zwanzig-Reguera-Rubí equation [R. Zwanzig, J. Phys. Chem. 96, 3926 (1992); D. Reguera and J. M. Rubí, Phys. Rev. E 64, 061106 (2001)] is derived and tested on the exactly solvable model, diffusion in a 2D linear cone. The effects of such correction are demonstrated and discussed on elementary nonstationary processes, a time dependent perturbation of the stationary flow and calculation of the mean first passage time.

Generalized method calculating the effective diffusion coefficient in periodic channels.

The method calculating the effective diffusion coefficient in an arbitrary periodic two-dimensional channel, presented in our previous paper [P. Kalinay, J. Chem. Phys. 141, 144101 (2014)], is generalized to 3D channels of cylindrical symmetry, as well as to 2D or 3D channels with particles driven by a constant longitudinal external driving force. The next possible extensions are also indicated. The former calculation was based on calculus in the complex plane, suitable for the stationary diffusion in 2D domains. The method is reformulated here using standard tools of functional analysis, enabling the generalization.

Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation

A point-like 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 1D Fokker-Planck (Kramers) 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/γ, γ denotes the friction coefficient. The corrections 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.

Response to “Comment on ‘Calculation of the mean first passage time tested on simple two-dimensional models’” [J. Chem. Phys.128, 197102 (2008)]

Bowles et al. interpreted their simulation results [J. Chem. Phys.121, 10668 (2004)] for two diffusing disks in his Comment. They stressed that there is no support for the −3∕2 exponent of the power law dependence of the hopping time on the width of the bottleneck there. As my calculations [J. Chem. Phys.126, 194708 (2007)] were not contested, the problem remains unresolved.