786 research outputs found
Time correlation functions of equilibrium and nonequilibrium Langevin dynamics: Derivations and numerics using random numbers
We study the time correlation functions of coupled linear Langevin dynamics
without and with inertia effects, both analytically and numerically. The model
equation represents the physical behavior of a harmonic oscillator in two or
three dimensions in the presence of friction, noise, and an external field with
both rotational and deformational components. This simple model plays pivotal
roles in understanding more complicated processes. The presented analytical
solution serves as a test of numerical integration schemes, its derivation is
presented in a fashion that allows to be repeated directly in a classroom.
While the results in the absence of fields (equilibrium) or confinement (free
particle) are omnipresent in the literature, we write down, apparently for the
first time, the full nonequilibrium results that may correspond, e.g., to a
Hookean dumbbell embedded in a macroscopically homogeneous shear or mixed flow
field. We demonstrate how the inertia results reduce to their noninertia
counterparts in the nontrivial limit of vanishing mass. While the results are
derived using basic integrations over Dirac delta distributions, we mention its
relationship with alternative approaches involving (i) Fourier transforms, that
seems advantageous only if the measured quantities also reside in Fourier
space, and (ii) a Fokker--Planck equation and the moments of the probability
distribution. The results, verified by numerical experiments, provide
additional means of measuring the performance of numerical methods for such
systems. It should be emphasized that this manuscript provides specific details
regarding the derivations of the time correlation functions as well as the
implementations of various numerical methods, so that it can serve as a
standalone piece as part of education in the framework of stochastic
differential equations and calculus.Comment: 35 pages, 5 figure
Topological analysis of polymeric melts: Chain length effects and fast-converging estimators for entanglement length
Primitive path analyses of entanglements are performed over a wide range of
chain lengths for both bead spring and atomistic polyethylene polymer melts.
Estimators for the entanglement length N_e which operate on results for a
single chain length N are shown to produce systematic O(1/N) errors. The
mathematical roots of these errors are identified as (a) treating chain ends as
entanglements and (b) neglecting non-Gaussian corrections to chain and
primitive path dimensions. The prefactors for the O(1/N) errors may be large;
in general their magnitude depends both on the polymer model and the method
used to obtain primitive paths. We propose, derive and test new estimators
which eliminate these systematic errors using information obtainable from the
variation of entanglement characteristics with chain length. The new estimators
produce accurate results for N_e from marginally entangled systems. Formulas
based on direct enumeration of entanglements appear to converge faster and are
simpler to apply.Comment: Major revisions. Developed near-ideal estimators which operate on
multiple chain lengths. Now test these on two very different model polymers
Key Epidemic Parameters of the SIRV Model Determined from Past COVID-19 Mutant Waves
Monitored infection and vaccination rates during past past waves of the coronavirus are used to infer a posteriori two-key parameter of the SIRV epidemic model, namely, the real-time variation in (i) the ratio of recovery to infection rate and (ii) the ratio of vaccination to infection rate. We demonstrate that using the classical SIR model, the ratio between recovery and infection rates tends to overestimate the true ratio, which is of relevance in predicting the dynamics of an epidemic in the presence of vaccinations
Epidemics Forecast From SIR-Modeling, Verification and Calculated Effects of Lockdown and Lifting of Interventions
Due to the current COVID-19 epidemic plague hitting the worldwide population it is of utmost medical, economical and societal interest to gain reliable predictions on the temporal evolution of the spreading of the infectious diseases in human populations. Of particular interest are the daily rates and cumulative number of new infections, as they are monitored in infected societies, and the influence of non-pharmaceutical interventions due to different lockdown measures as well as their subsequent lifting on these infections. Estimating quantitatively the influence of a later lifting of the interventions on the resulting increase in the case numbers is important to discriminate this increase from the onset of a second wave. The recently discovered new analytical solutions of Susceptible-Infectious-Recovered (SIR) model allow for such forecast. In particular, it is possible to test lockdown and lifting interventions because the new solutions hold for arbitrary time dependence of the infection rate. Here we present simple analytical approximations for the rate and cumulative number of new infections
Reasonable Limiting of 7-Day Incidence per Hundred Thousand and Herd Immunization in Germany and Other Countries
Based on hospital capacities, facts from past experience with the coronavirus disease 2019 (COVID-19) virus and the number of dark infections during the second wave (DII=2D2), a reasonable limiting value of 140/D2 for the 7-day incidence per 100,000 persons (MSDIHT) and a second wave herd immunization threshold fraction value of 0.26 in Germany were calculated. If the MSDIHT is held below this limiting value, the German hospital system can cope with the number of new seriously infected persons without any triage decisions. On the basis of the SIRV epidemics model, the classical threshold values for herd immunization were calculated for 18 countries. For these countries, the dates regarding when herd immunization against the second COVID-19 wave will be reached were estimated
Multi-Hamiltonian structure of the epidemics model accounting for vaccinations and a suitable test for the accuracy of its numerical solvers
We derive a generalized Hamiltonian formalism for a modified suscepti bleâinfectiousârecovered/removed (SIR) epidemic model taking into account the population V of vaccinated persons. The resulting SIRV model is shown to admit three possible functionally independent Hamiltonians and hence three associated Poisson structures. The reduced case of vanishing vaccinated sector shows a complete correspondence with the known Poisson structures of the SIR model. The SIRV model is shown to be expressible as an almost Nambu sys tem, except for a scale factor function breaking the divergenceless property. In the autonomous case with time-independent stationary ratios k and b, the SIRV model is shown to be a maximally super-integrable system. For this case we test the accuracy of numerical schemes that are suited to solve the stiff set of SIRV differential equations
Endocytosis of PEGylated nanoparticles accompanied by structural and free energy changes of the grafted polyethylene glycol
AbstractNanoparticles (NPs) are in use to efficiently deliver drug molecules into diseased cells. The surfaces of NPs are usually grafted with polyethylene glycol (PEG) polymers, during so-called PEGylation, to improve water solubility, avoid aggregation, and prevent opsonization during blood circulation. The interplay between grafting density Ïp and grafted PEG polymerization degree N makes cellular uptake of PEGylated NPs distinct from that of bare NPs. To understand the role played by grafted PEG polymers, we study the endocytosis of 8Â nm sized PEGylated NPs with different Ïp and N through large scale dissipative particle dynamics (DPD) simulations. The free energy change Fpolymer of grafted PEG polymers, before and after endocytosis, is identified to have an effect which is comparable to, or even larger than, the bending energy of the membrane during endocytosis. Based on self-consistent field theory Fpolymer is found to be dependent on both Ïp and N. By incorporating Fpolymer, the critical ligand-receptor binding strength for PEGylated NPs to be internalized can be correctly predicted by a simple analytical equation. Without considering Fpolymer, it turns out impossible to predict whether the PEGylated NPs will be delivered into the diseased cells. These simulation results and theoretical analysis not only provide new insights into the endocytosis process of PEGylated NPs, but also shed light on the underlying physical mechanisms, which can be utilized for designing efficient PEGylated NP-based therapeutic carriers with improved cellular targeting and uptake
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