The velocity autocorrelation function of an overdamped Brownian system with hard-core intraction

Hanna, Sarwat N.; Heß, W.; Klein, R.

doi:10.1088/0305-4470/14/12/004

Cited by 18 publications

(17 citation statements)

References 9 publications

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“…It is not possible to invert these Laplace transforms in general, but some special cases can be examined. The results have been derived independently by Ackerson & Fleishman (1982), Hanna et al (1981), Jones & Burfield (1982) and Felderhof & Jones (1983), and thereby confirm the formal correctness of the manipulations of $3. The interpretation of the results which follows differs somewhat from theirs.…”

Section: Solutions Of the Equationssupporting

confidence: 72%

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The effect of particle interactions on dynamic light scattering from a dilute suspension

Rallison

Hinch

1986

J. Fluid Mech.

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The calculation of dynamic laser-light scattering by dilute suspensions of Brownian particles is reviewed. It is shown that present theories of diffusion can provide approximations for the autocorrelation of the intensity of the scattered light that are only uniformly accurate for correlation times up to order (Do k2)-' where Do is the diffusivity of a single particle and k is the scattering wave vector. The meanings of, and connections between, down-gradient, self-and tracer diffusion for both short and long times are established and it is shown how these may be inferred from light-scattering experiments for optically monodisperse and polydisperse systems. For dilute systems, equations giving the time evolution of the intermediate and self-intermediate scattering functions, F ( k , t ) and Fs(k, t ) , accurate to first order in the volume concentration of particles are constructed, and are solved for suspensions of hard spheres with and without hydrodynamic interaction. For short and long times (semi-) analytic solutions are given; for intermediate times numerical results are presented. The formal correspondence of the limiting values of the time-dependent solutions with the results of Batchelor (1976, 1983) and others for steady sedimentation in polydisperse systems is established.good deal of attention. Table 1 gives a partial list of relevant papers. A detailed explanation of the symbols used is given in $2. The papers fall into two categories. Those influenced by suspension mechanical work on steady transport phenomena in two-phase flow have tended to use exact hydrodynamic data (which is an important ingredient of the problem), but have provided results appropriate only to special limiting cases of the light-scattering problem. Papers in the second category have attempted (more) complete solutions of the problem but have used approximate hydrodynamic data. For many papers in both categories it is not always clear to which regime in wavenumber and time the result applies, nor the extent of its accuracy and validity. In this account we seek to combine the better features of both approaches, to use exact hydrodynamics whenever possible, and to provide both a physical explanation for, and a mathematical proof of, the correspondence between the various results which provide partial solutions t o the full problem. The difference between 'exact ' and 'approximate ' hydrodynamics should be amplified since the term exact is intended to convey more than mere accuracy. The solution of two-and many-body problems in hydrodynamics can in general be achieved only numerically, and in consequence the final results are subject to rounding and cut off errors (e.g. see table 2). On the other hand, the analytic expressions given are exact, and can exhibit qualitative differences from results obtained using 'approximate ' data (see for example the discussion of $3.2). In $2 we delineate the various timescales of interest, and in $3 provide a full mathematical formulation for the dilute limit. In $54 and 5 we solve the equations ...

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Section: Solutions Of the Equationssupporting

confidence: 72%

“…Then, using the identities (4.2) is rapidly corrected. Pusey & Tough (1982) have also shown that the second cumulant l" is infinite at short times, and in the ka -4 1 limit Hanna et al (1981) have demonstrated that oc (Do t/a2))-t as t + 0 by an alternative method.…”

Section: Short-time Behaviour T 4 T 4 Tmentioning

confidence: 99%

The effect of particle interactions on dynamic light scattering from a dilute suspension

Rallison

Hinch

1986

J. Fluid Mech.

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“…which relates the force and the velocity autocorrelation functions, and whose functional form depends only on the properties of the evolution operator (the special case for equilibrium dynamics was derived in [56]). Once we identify the diffusivity with the velocity self-correlation function we get the Green-Kubo expression…”

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confidence: 99%

Linear Response Theory and Green-Kubo Relations for Active Matter

2019

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We address the question of how interacting active systems in a non-equilibrium steady-state respond to an external perturbation. We establish an extended fluctuation-dissipation theorem for Active Brownian Particles (ABP) which highlights the role played by the local violation of detailed balance due to activity. By making use of a Markovian approximation we derive closed Green-Kubo expressions for the diffusivity and mobility of ABP and quantify the deviations from the Stokes-Einstein relation. We compute the linear response function to an external force using unperturbed simulations of ABP and compare the results with the analytical predictions of the transport coefficients. Our results show the importance of the interplay between activity and interactions in the departure from equilibrium linear response.

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“…The latter observation can be rationalized by considering that small positional changes can give rise to large changes in the force for closely packed particles residing in regions of strong interaction-force gradient. The fact that H(t) is non-exponential is not surprising, given that it can be expressed in terms of the velocity autocorrelation function, a quantity which famously exhibits power law asymptotic behaviour ('long-time tails') [19,21]. Klein et al have shown analytically that for a dilute system of Brownian hard-spheres F (t) · F (0) eq,s ∼ t − 5 2 for long times.…”

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confidence: 99%

“…A clear next step is to investigate approximations to H(t) which enable predictions to be made from first-principles, without simulation input. Given the relation (19) it seems likely that existing approximations to the velocity autocorrelation function (e.g. projection operator approaches) could be usefully exploited.…”

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confidence: 99%

Communication: Green-Kubo approach to the average swim speed in active Brownian systems

Sharma

Brader

2016

The Journal of Chemical Physics

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We develop an exact Green-Kubo formula relating nonequilibrium averages in systems of interacting active Brownian particles to equilibrium time-correlation functions. The method is applied to calculate the density-dependent average swim speed, which is a key quantity entering coarse grained theories of active matter. The average swim speed is determined by integrating the equilibrium autocorrelation function of the interaction force acting on a tagged particle. Analytical results are validated using Brownian dynamics simulations.

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The velocity autocorrelation function of an overdamped Brownian system with hard-core intraction

Cited by 18 publications

References 9 publications

The effect of particle interactions on dynamic light scattering from a dilute suspension

The effect of particle interactions on dynamic light scattering from a dilute suspension

Linear Response Theory and Green-Kubo Relations for Active Matter

Communication: Green-Kubo approach to the average swim speed in active Brownian systems

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