Statistical Properties of Thermal Noise Driving the Brownian Particles in Fluids
EPJ Web of Conferences 108, 0 2 0 4 4 (2016 )
DOI: 10.1051/epjconf/ 2016 10 8 0 2 0 4 4
C Owned by the authors, published by EDP Sciences, 2016
Statistical Properties of Thermal Noise Driving the Brownian
Particles in Fluids
Jana Tóthová1 , a and Vladimír Lisý1,2 , b
1
Department of Physics, Faculty of Electrical Engineering and Informatics, Technical University of Košice,
Park Komenského 2, 042 00 Košice, Slovakia
2
Laboratory of Radiation Biology, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region,
Russia
Abstract. In several recent works high-resolution interferometric detection allowed to
study the Brownian motion of optically trapped microparticles in air and fluids. The
observed positional fluctuations of the particles are well described by the generalized
Langevin equation with the Boussinesq-Basset “history force” instead of the Stokes friction, which is valid only for the steady motion. Recently, also the time correlation
function of the thermal random force Fth driving the Brownian particles through collisions with the surrounding molecules has been measured. In the present contribution we
propose a method to describe the statistical properties of Fth in incompressible fluids.
Our calculations show that the time decay of the correlator Fth (t)Fth (0) is significantly
slower than that found in the literature. It is also shown how the “color” of the thermal
noise can be determined from the measured positions of the Brownian particles.
1 Introduction
Random processes in various fields of science are often described by phenomenological equations
of motion containing stochastic forces, the best known example being the Langevin equation (LE),
first designed to describe the Brownian motion (BM) of particles [1]. However, as it is known for
a long time theoretically [2–5] and reported in experiments (see, e.g., the review [6]), the LE fails
to describe the motion of the Brownian particles when their density is comparable to that of the
surrounding. Beginning with [7], the BM in fluids was successfully studied using the high-resolution
interferometry of optically trapped particles [8–10]. These experiments are in excellent agreement
with the prediction for the mean square displacement of the particle and related correlation functions
within the hydrodynamic theory of the BM of particles confined in a harmonic potential well [4].
The experimental access to short timescales [10] allowed also obtaining the time correlation function
Fth (t)Fth (0) of the thermal random force Fth (t), resulting from the collisions with the surrounding
molecules and thus responsible for the BM of the particles. The work [10] can be considered as the
first work where the “color” of the thermal noise has been measured. The method of calculation of
Fth (t)Fth (0) from the measured positional fluctuations of the particles proposed in [10, 11] leads
a e-mail:
b e-mail:
4
Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/201610802044
EPJ Web of Conferences
to a dependence Fth (t)Fth (0) ∼ t−3/2 . The discussion done below reveals shortcomings of this
method and gives two different ways to calculate the function Fth (t)Fth (0). In the frame of the linear
hydrodynamic theory of the BM in incompressible fluids our calculations are exact. At long times the
decay of Fth (t)Fth (0) is much slower than previously assumed in [6, 10–12].
2 Time correlations of the thermal noise
In the classical LE the resistance force against the motion of the Brownian particles is the Stokes friction force proportional to the particle velocity. In a more general hydrodynamic theory of the BM [2]
the Stokes force is replaced by the Boussinesq force [13], derived from the linearized Navier-Stokes
and continuity equations for incompressible fluids. This force describes hydrodynamic interactions of
a spherical tracer with the surrounding fluid [14]. These interactions appear at short time scales and,
as distinct from the more familiar Kubo’s generalization of the LE [15], the memory integral contains
the acceleration of the Brownian particle instead of its velocity. The LE for the velocity υ(t) = dx/dt
of Brownian particles then has the form
t
mυ̇ (t) + γυ (t) +
Γ t − t υ̇ t dt = Fth (t) ,
(1)
t0
where Γ(t) = γ(τf /πt)1/2 with τf = R2 ρf /η, R is the radius of the particle, ρf is the fluid density and
η its viscosity, γ = 6πηR is the Stokes friction coefficient, and m = mp + mf /2, with mf being the
mass of the fluid displaced by the particle of mass mp (we use the same notation as in [10]). The
particle is assumed to be in thermal equilibrium with the liquid. The time t0 denotes an initial moment
infinitely remote from t. When the particle is in an external harmonic field, the force −K x(t), where
x(t) is the particle displacement from the trap center, should be added in the right side of (1). In
the traditional LE the thermal noise force Fth (t) is white. This is not the case here since, due to the
fluctuation-dissipation theorem [15], the values of Fth (t) at different times correlate. The authors of
[10] have accessed the correlations in the colored Fth (t) by recording the positions of the particle and
calculating the autocorrelation function x(t)x(0). It was assumed that at long times the trapping
force dominates over friction. Ignoring the particle inertia, the LE was thus reduced to K x(t) ≈ Fth (t).
Then it was assumed that Fth (t)Fth (0) ≈ K 2 x(t)x(0). However, this requires that K x(0) ≈ Fth (0)
also holds, which is not true; in fact, at t → 0, K x(t) is less important than other terms in the LE. Here
we show that the calculation of Fth (t)Fth (0) is possible without these approximations.
Within the linear theory, the properties of Fth (t) do not depend on the external force [15]. To
find Fth (t)Fth (0), it is thus possible to use the LE without the term K x(t)
t and proceed as follows.
One can rewrite Eq. (1) for υ(t0 + t), t > 0, transform the integral to 0 , multiply this equation
by Fth (t0 ) = mυ̇(t0 ) + γυ(t0 ), and statistically average. For stationary processes υ̇(t0 )υ(t0 + t) =
− υ (t0 ) υ̇ (t0 + t) = −φ̇ (t) and all the terms in the resulting equation for Fth (t0 )Fth (t0 + t) can
be expressed through the velocity autocorrelation function φ(t) = υ(t)υ(0), e.g., υ̇(t0 )υ̇(t0 + t) =
(d/dt) υ̇(t0 )υ(t0 + t) = −φ̈(t). In the Laplace transformation we obtain
(2)
L {Fth (t) Fth (0)} = γ2 φ̃ (s) − m2 s + msΓ̃ (s) − γ Γ̃ (s) sφ̃ (s) − φ (0) .
Then, using φ̇(0) = 0 and φ(0) = kB T/m (the equipartition theorem), the Laplace-transformed equation for φ̃(s) = L {φ(t)} can be found from (1). In the theory of the hydrodynamic BM [3, 16–18]
−1
φ̃(s) = kB T γ + s m + Γ̃(s)
,
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