Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics
We analyze the transition from deterministic mean-field dynamics of several large particles and infinitely many small particles to a stochastic motion of the large particles. In this transition the small particles become the random medium for the large particles, and the motion of the large particle...
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2005
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| author | Kotelenez, P. Котеленез, П. |
| author_facet | Kotelenez, P. Котеленез, П. |
| author_sort | Kotelenez, P. |
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| datestamp_date | 2020-03-18T20:00:55Z |
| description | We analyze the transition from deterministic mean-field dynamics of several large particles and infinitely many small particles to a stochastic motion of the large particles. In this transition the small particles become the random medium for the large particles, and the motion of the large particles becomes stochastic. Under the assumption that the empirical velocity distribution of the small particles is governed by a probability density ?, the mean-field force can be represented as the negative gradient of a scaled version of ?. The stochastic motion is described by a system of stochastic ordinary differential equations driven by Gaussian space-time white noise and the mean-field force as a shift-invariant integral kernel. The scaling preserves a small parameter in the transition, the so-called correlation length. In this set-up, the separate motion of each particle is a classical Brownian motion (Wiener process), but the joint motion is correlated through the mean-field force and the noise. Therefore, it is not Gaussian. The motion of two particles is analyzed in detail and a diffusion equation is deduced for the difference in the positions of the two particles. The diffusion coefficient in the latter equation is spatially dependent, which allows us to determine regions of attraction and repulsion of the two particles by computing the probability fluxes. The result is consistent with observations in the applied sciences, namely that Brownian particles get attracted to one another if the distance between them is smaller than a critical small parameter. In our case, this parameter is shown to be proportional to the aforementioned correlation length. |
| first_indexed | 2026-03-24T02:46:18Z |
| format | Article |
| fulltext |
UDC 517.9+ 531.19
P. Kotelenez (Case Western Reserve Univ., Cleveland, USA)
CORRELATED BROWNIAN MOTIONS
AS AN APPROXIMATION TO DETERMINISTIC
MEAN-FIELD DYNAMICS
KOREL\OVANYJ BROUNIVS\KYJ RUX QK HRANYCQ
NABLYÛENNQ DETERMINISTS\KO} DYNAMIKY
SEREDN\OHO POLQ
We analyze the transition from deterministic mean-field dynamics of several large particles and infinitely many
small particles to a stochastic motion of the large particles. In this transition the small particles become the
random medium for the large particles and the motion of the large particles becomes stochastic. Assuming that
the empirical velocity distribution of the small particles is governed by a probability density ψ, the mean-field
force can be represented as the negative gradient of a scaled version of ψ. The stochastic motion is described
by a system of stochastic ordinary differential equations driven by Gaussian space-time white noise and the
mean-field force as a shift-invariant integral kernel. The scaling preserves a small parameter in the transition,
the so-called correlation length. In this set-up the separate motion of each particle is a classical Brownian motion
(Wiener process), but the joint motion is correlated through the mean-field force and the noise. Therefore, it is
not Gaussian. The motion of 2 particles is analyzed in detail and a diffusion equation is derived for the difference
in the positions of the 2 particles. The diffusion coefficient in the latter equation is spatially dependent, which
allows us to determine regions of attraction and repulsion of the two particles by computing the probability
fluxes. The result is consistent with observations in the applied sciences, namely that Brownian particles get
attracted to each other if the distance between them is smaller than a critical small parameter. In our case, this
parameter is shown to be proportional to the aforementioned correlation length.
Proanalizovano perexid vid determinists\ko] dynamiky seredn\oho polq dekil\kox velykyx çastynok ta
neskinçenno] kil\kosti malyx çastynok do stoxastyçnoho ruxu velykyx çastynok. Pid ças c\oho pere-
xodu malen\ki çastynky peretvorggt\sq u vypadkove seredovywe dlq velykyx çastynok, a rux velykyx
çastynok sta[ stoxastyçnym. Qkwo prypustyty, wo rozpodil empiryçno] ßvydkosti malyx çastynok
vyznaça[t\sq wil\nistg rozpodilu ψ, to sylu seredn\oho polq moΩna podaty qk vid’[mnyj hradi[nt
masßtabnoho peretvorennq ψ. Stoxastyçnyj rux opysano systemog stoxastyçnyx zvyçajnyx dyferen-
cial\nyx rivnqn\, kerovanyx haussovym prostorovo-çasovym bilym ßumom ta sylog seredn\oho polq qk
intehral\nym qdrom, invariantnym vidnosno zsuvu. Masßtabuvannq zberiha[ malyj parametr pry pere-
xodi (tak zvanu dovΩynu korelqci]). U danij postanovci okremyj rux koΩno] çastynky [ klasyçnym
brounivs\kym ruxom (vinerivs\kym procesom), ale spil\nyj rux korelg[t\sq sylog seredn\oho polq
i ßumom. Tomu vin ne [ haussivs\kym. Detal\no proanalizovano rux dvox çastynok i vyvedeno riv-
nqnnq dyfuzi] dlq riznyci poloΩen\ dvox çastynok. Koefici[nt dyfuzi] v ostann\omu rivnqnni [
prostorovo zaleΩnym, wo dozvolq[ vyznaçyty oblasti prytqhannq ta vidßtovxuvannq dvox çastynok
ßlqxom rozraxunku teçij imovirnostej. Rezul\tat uzhodΩu[t\sq iz spostereΩennqmy u prykladnyx
naukax, a same, z faktom, wo brounivs\ki çastynky prytqhugt\sq odna do odno], qkwo vidstan\ miΩ
nymy menßa za krytyçnyj malyj parametr. U vypadku, wo vyvça[t\sq, pokazano, wo cej parametr
proporcional\nyj zhadanij vywe dovΩyni korelqci].
1. Introduction: microscopic model and space-time scales. Kotelenez [1] analyzes a
model in which one can observe the transition from a completely deterministic motion
of a system of large and small particles with random initial conditions into a stochastic
motion. The small particles have very large velocities. The large particles’ momenta are
completely determined by the momenta of the small particles through a mean field type
interaction.
In a scaling limit Kotelenez (loc.cit.) shows that, under certain assumptions, the posi-
tions of the large particles become correlated Brownian motions, where the spatial corre-
lations can be computed through a transformation of the interaction force. A similar result
holds for the velocities in a Hamiltonian framework. To explain the transition from a de-
terministic to a stochastic motion the following assumptions and observations are made.
First, the state space of the particles is Rd with d ≥ 2. This allows small particles to
escape to infinity and to be replaced by other small particles. Secondly, assuming that
c© P. KOTELENEZ, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6 757
758 P. KOTELENEZ
the initial mean velocity of the small particles tends to infinity in the scaling limit, the
fast moving small particles interact with the large particles only for a short time before
escaping to infinity. Consequently, in different (small) time intervals the displacements
of the large particles are caused essentially by different small particles which started (by
assumption) independently. Therefore, these displacements become independent in the
limit.
The interactions between large particles can be neglected in this limit as their effects
can be included in the macroscale (after deriving approximate formulas for the action of
the small particles upon the large ones). The fractional step method (cf. [2]) provides a
rigorous framework to include these effects.
Microscopic and mesoscopic units are defined as fractions of macroscopic units. Let
Rd be partitioned into small cubes which are parallel to the axes. The cubes are denoted
by (R̄λ], where R̄λ is the center of the cube and λ ∈ N. These cubes are open on
the left and closed on the right (in the sense of d-dimensional intervals) and have side
length δR ≈ 1
n
, and the origin 0 is the center of a cell. The cells and their centers
will be used to “coarse-grain” the motion of the particles, placing the particles within a
cell at the midpoint (cf. [3]). Further, the small particles within a cell are grouped as
clusters, where particles in a cluster have similar initial velocities. Thus, we average
locally over the initial data of particles while maintaining globally a spatially distributed
structure of the particle systems. We may call δR a mesoscopic length unit (cf. [4, 5]
for the use of this terminology). Similarly, we choose a mesoscopic time unit δσ, during
which the interaction between small and large particles forms a pattern, and which can
be used to simplify the calculations (s. the following Remark 1.1). Here we choose
δσ ≈ 1
nd
, which follows from the need to control the variance of sums of independent
random variables and its generalization in Doob’s inequality, where the second moments
are ≈ nd, i.e., the number of cells in a unit cube. δσ then becomes a normalizing factor
at the forces acting on the large particle motion. Assuming, without loss of generality,
that the proportionality factors in the relations for δR and δσ equal 1, we now define
the mesoscopic and macroscopic time and spatial scales:
δσ =
1
nd
� 1, δR =
1
n
� 1. (1.1)
Remark 1.1. Stochastic approximations to elastic collisions have been obtained by
numerous authors for cases involving only one large particle and assuming no interaction
(collisions) between the small particles. Dürr, Goldstein and Lebowitz [6, 7] obtain an
Ornstein – Uhlenbeck approximation to the collision dynamics, generalizing a result of
Holley [8] from dimension d = 1 to dimension d = 3. Sinai and Soloveichik [9] obtain
an Einstein – Smoluchowski approximation in dimension d = 1 and prove that almost
all small particles collide with the large particle only finitely often. A similiar result
was obtained by Szász and Tóth [10, 11]. Szász and Tóth [11] obtain both Einstein –
Smoluchowski and Ornstein – Uhlenbeck approximations for the time evolution of one
large particle in dimension d = 1 (cf. also [12]).
Note that in the case of just one large particle, the fluid around that particle may look
homogeneous and isotropic, which leads to a relatively simple statistical description of
the displacement of that particle as a result of the “bombardment” of this large particle
by small particles. Further, whether or not the “medium” of small particles is spatially
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
CORRELATED BROWNIAN MOTIONS AS AN APPROXIMATION TO DETERMINISTIC . . . 759
correlated cannot influence the motion of only one large particle, as long as in the scaling
limit the time correlation length δs tends to 0.
In contrast, if there are at least two large particles and they move closely together,
the fluid around each of them will no longer be homogeneous and isotropic. In fact, the
fluid will get depleted (Asakura, and Oosawa [13], Goetzelmann, Evans, and Dietrich
[14] and the references therein as well as Kotelenez, Leitman, and Mann (Jr.) [15]). The
simplest argument to explain depletion is that if the large particles get closer together than
the diameter of a typical small particle, the space between the large particles must get
depleted (Goetzelmann, Evans, and Dietrich (loc.cit.)). Therefore, the forces, generated
by the collisions and acting on two different large particles, become statistically correlated
if the large particles move together closer than a critical length
√
ε > 0. To capture this
effect in the scaling limit, we choose the following δs � δσ as the microscopic time
unit as follows:
δs :=
√
ε
σ̄n
, (1.2)
where σ̄n is the mean velocity of the small particles.
Suppose that the time evolution of the particles is stationary (starting with some Gibbs
distribution).
Denote by Bε̂n
(q) the ε̂n-neighborhood of q, i.e., the ball of radius ε̂n and center
q. Let m̃ be the mass of a small particle. We first replace qλ,l, the position of the particle
starting in (R̄λ] with velocity
pλ,l
m̃
by the midpoint of the cell (R̄λ] (coarse graining).
Suppose there are approximately M̃n small (noninteracting) particles starting in the
same (R̄λ] and ndM̃n small particles in a unit cube. Let us make the following simpli-
fying assumptions:
(i) We assume that the small particles hit the large particle at its center, averaging
over all possible angles of contact. One might call this procedure “coarse graining” in
momentum space.
(ii) We consider collisions over a time interval of length δσ. The size of the “ob-
served” collisions is Ln ≤ ndM̃nε̂d−1
n
δσ
δs
, the average number of collisions during a
time of length δσ. If Ln � ndM̃nε̂d−1
n
δσ
δs
, we may consider Ln to be the size of a
sample. δσ is large enough so that most small particles will be able to escape to infin-
ity after colliding with the give large particle (cf. Sinai and Soloveichik (loc.cit.)) for the
one-dimensional case. Hence, we may assume that the “samples” in disjoint time intervals
[(l − 1)δσ, lδσ) are roughly independent. We take the rate of change in the velocity of
the large particle, VR(t), as a result of one collision to be proportional to δs. Adding up
those changes over a time interval of length δσ, we assume that the law of large numbers
(LLN) is applicable.
As a result of these assumptions, we first obtain that the (random) force acting on the
large particle R during the time interval δσ should be proportional
m̃
VR(t + δσ)− VR(t)
δσ
≈ m̃
cε,Ln
δσ
∑
λ
(R̄− R̄λ)
Ln∑
l=1
1{|R̄−R̄λ− pλ,lδs
m̃ |≤ε̂n}. (1.3)
Here, cε,Ln is some constant, depending on ε and Ln. Next, suppose that as a result of
the law of large numbers (LLN)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
760 P. KOTELENEZ
1
Ln
Ln∑
l=1
1{|R̄−R̄λ− pλ,lδs
m̃ |≤ε̂n} ≈
≈ E
1
Ln
Ln∑
l=1
1{|R̄−R̄λ− pλ,lδs
m̃ |≤ε̂n} = gε(R̄− R̄λ)ε̂d
n. (1.4)
Here, gε(R̄− R̄λ) is an even density of the velocities of a cluster of small particles with
start in the cell (R̄λ] and moving with approximately the same velocity. Set
Gε(R̄− R̄λ) :=
cε,Ln
Lnε̂d
n
δσ
(R̄− R̄λ)gε(R̄− R̄λ).
The procedure of replacing the collision effects by averages over small time intervals
may be called “coarse graining” in time.
Example 1.1. Let D > 0 be a diffusion coefficient and consider the rescaled
Maxwellian density
gε(q) :=
1
(2πε)
d
2
exp
(
−| q |
2
2ε
)
. (1.5)
Suppose
cε,LnLnε̂d
n
δσ
≈
√
D(4πε)
d
4
(
2
dε
) 1
2
.
Then, by (1.3)
cε,Ln
δσ
(R̄− R̄λ)
Ln∑
l=1
1{|R̄−R̄λ− pλ,lδs
m̃ |≤ε̂n} ≈
≈
√
D(4πε)
d
4
(
2
dε
) 1
2
(R̄− R̄λ)
1
(2πε)
d
2
exp
(
−| R̄− R̄λ |2
2ε
)
. (1.6)
We obtain
Gε(R̄− R̄λ) =
=
√
D(4πε)
d
4
(
2
dε
) 1
2 (
R̄− R̄λ
) 1
(2πε)
d
2
exp
(
−| R̄− R̄λ |2
2ε
)
. (1.7)
So, in the case of a Maxwellian velocity field, the displacement of the large particles
due to the collisions with (fast) moving small particles, analyzed in the mesoscale, is
approximately governed by the smooth interaction potential
Uε(R− q) := m̂D
(
2ε
d
) 1
2 1
(πε)
d
4
exp
(−|R− q|2
2ε
)
. (1.8)
In addition to the previous heuristic “derivation” of a mean-field force from short
range collisions, a mean-field dynamics, on a microscopic level, can result from long
range potentials, like a Coulomb potential or a smoothed Lenard – Jones potential, and an
appropriate mixture of both effects can lead to more complicated mean-field forces.
2. A deterministic mean-field model and its stochastic limit. The interaction be-
tween large and small particles is governed by a twice continuously differentiable odd
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
CORRELATED BROWNIAN MOTIONS AS AN APPROXIMATION TO DETERMINISTIC . . . 761
Rd-valued function Gε. We assume that all partial derivatives up to order 2 are square
integrable and that |Gε|m is integrable for 1 ≤ m ≤ 4, where “integrable” refers to the
Lebesgue measure on Rd. The function Gε will be approximated by odd Rd-valued
functions Gε,n with bounded supports. Further, we use the idea of coarse graining, as
described in the introduction. m̂ will be the mass of a large particle, and m denotes the
mass of a cluster of small particles within a cell. The velocities of the clusters in a cell
can be characterized as follows:
Hypothesis 2.1. There is a partitioning of the velocity space
Rd = ∪ι∈NBι,
and the velocities of each cluster take values in exactly one Bι, where for the sake of
simplicity we assume that all Bι are small cubic d-dimensional intervals (left open,
right closed), all with the same volume ≤ 1
nd
.
Set
Y (dQ, t) := m
∑
λ,ι
δQ̄n(t,λ,ι)(dQ), XN (dR, t) := m̂
N∑
j=1
δR̄j
n(t)(dR),
where R̄j
n(t) and Q̄n(t, λ, ι) are the positions at time t of the large and small particles,
respectively. “ ¯ ” means that the midpoints of those cells are taken, where the particles
are at time t, and the labels λ, ι mean that the small particle started at t = 0 in (R̄λ]
with velocity from Bι. Y and XN are called the empirical distributions of the small
and large particles, respectively. We include a friction coefficient βn for the large parti-
cles (cf. Uhlenbeck and Ornstein [16] for the rationale) 1 . Let “∨” denote “max”. The
assumptions on the most important parameters are put together in the following assump-
tion:
Hypothesis 2.2.
βn = np̃, d > p̃ > 0,
m = n−η, η ≥ 0,
σ̄n = np, p > (4d + 2) ∨ (2p̃ + 2η + 2d + 2).
The velocity of the ith large particle at time s, will be denoted V i
n(s), where
V i
n(0) = 0 ∀i. wλ,ι
0,n ∈ Bι will be the initial velocity of the small particle starting at
time 0 in the cell (R̄λ], ι ∈ N. Note that, for the infinitely many small particles, the
resulting friction due to the collision with the finitely many large particles should be neg-
ligible. Further, in a dynamical Ornstein – Uhlenbeck type model with friction βn the
“fluctuation force” has to be governed by a function G̃n. The relation to an Einstein –
Smoluchowski diffusion is given by
G̃n(R) ≈ βnGn(R),
as βn −→ ∞ (cf., e.g., [17, 18] or [19]). This factor will disappear, as we move from a
second order differential equation to a first order equation.
1 In the original draft of the derivation of Brownian motions friction was missing. I want to thank
J. A. Mann (Jr.) from the Chemical Engineering Department at CWRU for pointing out that the source of
Brownian motion is also the source of friction and had to be included.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
762 P. KOTELENEZ
Next, we identify the clusters in the small cells with velocity from Bι with ran-
dom variables. The empirical distributions of particles and velocities in cells define, in
a canonical way, probability distributions. The absence of interaction of the small ma-
terial particles among themselves leads to the assumption that their initial positions and
velocities are independent if modelled as random variables.
Let α ∈ (0, 1) be the expected average volume (in a unit cube) occupied by small
particles (for large n) and assume that the initial “density” function ϕn(q) for the small
particles satisfies:
0 ≤ ϕn(q) ≤ αnd.
We need a state to describe the outcome of finding no particle in the cell (the “empty
state”). Let � denote this empty state and set R̂d := Rd ∪{�}. The usual metric on Rd
is extended to R̂d by defining the distance between � and an arbitrary element r ∈ Rd
to be 1. The Borel sets in Rd and in R̂d will be denoted by Bd and B̂d, respectively.
Set
Ω := {R̂d ×Rd}N.
The velocity field of the small particles will be governed by a strictly positive proba-
bility density ψ(w) on Rd, which we rescale as follows:
ψn(w) :=
1
npd
ψ
( w
np
)
.
Define the initial velocities of the small particles starting in the cell (R̄λ] and from a
cluster, characterized by Bι, as random variables
Wλ,ι
0,n ∼ ψn(w)1Bι(w)
1∫
Bι
ψn(w)dw
. (2.1)
Further, let
ηλ,ι,n :=
∫
(R̄λ]
ϕn(q)dq
∫
Bι
ψn(w)dw
be the probability of finding a small particle at t = 0 in (R̄λ] and with velocity from the
cluster Bι. Define random variables ζλ,ι
n for the initial positions of the small particles
as follows:
ζλ,ι
n :=
{
R̄λ, with probability (w.p.) ηλ,ι,n,
�, w.p. 1− ηλ,ι,n.
(2.2)
Denote by µn,λ,ι and νn,λ,ι the distributions of ζλ,ι
n and Wλ,ι
0,n, respectively. Set
ωλ,ι := q̂λ,ι × wλ,ι ∈ R̂d ×Rd
and
Pn,λ,ι := µn,λ,ι ⊗ νn,λ,ι.
We assume the initial positions and velocities to be independent, i.e., we define the
initial joint probability distribution of positions and velocities on Ω to be the product
measure:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
CORRELATED BROWNIAN MOTIONS AS AN APPROXIMATION TO DETERMINISTIC . . . 763
Pn := ⊗λ∈N ⊗ι∈N Pn,λ,ι. (2.3)
Formally, the coarse-grained particle evolution in the mesoscale can be described by
the following Euler scheme:
Ri
n(t) = Ri
n(0) +
∑
s≤t
V i
n(s)δσ,
V i
n(s) =
∑
0<u≤s
exp[−βn(s− u)]
1
m̂m
∫
βnGn(R̄i
n(u−)− q)Y (dq, u)δσ, (2.4)
Qn(s, λ, ι) =
= R̄λ + wλ,ι
0,ns +
1
m̂m
∑
u<s
∑
v≤u
∫
βnGn(Q̄n(v, λ, ι)−R)XN (dR, v)(δσ)2,
if ζλ,ι
n = R̄λ.
Note that the empirical distribution of the small particles is not a priori finite on
bounded sets. In particular, we do not know whether or not infinitely many small par-
ticles interact with a given large particle at a given time u. Hence, we have to show
existence of the coarse-grained particle model:
Proposition 2.1. The Euler scheme (2.4) is defined for all s ≥ 0 a.s.
Proof. Use induction (for details, cf. [1]).
Let w(dq, ds) standard Gaussian white noise on Rd×R+ (which can be interpreted
as the increments of a space-time Brownian sheet), defined on the same probability space
as (R1(0), . . . , RN (0)) such that (R1(0), . . . , RN (0)) and w(dq, ds) are independent.
Consider the stochastic integral equations:
Ri(t) = Ri(0) +
√
α
t∫
0
∫
Gε(Ri(u)− q)w(dq, du), i = 1, . . . , N. (2.5)
The equations (2.5) are coupled only through the Brownian sheet w(dq, du). The
assumptions imply that the integrals define continuous square integrable martingales for
any adapted processes ri(·). Let us verify that (2.5) satisfies a uniform Lipschitz as-
sumption. Let ri, rj ∈ Rd two d-dimensional components of an (Nd)-vector r̂ and let
Gε,k, Gε,� denote the k-th and 3-th one-dimensional components of Gε, respectively.
In the following equation the process in brackets denotes the mutual quadratic variation
of the stochastic integrals. Set
Dk�,ij(r̂) :=
∫
Gε,k(ri − rj − q)Gε,�(−q)dq. (2.6)
The properties of the Brownian sheet imply:
=
[∫
Gε,k(ri − q)w(dq, du),
∫
Gε,�(rj − q)w(dq, du)
]
=
=
∫
Gε,k(ri − q)Gε,�(rj − q)dq = Dk�,ij(r̂)
by the shift invariance of the Lebesgue measure. Let “∧ ” denote “minimum”. It follows
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
764 P. KOTELENEZ
[∫
Gε,k(ri − q)−Gε,�(rj − q)w(dq, du)
]
=
= 2
∫
(Gε,k(−q)−Gε,k(ri − rj − q))Gε,�(−q)dq ≤ cG(|ri − rj | ∧ 1) (2.7)
by the smoothness and integrability assumptions of Gε, where cG < ∞. Therefore,
(2.5) has a unique solution and is a Markov process in RNd (cf. [20]). The follow-
ing theorem establishes the Einstein – Smoluchowski model as the approximation to the
evolution of the positions of the large particles.
Hypothesis 2.3. {R1
n(0), . . . , RN
n (0)} and
{
ζλ,ι
n ,Wλ,ι
n,0 : λ, ι ∈ N
}
are indepen-
dent;
(R1
n(0), . . . , RN
n (0)) =⇒ (R1(0), . . . , RN (0)), as n −→∞,
where (R1
n(0), . . . , RN
n (0)) are the initial positions of the large particles, (R1(0), . . .
. . . , RN (0)) the initial positions of (2.5) and “ =⇒ ” denotes weak convergence.
Theorem 2.1. Under Hypotheses 2.1 – 2.3
(R1
n(·), . . . , RN
n (·)) =⇒ (R1(·), . . . , RN (·))
in DRdN [0, t̂], as n→∞.
Here (R1(·), . . . , RN (·)) are the unique solutions of (2.5), (R1
n(·), . . . , RN
n (·)) are
the solutions of the Euler scheme (2.4) and DRdN [0, t̂] is the Skorokhod space of RdN-
valued cadlag functions.
The proof is given in [1].
3. Properties of the stochastic limit. We now consider the n-particle motion de-
scribed by (2.5), where we can incorporate the coefficient α into the definition of Gε.
Set
M i(t) :=
t∫
0
∫
Gε(Ri(u)− q)w(dq, du)
and
ck� :=
∫
Gε,k(q)Gε,�(q)dq.
Proposition 3.1. For each i = 1, . . . , N M i(·) is a d-dimensional Brownian mo-
tion with incremental covariance
c11 c12 . . . c1d
c21 c22 . . . c2d
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
cd1 cd2 . . . cdd
. (3.1)
Proof. The kernel Gε is spatially homogeneous. Therefore, the statement follows
from a d-dimensional generalization of Paul Levy’s theorem (cf. [21], Chapter 7, Theo-
rem 1.1).
The motion of the each large particle is Brownian, if we consider the appropriate
d-dimensional “marginal” process distribution of the motion described by (2.5) (with de-
terministic initial conditions). However, the joint motion is not Brownian. First of all,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
CORRELATED BROWNIAN MOTIONS AS AN APPROXIMATION TO DETERMINISTIC . . . 765
let us state the fact that two large particles never hit each other if they start in different
positions. The proof is due to Dawson [22].
Proposition 3.2. Suppose that i �= j and that
E|Ri(0)−Rj(0)|−2 <∞. (3.2)
Then
P
{
ω : ∃ t ∈ [0, T ] such that Ri(t, ω) = Rj(t, ω)
}
= 0. (3.3)
By (2.6) the mutual quadratic variation of the one-dimensional components of M i
and M j for i �= j :
[M i
k,M
j
� ] =
∫
Gε,k(Ri(s)− q)Gε,�(Rj(s)− q)ds =
=
∫
Gε,k(Ri(s)−Rj(s)− q)Gε,�(−q)ds. (3.4)
This implies that the joint quadratic variation depends on the distance of the particles.
In particular, assuming (3.2), the mutual quadratic variation does not become determinis-
tic. Hence, the joint motion cannot be Gaussian and, a fortiori, cannot be Brownian.
Next, we are going to derive a partial shift invariance of the solutions of (2.5). To this
end we define an infinite series expansion for w(dp, dt).
Let H0 be the space of measurable functions on Rd which are square integrable with
respect to the Lebesgue measure and let | · |0 be the usual L2-norm, which is induced by
the scalar product
〈f, g〉0 :=
∫
f(q)g(q)dq
for f, g ∈ H0. Let {φ̃n}n∈N be a complete orthonormal system (CONS) in H0 and
define an Md×d -valued function φn whose entries on the main diagonal are all φ̃n
and whose other entries are all 0. Let Rd,[0,∞) denote the space of Rd-valued adapted
continuous processes. Then for r(·) ∈ Rd,[0,∞)∫
Gε(r(t)− q)w(dq, dt) =
∞∑
n=1
∫
Gε(r(t)− q)φn(q)dqdβn(t), (3.5)
where βn(t) are Rd-valued i.i.d. standard Wiener processes. The right-hand side of (3.5)
defines the increment of an Rd-valued square integrable continuous martingale M.
If f and g are random variables with values in some measurable space, we will write
f ∼ g,
if f and g have the same distribution. Let h ∈ Rd and define the shifted Brownian
sheet by:
w−h(r, t) := w(r − h, t).
The representation (3.5) implies that w−h(dr, t) ∼ w(dr, t), considered as distribu-
tion valued Wiener processes, have the same distribution and that w−h(r, t) is itself a
Brownian sheet. The distribution space can be the Schwarz space of tempered distribu-
tions over Rd or a suitably chosen Hilbert subspace of the Schwarz space (cf. [23]).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
766 P. KOTELENEZ
In what follows we will focus on two particles, R1(·), R2(·), described by (2.5).
Denote the solutions of (2.5) with initial conditions Ri
0 and driving noise w(dr, dt) by
(R1(·, R1
0, w), R2(·, R2
0, w)).
For h ∈ Rd and i = 1, 2 we see that
Ri(t) + h = Ri(0) + h +
t∫
0
∫
Gε(Ri(u) + h− (q + h)w(dq, du) =
= Ri(0) + h +
t∫
0
∫
Gε(Ri(u) + h− (q)w−h(dq, du). (3.6)
Hence, the left-hand side of (3.6) is the solution
(
R1(·, R1
0 + h,w−h), R2(·, R2
0 +
+ h,w−h)
)
. Further, the assumptions on Gε imply that the solutions of (2.5) have a
version which is measurable in all parameters, i.e., in the initial conditions and in the
noise process, considered as a distribution-valued Wiener process (cf. [24]). This implies
that
(
R1(·, R1
0+h,w−h), R2(·, R2
0+h,w−h)
)
∼
(
R1(·, R1
0+h,w), R2(·, R2
0+h,w)
)
.
Thus, we obtain:
Proposition 3.3.(
R1(·, R1
0, w) + h,R2(·, R2
0, w) + h
)
∼
(
R1(·, R1
0 + h,w), R2(·, R2
0 + h,w)
)
, (3.7)
where the pair processes are considered as C([0,∞);R2d)-valued random variables.
Note that (3.7) is only correct if we shift both R1 and R2 by the same d-dimensional
vector h. Abbreviate r̂ := (r1, r2) ∈ R2d with d-dimensional coordinates r1 and r2,
respectively. Recalling (2.6), the generator, Âε, of the Markov pair process R̂(·) :=
:= (R1(·), R2(·)) is given by
Âε :=
1
2
d∑
k,�=1
2∑
i,j=1
Dk�,ij(r̂)
∂2
∂ri
k∂rj
�
. (3.8)
A core for this generator is, e.g., C2
0 (R2d,R), the space of twice continuously dif-
ferentiable real valued functions on R2d which vanish at infinity (cf. [25] (loc.cit.),
Ch. 8.2, Theorem 2.5). We are interested in local effects of the diffusion coefficient on
the distance |R2 − R1|. Let Id be the identity matrix in Rd and define the following
orthogonal transformation of coordinates:(
R1
R2
)
=
1√
2
(
Id − Id
Id Id
)(
u
v
)
. (3.9)
Let B be a Borel set in Rd and set ΓB := A(Rd × B), where A is the unitary
matrix in the right-hand side of (3.9). Consider the probability for the two-particle motion
to be at time t in ΓB , having started in r̂ :
P (t, r̂,ΓB) =
=
(
P
(
1√
2
(
R1(t, r1) + R2(t, r2)
)
,
1√
2
(
R2(t, r1)−R1(t, r2)
))
∈ Rd ×B
)
=
= P
(
1√
2
(R2(t, r1)−R1(t, r2)) ∈ B
)
. (3.10)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
CORRELATED BROWNIAN MOTIONS AS AN APPROXIMATION TO DETERMINISTIC . . . 767
Recalling (3.7), we easily see that (3.10) only depends on the difference a :=
1√
2
(r2−
−r1) and we may define the “marginal” probability distributions
P̃ (t, a, B) := P (t, r̂,ΓB) | 1√
2
(r2−r1)=a. (3.11)
Next, let f ∈ C2
0 (R2d,R). Set f̂(u, v) := (f ◦A)(u, v). We denote those f̂ which
do not depend on u by f̃ . Then f̃ can be considered as an element of C2
0 (Rd,R).
Define function with values in the nonnegative definite matrices by
D̄k�(
√
2a) := [Dk�,11(0)−Dk�,12(
√
2a)], k, 3 = 1, . . . , d, (3.12)
and let (σkj(a)) be the unique nonnegative definite square root of (D̄k�(
√
2a)).
It can be seen, that the “marginal” probability distributions, defined by (3.11) generate
a Feller – Markov process on Rd with generator
(Aεf̃)(a) :=
1
2
d∑
k,�=1
D̄k�(
√
2a)
(
∂2
∂ak∂a�
f̃
)
(a). (3.13)
Let βj be i.i.d. one-dimensional standard Brownian motions, j = 1, . . . , d. Then,
this Markov process can be represented as the unique solution to the following stochastic
Itô differential equation:
db(t) =
∑
j
σ·j(b)βj(dt), b(0) := a. (3.14)
Really, note that our assumptions on Gε imply that D̄k�(
√
2a) are twice continu-
ously differentiable with bounded partial derivatives. Therefore, (i) follows from Ethier
and Kurtz (loc.cit.) [21], Section 8.2, Theorem 2.5). Moreover, the diffusion matrix
(σkj(a)) is uniformly Lipschitz.
It is tempting to use the definition of the probability flux (cf. [5]) to compute regions
where there is a bias towards attraction between the two particles or a bias in favor of
repulsion. The problem with that definition is that the attractive and repulsive domains,
determined by the flux, depend on time t and the initial distribution of a density. How-
ever, if the density is near a point b approximately constant the sign of the flux at b is
completely determined by the diffusion coefficients. Set
−J̄u(b) :=
d∑
k,�=1
∂
∂b�
[(Dk�,11(0)−Dk�,12(
√
2b))],
E+ :=
{
b ∈ Rd :=
d∑
�=1
b� ≥ 0
}
, E− :=
{
b ∈ Rd :=
d∑
�=1
b� < 0
}
.
(3.15)
Since we can replace the derivatives by difference quotients, we can determine which
sign of −J̄u(b) implies a bias towards attraction and which one would indicate a bias
towards repulsion. Hence, we suggest the following:
Definition 3.1. A Borel set B is called “attraction-biased” if either B ⊂ E+ and
−J̄u(b) > 0 ∀b ∈ B or B ⊂ E− and −J̄u(b) < 0 ∀b ∈ B.
The set B is called a“repulsion-biased” if either B ⊂ E+ and −J̄u(b) < 0 ∀b ∈ B
or B ⊂ E− and −J̄u(b) > 0 ∀b ∈ B.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
768 P. KOTELENEZ
In the Maxwellian case from (1.7) there is a positive constant cε such that
−J̄u(b) =
[
d∑
k=1
bk
]
cε exp
(
−|b|
2
2ε
) [
d + 2− |b|
2
ε
]
. (3.16)
Hence, in the Maxwellian case a Borel set B is attraction-biased if and only if for all
b ∈ B we have
|b| <
√
ε(d + 2).
Further, B is repulsion-biased if and only if
|b| >
√
ε(d + 2).
Remarks 3.1. (i) We expect that for short times the distance between the two particles
in attraction-biased regions would decrease and in repulsion-biased regions this distance
would increase. In particular, for distances close to 0 the attraction bias would be consis-
tent with the depletion phenomena, which has been observed in fluids (cf. references in
the introduction). However, we do not claim that distances in an repulsion-biased region
will gradually get greater and perhaps converge to ∞, as t −→ ∞. A similar comment
holds for the attraction-biased case. Indeed, the proof of the following result can be found
in [25], Ch. 1.3: Assume d = 1 and let b(t, a) be the unique solution of (3.14) with
b(0) = a, where a �= 0. Then,
P
{
lim
t→∞
b(t, a) = 0
}
= 1. (3.17)
(3.17) means that the distance between both particles will eventually tend to 0. In
other words, the whole positive and negative real lines are attractive regions and the
repulsion-biased regions will have no long-term effect on the distances of the two parti-
cles. Recall that Brownian motion in R is recurrent. Now if the distance between the two
particles is much greater than
√
3ε, the process b(·) is very similar to a one-dimensional
Brownian motion which follows from the asymptotics of D̄(
√
2b), as b !
√
3ε in the
Maxwellian case. The recurrence, however, lets the distance become less than
√
3ε. In
this region the motion of the distance becomes skewed, favoring a decrease of the dis-
tance.
(ii) In dimension d ≥ 3 Brownian motion is transient and, therefore, attraction-biased
and repulsion-biased domains may play a different role in the long-time behavior of the
distance between the two particles than in the case d = 1. This will be investigated in a
future research.
Acknowledgement. The author wants to thank W. I. Skrypnik from the Institute of
Mathematics of the Ukrainian Academy of Sciences for informing him about the existence
of the papers [6, 7].
1. Kotelenez P. From discrete deterministic dynamics to stochastic kinematics — a derivation of Brownian
motions. – 2002. – Preprint # 02-150 (submitted).
2. Goncharuk N., Kotelenez P. Fractional step method for stochastic evolution equations // Stochast. Process.
and Appl. – 1998. – 73. – P. 1 – 45.
3. Ruelle D. Statistical mechanics: rigorous results. – New York, Tokyo: Addison Wesley, 1988.
4. Haken H. Advanced synergetics. – Berlin etc.: Springer, 1983.
5. Kampen N. G. van. Stochastic processes in physics and chemistry. – Amsterdam, New York: North
Holland Publ. Co., 1983.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
CORRELATED BROWNIAN MOTIONS AS AN APPROXIMATION TO DETERMINISTIC . . . 769
6. Dürr D., Goldstein S., Lebowitz J. L. A mechanical model of Brownian motion // Communs Math. Phys.
– 1981. – 78. – P. 507 – 530.
7. Dürr D., Goldstein S., Lebowitz J. L. A mechanical model for the Brownian motion of a convex body //
Z. Wahrscheinlichkeitstheor. und verw. Geb. – 1983. – 62. – P. 427 – 448.
8. Holley R. The motion of a heavy particle in an infinite one-dimensional gas of hard spheres // Ibid. – 1971.
– 17. – S. 181 – 219.
9. Sinai Ya. G., Soloveichik M. R. One-dimensional classical massive particle in the ideal gas // Communs
Math. Phys. – 1986. – 104. – P. 423 – 443.
10. Szász D., Tóth B. Bounds for the limiting variance of the “Heavy Particle” in R // Ibid. – 1986. – 104. –
P. 445 – 455.
11. Szász D., Tóth B. Towards a unified dynamical theory of the Brownian particle in an iIdeal gas // Ibid. –
1986. – 111. – P. 41 – 62.
12. Spohn H. Large scale dynamics of interacting particles. – Berlin etc.: Springer, 1991.
13. Asakura S, Oosawa F. J. Chem. Phys. – 1954. – 22. – P. 1255.
14. Goetzelmann B., Evans R., Dietrich S. Depletion forces in fluids // Phys. Rev. E. – 1998. – 57. – # 6.
15. Kotelenez P., Leitman M., Mann J. A. (Jr.) (manuscript in preparation).
16. Uhlenbeck G. E., Ornstein L. S. On the theory of the Brownian motion // Phys. Rev. – 1930. – 36. –
P. 823 – 841.
17. I’lin A. M., Khasminskii R. Z. On equations of Brownian motions // Probab. Theory and Appl. – 1964. –
9, # 3 (in Russian).
18. Nelson E. Dynamical theories of Brownian motions. – Princeton: Princeton Univ. Press, 1972.
19. Kotelenez P., Wang K. Newtonian particle mechanics and stochastic partial differential equations // Mea-
sure Valued Processes, Stochastic Partial Differential Equations and Interacting Systems / Ed. D. A. Daw-
son. – Centre Recherche Math. CRM Proc. and Lecture Notes. – 1994. – 5. – P. 130 – 149.
20. Kotelenez P. Class of quasilinear stochastic partial differential equations of McKean – Vlasov type with
mass conservation // Probab. Theory Relat. Fields. – 1995. – 102. – P. 159 – 188.
21. Ethier S. N., Kurtz T. G. Markov processes — characterization and convergence. – New York; Toronto:
Wiley and Sons, 1986.
22. Dawson D. A. Private communication. – 1993.
23. Kotelenez P. On the semigroup approach to stochastic evolution equations // Stochastic Space-Time Mod-
els and Limit Theorems / Eds L. Arnold, P. Kotelenez. – D. Reidel Publ., 1985. – P. 95 – 139.
24. Kotelenez P. Stochastic partial differential equations in the construction of random fields from particle
systems. Pt I: Mass conservation. – 1996. – Preprint. – P. 96 – 143.
25. Gikhman I. I., Skorokhod A. V. Stochastic differential equations and their applications. – Kiev: Naukova
Dumka, 1982 (in Russian).
Received 16.11.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
|
| id | umjimathkievua-article-3642 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:46:18Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/98d5ccac751cf8bd0da41b0526b30794.pdf |
| spelling | umjimathkievua-article-36422020-03-18T20:00:55Z Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics Корельований броунівський рух як границя наближення детерміністської динаміки середнього поля Kotelenez, P. Котеленез, П. We analyze the transition from deterministic mean-field dynamics of several large particles and infinitely many small particles to a stochastic motion of the large particles. In this transition the small particles become the random medium for the large particles, and the motion of the large particles becomes stochastic. Under the assumption that the empirical velocity distribution of the small particles is governed by a probability density ?, the mean-field force can be represented as the negative gradient of a scaled version of ?. The stochastic motion is described by a system of stochastic ordinary differential equations driven by Gaussian space-time white noise and the mean-field force as a shift-invariant integral kernel. The scaling preserves a small parameter in the transition, the so-called correlation length. In this set-up, the separate motion of each particle is a classical Brownian motion (Wiener process), but the joint motion is correlated through the mean-field force and the noise. Therefore, it is not Gaussian. The motion of two particles is analyzed in detail and a diffusion equation is deduced for the difference in the positions of the two particles. The diffusion coefficient in the latter equation is spatially dependent, which allows us to determine regions of attraction and repulsion of the two particles by computing the probability fluxes. The result is consistent with observations in the applied sciences, namely that Brownian particles get attracted to one another if the distance between them is smaller than a critical small parameter. In our case, this parameter is shown to be proportional to the aforementioned correlation length. Проаналізовано перехід від детерміністської динаміки середнього поля декількох великих частинок та нескінченної кількості малих частинок до стохастичного руху великих частинок. Під час цього переходу маленькі частинки перетворюються у випадкове середовище для великих частинок, а рух великих частинок стає стохастичним. Якщо припустити, що розподіл емпіричної швидкості малих частинок визначається щільністю розподілу ф, то силу середнього поля можна подати як від'ємний градієнт масштабного перетворення ф. Стохастичний рух описано системою стохастичних звичайних диференціальних рівнянь, керованих гауссовим просторово-часовим білим шумом та силою середнього поля як інтегральним ядром, інваріантним відносно зсуву. Масштабування зберігає малий параметр при переході (так звану довжину кореляції). У даній постановці окремий рух кожної частинки є класичним броунівським рухом (вінерівським процесом), але спільний рух корелюється силою середнього поля і шумом. Тому він не є гауссівським. Детально проаналізовано рух двох частинок і виведено рівняння дифузії для різниці положень двох частинок. Коефіцієнт дифузії в останньому рівнянні є просторово залежним, що дозволяє визначити області притягання та відштовхування двох частинок шляхом розрахунку течій імовірностей. Результат узгоджується із спостереженнями у прикладних науках, а саме, з фактом, що броунівські частинки притягуються одна до одної, якщо відстань між ними менша за критичний малий параметр. У випадку, що вивчається, показано, що цей параметр пропорціональний згаданій вище довжині кореляції. Institute of Mathematics, NAS of Ukraine 2005-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3642 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 6 (2005); 757–769 Український математичний журнал; Том 57 № 6 (2005); 757–769 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3642/4014 https://umj.imath.kiev.ua/index.php/umj/article/view/3642/4015 Copyright (c) 2005 Kotelenez P. |
| spellingShingle | Kotelenez, P. Котеленез, П. Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics |
| title | Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics |
| title_alt | Корельований броунівський рух як границя наближення детерміністської динаміки середнього поля |
| title_full | Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics |
| title_fullStr | Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics |
| title_full_unstemmed | Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics |
| title_short | Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics |
| title_sort | correlated brownian motions as an approximation to deterministic mean-field dynamics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3642 |
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