A Retarded Mean-Field Approach for Interacting Fiber Structures

1.2 MB·English

by  Raul Borsche

A RETARDED MEAN-FIELD APPROACH FOR INTERACTING
FIBER STRUCTURES
R. BORSCHE∗, A. KLAR∗†, C. NESSLER∗, A. ROTH∗, AND O. TSE∗
Abstract. We consider an interacting system of one-dimensional structures modelling fibers
with fiber-fiber interaction in a fiber lay-down process. The resulting microscopic system is inves-
tigated by looking at different asymptotic limits of the corresponding stochastic model. Equations
arisingfrommean-fieldanddiffusionlimitsareconsidered. Furthermore,numericalmethodsforthe
5 stochasticsystemanditsmean-fieldcounterpartarediscussed. Anumericalcomparisonofsolutions
1 correspondingtothedifferentscales(microscopic,mesoscopicandmacroscopic)isincluded.
0
Keywordsinteractingstochasticparticles, fibers, mean-fieldequations,retarded
2
potential, delay equations.
n
2010 AMS Subject Classification: 92D50, 35B40, 82C22, 92C15
a
J
1. Introduction. One-dimensional structures appear in various context of in-
6
dustrial applications. They are used, for example, in the modelling of polymers in
2
suspensions, composite materials, nanostructures, fiber dynamics in turbulent flows

] and, in particular, fiber lay-down in technical processes of non-woven materials. Fur-
S
thermore,suchstructureshavebeenmodelledondifferentlevelsofdescriptioninvolv-
D
ingdifferentscales. Besidesmicroscopicmodels,mesoscopickineticorFokker–Planck
h. equations have been widely used for a statistical description of the fiber or polymer
t distributions. We refer to [1, 2] and [18, 21] for concrete examples in the industry. In
a
this article, we consider non-woven materials, which are webs of long flexible fibers.
m
Production processes and models corresponding to the lay-down of such fibers have
[
been intensively investigated. See [4] and the above cited references.

1 Intheabovementionedinvestigations,theone-dimensionalstructures(fibers)un-
v der consideration were assumed to be mutually independent, which clearly does not
5
representreality. Therefore,thepresentworkaimsatincludingfiber-fiberinteraction,
6
thereby describing the size of each fiber and, simultaneously, the absence of intersec-
4
6 tion among fibers. This is achieved by simply including the interaction of structures
0 into a well investigated model described in non-woven production processes. Taking
1. intoaccounttheinteractionofthestructuresonthemicroscopiclevelleadstocoupled
0 systemsofstochasticdifferentialequations. Itsstatisticaldescriptionshouldalsotake
5 intoaccounttheinteractionsandwillconsequentlynolongerbebasedontheclassical
1
Fokker–Planck model.
:
v The new model makes use of a microscopic systems of retarded stochastic dif-
i ferential equations, and its mesoscopic description is obtained via formal mean-field
X
procedures. The mean-field limit is described by a McKean–Vlasov type equation
r
a with a delay term. We perform an analytical investigation of the mean field limit,
as well as a numerical comparison of microscopic, mean field and macroscopic equa-
tions. The analysis of the limit is based on the work in [5, 12, 15, 16, 17, 25, 31]. For
numerical methods for mean-field type equations we refer to [2, 26, 27, 30].
The paper is organized as follows: starting from a model for independent fibers,
we present in Section 2 a new model for interacting fibers, which takes into account
1FachbereichMathematik,TechnischeUniversit¨atKaiserslautern,Germany
{[email protected], [email protected], [email protected],
[email protected],[email protected]}
2FraunhoferITWM,Kaiserslautern,Germany
{[email protected],[email protected]}
1
the finite size of the fibers and prevents intersection of the fibers. The model is based
on a system of retarded stochastic differential equations with suitable interaction po-
tentials. Section3describesthemean-fieldequationandadiscussionofitsstationary
solutions. The core of this section is devoted to the proof of the mean-field limit for
the corresponding deterministic system, i.e., the rigorous derivation of the mean-field
equation from the system of retarded deterministic equations. Section 4 contains the
diffusion limit of the kinetic equation, while Section 5 contains a description of the
numericalmethodsusedforthemicroscopicandmean-fieldequations. Thenumerical
results include an investigation and comparison of stationary states, as well as a con-
vergence analysis of solutions to equilibrium for both the microscopic and mean-field
equations. We finally conclude in Section 6.
2. Interacting isotropic fiber models. We begin by reviewing a basic, mu-
tually independent fiber model for the lay-down process of fibers, described by a
stochastic dynamical system in dimensions d ≥ 2 (cf. [4, 21, 24]). Using the state
space M:=Rd×Sd−1, where Sd−1 denotes the unit sphere in Rd, a fiber is given by
a path of the following stochastic differential equation:
dx =τ dt,
t t
(cid:0) (cid:1) (cid:16) 1 (cid:17) (2.1)
dτ = I−τ ⊗τ ◦ − ∇ V(x )dt+AdW
t t t d−1 x t t
withinitialcondition(x ,τ )∈M. Here,V : Rd →Ristheso-calledcoilingpotential,
0 0
A a nonnegative diffusion coefficient and W the standard Brownian motion. We use
t
a coordinate free formulation, where ◦ denotes the Stratonovich integral. Note also
that (I−τ ⊗τ) is the projector of Rd onto the unit sphere Sd−1. We refer to [11] for
relatedwork. Weequipourstatespacewiththemeasuredxdν,wheredxdenotesthe
Lebesgue measure on Rd and dν the normalized surface measure on Sd−1. Note that
the dimensional scaling 1/(d−1) is introduced for convenience, in order to achieve a
stationary state which does not explicitly depend on the spatial dimension.
The fiber model above only describes the evolution of the center line of the fiber
anddoesnotcapturetheeffectsofthefinitesizeofthefibers. Hence,self-intersection
andintersectionamongfibersarenotpreventedinthemodel. Apossibilitytoremedy
this deficiency it to include hysteresis into the system, which enables the system to
prevent self-intersection, and for multiple fibers, intersection among them. Consider-
ations along this line of thought lead to the following interacting fiber model.
Consider N ∈ N fibers with position and velocity (xi,τi) ∈ M, for each i ∈
{1,...,N}. The interacting fiber model reads
dxi =τidt
t t
(cid:18)
dτi =(cid:0)I−τi⊗τi(cid:1)◦ − 1 ∇ V(xi)dt
t t t d−1 x t
(2.2)
1 (cid:20) 1 (cid:88)N 1(cid:90) t (cid:21) (cid:19)
− ∇ U(xi−xj)ds dt+AdWi ,
d−1 N t x t s t
j=1 0
with independent Brownian motions Wi. In comparison to the previous fiber model,
t
we include a scaled, nonlocal (in time) interaction term
1(cid:90) t
∇ U(xi−xj)ds,
t x t s
0
2
where U is an interaction potential, which is repulsive in our case, so as to avoid any
contact among the fibers. Note that a ’non-retarded’ version of this system would be
similar to a model for swarming with roosting potential (cf. [7, 8]). In addition to
the interaction term, our new model includes a delay term, i.e., an integration with
respect to time, to describe the fact that fibers interact with each other and with
itself on the whole fiber length. The factor 1/N leads to the so called ’weak coupling
scaling’(cf.[5,25]). Similiarly,thescaling1/tisanormalizationofthepotentialwith
respecttothetimeintegration. Alsonotethatthesummationdoesnotexcludei=j,
which accounts for self-interaction of a fiber with itself.
There is some freedom in the choice of the interaction potential. As a simple
example, one can use a mollifier type potential
(cid:16) (2R)2 (cid:17)
U(x)=U(|x|)=Cexp − , for |x|<2R,
(2R)2−|x|2
whereRisanonnegativeparameterrepresentingthefiberradiusandC >0isafixed
constant describing the strength of the interaction. Alternatively, a potential could
be described by a smoothed version of Heaviside type potential
U(x)=U(|x|)=CΘ(2R−|x|),
with Heaviside function Θ. A smooth version of such a potential may be given by
C
U(x)= , (2.3)
(cid:16) (cid:16) (cid:17)(cid:17)
1+exp −k 1− |x|2
(2R)2
for some regularizing parameter k >0.
In Figure 2.1 we compare the fiber curves for different noise amplitudes A. We
usethecoilingpotentialV(x)= 1|x|2 andtheinteractionpotentialU from(2.3),with
2
k =100, C =10andR=0.4. Sincethecomputationwasdoneforashortamountof
time, we neglect the scaling 1/t in front of the interaction part. We observe that the
non-intersecting fiber curves are increasingly altered with increasing noise amplitude.
Remark 1.
1. For U ≡0, one obtains a fully decoupled system for (xi,τi), and each fiber is
described by the mutually independent model given in (2.1).
2. Toconsiderinelasticinteractions,onecouldincludedampingtermsdepending
onthevelocityintheinteractionpotential. Thiswouldleadtoequationswhere
a dissipative force is included in the interaction term.
3. As in the case without interaction, it is also possible to formulate a smooth
version of the interacting fiber system. The basic idea is to replace the Brow-
nian motion on the sphere by an Ornstein–Uhlenbeck process (cf. [22]).
Remark 2. It is also possible to include reference curves γ into the model, which
describes,forexample,themotionofaconveyorbelt. Thiscanbedoneinthefollowing
way. We denote with ηi :R →Rd the actual fiber curves. Then (2.2) changes to
+
dηi =τidt
t t
(cid:18)
dτi =(cid:0)I−τi⊗τi(cid:1)◦ − 1 ∇ V(ηi−γ)dt
t t t d−1 x t
(2.4)
1 (cid:20) 1 (cid:88)N 1(cid:90) t (cid:21) (cid:19)
− ∇ U(ηi−ηj)ds dt+AdWi .
d−1 N t x t s t
j=1 0
3
Fig. 2.1. Comparison of the influence of the noise A for two interacting fibers.
A change of variables ξ :=ηi−γ describes the deviation of the fiber curves from the
i t
reference curve and thus, (2.4) may be written as
dξi =τidt−dγ
t t t
(cid:18)
dτi =(cid:0)I−τi⊗τi(cid:1)◦ − 1 ∇ V(ξi)dt
t t t d−1 x t
(2.5)
1 (cid:20) 1 (cid:88)N 1(cid:90) t (cid:21) (cid:19)
− ∇ U(ξi−ξj +γ −γ )ds dt+AdWi .
d−1 N t x t s t s t
j=1 0
Inthiscasetheforceduetotheinteractionpotentialdependsontherelativefiberpoint
position as well as on the difference γ −γ in the reference curve.
t s
In the relevant case of non-wovens on a conveyor belt moving with constant speed,
we consider d ∈ {2,3} and a reference curve given by γ = −v e t. Here v =
t ref 1 ref
v /v istheratiobetweenthespeedoftheconveyorbelt, v , andthespeedofthe
belt prod belt
production process, v , and e denotes the direction of belt movement (cf. [21]).
prod 1
In the following, we formulate a slightly more general model than (2.2). Since,
in reality, the fiber material is transported away by the belt, interaction will not take
placeforthefullhistoryofthefibers. Thus, itisreasonabletoconsideracut-offwith
4
a cut-off size H >0. Define
(cid:40)
t for t≤H
h(t)= , H ∈(0,∞).
H for t>H
Then the interacting fiber model with cut-off is given by
dxi =τidt
t t
(cid:18)
dτi =(cid:0)I−τi⊗τi(cid:1)◦ − 1 ∇ V(xi)dt
t t t d−1 x t
(2.6)
1 (cid:20) 1 (cid:88)N 1 (cid:90) t (cid:21) (cid:19)
− ∇ U(xi−xj)ds dt+AdWi .
d−1 N h(t) x t s t
j=1 t−h(t)
InthelimitH →0,weobtainanon-retardedinteractingparticlemodelwithconstant
speed, whereas, in the limit H →∞, we obtain the interacting fiber model in (2.2).
3. Mean-field equation. The associated mean-field equation for the distribu-
tion function f = f(t,x,τ) may be formally derived from the microscopic equations
following the procedure described, for example, in [6, 17]. The equation reads
∂ f +τ ·∇ f +Sf =Lf (3.1)
t x
with deterministic force term Sf =SVf +SUf, given by
(cid:18) (cid:19)
1
SVf =−∇ · f(I−τ ⊗τ) ∇ V ,
τ d−1 x
(cid:32) (cid:33) (3.2)
1 1 (cid:90) t (cid:90)
SUf =−∇ · f(I−τ ⊗τ) ∇ U(x−y)ρ(s,y)dyds ,
τ d−1h(t) x
t−h(t) Rd
where ∇ is the gradient on Sn−1, and diffusion operator
τ
A2
Lf = ∆ f, (3.3)
2 τ
where ∆ denotes the Laplace–Beltrami operator on Sd−1. In addition, we have the
τ
(cid:82) (cid:82)
normalization ρdx=1, where ρ denotes the zeroth-moment ρ= fdν.
Rd Sd−1
Remark 3. A stationary solution of the mean-field equation (3.1) is given by the
time-independent solution of
A2
τ ·∇ f +Sf = ∆ f.
x 2 τ
Looking for a solution independent of τ, we get
(cid:16) (cid:17)
ρ∇ lnρ+V +U (cid:63)ρ =0.
x
This leads to the integral equation
lnρ+V +U (cid:63)ρ=c, (3.4)
5
(cid:82)
where the constant c is fixed by the normalization ρdx=1. The integral equation
Rd
may be written in the equivalent fixed-point form
e−V−U(cid:63)ρ
ρ= (cid:82) . (3.5)
e−V−U(cid:63)ρdx
Rd
On the other hand, the stationary solution may also be characterised as the (unique)
minimizer of the ’energy functional’
(cid:90) (cid:90)
(cid:0) (cid:1)
F(ρ):= (lnρ−1)ρdx+ V +U (cid:63)ρ ρdx,
Rd Rd
where the first term describes the internal energy (entropy), and the second describes
the potential energy. In contrast to the case without interaction, the stationary solu-
tion has to be determined numerically.
The rest of this section is devoted to a rigorous proof of the mean-field limit for
the deterministic interacting fiber model
dxi
t =τi
dt t
dτti =(cid:0)I−τi⊗τi(cid:1)◦(cid:18)− 1 ∇ V(xi)
dt t t d−1 x t (3.6)
1 1 (cid:88)N 1 (cid:90) t (cid:19)
− ∇ U(xi−xj)ds ,
d−1N h(t) x t s
j=1 t−h(t)
towards the Vlasov type equation
∂ f +τ ·∇ f +Sf =0, (3.7)
t x
where S =SV +SU is as given in (3.2).
3.1. Mean-field limit of a retarded system. WeconsiderasystemofN ∈N
interacting particles with state Zi ∈Rm, m∈N at time t∈R
t +
dZi 1 (cid:88)N 1 (cid:90) t
t =a(Zi)+ B(Zi,Zj)ds, Zi =zi ∈Rm (3.8)
dt t N h(t) t s 0
j=1 t−h(t)
for each i∈{1,...,N}, where as above
(cid:40)
t for t≤H
h(t)= , H ∈(0,∞), or h(t)=t,
H for t>H
and a∈Lip(Rm), B ∈Lip (Rm×Rm;Rm) be globally Lipschitz, satisfying
b

sup |B(z ,zˆ)−B(z ,zˆ)|≤Lip(B)1∧|z −z |,
zˆ∈Rm 1 2 1 2
 sup |B(z,zˆ1)−B(z,zˆ2)|≤Lip(B)1∧|zˆ1−zˆ2|,
z∈Rm
whereweusethenotationx∧y =min{x,y}. Asin(2.2)and(2.6),(3.8)isaretarded
system of ordinary differential equations.
6
Denote by P (Rm), r >0, the set of Borel probability measures µ such that
r
(cid:90)
|z|rµ(dz)<∞.
Rm
For every probability measure µ∈P (Rm), we set
1
(cid:90)
Kµ(z):=µ(B(z,·))= B(z,zˆ)µ(dzˆ), z ∈Rm. (3.9)
Rm
Taking the empirical measure µN = 1 (cid:80)N δ(·−Zj)∈P (Rm) as µ in (3.9) gives
s N j=1 s 1
(cid:90) 1 (cid:88)N
KµN(Zi)= B(Zi,xˆ)µN(dxˆ)= B(Zi,Zj).
s t t s N t s
Rm
j=1
Therefore, we may consider a mean-field equation given by
dZ 1 (cid:90) t
t(z)=F(t,Z (z),{µ} ):=a(Z (z))+ Kµ (Z (z))ds, (3.10)
dt t t t h(t) s t
t−h(t)
withinitialconditionZ (z)=z, law(Z )=µ . Hereµ =Z (·) µ denotesthepush
0 0 0 t t # 0
forward of the measure µ ∈P (Rm), i.e.,
0 1
(cid:90) (cid:90)
ϕ(z)µ (dz)= ϕ(Z (z))µ (dz) ∀ϕ∈C (Rm),
t t 0 b
Rm Rm
under the flow Z ∈C(R ×Rm;Rm) generated by the mean-field equation, and {µ}
+ t
denotes the family of measures {µ ,s∈[0,t]} up to time t>0.
s
We begin with an existence and uniqueness result for solutions of (3.10). Our
results generalizes those given in [16], which follows the general scheme introduced
in [12] (cf. [5, 25]). The proof relies on a slight modification of the proof in [16,
Proposition 4.1]. For the sake of completeness, we include the proof in Appendix A.
Proposition 3.1. Let the assumptions on a and B above be satisfied. Then the
mean-field equation (3.10) admits a unique global solution Z ∈C(R ×Rm;Rm).
+
Remark 4. The family of measures {µ =Z (·) µ }⊂P (Rm) generated by the
t t # 0 1
flow Z ∈C(R ×Rm;Rm) provides a weak solution to the Vlasov equation
+
∂ µ +div(F(t,z,{µ} )µ )=0, (3.11)
t t t t
More precisely, for all h∈C∞(Rm), the functions µ (h) are differentiable,
0 t
dµ (h)
t =µ (F(t,·,{µ} )·∇h)
dt t t
for all t>0, and µ (h)→µ (h) for t→0 .
t 0 +
Next, we show that solutions to the mean-field equation (3.10) depend continu-
ously on the initial probability measures µ ∈P (Rm). To do so, we need to measure
0 1
the difference of two probability measures µ,ν ∈P (Rm). A convenient way is to use
1
the Monge–Kantorovich–Rubinstein distance W on P (Rm) defined in [20] (cf. [32]),
1 1
(cid:90)(cid:90)
W (µ,ν)= inf 1∧|x−y|π(dxdy),
1
π∈Π(µ,ν) Rm×Rm
7
where Π(µ,ν) is the set of Borel probability measures on Rm×Rm such that
(cid:90)(cid:90) (cid:90) (cid:90)
(φ(x)+ψ(y))π(dxdy)= φ(z)dz+ ψ(z)dz
Rm×Rm Rm Rm
for all φ,ψ ∈C (Rm). This distance is also called the Wasserstein distance.
b
Proposition 3.2. Let µj ∈ P (Rm), and Zj ∈ C(R ×Rm;Rm) be the corre-
0 1 +
sponding solution to the mean-field equation (3.10) with initial conditions
Zj(z)=z, law(Zj)=µj, for j =1,2
0 0 0
then for every T >0, the following stability estimate holds
sup W (µ1,µ2)≤c(1+T)ecT W (µ1,µ2),
1 t t 1 0 0
t∈[0,T]
for some constant c=c(F,H)>0, where µj :=Zj(·) µj
t t # 0
Proof. Let π ∈Π(µ1,µ2), and define
0 0 0
(cid:90)(cid:90)
D(t):= 1∧|Z1(z )−Z2(z )|π (dz dz )
t 1 t 2 0 1 2
Rm×Rm
Following the arguments in the proof of Proposition 3.1, we derive the estimate
(cid:90) t (cid:90) t 1 (cid:90) s
D(t)≤D(0)+Lip(F) D(s)ds+Lip(B) D(σ)dσds.
h(s)
0 0 s−h(s)
We only show the estimate for the case H = ∞, i.e. h(t) = t. The general case
H > 0 may be shown analogously. Similar to Proposition 3.1, by simple but tedious
computations, we have
(cid:90) t
D(t)≤D(0)+Lip(F) g(t,s)D(s)ds,
0
with g(t,s)=1+ln(t)−ln(s). Recursively, we obtain
(cid:18) (n+1)Tk(cid:19) (cid:88)k ((cid:96)+1)T(cid:96)
1−Lip(F)k sup D(t) ≤ Lip(F)(cid:96) D(0).
k! (cid:96)!
t∈[0,T]
(cid:96)=0
Therefore, passing to the limit k →∞ yields
sup D(t)≤c(1+T)ecT D(0),
t∈[0,T]
with c=max{1,Lip(F)}. Since
(cid:90)(cid:90)
D(t)= 1∧|z −z |π (dz dz ),
1 2 t 1 2
Rm×Rm
where π is the push forward measure of π under the map (Z1,Z2), and
t 0 t t
π ∈Π(µ1,µ2) =⇒ π ∈Π(µ1,µ2)
0 0 0 t t t
8
for any t≥0, we have
sup W (µ1,µ2)≤c(1+T)ecT D(0)
1 t t
t∈[0,T]
Finally, optimizing in π yields the required assertion.
0
Remark 5. For any N ∈ N, the family of empirical measures {µN,t ≥ 0} ⊂
t
P (Rm) defined by µN =Z (·) µN ∈P (Rm), where
1 t t # 0 1
N
1 (cid:88)
µN = δ(·−zj), zj ∈Rm, j =1,...,N
0 N
j=1
is a weak solution to the Vlasov equation (3.11) by construction. Therefore, if we
know that lim W (µ ,µN) → 0 for some µ ∈ P (Rm), then the stability result
N→∞ 1 0 0 0 1
given in Proposition 3.2 provides the convergence
lim W (µ ,µN)→0 for any t∈[0,T],
1 t t
N→∞
where µ =Z (·) µ , with Z ∈C([0,T]×Rm;Rm).
t t # 0
3.2. Application to the retarded fiber equations. We now use the results
above to show the mean-field limit of (3.6) towards (3.7). For this reason, we denote
Zi =(xi,τi)∈Rd×Rd, and write a=(a ,a ), B =(B ,B ) with
t t t 1 2 1 2
a (Zi)=τi, a (Zi)=− 1 (cid:0)I−τi⊗τi(cid:1)◦∇ V(xi),
1 t t 2 t d−1 t t x t
B (Zi,Zj)=0, B (Zi,Zj)=− 1 (cid:0)I−τi⊗τi(cid:1)◦∇ U(xi−xj).
1 t s 2 t s d−1 t t x t s
Obviously, we are unable to directly apply the results developed above, since a and
B do not satisfy the assumptions above. Consider B for the moment. Tedious but
2
simple computations yield
(cid:16) (cid:17)
|B (z ,zˆ)−B (z ,zˆ)|≤Lip(∇ U) (1+|τ |2)|x −x |+(|τ |+|τ |)|τ −τ |
2 1 2 2 x 1 1 2 1 2 1 2
|B (z,zˆ )−B (z,zˆ )|≤Lip(∇ U)(1+|τ|2)|xˆ −xˆ |
2 1 2 2 x 1 2
Hence, B is not globally Lipschitz. Fortunately, if we only consider B on the subset
M ⊂ Rd×Rd, then Lip(B) = 2Lip(∇ U). Consequently, B is globally Lipschitz on
x
M. Clearly, the same conclusion holds for a.
In order to ensure that Zi ∈M for all t≥0, we observe that
t
d 1
|τi|2 =0 for all t≥0, i=1,...,N.
dt2 t
Therefore,ifwestartwithτi ∈Sd−1,thenalsoτi ∈Sd−1 forallothertimest>0,and
t
we may apply our results obtained above to initial measures µ ∈ P (Rd×Rd) with
0 1
supp(µ )⊂M. Indeed, since Z (x,τ)∈M for any (x,τ)∈M, we may consider the
0 t
flow on M and the push forward µ =Z (·) µ ∈P (M), as soon as supp(µ )⊂M.
t t # 0 1 0
9
4. Large diffusion scaling. In this section we formally investigate situations
with large values for the noise amplitude on a diffusive time scale, i.e., we change
L˜ = (cid:15)L and t˜= (cid:15)t (cf. [3, 19]). Replacing the scaled operators in (3.1) and omitting
the tilde lead to the scaled equation
1
(cid:15)∂ f +τ ·∇ f +Sf = Lf. (4.1)
t x (cid:15)
We use a Hilbert expansion of the form f = f +(cid:15)f +... for (4.1). To order 1, we
0 1
simply get f =f (x)=ρ(x). To order (cid:15), one obtains
0 0
A2
τ ·∇ f +Sf = ∆ f ,
x 0 0 2 τ 1
which, due to the Fredholm alternative, gives
(cid:32) (cid:33)
2 1 (cid:90) t
f =− τ ·f ∇ lnf +V + (U (cid:63)f )(s,·)ds .
1 A2(d−1) 0 x 0 h(t) 0
t−h(t)
Integrating (4.1) with respect to dν gives
(cid:90) (cid:90)
(cid:15)∂ fdν+∇ · τfdν =0.
t x
Sd−1 Sd−1
Considering terms up to order (cid:15), we obtain
(cid:90)
∂ f +∇ · τf dν =0.
t 0 x 1
Sd−1
Therefore, inserting f and computing the integral over the tensor product yields
1
(cid:34) (cid:32) (cid:33)(cid:35)
2 1 (cid:90) t
∂ ρ= ∇ · ρ∇ lnρ+V + (U (cid:63)ρ)(s,·)ds .
t d(d−1)A2 x x h(t)
t−h(t)
This equation is similar to an equation derived in [4, 19, 24] for the case without an
interaction potential. The stationary equation reads
(cid:16) (cid:17) (cid:90)
ρ∇ lnρ+V +U (cid:63)ρ =0, ρdx=1,
x
Rd
which leads again to the integral equation (1.1) as for the mean-field case.
5. Numerical Method and Results. In this section we investigate the quali-
tative behavior of solutions corresponding to the interacting fiber model numerically.
More specifically, we consider the case of isotropic fibers in three spatial dimensions.
5.1. Numerical methods. We describe numerical solvers for the microscopic,
mean-field and macroscopic equations, respectively.
5.1.1. Microscopic model. To solve equations (2.2) or (2.6) numerically we
use the Euler–Maruyama method. Using Itˆo integration (2.2) is written as
dxi =τidt
t t
dτi =−1(cid:0)I−τi⊗τi(cid:1)(cid:18)∇ V(xi)+ 1 (cid:88)N 1(cid:90) t∇ U(xi−xj)ds(cid:19)dt
t 2 t t x t N t x t s
j=1 0
10