Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases
We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases...
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irk-123456789-1657482020-02-17T01:27:00Z Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases Lods, B. Toscani, G. Статті We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases by showing that the Homogenous Cooling State attracts any solution at an algebraic rate. Проаналізовано асимптотичну поведінку лінійних рівнянь Фоккера - Планка з коефіцієнтами, залежними від часу. Показано, що за явно обчислюваних умов відбувається релаксація до розподілу Максвелла з залежною від часу температурою. Цей результат застосовано до вивчення броунівського руху в гранульованих газах і показано, що однорідний охолоджуючий стан притягує будь-який розв'язок з алгебраїчною швидкістю. 2005 Article Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases / B. Lods, Toscani G. // Український математичний журнал. — 2005. — Т. 57, № 6. — С. 778–789. — Бібліогр.: 21 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165748 517.9 + 531.19 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Lods, B. Toscani, G. Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases Український математичний журнал |
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We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases by showing that the Homogenous Cooling State attracts any solution at an algebraic rate. |
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Lods, B. Toscani, G. |
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Lods, B. Toscani, G. |
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Lods, B. |
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Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases |
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Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases |
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Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases |
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Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases |
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Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases |
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long-time behavior of nonautonomous fokker-planck equations and cooling of granular gases |
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Інститут математики НАН України |
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Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases / B. Lods, Toscani G. // Український математичний журнал. — 2005. — Т. 57, № 6. — С. 778–789. — Бібліогр.: 21 назв. — англ. |
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Український математичний журнал |
| work_keys_str_mv |
AT lodsb longtimebehaviorofnonautonomousfokkerplanckequationsandcoolingofgranulargases AT toscanig longtimebehaviorofnonautonomousfokkerplanckequationsandcoolingofgranulargases |
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2025-07-14T19:48:11Z |
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1837653030645792768 |
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UDC 517.9+531.19
B. Lods (Politecnico Torino, Italy),
G. Toscani (Univ. Pavia, Italy)
LONG TIME BEHAVIOR
OF NONAUTONOMOUS FOKKER – PLANCK EQUATIONS
AND THE COOLING OF GRANULAR GASES
DOVHOÇASOVA POVEDINKA
NEAVTONOMNYX RIVNQN\ FOKKERA – PLANKA
TA OXOLODÛENNQ HRANUL\OVANYX HAZIV
We analyze the asymptotic behavior of linear Fokker – Planck equations with time-dependent coefficients. Re-
laxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly com-
putable conditions. We apply this result to the study of Brownian motion in granular gases, by showing that the
Homogenous Cooling State attracts any solution at an algebraic rate.
Proanalizovano asymptotyçnu povedinku linijnyx rivnqn\ Fokkera – Planka z koefici[ntamy, zaleΩ-
nymy vid çasu. Pokazano, wo za qvno obçyslgvanyx umov vidbuva[t\sq relaksaciq do rozpodilu Maks-
vella z zaleΩnog vid çasu temperaturog. Cej rezul\tat zastosovano do vyvçennq brounivs\koho ruxu
v hranul\ovanyx hazax i pokazano, wo odnoridnyj oxolodΩugçyj stan prytqhu[ bud\-qkyj rozv’qzok
z alhebra]çnog ßvydkistg.
1. Introduction. The linear Fokker – Planck equation (FPE)
∂ϕ
∂t
(v, t) = λ∇ · {vϕ(v, t) + θ∇ϕ(v, t)} , v ∈ R
N , N � 1, (1.1)
arises in many fields of applied sciences such that statistical mechanics, chemistry, math-
ematical finance (see the monographs [1] and [2] for a large account of applications to
the FPE). Such a drift-diffusion equation can be derived from the Langevin equation to
model the Brownian motion of particles in thermodynamical equilibrium. In this case, the
parameters λ and θ are two positive constants which represent respectively the friction
term and the temperature of the system. The qualitative analysis of equation (1.1) is well
documented in the literature. We refer to [1] for a precise description of the hilbertian and
spectral methods used in the study of (1.1) and to [3, 4] for the relatively recent approach
to the L1-theory by means of entropy-dissipation methods.
It is easy to notice that the set
M =
{
g ∈ L1(RN ), g � 0,
∫
RN
g(v)dv = 1,
∫
RN
vg(v)dv = 0,
∫
RN
v2g(v)dv = θ < ∞
}
is invariant under the action of the right-hand side of (1.1). Moreover, it is well-known that
(1.1) admits a unique steady state in M given by the Gaussian distribution (Maxwellian
function in the language of kinetic theory)
Mθ(v) = (2πθ)−N/2 exp
(
−v2/2θ
)
, v ∈ R
N .
Entropy-dissipation methods (see [5] for a review on recent results on the topic) provide
a precise picture of the asymptotic behavior of the solution to (1.1) for initial data in M.
Given f ∈ M , the (Boltzmann) relative entropy (finite or not) of f is defined as
c© B. LODS, G. TOSCANI, 2005
778 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
LONG TIME BEHAVIOR OF NONAUTONOMOUS FOKKER – PLANCK EQUATIONS . . . 779
H(f |Mf ) = H(f) −H(Mf ) =
∫
RN
f log
(
f
Mf
)
dv
where Mf is the unique Maxwellian distribution in M with the same temperature as
f . Given f0 ∈ M , with the assumption of bounded initial relative entropy
H(f0 |Mθ) < ∞,
it has been proven in [3] that the unique mass-preserving solution f(v, t) of (1.1) decays
exponentially fast with rate 2λ to Mθ(v) in relative entropy, i.e, the estimate
H(f |Mθ)(t) ≤ e−2λtH(f0 |Mθ)
holds. The classical Csiszar – Kullback inequality then allows to move convergence in
relative entropy to the more standard L1-setting. The following result is proved in [3]:
Theorem 1.1. Let λ and θ be two positive constants. Let us assume that f0 ∈ M
has a finite relative entropy. Then, there exists a constant C > 0 depending only on the
initial relative entropy such that the solution f(v, t) to (1.1) fulfills
‖f(·, t) −Mθ‖L1(RN ) � C exp (−λt)
for any t � 0 .
Main objective of this paper is to generalize this result allowing the friction term λ
and the temperature θ to fluctuate with time. In this case, the Fokker – Planck equation
reads:
∂ϕ
∂t
(v, t) = λ(t)∇ · {vϕ(v, t) + θ(t)∇ϕ(v, t)} (1.2)
where λ(t) and θ(t) are positive functions of time.
Nonautonomous Fokker – Planck equations arise for instance in the study of a period-
ically driven Brownian rotor [6] and in this case λ(t) and θ(t) are periodic functions of
time. In statistical mechanics, equation (1.2) arises as a natural generalization of equation
(1.1) in the context of nonequilibrium thermodynamics [7]. Among other models, equa-
tion (1.2) appears in the study of the tagged particle dynamics of a heavy particle in a gas
of much lighter inelastic particles. As observed by J. J. Brey, W. Dufty and A. Santos [8],
the large particles exhibit Brownian motion and the Boltzmann – Lorentz kinetic equation
satisfied by the distribution function of large particles can be reduced to a Fokker – Planck
equation whose coefficients depend on the temperature of the surrounding gas. Granular
gases being nonequilibrium systems, this temperature turns out to be time-dependent and
the Fokker – Planck equation derived in [8] is of the shape (1.2). Since the study of the
long time behavior of the solution to the Brey – Dufty – Santos model is one of the main
goals of our analysis, we will explain with much more details the approach of [8] in the
next section.
Because of the time-dependence of both λ(t) and θ(t) , Equation (1.2) does not
possess stationary states. Nevertheless, two natural questions arise:
Do they exist particular (self-similar) solutions to (1.2) which attract all other solutions
(as the Maxwellian does in the autonomous case)?
If such self-similar solutions exist, is it possible to reckon the rate at which they attract
the other solutions?
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
780 B. LODS, G. TOSCANI
We answer positively to these two questions under some reasonable conditions on
the time-behavior of friction and temperature. Our method is based upon suitable (time-
dependent) scalings which allow us to transform the nonautonomous equation (1.2) into a
Fokker – Planck equation of the form (1.1) and then to make use of Theorem 1.1. Clearly,
the self-similar profile is a Maxwellian with time-dependent temperature. In this context,
the Maxwellian distribution plays the role of the Barenblatt profile in the study of the
porous medium equation [9].
Application of this abstract result to our motivating example in kinetic theory of
granular gases, shows that the distribution of the Brownian particles relaxes towards a
Maxwellian distribution with time-dependent temperature. While this fact was already
been noticed by J. Javier Brey et al. [8] our result gives a precise estimate the rate of
convergence towards this self-similar profile (the so-called Homogeneous Cooling State)
which turns out to be only algebraic in time. Analogous results, which try to clarify the
role of the Homogeneous Cooling State in kinetic models of granular gases, have been
recently obtained for the case of the Boltzmann equation for inelastic Maxwell particles
[10]. We postpone a detailed discussion on this point in the conclusions of this note.
The organization of the paper is the following. In the next section, we present in
some details the derivation of the FPE in the context of Brownian motion for granular
gases. In Section 3, we deals with a general nonautonomous FPE and we answer to the
two aforementioned questions (Theorem 3.1). Finally, in Section 4 we turn back to our
motivated example of Brownian particles and we show how the abstract result of Section
3 allows us to estimate the rate of convergence towards the Homogeneous cooling state.
2. The Brownian motion in granular gases. The motion of heavy granular par-
ticles of mass m embedded in a low density gas whose particles have mass mg with
mg � m. has been considered in [8]. The particles under consideration are assumed to
be hard-spheres of R
3 and, for the sake of simplicity, the diameters of the particles of
both species are assumed to be equal and normalized to unit. The case of particles with
different diameter can be investigated as well, and does not lead to major supplementary
difficulties [8]. The collisions between the heavy particles and the fluid ones are partially
inelastic and are characterized by a coefficient of restitution ε ∈ (0, 1) . Assuming that
the concentration of heavy particles is small, one neglects the collision phenomena be-
tween them. Let us denote by f(ṽ, t) the distribution function of the heavy particles
having velocity ṽ ∈ R
3 at time t > 0 and by g(ṽ, t) the distribution function of the
surrounding gas where, for simplicity, it is assumed that these two quantities are indepen-
dent of the position. Then, the evolution of f(·, t) is given by the Boltzmann – Lorentz
equation, which in weak form reads
d
dt
∫
R3
f(ṽ, t)ψ(ṽ)dṽ =
∫
R3×R3×S2
|q · n|f(ṽ, t)g(w̃, t)[ψ(v ) − ψ(ṽ)]dṽdw̃dn (2.1)
for any test-function ψ(ṽ). Here q = ṽ − w̃ and (v ,w ) are the post-collisional
velocities:
v = ṽ − ∆(1 + ε)
1 + ∆
(q · n)n, w = w̃ +
∆(1 + ε)
1 + ∆
(q · n)n (2.2)
where ∆ is the mass ratio ∆ = mg/m . Note that, by assumption, ∆ � 1 .
To solve the linear equation (2.1), one has to make explicit g(ṽ, t) . Assuming that
the binary collisions between the surrounding particles are inelastic and characterized
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
LONG TIME BEHAVIOR OF NONAUTONOMOUS FOKKER – PLANCK EQUATIONS . . . 781
by a constant restitution coefficient 0 < εg < 1, g(ṽ, t) is given by a solution to the
(nonlinear) Boltzmann equation for granular hard-spheres [11, 12]. Leaving details to the
pertinent literature, the role of the equilibrium Maxwellian function in the elastic Boltz-
mann equation is here represented by the Homogeneous Cooling State (see References
[12, 13]) which implies that
g(ṽ, t) = vg(t)−3Φ
(
ṽ
vg(t)
)
where vg(t) is the thermal velocity of the gas particles defined as vg(t) =
= [2Tg(t)/mg]
1/2 . The temperature Tg(t) is defined in a standard way (see Section 3).
The self-similar profile Φ(·) is a stationary solution of some suitable steady Boltzmann
equation (see [11, 13]) and is not explicitly known. However, an important fact to be
noticed is that Φ(ṽ) is a function of εg which, in the quasielastic regime εg → 1 , con-
verges toward the Maxwellian distribution π−3/2 exp (−ṽ2). The temperature Tg(t) is
cooling because of the inelasticity of the collisions. Hereafter, we will assume Tg(t) to
obey the the so-called Haff’s law [14]:
Tg(t) = Tg(0) (1 + t/τ0)
−2 (2.3)
where τ0 > 0 is the characteristic time [8]:
τ−1
0 =
π(1 − ε2g)
12
√
Tg(0)
2mg
∫
R3×R3
Φ(ṽ)Φ(w̃)|ṽ − w̃|3dṽdw̃. (2.4)
The Boltzmann – Lorentz equation (2.1) can be reduced to a FPE with time-dependent
coefficients of the form (1.2) performing two asymptotic procedures:
The first procedure is a simple extension of the standard method for elastic particles
(grazing collisions asymptotics [15], see also [16] in the context of the dissipative linear
Boltzmann equation). Precisely, according to (2.2), one sees that, when a heavy particle
collides with a small one, the velocity of the heavy particle is only slightly altered:
|v − ṽ| =
∣∣∣∣∆(1 + ε)
1 + ∆
(q · n)n
∣∣∣∣ � 1
so that v � ṽ . Therefore, performing in (2.1) a formal expansion to leading order
in the mass ratio as ∆ → 0 , one obtains the following Fokker – Planck equation with
time-dependent coefficients:
∂f(ṽ, t)
∂t
= ∇ṽ ·
[
A(ṽ, t)f(ṽ, t) +
1
2
∇ṽ · (N(ṽ, t)f(ṽ, t))
]
(2.5)
where the vector A(ṽ, t) and, respectively, the tensor N(ṽ, t) are given by
A(ṽ, t) =
(1 + ε)∆
1 + ∆
π
2
∫
R3
g(w̃, t)q|q|dw̃,
and
Nij(ṽ, t) =
(
1 + ε∆
1 + ∆
)2
π
12
∫
R3
g(w̃, t)
(
|q|3δij + 3|q|qiqj
)
dw̃, i, j = 1, 2, 3.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
782 B. LODS, G. TOSCANI
We used above the standard notation
∇ṽ · (N(ṽ)f(ṽ)) =
3∑
j=1
∂
∂ṽj
Nij(ṽ, t)f(ṽ, t)
i=1,2,3
(see [8] (Appendix A) for a detailed derivation).
To simplify further the Fokker – Planck equation (2.5) one performs a second asymp-
totic procedure which consists in assuming that the thermal velocity of the heavy particles
v(t) = [2T (t)/m]1/2 is negligible with respect to the one of the surrounding particles
vg(t) : v(t) � vg(t). This leads to a formal expansion in T (t)∆/Tg(t) where T (t)
is the temperature of the Brownian particles (see Remark 2.1). In this case, the vector
A(ṽ, t) and the tensor N(ṽ, t) reduce to
A(ṽ, t) � α(t)ṽ and Nij(ṽ, t) � 2η(t)δij , i, j = 1, 2, 3,
where
α(t) = ζTg(t)1/2 and η(t) = ξTg(t)3/2. (2.6)
In (2.6) we used the notations
ζ =
2
√
2π
3√mg
(1 + ε)∆
∫
R3
Φ(w̃)|w̃|dw̃, (2.7)
and
ξ =
πm
−3/2
g
3
√
2
(1 + ε)2∆2
∫
R3
Φ(w̃)|w̃|3dw̃.
Taking these simplifications into account, equation (2.5) is replaced by the following
∂f
∂t
(ṽ, t) = ∇ṽ · {α(t)ṽ f(ṽ, t) + η(t)∇ṽf(ṽ, t)} . (2.8)
The Fokker – Planck equation (2.8) is of the form (1.2). The drift and diffusion coefficient
depend on time only through the surrounding gas temperature Tg(t) (see (2.6)).
Remark 2.1. The derivation of (2.8) form (2.1) is based upon a time-dependent
asymptotic procedure where it is assumed that
T (t)∆
Tg(t)
� 1. (2.9)
As shown in [8] this assumption requires ∆ → 0 and εg → 1 (quasielastic regime for
the surrounding gas) simultaneously. A further consequence of these assumptions is that
1
2
√
2∆
1 − ε2g
1 + ε
< 1. (2.10)
We will find again this condition hereafter.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
LONG TIME BEHAVIOR OF NONAUTONOMOUS FOKKER – PLANCK EQUATIONS . . . 783
3. Long time behavior of nonautonomous Fokker – Planck equations. In this
section we consider the general Fokker – Planck collision operator with time-dependent
coefficients written in the divergence form:
Qt(f)(v) = λ(t)∇v · {v f(v, t) + θ(t)∇vf(v, t)} , v ∈ R
N , N � 1, (3.1)
where λ(t) and θ(t) are two positive functions of time. We are concerned with the
large-time asymptotic behavior of the solution to the Cauchy problem
∂f
∂t
(v, t) = Qt(f)(v), v ∈ R
N , t > 0,
f(v, 0) = f0(v),
(3.2)
where the initial data f0 is assumed to be nonnegative and integrable,
f0 � 0, and f0 ∈ L1(RN ).
In accordance with the language of kinetic theory, we define the mass density #(t) , mean
velocity u(t) and temperature T (t) respectively as:
#(t) =
∫
RN
f(v, t)dv, u(t) =
1
#(t)
∫
RN
vf(v, t)dv,
and
T (t) =
1
N#(t)
∫
RN
|v − u(t)|2f(v, t)dv.
The number density is preserved by the (nonautonomous) Fokker – Planck operator (3.1)
while the mean velocity is preserved only if initially equal to zero. Precisely, if∫
Rn
vf0(v)dv = 0, then u(t) = 0 for any t > 0. In this case, the evolution of the
temperature is
dT (t)
dt
= −2λ(t) (T (t) − θ(t)) , t > 0. (3.3)
In order to find the intermediate asymptotic for (3.2), we look for a solution to (3.1) of the
shape:
f(v, t) = γ(t)−NF (v/γ(t), τ(t)) = γ(t)−NF (ṽ, τ)
where the new time scale τ = τ(t) is nonnegative and such that τ(0) = 0 , the scaled
velocity is
ṽ = v/γ(t)
and γ(·) is positive. Without loss of generality, one may assume that γ(0) = T0 :=
:=
∫
R3
v2f0(v)dv so that
F (ṽ, 0) = F (v/T0, 0) = f0(v/T0).
One sees immediately that
∂f
∂t
(v, t) =
τ̇(t)
γ(t)N
∂F
∂τ
(ṽ, τ) − γ̇(t)
γ(t)N+1
∇ṽ · (ṽF (ṽ, τ)) , (3.4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
784 B. LODS, G. TOSCANI
where the dot symbol stands for the time derivative. In the same way, one can show that
Qt(f)(v) =
λ(t)
γ(t)N
∇ṽ · (ṽF (ṽ, τ)) +
λ(t)θ(t)
γ(t)N+2
∇2
ṽF (ṽ, τ). (3.5)
This leads to the following evolution equation for F (·, τ) :
∂F
∂τ
(ṽ, τ) =
1
τ̇(t)
[
λ(t) +
γ̇(t)
γ(t)
]
∇ṽ · (ṽF (ṽ, τ)) +
λ(t)θ(t)
τ̇(t) γ(t)2
∇2
ṽF (ṽ, τ). (3.6)
One notes that (3.6) reduces to a “good” Fokker – Planck equation
∂F
∂τ
(ṽ, τ) = ∇ṽ · (ṽ F (ṽ, τ) + ∇ṽF (ṽ, τ)) , ṽ ∈ R
N , τ > 0, (3.7)
provided there exists some σ > 0 such that
1
τ̇(t)
[
λ(t) +
γ̇(t)
γ(t)
]
= 1 ∀t > 0, (3.8)
and
λ(t)θ(t)
τ̇(t) γ(t)2
= 1 ∀t > 0. (3.9)
Of course, to investigate the asymptotic behavior of F (·, τ) and apply Theorem 1.1, one
has to find conditions on λ(·) and θ(·) insuring that the time scale τ verifies
lim
t→∞
τ(t) = +∞.
Solving equation (3.8), (3.9) leads to
λ(t)θ(t)
γ2(t)
= τ̇(t) = λ(t) +
γ̇(t)
γ(t)
,
i.e.,
λ(t)θ(t) = γ2(t)λ(t) + γ(t)γ̇(t) = γ2(t)λ(t) +
1
2
d
dt
{γ2(t)}.
Since γ(0) =
√
T0 , one obtains
γ(t) = exp
(
−
t∫
0
λ(s)ds
){
T0 + 2
t∫
0
λ(s)θ(s) exp
(
2
s∫
0
λ(r)dr
)
ds
}1/2
, t > 0.
(3.10)
Now, from (3.9),
τ̇(t) =
λ(t)θ(t)
γ2(t)
=
λ(t)θ(t) exp
(
2
∫ t
0
λ(s)ds
)
T0 + 2
∫ t
0
λ(s)θ(s) exp
(
2
∫ s
0
λ(r)dr
)
ds
=
=
1
2
d
dt
log
[
T0 + 2
t∫
0
λ(s)θ(s) exp
2
s∫
0
λ(r)dr
ds
]
.
Solving this equation with the initial datum τ(0) = 0 one gets
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LONG TIME BEHAVIOR OF NONAUTONOMOUS FOKKER – PLANCK EQUATIONS . . . 785
τ(t) =
1
2
log
T0 + 2
t∫
0
λ(s)θ(s) exp
2
s∫
0
λ(r)dr
ds
, t > 0. (3.11)
Clearly,
lim
t→∞
τ(t) = +∞ if and only if
∞∫
0
λ(t)θ(t) exp
2
t∫
0
λ(s)ds
dt = ∞.
This leads to the following result.
Theorem 3.1. Let us assume that λ(·) and θ(·) are nonnegative functions on R+
satisfying
∞∫
0
λ(t)θ(t) exp
2
t∫
0
λ(s)ds
dt = ∞. (3.12)
Let us assume furthermore that f0 ∈ M has a finite relative entropy. Then, there exists
a constant C > 0 such that
‖f(·, t) − f∞(·, t)‖L1(RN
v ) �
≤ C{
T0 + 2
∫ t
0
λ(s)θ(s) exp
(
2
∫ s
0
λ(r)dr
)
ds
}1/2
, t > 0. (3.13)
The intermediate asymptotic profile f∞(v, t) is given by
f∞(v, t) = (2π T (t))−N/2 exp {−v2/2T (t)} = MT (t)(v)
where T (t) is the temperature of f(·, t) given by (3.10).
Proof. The proof reduces to the study of (3.6). Clearly, one may choose σ = 1
(this is equivalent to change τ by (3.11)). By (3.12), τ(t) → ∞ , and according to
Theorem 1.1,
‖F (·, τ) −M1(·)‖L1(RN
ṽ ) � C exp {−τ}, τ > 0, (3.14)
provided F (ṽ, 0) is of finite relative entropy. This is the case since F (ṽ, 0) = f0(v/T0) .
Now, turning back to the original variables, one gets the conclusion using the fact that
exp{−τ(t)} =
1 + 2
t∫
0
λ(s)θ(s) exp
2
s∫
0
λ(r)dr
ds
−1/2
, t > 0,
by virtue of (3.11).
The theorem is proved.
Remark 3.1. The intermediate asymptotic is given by the Maxwellian distribution
with the same temperature T (t) as the one of f(·, t) . Of course, this Maxwellian distri-
bution is a particular solution to (3.2). The most important feature of Theorem 3.1 is that
it provides the rate of convergence of any solution f(v, t) towards the self-similar profile
f∞(v, t) . This rate is explicit in terms of the known coefficients λ(t) and θ(t) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
786 B. LODS, G. TOSCANI
4. The homogeneous cooling state for the Brownian particles. We apply here the
results of Section 3 to the study of the so-called homogeneous cooling state for equa-
tion (2.8). Let f0(ṽ) be an element of M and let us consider the Cauchy problem
∂f
∂t
(ṽ, t) = λ(t)∇ṽ · {ṽ f(ṽ, t) + θ(t)∇ṽf(ṽ, t)} , ṽ ∈ R
3, t > 0,
f(ṽ, 0) = f0(ṽ),
(4.1)
where we transformed the right-hand side of equation (2.8) into a nonautonomous Fokker –
Planck operator of the form (3.1) by setting, according to (2.6),
λ(t) = α(t) = ζTg(t)1/2 and θ(t) =
η(t)
α(t)
= ξζ−1Tg(t). (4.2)
Accordingly, for any t > 0 and with the notations of Section 3
t∫
0
λ(s)θ(s) exp
2
s∫
0
λ(r)dr
ds = ξ
t∫
0
Tg(s)3/2 exp
2ζ
s∫
0
√
Tg(r)dr
ds =
= ξTg(0)3/2
t∫
0
(1 + s/τ0)−3 exp
2ζ
√
Tg(0)
s∫
0
dr
1 + r/τ0
ds.
Note that the last equality follows from Haff’s law (2.3). Clearly, for any s > 0
(1 + s/τ0)−3 exp
2ζ
√
Tg(0)
s∫
0
dr
1 + r/τ0
= (1 + s/τ0)ν
with ν = 2ζτ0
√
Tg(0) − 3. Consequently, condition (3.12) of Theorem 3.1 is verified
provided ν � −1. Note however that, if ν = −1 , then
T (t) = exp
−2
t∫
0
λ(s)ds
T0 + 2
t∫
0
λ(s)θ(s) exp
2
s∫
0
λ(r)dr
ds
=
= (1 + t/τ0)−2
(
T0 + ξTg(0)3/2τ0 log(1 + t/τ0)
)
,
so that
T (t)
Tg(t)
−→ ∞ as t → ∞.
In this case, assumption (2.9) is violated, and the Fokker – Planck equation (2.8) is mean-
ingless. Hence, condition (3.12) of Theorem 3.1 reduces to ν > 1 , i.e.,
ζτ0
√
Tg(0) > 1. (4.3)
Using (2.4) and (2.7) the above condition reads
16
(1 + ε)∆
1 − ε2g
∫
Φ(ṽ)|ṽ|dṽ∫
Φ(ṽ)Φ(w̃)|ṽ − w̃|3dṽdw̃
> 1.
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LONG TIME BEHAVIOR OF NONAUTONOMOUS FOKKER – PLANCK EQUATIONS . . . 787
Now, recall that the Fokker – Planck equation (3.1) turns out to be valid only for nearly
elastic surrounding particles (see Remark 2.1). Since in this quasielastic regime Φ(ṽ) ap-
proaches the Maxwellian distribution π−3/2 exp(−ṽ2) , one can reasonably approximate
the two moments
∫
R3
Φ(ṽ)|ṽ|dṽ and
∫
R3×R3
Φ(ṽ)Φ(w̃)|ṽ− w̃|3dṽdw̃ by their limits as
εg → 1 , obtaining ∫
R3
π−3/2 exp
(
−ṽ2
)
|ṽ|dṽ = 2/
√
π
and ∫
R3×R3
π−3 exp
(
−ṽ2 − w̃2
)
|ṽ − w̃|3dṽdw̃ = 16/
√
2π.
Using these approximation, equation (4.3) turns out to be equivalent to
2
√
2∆
1 + ε
1 − ε2g
> 1.
Once again, we find the same condition (2.10) of validity of the nonautonomous Fokker –
Planck equation (2.8).
Remark 4.1. Note that, here again, we only assume that the surrounding gas particles
suffer nearly elastic collisions (i.e., 1 − εg � 1 ) but we not assume εg to be equal to
one. As a consequence, we do not replace the cooling state profile Φ by the Maxwellian
distribution, but we only assume that its moments do not differ to much from the ones of
the Maxwellian distribution.
The previous reasoning leads to the following theorem.
Theorem 4.1. Let us assume that
1
2
√
2∆
1 − ε2g
1 + ε
−→ β−1 < 1 as ∆ → 0, εg → 1.
Let f0 ∈ M be of finite relative entropy. Then, the solution f(ṽ, t) to the Fokker –
Planck equation (3.1) converges towards the cooling Maxwellian
f∞(ṽ, t) = (2πT (t))−3/2 exp
{
−ṽ2/2T (t)
}
,
and the following bound holds
‖f(·, t) − f∞(·, t)‖L1(R3) = O(t1−β), t → ∞.
The temperature of the Maxwellian is given by
T (t) =
(1 + ε)∆Tg(0)
2mg(1 − β−1)
(1 + t/τ0)
−2 +
(
T (0) − (1 + ε)∆Tg(0)
2mg(1 − β−1)
)
(1 + t/τ0)
−2β
.
Proof. The proof is a straightforward application of Theorem 3.1. Here we use the
fact that
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
788 B. LODS, G. TOSCANI
t∫
0
λ(s)θ(s) exp
2
s∫
0
λ(r)dr
ds = ξTg(0)3/2
t∫
0
(1 + s/τ0)2β−3ds =
=
τ0
2β − 2
ξTg(0)3/2
{
(1 + t/τ0)2β−2 − 1
}
.
Moreover, by (3.10), the temperature of the heavy particles is
T (t) = exp
−2
t∫
0
λ(s)ds
T (0) + 2
t∫
0
λ(s)θ(s) exp
2
s∫
0
λ(r)dr
ds
.
This last quantity is explicitly computable using the expressions of τ0 and ξ .
Remark 4.2. One notes that T (t) obeys asymptotically Haff’s law since the decay
of temperature of the heavy particles for large t is in O(1 + t/τ0)−2 . Actually,
T (t)
Tg(t)
−→ (1 + ε)∆
2mg(1 − β−1)
as t → ∞.
An interesting feature is that, depending on the values of the parameters ε, ∆ , mg and
β , the temperature of the heavy particles is greater or smaller than the one of the sur-
rounding gas. This contrasts the classical case of elastic particles in equilibrium. Indeed,
in this case, according to Theorem 1.1, the distribution function relaxes to a Maxwellian
distribution whose temperature θ is exactly the one of the surrounding bath (see [3] for
more details). We refer the reader to [8] for a discussion of the competing effects which
imply the asymptotic difference between T (t) and Tg(t) .
Remark 4.3. We point out that, whereas for systems in equilibrium, the relaxation
rate is exponential (Theorem 1.1), one notes here that the Homogeneous Cooling State
f∞(ṽ, t) attracts the distribution function f(ṽ, t) only with an algebraic rate.
5. Concluding remarks. We discussed in this paper the intermediate asymptotics of
a linear Fokker – Planck equation with time-dependent coefficients of the form (1.2). We
showed that, under some reasonable assumptions on the drift and diffusion coefficients,
any solution f(v, t) to (1.2) relaxes towards a Maxwellian distribution function whose
(time-dependent) temperature is the one of f(v, t) . More important is the fact that the
rate of convergence towards this self-similar solution is explicitly computable in terms of
the coefficients and the initial temperature.
We applied our result to the motivating example of Brownian motion in granular flu-
ids, already addressed in [8]. For such a model, the Fokker – Planck equation (1.2) is
an approximation of the Boltzmann – Lorentz equation. According to our general result
(Theorem 3.1), the so-called Homogeneous Cooling State for this model is a Maxwellian
distribution whose temperature obeys asymptotically the Haff’s law. Moreover, the rate
of convergence towards this self-similar solution is algebraic in time. We wish to empha-
size here the fact that the question of the rate of convergence towards the Homogenous
Cooling State (HCS) is of primary importance in the kinetic theory of gases. We recall
here that any solution to the nonlinear Boltzmann equation for inelastic interactions re-
laxes towards a Dirac mass because of the dissipation of the kinetic energy. It has been
conjectured however by Ernst and Brito [13, 17] that the HCS attracts any solution faster
than the Dirac mass does. For hard-spheres interactions, only few results support this
conjecture. For nearly elastic flows in one-dimension, the nonlinear Boltzmann equation
reduces to a nonlinear friction equation [18, 19] and it has been shown in [20] that, in this
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LONG TIME BEHAVIOR OF NONAUTONOMOUS FOKKER – PLANCK EQUATIONS . . . 789
case, the HCS does not attract much faster than the Dirac mass since the improvement in
the rate of convergence is only logarithmic in time. This question has also been addressed
recently in [21].
Acknowledgement. Part of this research was carried out during the stay of B. Lods
at the IMATI-CNR/Department of Mathematics of the University of Pavia. He would
like to express his sincere gratitude to P. Pietra for her kind hospitality during this period.
The work of B. L. at Pavia was supported by the IHP project “HYperbolic and Kinetic
Equations”, No. HPRN-CT-2002-00282, funded by the EC. G. T. acknowledges financial
supports both from the project “HYperbolic and Kinetic Equations”, funded by the EC.,
and from the Italian MURST, project “Mathematical Problems in Kinetic Theories”.
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– P. 407 – 428.
Received 06.10.2004
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