ON THE TREND TO EQUILIBRIUM FOR THE FOKKER PLANCK EQUATION AN INTERPLAY

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ON THE TREND TO EQUILIBRIUM FOR THE FOKKER-PLANCK EQUATION : AN INTERPLAY BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS P. A. MARKOWICH? AND C. VILLANI?? Abstract. We present connections between the problem of trend to equilibrium for the Fokker-Planck equation of statistical physics, and several inequalities from functional analysis, like logarithmic Sobolev or Poincare inequalities, together with some inequalities arising in the context of concentration of measures, introduced by Talagrand, or in the study of Gaussian isoperimetry. Contents 1. The Fokker-Planck equation 2 2. Trend to equilibrium 3 3. Entropy dissipation 4 4. Logarithmic Sobolev inequalities 6 5. The Bakry-Emery reversed point of view 7 6. Log Sobolev ? Poincare 8 7. The nonuniformly convex case 9 8. Generalizations to other physical systems 10 9. An example : generalized porous medium equations 12 10. Gaussian isoperimetry 14 11. Talagrand inequalities and concentration of the Gauss measure 15 12. Log Sobolev ? Talagrand ? Poincare 16 13. Related PDE's 18 13.1. The Monge-Ampere equation 18 13.2. The Hamilton-Jacobi equation 19 13.3. The sticky particles system 19 14. HWI inequalities 19 15. Displacement convexity 20 References 22 Acknowledgement: The authors acknowledge support by the EU-funded TMR- network ‘Asymptotic Methods in Kinetic Theory' (Contract _ ERB FMRX CT97 0157) and from the Erwin-Schrodinger-Institute in Vienna.

• elemen- tary inequality


• distance between


• diffusion equations


• relative entropy


• diffusion


• entropy dissipation


• e?v


• known ?


• space ? ?


• lyapunov functional

Sujets

• Analysis
• Vienna
• Kullback?Leibler divergence
• Diffusion

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 21
• Langue: English

Extrait

ON THE TREND TO EQUILIBRIUM FOR THE
FOKKER-PLANCK EQUATION : AN INTERPLAY
BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS
⁄ ⁄⁄P. A. MARKOWICH AND C. VILLANI
Abstract. Wepresentconnectionsbetweentheproblemoftrend
toequilibriumfortheFokker-Planckequationofstatisticalphysics,
and several inequalities from functional analysis, like logarithmic
Sobolev or Poincar´e inequalities, together with some inequalities
arising in the context of concentration of measures, introduced by
Talagrand, or in the study of Gaussian isoperimetry.
Contents
1. The Fokker-Planck equation 2
2. Trend to equilibrium 3
3. Entropy dissipation 4
4. Logarithmic Sobolev inequalities 6
5. The Bakry-Emery reversed point of view 7
6. Log Sobolev) Poincar´e 8
7. The nonuniformly convex case 9
8. Generalizations to other physical systems 10
9. An example : generalized porous medium equations 12
10. Gaussian isoperimetry 14
11. Talagrand inequalities and concentration of the Gauss
measure 15
12. Log Sobolev) Talagrand) Poincar´e 16
13. Related PDE’s 18
13.1. The Monge-Amp`ere equation 18
13.2. The Hamilton-Jacobi 19
13.3. The sticky particles system 19
14. HWI inequalities 19
15. Displacement convexity 20
References 22
Acknowledgement: The authors acknowledge support by the EU-funded TMR-
network ‘Asymptotic Methods in Kinetic Theory’ (Contract # ERB FMRX CT97
0157) and from the Erwin-Schr odinger-Institute in Vienna.
12 P. A. MARKOWICH AND C. VILLANI
1. The Fokker-Planck equation
The Fokker-Planck equation is basic in many areas of physics. It
reads
¡ ¢@‰ n(1) =r¢ D(r‰+‰rV) ; t‚0; x2R ;
@t
where D = D(x) is a symmetric, locally uniformly positive definite
(diffusion) matrix, and V = V(x) a confining potential. Here the un-
known ‰ = ‰(t;x) stands for the density of an ensemble of particles,
and without loss of generality can be assumed to be a probability dis-
ntribution onR since the equation (1) conserves nonnegativity and the
nintegral of the solution over R . The phase space can be a space of
position vectors, but also a space of velocities v; in the latter case the
2potential V is usually the kinetic energyjvj =2.
We refer to [32] for a phenomenological derivation, and a lot of basic
references. TheFokker-Planckequationcanbesetonanydifferentiable
structure, in particular on a Riemannian manifold M, rather than on
nEuclidean spaceR . It can also be considered in a bounded open set,
with (say) a vanishing out-flux condition at the boundary.
Far from aiming at a systematic study of equation (1), our intention
here is to focus on some tight links between this equation, and several
functional inequalities which have gained interest over the last decade,
and especially in the last years. In order to simplify the presentation,
we restrict to the case when the diffusion matrix is the identity – but
in order to keep some generality in (1), we allow any underlying Rie-
mannian structure. Thus we shall study
@‰
n(2) =r¢(r‰+‰rV); t‚0; x2R or M:
@t
Moreover, wedonotaddressregularityissues, andshallalwaysassume
2that V is smooth enough, say C , perform formal calculations and do
not deal with their rigorous justifications in this paper.
As dictated by physical intuition, we mention that the stochastic
differential equation underlying (2) is
dX =dW ¡rV(X )dt;t t t
with W a standard Wiener process (or Brownian motion). Thus thet
Fokker-Planck equation models a set of particles experiencing both
diffusionand drift. The interplaybetween these twoprocesses is at the
basis of most of its properties.THE FOKKER-PLANCK EQUATION 3
2. Trend to equilibrium
Let us begin an elementary study of the Fokker-Planck equation.
¡VFrom (2) we see that there is an obvious stationary state : ‰ = e
¡V(adding a constant to V if necessary, one can always assume that e
is a probability distribution). It is then natural to change variables by
¡Vsetting ‰=he . Then we obtain for (2) the equivalent formulation
@h n(3) =Δh¡rV ¢rh; t‚0; x2R or M:
@t
¡VThe operator L=Δ¡rV ¢r is self-adjoint w.r.t. the measure e .
More precisely,
(4) hLh;gi ¡V =¡hrh;rgi ¡V:e e
2(weusetheobviousnotationforweightedL -scalarproductsandnorms).
In particular,
2hLh;hi ¡V =¡krhk ;2 ¡Ve L (e )
so that L is a nonpositive operator, whose kernel consists of constants
¡V(since e is a positive function). This shows that the only acceptable
¡Vequilibria for (2) are constant multiples of e – the constant being
1 ¡Vdetermined by the norm of h in L (e ), which is preserved.
Now, consider the Cauchy problem for the Fokker-Planck equation,
which is (2) supplemented with an initial condition
Z
‰(t=0;¢)=‰ ; ‰ ‚0; ‰ =1:0 0 0
We expect the solution of the Cauchy problem to converge to the equi-
¡Vlibrium state e , and would like to estimate the rate of convergence
in terms of the initial datum. Let us work with the equivalent formu-
Vlation (3), with the initial datum h = ‰ e . Since L is a nonpositive0 0
self-adjoint operator, we would expect h(t;¢) to converge exponentially
fastto1ifLhasaspectralgap‚>0. Thiseasilyfollowsbyelementary
spectral analysis, or by noting that the existence of a spectral gap of
¡Vsize ‚ for L is equivalent to the statement that e satisfies a Poincar´e
inequality with constant ‚, i.e
(5) •Z Z Z ‚
2 ¡V ¡V 2 ¡V 2 ¡V8g2L (e ); ge dx=0=) jrgj e ‚‚ g e :
Indeed, knowing (5), and using (3), one can perform the computation
Z Z Z
d 2 ¡V 2 ¡V 2 ¡V(6) ¡ (h¡1) e =2 jrhj e ‚2‚ (h¡1) e ;
dt4 P. A. MARKOWICH AND C. VILLANI
which entails
Z Z
2 ¡V ¡2‚t 2 ¡V(h¡1) e •e (h ¡1) e :0
Thus, if h solves (3) with initial datum h ,0
2 ¡V ¡‚th 2L (e ))kh(t;¢)¡1k 2 ¡V •e kh ¡1k 2 ¡V :0 L (e ) 0 L (e )
Equivalently, if ‰ solves (2) with initial datum ‰ ,0
2 V ¡V ¡‚t ¡V(7) ‰ 2L (e ))k‰(t;¢)¡e k 2 V •e k‰ ¡e k 2 V :0 0L (e ) L (e )
Thisapproachisfastandeffective,buthasseveraldrawbacks, which
are best understood when one asks whether the method may be gen-
eralized :
1) Note that the functional space which is natural at the level of (3)
2 ¡V(h2 L (e )) is not at all so when translated to the level of (2) (‰2
2 VL (e )). Formathematicalandphysicalpurposes,itwouldbedesirable
1to be as close as possible to the space ‰ 2 L (which corresponds to
the assumption of finite mass).
2)Thephysicalcontentoftheestimate(7)isquiteunclear. Strongly
based on the theory of linear operators, this estimate turns out to
be very difficult, if not impossible, to generalize to nonlinear diffusion
equations(likeporousmediumequations,ortheFokker-Planck-Landau
equations) which arise in many areas of physics, cf. Section 8.
3)Also,itisoftenquitedifficulttofindexplicitvaluesofthespectral
gap of a given linear operator. Many criteria are known, which give
existence of a spectral gap, but without estimate on its magnitude the
results obtained in this manner are of limited value only.
3. Entropy dissipation
2 ¡VInsteadofinvestigatingthedecayinL (e )normforh,wecouldas
well consider a variety of functionals controlling the distance between
hand1. Actually,whenever`isaconvexfunctiononR,onecancheck
that
Z Z ‡ ·
‰¡V ¡V(8) `(h)e dx= ` e dx
¡Ve
defines a Lyapunov functional for (3), or equivalently for (2). Indeed,
Z Z
d ¡V 00 2 ¡V(9) `(h)e dx=¡ ` (h)jrhj e dx:
dtTHE FOKKER-PLANCK EQUATION 5
2Our previous computation in the L norm corresponds of course to
2thechoice`(h)=(h¡1) ;buttoinvestigatethedecaytowardsequilib-
rium, we could also decide to consider any strictly convex, nonnegative
function ` such that `(1)=0.
For several reasons, a very interesting choice is
(10) `(h)=hlogh¡h+1:
R
¡VIndeed, in this case, taking into account the identity (h¡1)e =0,
we find
Z Z Z
‰¡V(11) `(h)e = ‰log = ‰(log‰+V):
¡Ve
This functional is well-known. In kinetic theory it is often called the
free energy, while in information theory it is known as the (Kullback)
¡Vrelative entropy of ‰ w.r.t. e (strictly speaking, of the measure ‰dx
¡Vw.r.t. the measure e dx).
As we evoke in section 8, the occurrence of the relative entropy is
rather universal in convection-diffusion problems, linear or nonlinear.
Soanassumptionofboundednessoftheentropy(fortheinitialdatum)
issatisfactorybothfromthephysicalandfromthemathematicalpoint
¡Vof view. We shall denote the relative entropy (11) by H(‰je ), which
is reminiscent of the standard notation of Boltzmann’s entropy.
The relative entropy is an acceptable candidate for controlling the
distance b