Cercignani’s conjecture

Carlo Cercignani (1983)

Carlo Cercignani 1983 (Photo credit Kazuo Aoki)

Soon will happen a Conference in memory of Carlo Cercignani at IHP, Paris 9-11 february which I am co-organizing together with François Bolley, Laurent Desvillettes and Silvia Lorenzani. Moreover several works quoted for the Fields medal 2010 of Cédric Villani are directly related and somehow motivated by the so-called Cercignani’s conjecture in kinetic theory.

As a tribute to Carlo Cercignani, one of founder of the modern mathematical kinetic theory and a great scientist, here is a short presentation about Cercignani’s conjecture. More details (including references and latest results about the conjecture) can be found in this review paper.

Boltzmann equation

The Boltzmann equation describes the behavior of a rarefied gas when the only interactions taken into account are binary collisions. In the case when this distribution function is assumed to be independent of the position, it reduces to the spatially homogeneous Boltzmann equation

\displaystyle   \frac{\partial f}{\partial t}(t,v) = Q(f,f)(t,v), \quad v \in \mathbb{R}^d, \quad t \geq 0

where {Q} is the quadratic Boltzmann collision operator, defined by the bilinear form

{Q(g,f) = \int   B (g'_* f' - g_* f) \, dv_* \, d\sigma}

with the shorthands {f'=f(v')}, {g_*=g(v_*)} and {g'_*=g(v'_*)}, where

\displaystyle {v' = (v+v_*)/2 + \sigma |v-v_*|/2 }

and

\displaystyle {v'_* = (v+v_*)/2 - \sigma |v-v_*|/2 }

stand for the pre-collisional velocities of particles which after collision have velocities {v} and {v_*}.  The function {B} is called the Boltzmann collision kernel and it is determined by physics (it is related to the physical cross-section {\Sigma(v-v_*,\sigma)} by the formula {B=|v-v_*| \, \Sigma}). On physical grounds (in particular galilean invariance), it is assumed that {B \geq 0} and {B} is a function of {|v-v_*|} and {\cos\theta} only.

The collision kernel

In the theory of Maxwell and Boltzmann, the interaction between particles is reflected in the formula for the collision kernel {B}. It may be short-range or long-range. The most important case of short-range interaction is the hard spheres model, where particles are spheres interacting by contact. In that case, {B=|v-v_*|} in dimension {d=3} (constant cross-section). Typical models of long-range interactions are given by inverse power-law forces. In dimension {d=3}, if the intermolecular force scales like {r^{-s}} with {s > 2}, then

\displaystyle  B(|v-v_*|, \cos \theta) = |v-v_*|^\gamma \, b(\theta), \quad \theta \in [0,\pi],

where {b} is smooth except near {\theta =0},

\displaystyle  b(\theta) \sim_{\theta =0} \mbox{cst} \, \theta^{-2 - \nu},

and

\displaystyle  \gamma = \frac{s-5}{s-1}, \qquad \nu = \frac{2}{s-1}.

Conserved quantities and entropy structure

Boltzmann’s and Landau’s collision operators have the fundamental properties of conserving mass, momentum and energy

\displaystyle \int Q(f,f)(v) \, \phi(v)\,dv = 0

for \phi(v)=1,v,|v|^2/2 and satisfying (the first part of) Boltzmann’s {H} theorem, which can be formally written as

\displaystyle {\mathcal D}(f):= - \int  Q(f,f)\log f \, dv \ge 0

where Boltzmann’s so-called “{{\mathcal H}} functional”

\displaystyle   {\mathcal H}(f) = \int f \, \log f\, dv

is the opposite of the entropy of the gas. The second part of Boltzmann’s {H} theorem states that under appropriate boundary conditions, any equilibrium distribution function (that is, such that {v\cdot\nabla_x f = Q(f,f)}) satisfies {{\mathcal D}(f)=0}, or equivalently {Q(f,f)=0}, and takes the form of a Maxwellian (gaussian) distribution associated with the parameters {\rho= 0}, {u\in\mathbb{R}^d} and {T\geq 0} which are interpreted as respectively the density, mean velocity and temperature of the gas.

The linearized collision operators

We introduce the fluctuation around the Maxwellian equilibrium {M} computed above:

\displaystyle   f = M + Mh, \quad v \mapsto h(v) \in L^2(M)

where {L^2(M)} denotes the Lebesgue space {L^2} on {\mathbb{R}^d} with reference measure {M(v) \, dv}. Then the linearized collision operator writes

\displaystyle   Lh = M^{-1} \left[ Q(Mh,M) + Q(M,Mh) \right].

It is easy to check that {L} is symmetric in the Hilbert space {L^2(M)} and that it is non-positive in this space (this is the linearized form of the {H} theorem). The dissipation of squared {L^2} norm, that is the opposite of the Dirichlet form associated with {L}, is

\displaystyle   D(h) \frac{1}{4} \int (h' + h'_* - h - h_* )^2 \, B \, M \,  M_* \, dv\, dv_* \, d\sigma \ge 0.

It is straightforward from this formula that the null space of {L} has dimension {d+2}, and is spanned by the so-called collisional invariants {1,v_1, \dots, v_d, |v|^2}.

Comparison with usual differential operators and classification

The Boltzmann collision operators are a priori extremely intricate, partly due to their integral or integro-differential nature (and partly of course due to their nonlinear nature!). Therefore it is useful, in order to grab an intuition of these operators, to draw a parallel with usual differential operators which are more familiar.

For the Boltzmann collision operators, say with collision kernel of the form {B = \Phi(|v-v_*|) \, b(|\theta|)}, the most important two “parameters” interplaying and determining its behavior are (1) the growth or decay of {\Phi}, and (2) the singularity of {b} at grazing collisions {\theta \sim 0}. To be more precise, let us consider the model case {\Phi(z)=z^\gamma}, {\gamma \in  (-d,+\infty)} and {b(|\theta|) \sim  \theta^{-(d-1)-\nu}}, {\nu \in  (-\infty,2)} as {\theta \sim 0}, for the Boltzmann collision operator. Then the order of singularity (2) plays the role of the order (highest number of derivatives) in a differential operator. For instance {\nu <0} in the model means a zero order operator, whereas {\nu \in (0,2)} means a fractional differential operator with order {\nu}. And the growth or decay of {\Phi} (1) plays the role of the growth or decay of the coefficients in a differential operator. Therefore {\gamma=0} (the so-called Maxwell or pseudo-Maxwell molecules cases) would correspond to a constant coefficients differential operator, and {\gamma  =1} (similar to hard spheres) would correspond to unbounded polynomially growing coefficients. From this comparison it becomes natural to consider the Landau collision operator with {\Phi(z)=z^\gamma} formally as the limit case {\nu=2} in the above classification.

Estimating relaxation to equilibrium

The relaxation to equilibrium has been studied since the works of Boltzmann and it is at the core of kinetic theory. The motivation is to provide an analytic basis for the second principle of thermodynamics for a statistical physics model of a gas out of equilibrium. Indeed, Boltzmann’s famous {H} theorem gives an analytic meaning to the entropy production process and identifies possible equilibrium states. In this context, proving convergence towards equilibrium is a fundamental step to justify Boltzmann model, but cannot be fully satisfactory as long as it remains based on non-constructive arguments. Indeed, as suggested implicitly by Boltzmann when answering critics of his theory based on Poincaré’s recurrence Theorem, the validity of the Boltzmann equation breaks for very large time for a discussion). It is therefore crucial to obtain constructive quantitative informations on the time scale of the convergence, in order to show that this time scale is much smaller than the time scale of validity of the model. Cercignani’s conjecture is an attempt to provide such constructive quantitative estimates. In the words of Cercignani: “The present contribution is intended as a step toward the solution of the first main problem of kinetic theory, as defined by Truesdell and Muncaster, i.e. “to discover and specify the circumstances that give rise to solutions which persist forever”.” It is inspired by the entropy – entropy production method, that we now briefly describe.

The entropy – entropy production method

This method was first used in kinetic theory for the Fokker-Planck equation

\displaystyle    \partial_t f = \nabla_v \cdot ( \nabla f + v\, f), \quad v \in \mathbb{R}^d, \quad  \int_{\mathbb{R}^d} f(w) \,dw =1.

In that case, the equilibrium {M} is given by the formula

\displaystyle    M(v) = (2\pi)^{-d/2} \,   e^{- |v|^2/2}

and the entropy production is

\displaystyle    {\mathcal D}_{FP}(f) = \int_{\mathbb{R}^d} f(v)\, \bigg| \nabla \log \frac{f}{M} \bigg|^2\,  dv.

The exponential convergence is then obtained thanks to the logarithmic Sobolev inequality, which exactly means in this setting

\displaystyle    {\mathcal D}_{FP}(f) \ge 2\, \bigg[ {\mathcal H}(f) - {\mathcal H}(M) \bigg].

Consider the more general case of an equation for which a Lyapunov functional {\mathcal{H}_*} exists, that is

\displaystyle  {\mathcal D}_{*}(f(t)):= - \frac{d}{dt} {\mathcal H}_{*}(f(t)) \ge 0

and assume that the entropy {-{\mathcal H}_{*}} is maximal for a unique function {M_*} (among the functions belonging to a space depending on the conserved quantities in the equation). As seen in the previous section, this structure is provided by the {H} theorem for Boltzmann and Landau equations. The entropy – entropy production method consists in looking for (functional) estimates like

\displaystyle   {\mathcal D}_*(f) \ge \Theta\bigl( {\mathcal H}_*(f) - {\mathcal H}_*(M_*) \bigr),

where {\Theta : \mathbb{R}_+ \rightarrow \mathbb{R}_+} is a function such that

\displaystyle \Theta(x) = 0 \qquad \iff \qquad x=0.

The stronger {\Theta} increases near {0} the better the rate of relaxation to equilibrium, since the differential inequality

\displaystyle    \frac{d}{dt} \bigl({\mathcal H}_*(f) - {\mathcal H}_*(M_*) \bigr) \le -\,   \Theta\bigl( {\mathcal H}_*(f) - {\mathcal H}_*(M_*) \bigr)

leads to

\displaystyle  {\mathcal H}_*(f(t)) - {\mathcal H}_*(M_*) \le R(t),

where {R} is the reciprocal of a primitive of {{ - 1/\Theta}}. Then, if the relative entropy {{\mathcal H}_*(f) - {\mathcal  H}_*(M_*)} is coercive in the sense that it controls from below some distance or some norm (denoted by {\|\,\,\|_*}) between {f} and its associated equilibrium distribution {M_*} (for the Boltzmann entropy this is precisely provided by the so-called Czizsar-Kullback-Pinsker inequality, we obtain

\displaystyle \| f(t) - M_* \|_* \le S(t),

where (generically) S(t) = C \sqrt{R(t)}. In the particular case {\Phi(x) = C x} (like in the case of the Fokker-Planck equation), one gets

\displaystyle R(t) \le C e^{-C't}

i.e., exponential convergence towards equilibrium (with explicit rate). In the slightly worse case {\Phi(x) = C_{\varepsilon}x^{1+\varepsilon}} for some (or all) {\varepsilon>0} we can deduce

\displaystyle   R(t) \le C_{\varepsilon} ' t^{-1/\varepsilon},

and we thus get algebraic convergence towards equilibrium (with explicit rate). When {\varepsilon>0} can be taken as small as one wishes, we speak of almost exponential convergence.

Cercignani’s conjecture

The original Cercignani’s conjecture is written in the following form: for any {f} and its associated maxwellian state {M} with same mass, momentum and temperature

\displaystyle    {\mathcal D}(f) \ge \lambda \, \rho \, \big[ {\mathcal H}(f) - {\mathcal H}(M) \big],

where {\rho} is the density (mass of {f}) and {\lambda >0} is a “suitable constant”. We shall now develop this very general statement into a layer of more specified conjectures. Let us fix {\rho=1} without loss of generality. In the case when the constant {\lambda} only depends on the collision kernel {B}, the temperature of {M} (or {f}), and some bound on the entropy of {f} (i.e., only the basic physical a priori estimates), we shall call this inequality the strong form of Cercignani’s conjecture. In the case when the constant {\lambda} also depends on some additional bounds on {f} (typically of smoothness, moments and lower bounds), we shall call such an inequality the weak form of Cercignani’s conjecture. Let us point out that it is of crucial importance to know whether the bounds used can be shown to be propagated by the Boltzmann equation, in order to be able to “apply” the weak form of Cercignani’s conjecture to the relaxation to equilibrium of its solutions. This of course guides which bounds are natural or not. In the slightly different case when the following inequality holds

\displaystyle    {\mathcal D}(f) \ge \lambda_\varepsilon \, \big[ {\mathcal H}(f) - {\mathcal H}(M)  \big]^{1+\varepsilon}, \quad \varepsilon >0

we shall speak of the {\varepsilon}-polynomial Cercignani’s conjecture and it can be divided again into weak and strong versions according to how much the constant {\lambda_\varepsilon} depends on {f}. Finally a strictly similar hierarchy of conjectures can be formulated on the Landau entropy production functional, and we shall call it Cercignani’s conjecture for the Landau equation.

A linearized counterpart to the conjecture

A natural linearized counterpart of Cercignani’s conjecture for the Boltzmann or Landau equation consists in replacing the entropy production functional and the Boltzmann entropy by their linearized approximation, i.e., respectively the Dirichlet form of the collision operators discussed above and the {L^2(M)} norm. This spectral gap question was already well-known for a long time and used by Cercignani as an inspiration and supportive argument for his conjecture. So let us call this the linearized Cercignani’s conjecture:

\displaystyle    D(h) \ge \lambda \, \| h - \Pi(h)\|_{L^2(M)} ^2,

where {\Pi} denotes the orthogonal projector in {L^2(M)} onto the null space of the linearized collision operator, and {\lambda} only depends on the collision kernel {B} and the temperature of {M}. Note that due the linear homogeneity of this relation, no weak version (with constant depending on the function {h}) would make sense. Again obviously the same question can be asked on the Dirichlet form of the Landau collision operators and leads to the linearized Cercignani’s conjecture for the Landau equation.

Comparison with differential operators

In the light of the comparison we have made with usual differential operators, a functional inequality interpretation of Cercignani’s conjecture is the following. Its nonlinear form is an intricate (because of strong nonlinearity and average over additional angular variables) amplified version of a logarithmic Sobolev inequality. Its linearized form is an intricate (because again of average over additional angular variables) amplified version of a Poincaré inequality.

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