Interacting Particle Systems and Conformally invariant operators: Difference between pages

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The (second order) integro-differential equation
On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by
\[ \begin{array}{rl}
\[
\partial_t \rho &= \text{div} \left( D(\rho) \nabla \rho+\sigma(\rho) \nabla V[\rho]\right )\\
A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M),
V[\rho] & = J * \rho
\]
\end{array}\]
where $a, b$ are constant.
describes at the macroscopic scale the phase segregation in  a gas whose particles are interacting at long ranges, as shown by Giacomin and Lebowitz <ref name="GL97"/>.  More concretely, the above equation is the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics.


Examples of conformally invariant operators include:


== The interacting particle system ==  
* The conformal Laplacian:
\[
L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g,
\]
where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.


At the microscopic level, the system is described by a function
* The Paneitz operator <ref name="paneitz1983quartic"/> <ref name="paneitz2008quartic"/>:
\[
P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q,
\]
where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor,
\[
Q=c_n|Ric_g|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR
\]
and
\[
a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}.
\]
This is a fourth order operator with leading term $(-\Delta_g)^2$.


\[ \eta  : \Lambda_\gamma \to \{ 0,1\} \]
* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl <ref name="Juhl1"/><ref name="Juhl2"/> (see also Fefferman-Graham<ref name="FG13"/> ). The formula are more explicit when they are on the standard spheres.


where  $\gamma>0$ taken very small represents the spatial scale and  $\Lambda_\gamma $ denotes the finite $d$-dimensional lattice
*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.


\[ \Lambda_\gamma = \{ 1,2,...,[\gamma^{-1}]\}^d\]


in other words, a cube inside $\mathbb{Z}^d$ with sides given by $[\gamma^{-1}]$, as $\gamma \to 0$, this exhausts all of $\mathbb{Z}^d$. The set of all posible configurations $\eta$ will be denoted by $\Omega_\gamma$, this is the state space where the (microscopic scale) dynamics takes place. As $\gamma \to 0$ we expect to recover the above as a limiting dynamical system the integro-differential equation listed above, of course first we have to describe the microscopic dynamics.
Special cases:


Given any initial condition $\eta_0 : \Lambda_\gamma \to \{0,1\}$, we consider a stochastic Poisson jump $\{ \eta_t\}_{t\in\mathbb{R}_+}$ process with values in $\Lambda_\gamma$  and which is generated by the operator
* On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.


\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\]
* On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator <ref name="branson1987group"/> of explicit formula:
\[
P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2},
\]
where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$
* is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,


Here  $\eta^{x,y}$ denotes the state $\eta$ where the values at $x$ and $y$ have been interchanged and the kernel $c_\gamma(x,y;\eta)$ is defined as
* has the eigenfunctions of spherical harmonics, and  


\[c_\gamma(x,y;\eta) = \left \{ \begin{array}{rl}
* is the inverse of a spherical Riesz potential.
\Phi \left ( \beta\left [ H(\eta^{x,y})-H(\eta) \right ] \right) & \text{ if }\; |x-y|=1\\
0 & \text{ otherwise }
  \end{array}\right.\]




== The Hydrodynamic limit ==  
== References ==
{{reflist|refs=
 
<ref name="branson1987group">{{Citation | last1=Branson | first1= Thomas P | title=Group representations arising from Lorentz conformal geometry | journal=Journal of functional analysis | year=1987 | volume=74 | pages=199--291}}</ref>
 
<ref name="CSextension">{{Citation | last1=Caffarelli | first1= Luis | last2=Silvestre | first2= Luis | title=An extension problem related to the fractional Laplacian | journal=Communications in Partial Differential Equations | year=2007 | volume=32 | pages=1245--1260}}</ref>
 
<ref name="chang2011fractional">{{Citation | last1=Chang | first1= Sun-Yung Alice | last2=González | first2= Maria del Mar | title=Fractional Laplacian in conformal geometry | journal=Advances in Mathematics | year=2011 | volume=226 | pages=1410--1432}}</ref>
 
<ref name="FG13">{{Citation | last1=Fefferman | first1= Charles | last2=Graham | first2= C | title=Juhl’s formulae for GJMS operators and 𝑄-curvatures | journal=Journal of the American Mathematical Society|year=2013 | volume=26 | pages=1191--1207}}</ref>


If $\mu_\gamma \in \Omega_\gamma$ is such that $\mu_\gamma \to \rho \in L^1(\mathbb{T}^d)$ and if $\rho(x,t)$ solves the integro-differential equation with initial data $\rho_0$, then
<ref name="gover2004conformally">{{Citation | last1=Gover | first1= A | last2=Hirachi | first2= Kengo | title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem | journal=Journal of the American Mathematical Society |year=2004 |volume=17 | pages=389--405}}</ref>


\[ \lim \limits_{\gamma \to 0} P^{\mu_\gamma}_\gamma  \left ( \left | \gamma^d \sum\limits_k \phi(\gamma k)\eta_{t\gamma^{-2}}(k)-\int_{\mathbb{T}^d}\phi(x)\rho(x,t)dx \right |>\delta \right ) =0\]
<ref name="GJMS">{{Citation | last1=Graham | first1= C Robin | last2=Jenne | first2= Ralph | last3=Mason | first3= Lionel J | last4=Sparling | first4= George AJ | title=Conformally invariant powers of the Laplacian, I: Existence | journal=Journal of the London Mathematical Society | year=1992 | volume=2 | pages=557--565}}</ref>


<ref name="graham2003scattering">{{Citation | last1=Graham | first1= C Robin | last2=Zworski | first2= Maciej | title=Scattering matrix in conformal geometry | journal=Inventiones mathematicae | year=2003 | volume=152 | pages=89--118}}</ref>


== References ==
<ref name="Juhl1">{{Citation | last1=Juhl | first1= Andreas | title=On the recursive structure of Branson’s Q-curvature | journal=arXiv preprint arXiv:1004.1784}}</ref>
{{reflist|refs=
 
<ref name="Juhl2">{{Citation | last1=Juhl | first1= Andreas | title=Explicit formulas for GJMS-operators and Q-curvatures | journal=Geometric and Functional Analysis | year=2013|volume=23 | pages=1278--1370}}</ref>
 
<ref name="paneitz1983quartic">{{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds |year=1983 | journal=preprint}}</ref>


<ref name="GL97"> {{Citation | last1=Lebowitz | first1=Joel | last2=Giacomin | first2=Giambattista | title=Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits | doi=10.1007/BF02181479 | year=1997 | journal=Journal of Statistical Physics | issn=0022-4715 | volume=87 | issue=1 | pages=37–61}} </ref>
<ref name="paneitz2008quartic">{{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary) | url=http://dx.doi.org/10.3842/SIGMA.2008.036 | doi:10.3842/SIGMA.2008.036 | year=2008 | journal=SIGMA Symmetry Integrability Geom. Methods Appl. | issue=4 | Paper=036}}</ref>


}}
}}

Revision as of 22:05, 23 September 2013

On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by \[ A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M), \] where $a, b$ are constant.

Examples of conformally invariant operators include:

  • The conformal Laplacian:

\[ L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g, \] where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.

\[ P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q, \] where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor, \[ Q=c_n|Ric_g|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR \] and \[ a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}. \] This is a fourth order operator with leading term $(-\Delta_g)^2$.

  • GJMS operators [3]: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in [4] for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl [5][6] (see also Fefferman-Graham[7] ). The formula are more explicit when they are on the standard spheres.
  • Scattering operators [8], or the conformally invariant fractional powers of the Laplacian [9]: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors [9] reconciled the way of defining $P_\sigma$ in [8] and the localization method of Caffarelli-Silvestre [10] for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.


Special cases:

  • On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.
  • On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator [11] of explicit formula:

\[ P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2}, \] where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$

  • is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,
  • has the eigenfunctions of spherical harmonics, and
  • is the inverse of a spherical Riesz potential.


References

  1. Paneitz, S (1983), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds", preprint 
  2. Paneitz, S (2008), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)", SIGMA Symmetry Integrability Geom. Methods Appl. (4), http://dx.doi.org/10.3842/SIGMA.2008.036 
  3. Graham, C Robin; Jenne, Ralph; Mason, Lionel J; Sparling, George AJ (1992), "Conformally invariant powers of the Laplacian, I: Existence", Journal of the London Mathematical Society 2: 557--565 
  4. Gover, A; Hirachi, Kengo (2004), "Conformally invariant powers of the Laplacian—a complete nonexistence theorem", Journal of the American Mathematical Society 17: 389--405 
  5. Juhl, Andreas, "On the recursive structure of Branson’s Q-curvature", arXiv preprint arXiv:1004.1784 
  6. Juhl, Andreas (2013), "Explicit formulas for GJMS-operators and Q-curvatures", Geometric and Functional Analysis 23: 1278--1370 
  7. Fefferman, Charles; Graham, C (2013), "Juhl’s formulae for GJMS operators and 𝑄-curvatures", Journal of the American Mathematical Society 26: 1191--1207 
  8. 8.0 8.1 Graham, C Robin; Zworski, Maciej (2003), "Scattering matrix in conformal geometry", Inventiones mathematicae 152: 89--118 
  9. 9.0 9.1 Chang, Sun-Yung Alice; González, Maria del Mar (2011), "Fractional Laplacian in conformal geometry", Advances in Mathematics 226: 1410--1432 
  10. Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian", Communications in Partial Differential Equations 32: 1245--1260 
  11. Branson, Thomas P (1987), "Group representations arising from Lorentz conformal geometry", Journal of functional analysis 74: 199--291