Showing papers by "David Ruelle published in 2008"

Differentiation of SRB states for hyperbolic flows

Abstract: Let the ${\cal C}^3$ vector field ${\cal X}+aX$ on $M$ define a flow $(f^t_a)$ with an Axiom A attractor $\Lambda_a$ depending continuously on $a\in(-\epsilon,\epsilon)$. Let $\rho_a$ be the SRB measure on $\Lambda_a$ for $(f^t_a)$. If $A\in{\cal C}^2(M)$, then $a\mapsto\rho_a(A)$ is ${\cal C}^1$ on $(-\epsilon,\epsilon)$ and $d\rho_a(A)/da$ is the limit when $\omega\to0$ with ${\rm Im}\omega>0$ of $$ \int_0^\infty e^{i\omega t}dt \int\rho_a(dx) X(x)\cdot abla_x(A\circ f_a^t) $$

What physical quantities make sense in nonequilibrium statistical mechanics

Abstract: Statistical mechanics away from equilibrium is in a formative stage, where general concepts slowly emerge. We restrict ourselves here to the study of transport phenomena, especially heat transport, and consider several well-studied classes of systems: finite systems with isokinetic thermostats, infinite classical Hamiltonian systems, infinite quantum spin systems. For those classes we discuss how various physical quantities can be defined. Our review however leaves several basic questions quite open. * Math. Dept., Rutgers University, and IHES, 91440 Bures sur Yvette, France. email: ruelle@ihes.fr

Characterization of Lee-Yang polynomials

Abstract: The Lee-Yang circle theorem describes complex polynomials of degree $n$ in $z$ with all their zeros on the unit circle $|z|=1$. These polynomials are obtained by taking $z_1=...=z_n=z$ in certain multiaffine polynomials $\Psi(z_1,...,z_n)$ which we call Lee-Yang polynomials (they do not vanish when $|z_1|,...,|z_n| 1$). We characterize the Lee-Yang polynomials $\Psi$ in $n+1$ variables in terms of polynomials $\Phi$ in $n$ variables (those such that $\Phi(z_1,...,z_n) e0$ when $|z_1|,...,|z_n|<1$). This characterization gives us a good understanding of Lee-Yang polynomials and allows us to exhibit some new examples. In the physical situation where the $\Psi$ are temperature dependent partition functions, we find that those $\Psi$ which are Lee-Yang polynomials for all temperatures are precisely the polynomials with pair interactions originally considered by Lee and Yang.