arXiv subjects
Publications and source records attributed to David Ruelle.
We claim that human mathematics is only a limited part of the consequences of the chosen basic axioms. Properly human mathematics varies with time but appears to have universal features which we try to analyze. In particular the functioning of the human brain privileges concept naming and short formulations. This leads to organizing mathematical knowledge structurally. We consider briefly the problem of non-mathematical sciences.
Let $z$ be the activity of point particles described by classical equilibrium statistical mechanics in ${\bf R}^\nu$. The correlation functions $\rho^z(x_1,\dots,x_k)$ denote the probability densities of finding $k$ particles at $x_1,\dots,x_k$. Letting $\phi^z(x_1,\dots,x_k)$ be the cluster functions corresponding to the $\rho^z(x_1,\dots,x_k)/z^k$ we prove identities of the type $$ \phi^{z_0+z'}(x_1,\dots,x_k) $$ $$ =\sum_{n=0}^\infty{z'^n\over n!}\int dx_{k+1}\dots\int dx_{k+n}\,\phi^{z_0}(x_1,\dots,x_{k+n}) $$ It is then non-rigorously argued that, assuming a suitable cluster property (decay of correlations) for a crystal state, the pressure and the translation invariant correlation functions \- $\rho^z(x_1,\dots,x_k)$ are real analytic functions of $z$.
If ${\cal F}$ is a set of subgraphs $F$ of a finite graph $E$ we define a graph-counting polynomial $$ p_{\cal F}(z)=\sum_{F\in{\cal F}}z^{|F|} $$ In the present note we consider oriented graphs and discuss some cases where ${\cal F}$ consists of unbranched subgraphs $E$. We find several situations where something can be said about the location of the zeros of $p_{\cal F}$.
This note presents a non-rigorous study of the linear response for an SRB (or `natural physical') measure $\rho$ of a diffeomorphism $f$ in the presence of tangencies of the stable and unstable manifolds of $\rho$. We propose that generically, if $\rho$ has no zero Lyapunov exponent, if its stable dimension is sufficiently large (greater than 1/2 or perhaps 3/2) and if it is exponentially mixing in a suitable sense, then the following formal expression for the first derivative of $\rho(\phi)$ with respect to $f$ along $X$ is convergent: $$ \Psi(z)=\sum_{n=0}^\infty z^n\int\rho(dx)\,X(x)\cdot\nabla_x(\phi\circ f^n)\qquad{\rm for}\qquad z=1 $$ This suggests that an SRB measure may exist for small perturbations of $f$, with weak differentiability.
In earlier papers we have studied the turbulent flow exponents $\zeta_p$, where $\langle|\Delta{\bf v}|^p\rangle\sim\ell^{\zeta_p}$ is the contribution to the fluid velocity at small scale $\ell$. Using ideas of non-equilibrium statistical mechanics we have found $$ \zeta_p={p\over3}-{1\over\ln\kappa}\ln\Gamma({p\over3}+1) $$ where $1/ln\kappa$ is experimentally $\approx 0.32\pm 0.01$. The purpose of the present note is to propose a somewhat more physical derivation of the formula for $\zeta_p$. We also present an estimate $\approx 100$ for the Reynolds number at the onset of turbulence.
This note presents a minimal approach to the origin of life, following standard ideas. We pay special attention to the point of view of non-equilibrium statistical mechanics, and in particular to detailed balance. As a consequence we propose a characterization of pre-biological states.
Given a system $M$ in a thermal bath we obtain a generalized detailed balance relation for the ratio $r=\pi_\tau(K\to J)/\pi_\tau(J\to K)$ of the transition probabilities $M:J\to K$ and $M:K\to J$ in time $\tau$. We assume an active bath, containing solute molecules in metastable states. These molecules may react with $M$ and the transition $J\to K$ occurs through different channels $\alpha$ involving different reactions with the bath. We find that $r=\sum p^\alpha r^\alpha$, where $p^\alpha$ is the probability that channel $\alpha$ occurs, and $r^\alpha$ depends on the amount of heat (more precisely enthalpy) released to the bath in channel $\alpha$.
This note analyzes the physical basis of J.R. England's paper "Statistical physics of self-replication." [J. Chem. Phys. {\bf 139}, 121923(2013)]. We follow England's use of time-reversal symmetry but replace stochastic by deterministic dynamics, and introduce a definition of metastable states based on equilibrium statistical mechanics. We rederive England's detailed balance relation and obtain another similar relation which appears more natural and remains valid for quantum systems. The detailed balance relations are based on serious physical ideas, and either of them can be used for England's biological discussion. This biological discussion does of course deserve further scrutiny.
The macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow for a suitable mechanical system. Turbulent fluctuations (intermittency) then correspond to thermal fluctuations, and this allows to estimate the exponents tau_p and zeta_p associated with moments of dissipation fluctuations and velocity fluctuations. This approach, initiated in an earlier note, is pursued here more carefully. In particular we derive probability distributions at finite Reynolds number for the dissipation and velocity fluctuations, and the latter permit an interpretation of numerical experiments. Specifically, if p(z)dz is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number increases, log p(z) passes from a concave to a linear then to a convex profile for large z as observed. We show that the central limit theorem applies to the dissipation and velocity distribution functions, so that a logical relation with the lognormal theory of Kolmogorov and Obukhov is established. We find however that the lognormal behavior of the distribution functions fails at large value of the argument, so that a lognormal theory cannot correctly predict the exponents tau_p and zeta_p.
Present day mathematics is a human construct, where computers are used more and more but do not play a creative role. This situation may change however: computers may become creative, and since they function very differently from the human brain they may produce a very different sort of mathematics. We discuss what this post-human mathematics may look like, and the philosophical consequences that this may entail.
We reformulate the problem of hydrodynamic turbulence as a heat flow problem. We obtain thus a prediction $$ \zeta_p={p\over3}-{1\over\ln\kappa}\ln\Gamma({p\over3}+1) $$ for the exponents of the structure functions ($<|\Delta_rv|^p>=r^{\zeta_p}$). The meaning of the adjustable parameter $\kappa$ is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(\ln\kappa)^{-1}=.32\pm.01$ is compatible with experimental data. This agreement lends supports to our physically motivated picture of turbulence.
We consider a class of Ising spin systems on a set \Lambda of sites. The sites are grouped into units with the property that each site belongs to either one or two units, and the total internal energy of the system is the sum of the energies of the individual units, which in turn depend only on the number of up spins in the unit. We show that under suitable conditions on these interactions none of the |\Lambda| Lee-Yang zeros in the complex z = exp{2\beta h} plane, where \beta is the inverse temperature and h the uniform magnetic field, touch the positive real axis, at least for large values of \beta. In some cases one obtains, in an appropriately taken \beta to infinity limit, a gas of hard objects on a set \Lambda'; the fugacity for the limiting system is a rescaling of z and the Lee-Yang zeros of the new partition function also avoid the positive real axis. For certain forms of the energies of the individual units the Lee-Yang zeros of both the finite- and zero-temperature systems lie on the negative real axis for all \beta. One zero-temperature limit of this type, for example, is a monomer-dimer system; our results thus generalize, to finite \beta, a well-known result of Heilmann and Lieb that the Lee-Yang zeros of monomer-dimer systems are real and negative.
Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier's law for heat conduction from the laws of mechanics remains thus a major unsolved problem. In this note we present a deterministic mechanical model of a heat-conducting chain with nontrivial interactions, where kinetic energy fluctuations at the nodes of the chain are removed. In this model the derivation of Fourier's law can proceed rigorously.
Using an extended Lee-Yang theorem and GKS correlation inequalities, we prove, for a class of ferromagnetic multi-spin interactions, that they will have a phase transition(and spontaneous magnetization) if, and only if, the external field $h=0$ (and the temperature is low enough). We also show the absence of phase transitions for some nonferromagnetic interactions. The FKG inequalities are shown to hold for a larger class of multi-spin interactions.
Let $\rho$ be an SRB (or "physical"), measure for the discrete time evolution given by a map $f$, and let $\rho(A)$ denote the expectation value of a smooth function $A$. If $f$ depends on a parameter, the derivative $\delta\rho(A)$ of $\rho(A)$ with respect to the parameter is formally given by the value of the so-called susceptibility function $\Psi(z)$ at $z=1$. When $f$ is a uniformly hyperbolic diffeomorphism, it has been proved that the power series $\Psi(z)$ has a radius of convergence $r(\Psi)>1$, and that $\delta\rho(A)=\Psi(1)$, but it is known that $r(\Psi)<1$ in some other cases. One reason why $f$ may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for $(f,\rho)$. The present paper gives a crude, nonrigorous, analysis of this situation in terms of the Hausdorff dimension $d$ of $\rho$ in the stable direction. We find that the tangencies produce singularities of $\Psi(z)$ for $|z|<1$ if $d<1/2$, but only for $|z|>1$ if $d>1/2$. In particular, if $d>1/2$ we may hope that $\Psi(1)$ makes sense, and the derivative $\delta\rho(A)=\Psi(1)$ has thus a chance to be defined
The classical theory of linear response applies to statistical mechanics close to equilibrium. Away from equilibrium, one may describe the microscopic time evolution by a general differentiable dynamical system, identify nonequilibrium steady states (NESS), and study how these vary under perturbations of the dynamics. Remarkably, it turns out that for uniformly hyperbolic dynamical systems (those satisfying the "chaotic hypothesis"), the linear response away from equilibrium is very similar to the linear response close to equilibrium: the Kramers-Kronig dispersion relations hold, and the fluctuation-dispersion theorem survives in a modified form (which takes into account the oscillations around the "attractor" corresponding to the NESS). If the chaotic hypothesis does not hold, two new phenomena may arise. The first is a violation of linear response in the sense that the NESS does not depend differentiably on parameters (but this nondifferentiability may be hard to see experimentally). The second phenomenon is a violation of the dispersion relations: the susceptibility has singularities in the upper half complex plane. These "acausal" singularities are actually due to "energy nonconservation": for a small periodic perturbation of the system, the amplitude of the linear response is arbitrarily large. This means that the NESS of the dynamical system under study is not "inert" but can give energy to the outside world. An "active" NESS of this sort is very different from an equilibrium state, and it would be interesting to see what happens for active states to the Gallavotti-Cohen fluctuation theorem.
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 $Ψ(z_1,...,z_n)$ which we call Lee-Yang polynomials (they do not vanish when $|z_1|,...,|z_n|<1$ or $|z_1|,...,|z_n|>1$). We characterize the Lee-Yang polynomials $Ψ$ in $n+1$ variables in terms of polynomials $Φ$ in $n$ variables (those such that $Φ(z_1,...,z_n)\ne0$ 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 $Ψ$ are temperature dependent partition functions, we find that those $Ψ$ which are Lee-Yang polynomials for all temperatures are precisely the polynomials with pair interactions originally considered by Lee and Yang.
By using a suitable Banach space on which we let the transfer operator act, we make a detailed study of the ergodic theory of a unimodal map $f$ of the interval in the Misiurewicz case. We show in particular that the absolutely continuous invariant measure $ρ$ can be written as the sum of 1/square root spikes along the critical orbit, plus a continuous background. We conclude by a discussion of the sense in which the map $f\mapstoρ$ may be differentiable.