A Theory of Hydrodynamic Turbulence Based on Non-equilibrium Statistical Mechanics

In earlier papers, we have studied the turbulent flow exponents \(\zeta _p\), where \(\langle |\Delta \mathbf{v}|^p\rangle \sim \ell ^{\zeta _p}\) and \(\Delta \mathbf{v}\) is the contribution to the fluid velocity at small scale \(\ell \). Using ideas of non-equilibrium statistical mechanics we have found

$$\begin{aligned} \zeta _p={p\over 3}-{1\over \ln \kappa }\ln \Gamma \left( {p\over 3}+1\right) \end{aligned}$$

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.

     

Non-Equilibrium Statistical Mechanics of Turbulence

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 (Ruelle, PNAS 109:20344–20346, 2012). 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 (Ruelle, 2012), 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 (Schumacher, Preprint, 2014). Specifically, if \(p(z)dz\) is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number \(\mathcal{R}\) increases, \(\ln p(z)\) passes from a concave to a linear then to a convex profile for large \(z\) as observed in (Schumacher, 2014). 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 (J. Fluid Mech. 13:82–85, 1962) 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 \(\zeta _p\) .     

Entropy Production in Quantum Spin Systems

We consider a quantum spin system consisting of a finite subsystem connected to infinite reservoirs at different temperatures. In this setup we define nonequilibrium steady states and prove that the rate of entropy production in such states is nonnegative. Received: 7 June 2000 / Accepted: 5 November 2000     

Spectral properties of a class of operators associated with conformal maps in two dimensions

Iff is a rational map of the Riemann sphere, define the transfer operator by Let also be the Banach space of functions for which the second derivatives are measures. Ifg andg satisfies a simple integrability condition (implying thatg vanishes at critical points and multiple poles off) then is a bounded linear operator on . The essential spectral radius of can be estimated and, under suitable conditions, proved to be strictly less than the spectral radius. Similar estimates for more general operators are also obtained.     

Functional equation for dynamical zeta functions of Milnor-Thurston type

A Milnor-Thurston type dynamical zeta function _L (Z) is associated with a family of maps of the interval (–1, 1). Changing the direction of time produces a new zeta function _L (Z). These zeta functions satisfy a functional equation _L (Z) _L (Z)= ₀(Z) (where amounts to sign changes and, generically,₀1). The functional equation has non-trivial implications for the analytic properties of _L (Z).     

Positivity of entropy production in the presence of a random thermostat

We study nonequilibrium statistical mechanics in the presence of a thermostat acting by random forces, and propose a formula for the rate of entropy productione() in a state . When is a natural nonequilibrium steady state we show thate()0, and sometimes we can provee()>0.     

Modular invariants for affine\widehat{SU}(3) theories at prime heights

A proof is given for the existence of two and only two modular invariant partition functions in affine theories at heightsn=k+3 which are prime numbers. Arithmetic properties of the ring of algebraic integers () which is related toSU(3) weights are extensively used.Chercheur IISN, Address after October 1: Dublin Institute for Advanced Studies, Dublin, IrelandChercheur IISN     

Occurrence of strange AxiomA attractors near quasi periodic flows onT m,m≧3

It is shown that by a smallC ² (resp.C ^∞) perturbation of a quasiperiodic flow on the 3-torus (resp. them-torus,m>3), one can produce strange AxiomA attractors. Ancillary results and physical interpretation are also discussed.