1 Viscosity, Reversibillity, Chaotic Hypothesis, smooth,1 in SF, [3, 43], i.e. SF is an invariant surface Fluctuation Theorem and Lyapunov Pairing in F. And the equations of motion that H generates can Giovanni Gallavotti be written (using incompressibility of (q, p) ∈ SF) as: August 22, 2021 p˙ξ = −∂qξQ(q, p)ξ, q˙ξ = pξ (1.2) Abstract: Incompressible fluid equations are studied with UV cut-off and in periodic boundary conditions. where Qξ is q-dependent and quadratic in p, see Ap- Properties of the resulting ODEs holding uniformly in pendix A. the cut-off are considered and, in particular, are conjec- Sincep ˙ξ = ∂tpξ + (pξ · ∂pξ), setting x = qξ,u(x)= pξ tured to be equivalent to properties of other time reversible and P (x)= ∂qξQ(q, p)ξ, the equations become: equations. Reversible equations with the same regulariza- e e tion and describing equivalently the fluid, and the fluc- q˙ξ = pξ, ∂tu(x) + ((u · ∂)u)(x)= −∂P (x) (1.3) tuations of large classes of observables, are examined in the context of the “Chaotic Hypothesis”, “Axiom C” and with ∂ · u = 0, and P as above. The Lagrangian form the “Fluctuation Theorem”. e e of Euler’s equations, Eq.(1.2) or (1.3), will be called E∗. See Appendix A. The above “geodesic” formulation of E, E∗ will be used I. ON THE EQUATIONS to exhibit symmetry properties of Euler’s equation which may be relevant also for the IN (irreversible Navier- The incompressible Euler equation, denoted E, in a Stokes) equations: periodic container T d = [0, 2π]d, d = 2, 3, for a smooth velocity field u(x), x ∈ T d is: ∂tu(x) + ((u · ∂)u)(x)= ν∆u(x) − ∂P (x)+ f(x) (1.4) with the conditionse e ∂ · u = 0, T d u = 0. ˙u(x)=−(u(x) · ∂x)u(x)− ∂xP (x), R (1.1) II. ULTRAVIOLET REGULARIZATION ∂x · u(x)=0e , e dx u(x)= 0 ZT d Here we study the regularized version, see below, of d −1 where P = − i,j=1 ∆ (∂iuj ∂j ui) is the ’pressure’ and E or IN, Eq.(1.1),(1.4), obtained by requiring that the ∆=Laplace operator.P Fourier’s transform uk of u does not vanish only for It is also useful to consider the E equations from the modes k with components ≤ N. “Lagrangian viewpoint”: a configuration of the fluid is We shall focus on properties of the solutions which hold described by assigning the dispacement x = qξ of a fluid uniformly in the cut-off N: the space of such u’s with 0 element, from the reference position ξ ∈ T d, and the divergence (∂ · u = 0) and 0 average ( T d u(x) = 0) will velocityq ˙ξ of the same fluid element. So the state of be denoted CN . R the fluid is (q, ˙q) where q is a smooth map of T d to Therefore the equation in dimension d = 2, 3 is ex- d itself and ˙q is a smooth vector field on T with 0 aver- pressed in terms of complex scalars uβ,k = uβ,−k, β = d age. Denote F the space of the dynamical configurations 1, . , d, k ∈ Z , |kβ| ≤ N: thus the number of real co- (q, ˙q) ∈ Dif(T d) × Lin(T d)= F where Dif(T d) is the ordinates is N = 4N(N + 1) in 2D and N = 2(4N 3 + set of diffeomorphisms of T d and Lin(T d) the space of 6N 2 +3N) in 3D and N will be the dimension of the d the vector fields on T . phase space CN . Actually we concentrate on the subspace of (q, ˙q) ∈ For instance in 3D choose, for each k 6= 0, two unit def (SDif(T d) × SLin(T d)) = SF ⊂F where the evolu- vectors eβ(k)= −eβ(−k), β =1, 2, mutually orthogonal tion of an incompressible fluid takes place: SDif(T d) be- and orthogonal to k; data are combined to form a velocity ing the volume preserving diffeomorphisms and SLin(T d) field: the 0-divergence vector fields. −ik·x u(x)= uke , k = (kβ)β=1,2,3 A (q, ˙q) = {qξ, q˙ξ}ξ∈T d ∈ F should be regarded as a 0<X|k|≤N set of Lagrangian coordinates labeled by ξ ∈ T d. And (2.1) the equations Eq.(1.1) can be derived from a Hamil- uk = iuβ,k eβ(k), k · eβ(k)=0 tonian in canonical coordinates (q, p) ∈ F which is βX=1,2 quadratic in p and which generates motions in F evolv- ing leaving SF invariant. Therefore the motion in F is a with |k| = maxj |kj |,u−k,j = uk,j. “geodesic motion” (i.e. a motion generated by a Hamil- tonian quadratic in the momenta). A key remark is that the motions that follow initial 1 data in SF remain, as long as the evolution is defined and This might be a very short time. 1 2 β1,β2,β e k k e k e k ϕ Define Dk1,k2,k = −( β1 ( 1) · 2)( β2 ( 2) · β( )). In- that vanish on the divergenceless fields ): and in the end −ik·x troduce also forcing f = ifk,βeβ(k)e and vis- its symmetric part is P times the multiplication operator, k,β d d d 2 on 0-divergence fields (ϕj (x)) = ϕ ∈ L2(T )×R , by: cosity in the form −νk uPk. j=1 2def 2 2 2 The IN equations Eq.(1.4) become, if k = k1 +k2 +k3 1 Wi,j (x)= ∂xj ui(x)+ ∂xi uj(x) (2.6) and the sum is restricted to |k|, |k1|, |k2|≤ N: 2 β1,β2,β 2 i.e. P times the operator (Jϕ)i(x)= j Wi,j (x)ϕj (x). u˙ β,k = D uβ ,k uβ ,k − νk uβ,k + fβ,k (2.2) k1,k2,k 1 1 2 2 P βX1,β2 Introducing also viscosity (and forcing, which how- k=k1+k2 ever does not contribute) Eq.(2.6) immediately leads which will define the regularized IN equation. to express the symmetric part of the Jacobian of the The 2D case is similar but simpler: no need for the regularized IN, irreversible Navier-Stokes, as P times ⊥ ν k J = νδi,j ∆+ Wi,j . labels β, and e(k) can be taken ||k|| . i,j d d β1,β2,β2 2 −1 2 The coefficients D can be used to check that if Defining, for d = 2, 3, w(x) = 4d i,j=1 Wi,j (x) k1,k2,k3 ν 2 the inequality J ≤ ν∆+w(x), derived inP [28, 33] for the ν =0, f = 0 then for all u ∈CN : nonregularized IN equation, remains valid for the regu- d d larized one and leads to the bound, [28, 33]: u(x)2dx =0, u(x) · (∂ ∧ u(x))dx = 0 (2.3) dt ZT d dt ZT d Theorem: the sum of the averages of the first p eigen- As is well known, the first of Eq.(2.3) leads to the a priori, values of the (Schr¨odinger operator) νD + w(x) yields an upper bound to the sum of the first p Lyapunov exponents N-independent, bounds for the solutions of the E and IN 4 equations: (of any invariant distribution on F) of the flow St. F ||uX,N (t)||2 ≤ max(E , ( 0 )2), X = E,IN (2.4) 2 0 ν III. REVERSIBLE EQUATIONS X,N def X,N satisfied (for all X) by solutions t → u (t) = St u, The theory of nonequilibrium fluctuations has led to 2 2 in terms of E0 = ||u(0)||2 = β,k |uβ,k| and F0 = ||f||2. studying phenomena via equations considered equivalent From now on the cut-offP N will be kept constant and (at least for some of the purposes of interest) to the “fun- the solution of the equations will be denoted simply Stu damental” ones. dropping the X,N as superscript of the solution map St. Thus new non-Newtonian forces have been added to Only when not clear from the context a superscript E or systems of particles claiming that the values of important IN or a label N will be added to clarify whether reference quantities would have the same values as those implied is made to the evolution, or to its properties, following E by the fundamental equations, even in cases in which the or IN equation with cut-off N. modification was drastic: with the advantage, in several By scaling, the equation can and will be written in a cases, of greatly facilitating simulations, [11, 25, 30]. fully dimensionless form in which ||f||2 = 1. At the same time the idea that modification of the E,N equations would not affect, at least in some important The Jacobian of the Euler flow St u with UV cut- off N is more easily written, without using the Fourier’s cases, most of their predictions arose in other domains: transform representation of u, directly from Eq.(1.1) and, it appeared for instance, in [39], to show that the Navier- see Appendix B, is the sum of the following convolution Stokes (IN above) equation could be modified, into new d d d reversible equations, still remaining consistent with se- operator on (ϕj (x))j=1 = ϕ ∈ L2(T ) × R : lected predictions of the Obukov-Kolmogorov theory. ∂u˙ i(x) In [14, 15] an attempt was presented to link empirical = −Pδ(x − y)∂yj ui(y), i,j =1, .., d, (2.5) equivalence observations to the well established theory of ∂uj(y) the equivalence of ensembles in Statistical Mechanics.