Equilibrium Statistical Mechanics for the Truncated Burgers-Hopf Equations

Transcription

1 Equilibrium Statistical Mechanics for the Truncated Burgers-Hopf Equations Di Qi, and Andrew J. Majda Courant Institute of Mathematical Sciences Fall 2016 Advanced Topics in Applied Math Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

2 Introduction Consider large non-linear systems of ordinary differential equations dx dt = F ( ) X, X R N, F = (F 1,, F N ), N 1, X t=0 = X 0. Common features of these systems: The dynamics are highly chaotic, thus a single trajectory is highly unpredictable; Observations as well as physical and numerical experiments indicate the existence of coherent patterns; It makes sense to study the ensemble behaviour of trajectories rather than a single trajectory. Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

3 Overview about equilibrium statistical mechanics We study the ensembles of solutions in probability measures or statistical solutions on the phase space R N. The Liouville property is equivalent to volume preserving flow map. If an ensemble of initial data in the phase space is stretched in some direction, it must be compensated for by squeezing in some other direction under the flow map and vice versa. The existence of conserved quantities offer constraints on the probability distributions. The evolution of the probability measures on phase space enables us to define the associated information-theoretic entropy which sets the stage for the maximum entropy principle. The most probable states (probability distributions), or the Gibbs measure is obtained with the Liouville property and conserved quantities. The Gibbs measure turns out to be an invariant measure of the flow maps, or a stationary statistical solution; The dynamics is chaotic and instability is abundant so that there is some sort of ergodicity of the system 1. 1 Majda & Tong, Ergodicity of truncated stochastic Navier-Stokes with deterministic forcing and dispersion, JNS, Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

4 Lecture 8: Equilibrium statistical mechanics for the truncated Burgers-Hopf equations 1 Review: Introduction to statistical mechanics for ODEs 2 Statistical mechanics for the truncated Burgers-Hopf equations The truncated Burgers-Hopf system The Gibbs measure and the prediction of equipartion of energy Numerical evidence of the validity of the statistical theory TBH as a model with statistical features with atmosphere 3 Statistically relevant conserved quantities for the truncated Burgers-Hopf equation Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

5 Outline 1 Review: Introduction to statistical mechanics for ODEs 2 Statistical mechanics for the truncated Burgers-Hopf equations The truncated Burgers-Hopf system The Gibbs measure and the prediction of equipartion of energy Numerical evidence of the validity of the statistical theory TBH as a model with statistical features with atmosphere 3 Statistically relevant conserved quantities for the truncated Burgers-Hopf equation Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

6 DiProof: Qi, and Andrew The J. Majda proof (CIMS) follows fromtruncated the calculus Burgers-Hopf identity Equations (cf. Majda and Bertozzi, Oct. 27, / 36 The WeLiouville first introduce property the Liouville property. Definition 7.1 (Liouville property) A vector field F X is said to satisfy the Liouville property if it is divergence free, i.e. div X F = N F j = 0 (7.2) X j j=1 An important consequence of the Liouville property is that it implies the flow map associated with (7.1) is volume preserving or measure preserving on the 222 Equilibrium statistical mechanics for systems phase space. For this purpose we define the flow map t of ODEs X t 0, t N N associated The divergence with the freefinite-dimensional nature of the vector ODEfield system F gives (7.1) anbyimportant property to the flow map t d X, i.e. the flow map t X is volume preserving or measure preserving on the phase dt t space. X = F This t property X can be t X shown t=0 = exactly X 0 in the same(7.3) way as we prove that the flow map of an incompressible fluid is volume preserving. Proposition 7.1 t X is volume preserving or measure preserving on the phase space, i.e. J t def ( ) = det X t X 1 (7.4) for all time t 0.

7 DiProof: Qi, and Andrew The J. Majda proof (CIMS) follows fromtruncated the calculus Burgers-Hopf identity Equations (cf. Majda and Bertozzi, Oct. 27, / 36 The WeLiouville first introduce property the Liouville property. Definition 7.1 (Liouville property) A vector field F X is said to satisfy the Liouville property if it is divergence free, i.e. div X F = N F j = 0 (7.2) X j j=1 An important consequence of the Liouville property is that it implies the flow map associated with (7.1) is volume preserving or measure preserving on the 222 Equilibrium statistical mechanics for systems phase space. For this purpose we define the flow map t of ODEs X t 0, t N N associated The divergence with the freefinite-dimensional nature of the vector ODEfield system F gives (7.1) anbyimportant property to the flow map t d X, i.e. the flow map t X is volume preserving or measure preserving on the phase dt t space. X = F This t property X can be t X shown t=0 = exactly X 0 in the same(7.3) way as we prove that the flow map of an incompressible fluid is volume preserving. Proposition 7.1 t X is volume preserving or measure preserving on the phase space, i.e. J t def ( ) = det X t X 1 (7.4) for all time t 0.

8 0 t + F X p = 0 Utilizing the measure preserving ( ) property of the flow map t X we may verify This proves Liouville s theorem. Define that athis probability definition density indeed p yields X, t by a the probability pull-backmeasure. of the initial In fact, probability we candensity prove ( ) that p 0 X Tothe seeone that parameter p t is afamily probability of probability density measures function, p t we notice satisfies that thep Liouville X t is non-negative equation, or, by equivalently, equation (7.6) ( thatand the ) the probability assumption ( (Φ p X, t t = p ) density ( )) functions are transported 1 that p 0 X 0 is a probability density by the vector field F., function. Next we set = N in (7.10) and we have ) Proposition 7.2 p X t = p p 0 X ( t d t X 1 = X pis 0 transported X d X = 1 by the vector field (7.12) F, i.e. it satisfies the Liouville s N equation N and therefore p X t defines a p probability t + density for t>0. This concludes the proof of the proposition. F X p = 0 (7.7) Notice that the Liouville equation (7.7) is a linear equation in p; it then follows that and any hence function p X t of is p a also probability satisfies density the Liouville function equation. for all time. Thus we have: Evolution of probability measures Corollary 7.1 Let G p be any function of the probability density p. Then G p + F t X G p = 0 (7.13) 224 Equilibrium statistical mechanics for systems of ODEs i.e. G p satisfies Liouville s equation. This further implies that d G p X t = 0 (7.14) dt N Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

9 7.2 Introduction to statistical mechanics for ODEs 223 Proof: From the transport theorem (cf. Majda and Bertozzi, 2001 or Chorin and Marsden, 1993), we know that for any function f X t in phase space ( ) f f X t d X = t t + div f F d X (7.8) t t and since F X t is divergence free, this equation reduces to ( ) f f X t d X = t t + F X f d X (7.9) t t Next we let f X t = p X t in the last equation. From Definition 7.6 on the pull-back probability density function p, we have ) p X t d X = p 0 ( t 1 X d X t = t p 0 Y det ( ) Y t Y d Y = p 0 Y d Y (7.10) where we made the change of variables X = t Y in the equation, and utilized the fact that the Jacobian determinant of the flow map is identically equal to one. This implies that the left-hand side of (7.9) is identically zero, and thus Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

10 7.2.3 Conserved quantities and their ensemble averages Conserved quantities We now introduce Conserved the secondquantities ingredientand for doing their ensemble equilibriumaverages statistical mechanics for nowodes. introduce We assume the second thatingredient the system for (7.1) doing possesses equilibrium L conserved statisticalquantities mechan- We ics E l for X t, ODEs. i.e. We assume that the system (7.1) possesses L conserved quantities E l X t, i.e. E l X t = E l X 0 1 l L (7.17) E l X t = E l X 0 1 l L (7.17) These conserved quantities could be the truncated energy, enstrophy, or higher These moments conserved for the truncated quantitiesquasi-geostrophic could be the truncated equations, energy, or the enstrophy, truncated orenergy higher moments and For linear the for momentum truncated quasi-geostrophic the truncated for thequasi-geostrophic truncated Burgers Hopf equations: truncated equations, equation, or theor energy, truncated the Hamiltonian enstrophy, energy or higher moments; and linear angular momentum momentum for for the truncated the point-vortex Burgers Hopf systemequation, etc. The or ensemble the Hamiltonian average For the truncated Burgers-Hopf equation: the truncated energy, linear and of these angular conserved momentum quantities for the with point-vortex respect to system a probability etc. The density ensemble function average p is momentum, or Hamiltonian. of defined theseas conserved quantities with respect to a probability density function p is defined as E l = E l p E l X p X d X 1 l L (7.18) E l = E l p E l X p X d X 1 l L (7.18) We naturally expect that these ensemble N averages are conserved in time. Indeed, We we have naturally the following expect that result: these ensemble averages are conserved in time. Indeed, we have the following result: Proposition 7.3 Proposition 7.3 E l p t = E l p0 for all t (7.19) E l p t = E l p0 for all t (7.19) Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

11 the definition of information theoretic entropy on the real line from Section 6.4 to this case of an arbitrary Euclidean space N. We define the ( Shannon ) entropy The forshannon the probability entropy density S for the function probability p X density on N function that is absolutely p X on Rcontinuous N is absolutely with respect continuous to the Lebesgue with respect measure to the by Lebesgue measure by p = p X ln p X d X (7.20) N Maximum entropy principle Note that the Shannon entropy is exactly identical with the Boltzmann entropy in the statistical mechanics of gas particles. Furthermore, by setting G p = p ln p in Corollary 7.1, we can see that the Shannon entropy is conserved in time p t = p 0 (7.21) The question now is how to pick our probability measure p with the least bias maximum for doing future entropy measurements. principle We invoke the maximum entropy principle and claim that the probability density S (p function ) = maxp with the least bias for conducting further measurements should be the one that S (p) maximizes, p C the Shannon entropy (7.20), subject to the subject constraint to theset constraint of measurements of measurements (conserved quantities) as defined through { (7.18). More precisely, let { ( ) ˆ ( ) } } C = p X 0, p X dx = p X 0 p X d X = 1, E l = E l, 1 l L. R N = 1 E l p = E l 1 l L (7.22) N Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

12 Gibbs measure in the 1 + most ln p = probable 0 + state L l E l X (7.26) Variational derivative with respect to the probability density function where 0 is the Lagrange multiplier for the constraint that δs δe ( ) p must be a probability density function, and l l is = the (1 Lagrange + ln p), multiplier δp δp = for E l E X l for. each l respectively. Equation (7.26) can be rewritten as ( ) L p X = c exp l E l X or equivalently, we may write the most probable probability density function in the form analogous to the Gibbs measure in statistical mechanics for gas particle systems ( ) L p X = X = C 1 exp l E l X (7.27) provided that the constraints are normalizable, i.e. ( ) L C = exp l E l X d X < (7.28) N and the l s are the Lagrange multipliers so that satisfies (7.18). l=1 Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36 l=1 l=1 l=1

13 7.2 Introduction to statistical J mechanics for ODEs 227 F X G E 1 E J = F X E G j = 0 (7.30) Proposition 7.4 Let E j 1 j J be conserved quantities E j=1 j of the ODE (7.1). For any smooth function G E 1 E J, G E 1 E J is a steady state solution Finally setting G = in (7.30) we deduce (7.29). This completes the proof of to the Liouville equation. In particular, the most probable probability density the proposition. function given by (7.27) is a steady state solution to the Liouville equation, i.e. An important consequence of a probability F X density function satisfying the steady = 0 (7.29) state Liouville equation is that the associated probability measure is an invariant Proof: probability Since measure E j X t on is the conserved phase space. in time, we have Definition 7.2 (Invariant 0 = d measure) dt E j X t = E A measure on N is said to be invariant j under the flow map t if t + F X E j = F X E j for all j Thus t 1 = (7.31) J for all (measurable) F set X G E 1 E N and J all = time F t. X E G j = 0 (7.30) E j=1 j An easy consequence of this definition and proposition 7.4 is Finally setting G = in (7.30) we deduce (7.29). This completes the proof of the Corollary proposition. 7.2 The Gibbs measure is an invariant measure of the dynamical system (7.1), i.e. it is a stationary statistical solution to (7.1). An important consequence of a probability density function satisfying the steady state Proof: Liouville The proof equation is a straightforward is that the associated calculation. probability For any measure (measurable) is an set invariant, we have Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

14 dictions of behavior of ensemble averages of solutions to the ODE system (7.1). The natural question then is if such a theory is valid. In other words, we want to know if a typical ensemble of solutions does behave as predicted by the statistical theory. Taking the ensemble average of exact solutions of (7.1) analytically is an extremely Taking thechallenging ensemble average problem. of Aexact practical solutions approach analytically is to is utilize an extremely numerical simulations. challenging Theproblem; obvious direct numerical approach is to simulate a very large ensemble Directof numerical trajectories approach with initial to simulate distribution a very satisfying large ensemble the Gibbs of trajectories measure and is thenvery takecostly their average in realistic at asystems; later (large) time. This is usually very costly in realisticinstead, systemswe forusually the atmosphere, assume that andthe ocean Gibbsand measure such ensemble is ergodicpredictions so that thewill be studied spatial in average Chapter is equivalent 15. Instead, towe theusually time average assume that the Gibbs measure is ergodic so that the spatial average is equivalent to the time average, i.e. g X = g X N 1 T0 +T X d X = lim g X t dt (7.32) T T T 0 Ergodicity and time averaging for almost all trajectories X t. Such an ergodicity assumption allows us to verify the statistical prediction utilizing the long time average of one typical trajectory A simple example violating the Liouville property Clearly, the Liouville property in (7.2) for the vector field F is central for the Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

15 p t + div X Fp t = 0 Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36 Clearly, the Liouville property in (7.2) for the vector field F is central for the above theory. In order to appreciate the importance of this assumption, we present a dynamical system, which has a family of conserved quantities, but violates the The Liouville Liouville property property (7.2) for and thehas vector no invariant field F is measure centralthat for the is absolutely above theory. continuous with respect to the Lebesgue measure. Let X = X 1 X 2 T and F T = X 1 X = X 1 X 2 X1 2, i.e. consider the following dynamical system A simple example violating the Liouville property dx 1 dx = X dt 1 X 2 2 = X1 2 dt 7.3 Truncated Burgers Hopf equations 229 The system has the family of conserved quantities E X X for an arbitrary smooth function, E s. Also, the vector field F violates the Liouville property since div X F = X 2 0 Statistical solutions of this equation satisfy the Fokker Planck equation

16 p t Statistical solutions of this equation t + div satisfy X Fp the t = Fokker Planck 0 equation and invariant measures like the p t Gibbs t + div measure, X Fp t = which 0 are functions of the conserved quantities, necessarily are steady solutions of this equations, i.e. and invariant measures like the Gibbs measure, which are functions of the conserved quantities, necessarily are steady div X Fp solutions t = 0 of this equations, i.e. Now it is straightforward to verify that divx for any smooth conserved quantity E X X Fp t = 0 div Now it is straightforward to verify that X FE 0 for any smooth conserved quantity E X X so that none of the non-trivial functions divx of the conserved quantities defines an FE 0 invariant measure. so In that fact, none it isof easy thetonon-trivial see that all functions solutionsofconverge the conserved to the vertical quantities axis defines (X 1 = 0) an invariant and thus all measure. invariant measures must be supported on the vertical axis, and hence not Inabsolutely fact, it is easy continuous to see that withall respect solutions to the converge Lebesgue to the measure. vertical Indeed, axis (Xwe 1 = can 0) All solutions converge to the vertical axis (X and showthus that all all invariant measuresmust for this be supported dynamical 1 = 0), and all invariant measures onsystem the vertical must be axis, given and in hence the must be supported on the vertical axis not formabsolutely of continuous with respect to the Lebesgue measure. Indeed, we can show that all invariant X measures 1 Xfor 2 thisfor dynamical any PM system must be given in the form of and thus it solves the Fokker Planck equation in the sense of distribution and is not absolutely continuous X 1 with X respect 2 to for the any Lebesgue PM measure. and thus it solves the Fokker Planck equation in the sense of distribution and is Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

17 p t Statistical solutions of this equation t + div satisfy X Fp the t = Fokker Planck 0 equation and invariant measures like the p t Gibbs t + div measure, X Fp t = which 0 are functions of the conserved quantities, necessarily are steady solutions of this equations, i.e. and invariant measures like the Gibbs measure, which are functions of the conserved quantities, necessarily are steady div X Fp solutions t = 0 of this equations, i.e. Now it is straightforward to verify that divx for any smooth conserved quantity E X X Fp t = 0 div Now it is straightforward to verify that X FE 0 for any smooth conserved quantity E X X so that none of the non-trivial functions divx of the conserved quantities defines an FE 0 invariant measure. so In that fact, none it isof easy thetonon-trivial see that all functions solutionsofconverge the conserved to the vertical quantities axis defines (X 1 = 0) an invariant and thus all measure. invariant measures must be supported on the vertical axis, and hence not Inabsolutely fact, it is easy continuous to see that withall respect solutions to the converge Lebesgue to the measure. vertical Indeed, axis (Xwe 1 = can 0) All solutions converge to the vertical axis (X and showthus that all all invariant measuresmust for this be supported dynamical 1 = 0), and all invariant measures onsystem the vertical must be axis, given and in hence the must be supported on the vertical axis not formabsolutely of continuous with respect to the Lebesgue measure. Indeed, we can show that all invariant X measures 1 Xfor 2 thisfor dynamical any PM system must be given in the form of and thus it solves the Fokker Planck equation in the sense of distribution and is not absolutely continuous X 1 with X respect 2 to for the any Lebesgue PM measure. and thus it solves the Fokker Planck equation in the sense of distribution and is Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

18 Outline 1 Review: Introduction to statistical mechanics for ODEs 2 Statistical mechanics for the truncated Burgers-Hopf equations The truncated Burgers-Hopf system The Gibbs measure and the prediction of equipartion of energy Numerical evidence of the validity of the statistical theory TBH as a model with statistical features with atmosphere 3 Statistically relevant conserved quantities for the truncated Burgers-Hopf equation Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

19 The inviscid Burgers-Hopf equation Since the atmosphere/ocean dynamics is extremely complicated, it makes sense to study simplified models that capture some specific features of the atmosphere/ocean dynamics. The inviscid Burgers-Hopf equation u t + 1 ( u 2 ) 2 x = 0, is one of the extensively studied models in applied mathematics. One of the prominent features of this equation is the formation and propagation of shocks (discontinuity) from smooth initial data. Our interest here is models that capture certain salient features of the atmosphere/ocean dynamics; Consider spectral truncations of the Burgers-Hopf equation which conserves energy; Such approximation would be disastrous since shock waves form and dissipate energy. Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

20 The truncated Burgers-Hopf systems Suppose the Burgers-Hopf equation is equipped with periodic boundary condition X 1 (uλ )t + PΛ uλ2 x = 0, uλ PΛ u = u k (t) e ikx. 2 k Λ The equation can be written equivalently for the amplitudes u k with k Λ X ik d u k = u p u q. dt 2 k=p+q, p, q Λ Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

21 non-linear ordinary differential equations for the complex amplitudes û k t with k d dt ûk = ik û 2 p û q (7.37) Conservation of momentum and energy k=p+q p q It is elementary to show that solutions of the equations in either (7.35) or (7.37) conserve both momentum and energy, i.e. M = u t =û 0 (7.38) and 7.3 Truncated Burgers Hopf equations 231 E = 1 u 2 2 dx = 1 2 û û k 2 (7.39) are constant in time for solutions of (7.35) or (7.37). The proof for conservation of energy is as follows u 2 t = 1 ( ) u 2 P u 2 x = 1 ( ) u 2 u 2 (u x = 1 ) 3 3 x = 0 The momentum constraint in (7.38) is associated with trivial dynamical behavior and thus after a Galilean transformation, we have M = 0, so that û 0 t 0 in the formula for the energy, E, in (7.39). Also, all of the sums in (7.37) involve only Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36 k=1

22 The quantity, H, is actually the Hamiltonian for the truncated Burgers Hopf system (see Abramov et al., 2003). It is well known that non-trivial smooth solutions of (7.33) develop discontinuities in finite time and thus exhibit a transfer of energy from large scales to small scales. For functions with û t identically zero instantaneously for k> Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, 2016, 20 / 36 of energy is as follows u 2 t = 1 ( ) u 2 P u 2 x = 1 ( ) u 2 u 2 (u x = 1 ) 3 3 x = 0 Conservation of Hamiltonian The momentum constraint in (7.38) is associated with trivial dynamical behavior and thus after a Galilean transformation, we have M = 0, so that û 0 t 0 in the formula for the energy, E, in (7.39). Also, all of the sums in (7.37) involve only k with 1 k. In addition to the conserved quantities in (7.38) and (7.39), it is easy to check that the third moment H = is also conserved. Indeed d u 3 dt dx = 3 u 2 u t dx = 3 2 = 3 2 u 3 dx (7.40) P u 2 P ( ) u 2 x = 0 u 2 P ( ) u 2 x

23 232 Equilibrium statistical mechanics for systems of ODEs Checking the Liouville property form as in (7.1). Let û k 1 k be the defining modes for the equations in Let(7.37), û k, 1 and k let Λ be the defining modes û k = a k + ib k 1 k (7.41) with a k and b k being the real and imaginary part of û k respectively. Thus we have N = 2 number of unknowns. Let X = a 1 b 1 a b X 2k 1 = a k X 2k = b k 1 k (7.42) Equation (7.37) can then be written in a compact form d X dt = F X with the vector field F satisfying the property X 0 = X 0 F 2k 1 X = k X 2k X 4k 1 X 2k 1 X 4k + F 2k 1 X 1 X 2k 2 X 2k+1 X N F 2k X = k X 2k 1 X 4k 1 + X 2k X 4k + F 2k X 1 X 2k 2 X 2k+1 X N (7.43) Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

24 where Var denotes the variance. The canonical Gibbs ensemble predicts a zero mean state and a spectrum with equi-partition of energy in all modes according to (7.44). Next we ask whether the same procedure would work for the conserved quantity given by H = u 3 dx from (7.40). This is an odd function of u and grows cubically at infinity so that the associated Gibbs measure is infinite. Thus, this Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / The Gibbs measure and the prediction of equipartition of energy The Gibbs measure from the energy First, assume that the momentum, M 0 = 0, and that the energy E in (7.39) is the relevant statistical quantity for further measurements. Combining the Liouville property together with the conserved energy and the general framework from Assume that the momentum, M 0 = 0, the most probable distribution, or the Section 7.2 we deduce from (7.27) that the most probable distribution, or the canonical Gibbs measure is given by canonical Gibbs measure, is given by ( ) = C 1 exp û 2 k 2 > 0 (7.44) k=1 This is a product of identical Gaussian measures. For a given value of the mean energy E, the Lagrange multiplier is given by, thanks to (7.39) = E and Var a k = Var b k = 1 = E (7.45) 7.3 Truncated Burgers Hopf equations 233

25 Gibbs measure from the Hamiltonian 7.3 Truncated Burgers Hopf equations 233 where Var denotes the variance. The canonical Gibbs ensemble predicts a zero mean state and a spectrum with equi-partition of energy in all modes according to (7.44). Next we ask whether the same procedure would work for the conserved quantity given by H = u 3 dx from (7.40). This is an odd function of u and grows cubically at infinity so that the associated Gibbs measure is infinite. Thus, this conserved quantity provides an example which is not statistically normalizable so that the intergal in (7.28) is infinite even though it is the Hamiltonian for this system. We discuss the statistical relevance of this conserved quantity for the dynamics briefly later in this book (see Abramov et al., 2003) Numerical evidence of the validity of the statistical theory We show below that the statistical prediction (7.44) of equi-partition of energy in all modes is satisfied with surprising accuracy for of moderate size. For all the numerical results reported here (Majda and Timofeyev, 2000, 2002), a pseudo-spectral method of spatial integration combined with fourth-order Runge Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

26 Kutta time stepping is utilized for (7.37). We can anticipate from (7.44), which Numerical predicts energy set-up equi-partition, for the thattruncated in dynamic simulations Burgers-Hopf both the high equation and low spatial wave numbers are equally important, so increased spatial resolution in the pseudo-spectral algorithm is needed. This is achieved by increasing the length of We the show array the containing statistical the prediction discrete of Fourier equi-partition coefficients of energy by adding in all zeros modes for for wave Λ of moderate numbers size. k with <k and then performing the discrete Fourier transform to Athepseudo-spectral x-space on thismethod bigger array of spatial for pseudo-spectral integration iscomputations. used; The choice Fourth-order 4 works well Runge-Kutta in practicetime and is stepping utilizedis below. utilized Thewith typical typical time time step for stepthe Runge Kutta t = 2 10 time 4 ; integrator is t = , with the necessity to use smaller time The steps initial foraveraging larger values value, of the T 0 total = 100, energy and E averaging (smaller window, and/or T larger = 5000, ). Inare allutilized. simulations presented below the energy computed from (7.39) is conserved within 10 4 %, a relative error of All statistical quantities are computed as time averages, i.e. we assume that the Gibbs measure (7.44) is ergodic and hence (7.32) applies. In particular, the energy in the kth mode is computed by û k 2 = 1 T T+T 0 T 0 û k t 2 dt (7.46) In the simulation presented here, the initial averaging value, T 0 = 100, and averaging window, T = 5000, were utilized. This is a severe test because only the micro-canonical statistics for an individual solution of (7.37) are utilized rather Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

27 to satisfy the energy constraint given by E. A wide range of simulations of (7.37) have been performed in several parameter regimes. Here we discuss the case = 10 and = 100. Truncations varying both and (Majda and Timofeyev, 2000, 2002) exhibit qualitatively similar behavior and we focus our attention on the numerical simulations with = 10, Di = Qi, 100. and Andrew Many J. Majda more (CIMS) results of this Truncated type Burgers-Hopf can be Equations found in the two papers Oct. mentioned 27, / 36 Numerical set-up for the truncated Burgers-Hopf equation Here we discuss the case with θ = 10, Λ = Equilibrium statistical mechanics for systems of ODEs The initial data are selected at random with Fourier coefficients u k,1 k< 15 sampled from a Gaussian distribution with mean zero and variance The tail Fourier coefficients with 15 k are initialized with random phases and equal amplitudes ( ) u k 2 = 1 16 E u 15 j 2 15 k j=1

28 Figure 7.1 Galerkin truncation with = 10 = 100. Energy spectrum; circles Numerics, solid line analytical predictions. Note the small vertical scale consistent with errors of at most a few percent. Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36 The energy spectrum for the real and imaginary parts of the Fourier modes is presented in the top and the bottom parts of Figure 7.1, respectively. The Energy straight spectrum lines in Figure for the 7.1 correspond real and to the imaginary theoretically parts predictedof value the Fourier modes

29 To check the Gaussianity of the dynamics we compute the relative error between the analytical predictions in (7.47) and numerical estimates given by Rel. Err. 4th Moment = Reû k Rel. Err. 6th Moment = Reû k (7.48) Rel. Err. 8th Moment = Reû k Truncated Burgers Hopf equations 235 In Figure 7.2, the relative error in (7.48) for the fourth, sixth, and eighth moments 0.1 is computed as a function of the wave number from the simulations of (7.37) with = 10 and = 100. Relative errors in the fourth moment prediction (the 0.05 top part of Figure 7.2) are less than 2% for almost all wave numbers and never exceed 3.5%, with 0 the largest errors for the low wave numbers. For the sixth moments (the middle part of Figure 7.2) the relative errors are less than 5% for 0.1 most of the wave numbers and do not exceed 9% overall. For the eighth moment (the bottom part 0.05 of Figure 7.2) the relative errors are about 7 8% for most of the Fourier modes. Thus, the higher-order statistics agree with the predictions of the invariant measure 0 in 10 (7.44), 20 and 30 the40 Gaussianity of70 the dynamics of 100the equations in (7.37) is confirmed with surprising accuracy Truncated Burgers Hopf equation as a model with statistical features in 0 common with atmosphere In this section, we show that the truncated Burgers Hopf equation is a simple Di Qi, and Andrew J. Majda Figure (CIMS) 7.2 Galerkin truncation Truncated with Burgers-Hopf = 10, Equations = 100. Relative errors for numer- Oct. 27, / 36

30 TBH as a model with statistical features with atmosphere The truncated Burgers-Hopf equation is a simple model with many degrees of freedom with many statistical properties similar to those occurring in dynamical systems relevant to the atmosphere. Long time correlated large-scale modes for low frequency variability; Short time correlated weather modes at small scales; A simple correlation scaling theory for the modes is developed. Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

31 relative entropy. A scaling theory for temporal correlations A scaling theory for temporal correlations It is a simple matter to present a scaling theory which predicts that the temporal correlation times of the large-scale modes are longer than those for the small-scale modes. Recall from (7.45) that the statistical predictions for the energy per mode 7.3 Truncated Burgers Hopf equations 237 is E/ = 1 ; since E/ has units length 2 /time 2, and the wave number k has units length 1, the predicted eddy turnover time for the kth mode is given by ( ) 1/2 1 E k = k 1 k If the physical assumption is made that the kth mode decorrelates on a time scale proportional to the eddy turnover time with a universal constant of proportionality, C 0, a simple plausible scaling theory for the dynamics of the equation in (7.35) or (7.37) emerges and predicts that the correlation time for the kth mode, T k, is given by T k = C 0 1 k (7.49) k Thus, the scaling theory implied by (7.49) shows that the larger-scale modes in Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

32 Numerical evidence for the correlation scaling theory Numerical evident for the correlation scaling theory For the numerical results reported here, the correlation function of the kth mode, X k = Re û k, is computed from time averaging 238 asequilibrium in (7.46), statistical mechanics i.e. for systems anyof ODEs time, the correlation function C k is computed by 1.2 C k = 1 T T0 +T T X k t X k t dt (7.50) with T picked sufficiently large. We also define the correlation time for the kth 0.4 mode through T k = 0 C k d 0 (7.51) Here results are presented for the same case, 10 = 100, studied earlier to confirm equipartition of energy. The time correlations of Re û k for Here results are presented for the same case θ = modes 10, k = 1 Λ 2 3 = Equilibrium statistical mechanics for systems of ODEs k = are presented in Figure 7.3 illustrating the wide range of time scales present in the system. The elementary scaling theory for correlation times developed in (7.49) is compared with the numerically computed correlation times in Figure 7.4. There the scaling formula in (7.49) is multiplied by a normalization 0.6 constant, C 0, to exactly match 0.4the correlation time for the mode with 0.4 k = 1. As shown in Figure 7.4, the simple 0.3 theory proposed in (7.49) is an Figure 7.3 Galerkin truncation with = 10, = 100. Correlation functions for (a) Correlation functions for k = 1, 2, 3, 10, 15, 20 Figure 7.3 Galerkin truncation with = 10, = 100. Correlation functions for modes k = (b) Correlation time v. wavenumber k. DNS & scaling theory Figure 7.4 Galerkin truncation with = 10, = 100. Correlation times v. wave number k. Circles DNS, solid line predictions of the scaling theory. Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

33 Outline 1 Review: Introduction to statistical mechanics for ODEs 2 Statistical mechanics for the truncated Burgers-Hopf equations The truncated Burgers-Hopf system The Gibbs measure and the prediction of equipartion of energy Numerical evidence of the validity of the statistical theory TBH as a model with statistical features with atmosphere 3 Statistically relevant conserved quantities for the truncated Burgers-Hopf equation Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

34 Statistically relevant conserved quantities for TBH The TBH equations (u Λ ) t P ( ) Λ u 2 Λ = 0, k Λ, x have three conserved quantities, the momentum M, energy E, and Hamiltonian H ˆ M = u Λ, E = 1 ˆ ˆ ( ) ( ) P Λ u 2 Ω 2 Λ, H = P Λ u 3 Λ. Ω Ω We showed a non-trivial Gibbs measure with equipartition of energy associated the energy E alone; The question arises regarding the statistical significance of the conserved quantity H 2. 2 Abramov, Kovacic, Majda, Hamiltonian structure and statistically relevant conserved quantities for the truncated Burgers-Hopf equation, CMS Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

35 Spectral tilt for statistical relevant values of Hamiltonian For statistically relevant values of the Hamiltonian, the direct numerical simulations showed a surprising spectral tilt rather than equipartition of energy. This spectral tilt was predicted and confirmed independently by Monte-Carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the spectral tilt, 1 û k 2 = E 2 Λ ( H 2 (Λ + 1) /2 k 5 E 3 Λ ), (78) where Λ denotes the number of degrees of freedom, E is energy, and H is Hamiltonian. Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

36 186 R. V. ABRAMOV AND A. J. MAJDA The linear spectral tilt (Strong Mixing) 1 x The energy spectrum for the simulation with Λ =100, H =0.01. Solidline linearfitfoundbyleast squares, dash-dotted line linear fit (78), dashed line shows equipartition The time correlation function for the Fourier mode k =1, Λ =100, H =0.01 Fig. 15. The spectral tilt and the time correlation function for the case Λ =100, H =0.01. The correlation function confirms mixing, the spectral tilt is linear, and the predicted tilt coincides with the actual slope Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

37 The linear spectral tilt (Weak Mixing) 1 x The energy spectrum for the simulation with Λ =100, H =0.15. Solidline linearfitfoundbyleast squares, dash-dotted line linear fit (78), dashed line shows equipartition The time correlation function for the Fourier mode k =1, Λ =100, H =0.15 Fig. 20. The spectral tilt and the time correlation function for the case Λ =100, H =0.15. Long correlations show failure of mixing, the spectral tilt is nonlinear and the predicted tilt is much more steeper than the actual slope Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36

38 Questions & Discussions Di Qi, and Andrew J. Majda (CIMS) Truncated Burgers-Hopf Equations Oct. 27, / 36