The The FermiFermi--PastaPasta--UlamUlamproblemproblem

The birth of The birth of nonlinearnonlinear physicsphysics and and dynamicaldynamical chaoschaos

Paolo Paolo ValentiniValentini

SUMMARYSUMMARY

The The FermiFermi--PastaPasta--UlamUlam experimentexperiment on on MANIAC at LANL in 1955MANIAC at LANL in 1955The The ergodicergodic hypothesishypothesisThe 32The 32--atomatom--fixedfixed--end end chainchain: : linearlinear and and nonlinearnonlinear springsspringsConclusionsConclusions

SUMMARYSUMMARY

The The FermiFermi--PastaPasta--UlamUlam experimentexperiment on on MANIAC at LNL in 1955MANIAC at LNL in 1955The ergodic hypothesisThe 32-atom-fixed-end chain: linear and nonlinear springsConclusions

The FPU The FPU experimentexperiment1D 1D chainchain of of particlesparticles withwith nearestnearest neighborneighborinteractionsinteractionsParticlesParticles are are connectedconnected withwith springssprings thatthat havehave a a linearlinear and a and a weaklyweakly nonlinearnonlinear componentcomponentParticlesParticles werewere givengiven anan initialinitial displacementdisplacementaccordingaccording toto a a fundamentalfundamental mode (or a mode (or a superposition of superposition of modesmodes) of the ) of the linearlinear systemsystemNewtonNewton’’s s equationsequations of of motionmotion wouldwould thenthen bebeintegratedintegrated on the computer MANIAC (on the computer MANIAC (i.ei.e., ., anan MD MD simulationsimulation))

ImplementationImplementation detailsdetailsAllAll atomsatoms havehave a a unitunit mass and the mass and the elasticelasticconstantconstant hashas beenbeen takentaken one, one, asas in the in the originaloriginalexperimentexperimentTwoTwo modelsmodels are are usuallyusually consideredconsidered in the in the literatureliterature: the : the αα-- and and ββ-- modelmodel

ImplementationImplementation detailsdetails 22The The equationsequations of of motionmotion havehave beenbeen integratedintegrated withwith a a secondsecond orderorder schemescheme ((VelocityVelocity VerletVerlet) and a ) and a fourthfourth orderorderschemescheme ((ForestForest--RuthRuth))In the In the absenceabsence of of nonlinearnonlinear forcesforces, the , the normalnormal modesmodescan can bebe expressedexpressed asas a a FourierFourier representationrepresentation of the of the displacementsdisplacements. . ForFor exampleexample forfor fixedfixed endsends::

SUMMARYSUMMARY

The Fermi-Pasta-Ulam experiment on MANIAC at LNL in 1955The ergodic hypothesisThe 32-atom-fixed-end chain: linear and nonlinear springsConclusions

The The ergodicergodic hypothesishypothesisStated by Ludwig Stated by Ludwig BoltzmannBoltzmann in 1871, it is at basis of in 1871, it is at basis of statistical mechanics (or is it?)statistical mechanics (or is it?)The hypothesis: given sufficient time, the dynamical The hypothesis: given sufficient time, the dynamical trajectory of a system will visit all microstates consistent trajectory of a system will visit all microstates consistent with the applied constraintswith the applied constraintsAssuming the Assuming the ergodicityergodicity of the system, then the phase of the system, then the phase average of a function defined over phase space is the average of a function defined over phase space is the same as its time averagesame as its time averageFPU believed that a 32 particle system would reveal FPU believed that a 32 particle system would reveal ergodicergodic behavior, provided that the nonlinearity is not behavior, provided that the nonlinearity is not extremely small: to the great surprise of FPU, the result extremely small: to the great surprise of FPU, the result was the opposite!was the opposite!

SUMMARYSUMMARY

The Fermi-Pasta-Ulam experiment on MANIAC at LNL in 1955The ergodic hypothesisThe 32-atom-fixed-end chain: linear and nonlinear springsConclusions

Simulation resultsSimulation results

The ends of the chain are kept fixed: only The ends of the chain are kept fixed: only sinusoidal modes are normal modes for sinusoidal modes are normal modes for the linear chain (i.e. no nonlinearity the linear chain (i.e. no nonlinearity present)present)Only the ground mode (k = 1) was excited Only the ground mode (k = 1) was excited and numerical experiments were and numerical experiments were performed varying performed varying ββ єє {0.3, 1, 3}{0.3, 1, 3}

Simulation results: Simulation results: ββ = 0.3= 0.3

Simulation results: Simulation results: ββ = 1= 1

Simulation results: Simulation results: ββ = 3= 3

The FPU paradoxThe FPU paradoxThe first attempts to explain the FPU paradox The first attempts to explain the FPU paradox questioned the accuracy of the questioned the accuracy of the numericsnumericsA fourth order integration scheme (the ForestA fourth order integration scheme (the Forest--Ruth Ruth algorithm) has been implementedalgorithm) has been implemented

The FPU paradox: the threshold of The FPU paradox: the threshold of stochasticitystochasticity

Many studies therefore demonstrated that the FPU could Many studies therefore demonstrated that the FPU could not be trivialized accusing the low accuracy of the not be trivialized accusing the low accuracy of the integration scheme employed by FPUintegration scheme employed by FPUA successful approach to resolve the paradox is due to A successful approach to resolve the paradox is due to ChirikovChirikov: he introduced the notion of : he introduced the notion of stochasticitystochasticity (or (or dynamical chaos)dynamical chaos)He found analytically a stochastic threshold, above He found analytically a stochastic threshold, above which the FPU system was shown to behave in which the FPU system was shown to behave in accordance with the original expectations of FPU, accordance with the original expectations of FPU, revealing strong statistical properties, such as energy revealing strong statistical properties, such as energy equipartitionequipartition among the linear modesamong the linear modes

Simulation results: three modes Simulation results: three modes (k = 1, 3, 5) are excited and (k = 1, 3, 5) are excited and ββ=0=0.003.003

Simulation results: three modes Simulation results: three modes (k = 1, 3, 5) are excited and (k = 1, 3, 5) are excited and ββ=0=0.03.03

The threshold of The threshold of stochasticitystochasticity

When higher modes are initially excited, the FPU When higher modes are initially excited, the FPU system is above the threshold of system is above the threshold of stochasticitystochasticity, , therefore lower values for therefore lower values for ββ are are necessarynecessary totorevealreveal a a stochasticstochastic behaviorbehavior of the systemof the systemThisThis can can bebe shownshown analiticallyanalitically ((ChirikovChirikov, 1959), 1959)

The threshold of The threshold of stochasticitystochasticityThe FPU was then explained. The original The FPU was then explained. The original simulations were performed by FPU for the simulations were performed by FPU for the lowest value of k = 1lowest value of k = 1For that value of k, only strong perturbations For that value of k, only strong perturbations (nonlinearity) could reveal a non(nonlinearity) could reveal a non--recurrent recurrent behaviorbehaviorIndeed, the nonlinear energy in numerical Indeed, the nonlinear energy in numerical studies of FPU had never exceeded 10% the studies of FPU had never exceeded 10% the total energy, therefore the system was well total energy, therefore the system was well below the below the stochasticitystochasticity borderborder

SUMMARYSUMMARY

The Fermi-Pasta-Ulam experiment on MANIAC at LNL in 1955The ergodic hypothesisThe 32-atom-fixed-end chain: linear and nonlinear springsConclusions

ConclusionsConclusionsSimulations to study the FPU problems have been runSimulations to study the FPU problems have been runIncreasing the order of accuracy of the integration Increasing the order of accuracy of the integration schemes adopted does not prove successful in schemes adopted does not prove successful in explaining the FPU paradox: Velocity explaining the FPU paradox: Velocity VerletVerlet (2(2°° order) order) and Forestand Forest--Ruth (4Ruth (4°° order) have been implemented and order) have been implemented and revealed that the recurrent/nonrevealed that the recurrent/non--recurrent behavior is recurrent behavior is determined by intrinsic properties of the FPU systemdetermined by intrinsic properties of the FPU systemThe The stochasticitystochasticity threshold successfully explains the threshold successfully explains the behavior of the FPU, revealing a dependence of the behavior of the FPU, revealing a dependence of the stochastic behavior (stochastic behavior (ergodicityergodicity) on the initial conditions) on the initial conditionsThe criterion due to The criterion due to ChirikovChirikov has shown that when the has shown that when the system is excited in higher modes, lower values of the system is excited in higher modes, lower values of the nonlinearity parameter nonlinearity parameter ββ are are necessarynecessary toto trigger a trigger a chaoticchaotic behaviorbehavior ((dynamicaldynamical chaoschaos))

ReferencesReferencesBerman and Berman and IzrailevIzrailev, , The FermiThe Fermi--PastaPasta--UlamUlamproblem: 50 years of progressproblem: 50 years of progress, Chaos, 15 (2005), Chaos, 15 (2005)Ford and Waters, Ford and Waters, Computer studies of energy Computer studies of energy sharing and sharing and ergodicityergodicity for nonlinear oscillator for nonlinear oscillator systemssystems, Journal of Mathematical Physics, 4 , Journal of Mathematical Physics, 4 10 10 (1963)(1963)TadmorTadmor E. B., E. B., AtomsticAtomstic and and multiscalemultiscale modeling modeling of materials, Lecture notesof materials, Lecture notesChirikovChirikov B. V., B. V., A universal instability of manyA universal instability of many--dimensional oscillator systemsdimensional oscillator systems, Physics Reports , Physics Reports (Review Section of Physics) 52, No. 5 (1979), (Review Section of Physics) 52, No. 5 (1979), 263263--379379

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