Fluctuating hydrodynamics for a chain of nonlinearly coupled rotators
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Fluctuating hydrodynamics for a chain of nonlinearlycoupled rotators
Herbert Spohn
Zentrum Mathematik and Physik Department, TU M¨unchen,Boltzmannstr. 3, D-85747 Garching, Germanyemail: [email protected]
Abstract.
We study chains of rotators from the perspective of nonlinear fluctuatinghydrodynamics. As confirmed by previous MD simulations, at intermediate temperaturesdiffusive transport is predicted. At low temperatures we obtain the FPU scenario withsuppressed heat peak.s recognized for some time, one-dimensional anharmonic chains may have anomaloustransport properties [1, 2, 3, 4]. For example, a small disturbance of the equilibrium statewill spread super-diffusively. But there is another large class of chains which exhibitsregular diffusive transport. Thus one central question is how to characterize these twouniversality classes. (There is also the class of integrable chains, as Toda and harmonic,which have ballistic transport.) As one crucial feature, momentum conservation has beenidentified. Momentum conservation holds for chains for which the interaction potentialdepends only on positional differences, the most prominent example being the Fermi-Pasta-Ulam (FPU) chains. On the other hand, if there is an on-site potential, thenmomentum is not conserved and the scattering from the underlying lattice potential forcesregular transport.A chain of nonlinearly coupled rotators seems to violate the above dichotomy. Theinteraction depends only on the difference in angles. Thus angular momentum is con-served. Nevertheless, through molecular dynamics (MD) simulations diffusive behaviorhas been convincingly demonstrated [5]. The purpose of my note is to explain why, fromthe perspective of nonlinear fluctuating hydrodynamics, regular transport is indeed thenatural option.The hamiltonian of the rotator chain reads H = N X j =1 (cid:0) p j + V ( ϕ j +1 − ϕ j ) (cid:1) (1)with periodic boundary conditions, ϕ N + j = ϕ j . The ϕ j ’s are angles and the p j ’s angularmomenta. Hence the phase space is ([ − π, π ] × R ) N . The standard choice for V is V ( ϑ ) = − cos ϑ , but in our context any 2 π -periodic potential is admitted. The equations of motionare ddt ϕ j = p j , ddt p j = V ′ ( ϕ j +1 − ϕ j ) − V ′ ( ϕ j − ϕ j − ) . (2)Obviously angular momentum is locally conserved with the angular momentum current J ( j, t ) = − V ′ ( ϕ j ( t ) − ϕ j − ( t )) . (3)As local energy we define e j = p j + V ( ϕ j +1 − ϕ j ). Then e j is locally conserved, since ddt e j = p j +1 V ′ ( ϕ j +1 − ϕ j ) − p j V ′ ( ϕ j − ϕ j − ) , (4)from which one reads off the energy current J ( j, t ) = − p j ( t ) V ′ ( ϕ j ( t ) − ϕ j − ( t )) . (5)The hamiltonian for FPU type chains has the same structure as (1), H FPU = N X j =1 (cid:0) p j + V FPU ( q j +1 − q j ) (cid:1) , (6)2nly now the positions q j ∈ R and V FPU should have at least a one-sided increase toinfinity. FPU type chains have three conservation laws, momentum, energy, and stretch r j = q j +1 − q j . Also for angles one can define the difference ˜ r j = ϕ j +1 − ϕ j mod 2 π .Because of the modulo 2 π such stretch is not conserved. A rotator chain has only twoconserved fields.According to nonlinear fluctuating hydrodynamics, the transport anomaly is relatedto the macroscopic Euler currents being nonlinear functions of the conserved fields [6].Thus for the rotator chain we have to compute the Euler currents in equilibrium. Sincethere are two conserved fields the canonical equilibrium state reads1 Z N N Y j =1 exp (cid:2) − β (cid:0) ( p j − u ) + V ( ϕ j +1 − ϕ j ) (cid:1)(cid:3) dϕ j dp j , (7)where β > u ∈ R the average angular momentum. Now hJ ( j ) i N = −h V ′ ( ϕ j − ϕ j − ) i N , hJ ( j ) i N = − u h V ′ ( ϕ j − ϕ j − ) i N , (8)average with respect to the canonical ensemble (7). We claim thatlim N →∞ h V ′ ( ϕ j − ϕ j − ) i N = 0 . (9)For this purpose we expand in Fourier series as e − V ( ϑ ) = X m ∈ Z a ( m ) e − i mϑ , f ( ϑ ) = X m ∈ Z ˆ f ( m ) e − i mϑ . (10)Then, working out all Kronecker deltas from the integration over the ϕ j ’s, one arrives at h f ( ϕ j +1 − ϕ j ) i N = (cid:16) X m ∈ Z a ( m ) N (cid:17) − X m ∈ Z a ( m ) N (cid:16) a ( m ) − X ℓ ∈ Z ˆ f ( ℓ − m ) a ( ℓ ) (cid:17) . (11)Since a (0) > | a ( m ) | for all m = 0,lim N →∞ h f ( ϕ j +1 − ϕ j ) i N = a (0) − X ℓ ∈ Z ˆ f ( ℓ ) a ( ℓ ) = 1 Z Z π − π dϑf ( ϑ ) e − βV ( ϑ ) . (12)For f ( ϑ ) = V ′ ( ϑ ), the latter integral vanishes because of periodic boundary conditions in ϑ . We conclude that both currents vanish on average.On a large scale the conserved fields are expected to be governed by fluctuating hy-drodynamics [7]. Since the Euler currents vanish, the Langevin equations are linear withwhite noise currents and a dissipative second order drift term. There are two conservedfields and we form the equilibrium time correlations as S αα ′ ( j, t ) = h g α ( j, t ) g α ′ (0 , i , ~g ( j, t ) = (cid:0) p j ( t ) , e j ( t ) (cid:1) (13)for the infinite lattice. p j is odd and e j is even under time reversal. Hence in the Green-Kubo formula the cross term vanishes. Thus, for large j, t , S αα ′ ( j, t ) = δ αα ′ (4 πD α t ) − / f G ((4 πD α t ) − / j ) , (14)3here f G is the unit Gaussian. D α is the diffusion coefficient of mode α . Of course, itcan be written as a time-integral over the corresponding total current-current correlation,but its precise value has to be determined numerically. Energy diffusion is well confirmedin MD simulations [5]. The total energy current correlation decays exponentially and thethermal conductivity depends on temperature as e c β with some constant c for sufficientlysmall temperatures. Momentum diffusion has been noted only recently [8].At low temperatures there is a more interesting scenario. If we assume that V hasa unique global minimum at some angle ϑ , then the states of minimal energy are ϕ j = jϑ + ϑ , p j = 0. By redefining the angles one can always achieve ϑ = 0 . The angle ϑ ∈ [ − π, π ] labels the broken rotational symmetry. Let us now fix some large N andconsider the initial state, where ϕ = 0 and ϕ j +1 − ϕ j are independent Gaussian randomvariables with variance 1 /βV ′′ (0), V ′′ (0) > β is taken so large that β − N / ≪
1. This initial state is not stationary under the dynamics, but it has a verylong life time. It is a rare event for the angle differences to change by 2 π . But then onecan expand H relative to the ground state configuration with the result H = N X j =1 (cid:0) p j + ˜ V ( q j +1 − q j ) (cid:1) . (15)Here ˜ V is given by the Taylor expansion of V ( ϑ ) at 0 up to the first even power, larger than2, which has a positive Taylor coefficient. On the low temperature scale, the deviations q j +1 − q j take real values. Thus, in approximation, we have regained the three conservationlaws of the FPU chain.One can now apply the results from [6]. The equilibrium time correlations have athree peak structure, two symmetric sound peaks which move with the speed of soundand broaden according to the KPZ scaling function, and a central heat peak which isstanding still and broadens according to the -Levy distribution. At low temperaturesthe Landau-Placzek ratio for the heat peak is much smaller than the one for the soundpeaks [10]. Thus in a MD simulation one will see only the two sound peaks. At first,they will move ballistically, but then slow down to eventually cross over to the Gaussianscaling (14). The precise cross over still needs to be investigated.In closing, we mention that the non-linear Sch¨odinger equation on the one-dimensionallattice has a comparable structure. In this case the lattice field is ψ j ∈ C , where real andimaginary part are the canonically conjugate fields. The hamiltonian reads H = N X j − (cid:0) | ψ j +1 − ψ j | + λ | ψ j | (cid:1) (16)with coupling λ >
0. This chain is non-integrable and the locally conserved fields are thenumber density n j = | ψ j | and the local energy e j = | ψ j +1 − ψ j | + λ | ψ j | . The Eulercurrents vanish and both modes should have diffusive transport. This has been confirmedin [11, 12], where also the two transport coefficients are measured in their dependence on β and the chemical potential µ . However, at low temperatures one observes sound peakswith KPZ scaling [13, 14, 15]. The mechanism explained before is at work. We fix the4otal number as P Nj =1 | ψ j | = N . In the ground state the global phase is broken and theminimizing field configuration equals ψ G j = ρ e i ϑ with ρ = 1. Expanding the hamiltonian(16) at ψ G j , momentum conservation is regained in approximation. The effective lowtemperature hamiltonian is similar to (15), but the leading correction to the Gaussiantheory involves terms which couple the p ’s and q ’s. Stretch, momentum, and energy areconserved, but the heat peak is hardly visible. The correlations have two symmetricallylocated sound peaks which broaden according to KPZ scaling. References [1] S. Lepri, R. Livi, and A. Politi, Heat conduction in chains of nonlinear oscillators.Phys. Rev. Lett. , 1896–1899 (1997).[2] S. Lepri, R. Livi, and A. Politi, Thermal conduction in classical low-dimensionallattices. Physics Reports , 1–80 (2003).[3] A. Dhar, Heat transport in low-dimensional systems. Adv. Physics , 457–537(2008).[4] H. van Beijeren, Exact results for anomalous transport in one-dimensional Hamilto-nian systems. Phys. Rev. Lett. , 180601 (2012).[5] C. Giardin`a, R. Livi, A. Politi, and M. Vassalli, Finite thermal conductivity in 1Dlattices, Phys. Rev. Lett. , 2144–2147 (2000).[6] H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. , 1191–1227 (2014).[7] L.D. Landau and E.M. Lifshitz, Fluid Dynamics . Pergamon Press, New York, 1963.[8] Yunyun Li, Sha Liu, Nianbei Li, P. H¨anggi, Baowen Li, 1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport,arXiv:1407.1161v2 (2014).[9] Ch. B. Mendl and H. Spohn, Equilibrium time-correlation functions for one-dimensional hard-point systems. Phys. Rev. E , 012147 (2014).[10] M. Straka, KPZ scaling in the one-dimensional FPU-model. Master Thesis, Univer-sity of Florence, Italy (2013).[11] S. Iubini, S. Lepri, A. Politi, The nonequilibrium discrete nonlinear Schr¨odingerequation, Phys. Rev. E , 011108 (2012).[12] S. Iubini, S. Lepri, R. Livi, A. Politi, Off-equilibrium Langevin dynamics of thediscrete nonlinear Schr¨odinger chain, J. Stat. Mech. (2013) P08017.513] M. Kulkarni and A. Lamacraft, Finite-temperature dynamical structure factor ofthe one-dimensional Bose gas: From the Gross-Pitaevskii equation to the Kardar-Parisi-Zhang universality class of dynamical critical phenomena. Phys. Rev. A88