Brownian regime of finite-N corrections to particle motion in the XY hamiltonian mean field model
BBrownian regime of finite- N correctionsto particle motion in the XY hamiltonian mean field model Bruno V. Ribeiro, ∗ Marco A. Amato, and Yves Elskens Instituto Federal de Educa¸c˜ao, Ciˆencia e Tecnologia de Goi´as - Cˆampus Valpara´ıso, BR 040, Km 6, ´Area 8, Gleba E,Fazenda Saia Velha, Parque Esplanada IV - Valpara´ıso de Goi´as - GO, Brasil Instituto de F´ısica, Universidade de Bras´ılia, CP: 04455, 70919-970 - Bras´ılia - DF, BrasilInternational Center for Condensed Matter Physics, Universidade de Bras´ılia, DF - Brasil Equipe turbulence plasma, case 322, PIIM, UMR 7345 CNRS,Aix-Marseille Universit´e, campus Saint-J´erˆome, FR-13397 Marseille cedex 13, France
We study the dynamics of the N -particle system evolving in the XY hamiltonian mean field(HMF) model for a repulsive potential, when no phase transition occurs. Starting from a homo-geneous distribution, particles evolve in a mean field created by the interaction with all others.This interaction does not change the homogeneous state of the system, and particle motion is ap-proximately ballistic with small corrections. For initial particle data approaching a waterbag, it isexplicitly proved that corrections to the ballistic velocities are in the form of independent browniannoises over a time scale diverging not slower than N / as N → ∞ , which proves the propagationof molecular chaos. Molecular dynamics simulations of the XY-HMF model confirm our analyticalfindings. Keywords : deterministic chaos, stochasticity, mean-field models, propagation of chaos, asymptoticindependence, brownian limit, finite N noise, long-range systemINTRODUCTION
The dynamics of long-ranged interacting systems are atopic of active investigation due to their intriguing prop-erties [4]. As particles interact with every other onesin these systems, collective behaviour is prone to dom-inate over collisional processes during long times. Thisinterplay between collisional relaxation and collective be-haviour is responsible for the richness of phenomena inlong-ranged systems, as well as for their unusual relax-ation towards equilibrium. For example, starting froman initial nonequilibrium configuration, these systemsrapidly evolve to a quasistationary state (QSS) wherethey are trapped for long times [1, 3]. These times scalewith the number of particles in the system and are fol-lowed by a proper relaxation towards thermodynami-cal equilibrium. For the complexity of their dynamics,these systems have raised interest in various fields suchas plasma physics, astrophysics, statistical mechanics andapplied mathematics (see [4, 6, 7, 13] for reviews).Dynamical properties, including the intricate collec-tive behaviour of constituents, of these systems maybe unveiled through one-dimensional finite N models ofmean-field type [4]. Despite their simplicity, these one-dimensional models present many of the rich dynamicalproperties of long-ranged systems, such as the emergenceof QSS, where the time average of macroscopic quantitiesdiffers from their ensemble averages in statistical equilib-rium. One such system is the XY hamiltonian mean fieldmodel (HMF), in which N identical particles interact viaa infinite-ranged potential and have their motion con-fined to a unit circle. This system evolves in phase-space S N π × R N according to the hamiltonian [1] H = N (cid:88) i =1 p i − N N (cid:88) i,j =1 V cos( q j − q i ) , (1)where the i -th particle has unit mass, position q i and mo-mentum p i , S π = R / π and the constant V defines thenature of the interaction. In this system, each particleinteracts with all other ones through a force field that is,at each instant, the sum of the individual fields producedby all particles. The interaction term in the hamiltonianis equivalent to the interaction term in the XY Heisen-berg model (or “rotator model”) in the mean-field ap-proach. The model with positive interactions ( V >
V <
0) corresponds to the antiferromagneticcase.One can understand the complexity of the dynamics ofthis seemingly simple system by analysing the equationsof motion ˙ q i = p i (2)˙ p i = − VN N (cid:88) j =1 sin( q i − q j ) , (3)which correspond to a set of N fully coupled pendula .Introducing the mean-field quantity (as a reference tothe Heisenberg model we call it “magnetization”) M = 1 N N (cid:88) i =1 e i q i ≡ M e i ϕ , (4)we write the equations of motion (e.o.m.)¨ q i = − V M sin( q i − ϕ ) . (5) a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Therefore, the motion of each particle is determined bya self-consistent interaction with a mean-field M , whichdepends on time implicitly through the instantaneousvalues of particle positions. The average energy per par-ticle is given by U = HN = (cid:10) p (cid:11) V − M ) , (6)where (cid:104)·(cid:105) represents average over all particles. From thislast expression, we can use M as an order parameter tocharacterize the phases of the system. For the ferromag-netic case ( V = 1), the ground state corresponds to astate in which all particles are clustered, giving a highvalue for M ; in the antiferromagnetic state ( V = − M .Equilibrium statistical mechanics calculations in thecanonical ensemble predict a phase transition in the fer-romagnetic case between a clustered phase, with non-zero values for M , and a homogeneous phase, in which M vanishes [1, 2]. For the antiferromagnetic case, theonly equilibrium solution is the homogeneous state inwhich M = 0. Though microcanonical calculations aremore difficult, it has been proved that both ensemblesare equivalent for (1) [4], and it has been numericallychecked that, for the HMF model, ensembles are equiv-alent for high enough energies[20] [5]. Since moleculardynamics (MD) for the system (2)-(3), as we considerbelow, involve no heat bath, the relevant Gibbs ensembleis the microcanonical one.In the N → ∞ limit, the description of the systemis rigorously given by the Vlasov equation [17]. Thisequation governs the evolution of the single particle dis-tribution function f ( q, p, t ). For long-ranged interactingsystems, the QSSs that follow the initial nonequilibriumconfiguration represent stable steady states of the under-lying Vlasov equation.However, for finite systems, in the antiferromagneticcase, MD simulations show that starting from an initialhomogeneous configuration, the value of M ( t ) fluctuatesaround small values due to density fluctuations of parti-cles in the circle S π . Fluctuations as this are, usually,assumed to have a diffusive nature based on empiricaldata from molecular dynamics simulations [9].In this note, we work out the dynamical details thatlead to the fluctuations in the mean-field quantity M ,showing the nature of the underlying process driving themotion of the particles. Thanks to the simplicity of theXY-HMF model, we provide a direct, short proof thatthese fluctuations generate independent diffusion of par-ticles on long time-scales. BALLISTIC MOTION AND CORRECTIONS
Because there is no cluster formation in the stable stateof the repulsive case, a first approximation to the particle motion is the ballistic motion q j ( t ) = q j + v j t + Q j , (7)where v j is a constant velocity associated with particle j and Q j is a small correction to ballistic motion ( Q j (cid:28) π ), which we will neglect for the moment. So, we rewrite M e i ϕ (cid:39) M (0) e i ϕ (0) := 1 N N (cid:88) j =1 e i( q j + v j t ) (8)and ¨ q j (cid:39) M (0) sin( q j − ϕ (0) ) . (9)To study the nature of the evolution of the system, ourgoal is to write the evolution equation (9) as a differen-tial equation driven by a known (possibly wild, rough orstochastic) process. As we expect a small noise, its ef-fects will show up for large times, so we rescale time to t (cid:48) = t/N (cid:48)(cid:48) with some large N (cid:48)(cid:48) ; simultaneously, assumewe can write the number of particles as the product oftwo integers N = N (cid:48) · N (cid:48)(cid:48) and consider M d t = 1 N N (cid:88) j =1 e i( q j + v j t ) d t = N (cid:48)− α √ π d W Nt (cid:48) , (10)where we introduce the complex-valued processd W Nt (cid:48) = 1 √ πN (cid:48) − α N (cid:88) j =1 e i q j e i N (cid:48)(cid:48) v j t (cid:48) d t (cid:48) (11)with an exponent 0 < α < q i = N (cid:48)(cid:48) p i d t (cid:48) , (12)d p i = √ πN (cid:48)− α [sin( q i ) d (cid:60) ( W Nt (cid:48) ) − cos( q i ) d (cid:61) ( W Nt (cid:48) )] . (13)To keep the evolution of rescaled quantities independentof system size, we rescale momentum to p (cid:48) i = N (cid:48)(cid:48) p i andwe balance scalings N (cid:48)(cid:48) = N (cid:48) α to getd q i = p (cid:48) i d t (cid:48) , (14)d p (cid:48) i = √ π [sin( q i ) d (cid:60) ( W Nt (cid:48) ) − cos( q i ) d (cid:61) ( W Nt (cid:48) )] . (15)Note that the interaction potential in (1) is well be-haved, so that microscopic motion is smooth for finite N , thus (14)-(15) involve ordinary calculus with differ-entials d (cid:60) ( W Nt (cid:48) ) and d (cid:61) ( W Nt (cid:48) ).Consistently, we set v (cid:48) j = N (cid:48)(cid:48) v j , (16)so that d W Nt (cid:48) = N (cid:48)(cid:48) / N (cid:48) − α d W N (cid:48) N (cid:48)(cid:48) ( t (cid:48) ) (17)with the rescaled process defined byd W N (cid:48) N (cid:48)(cid:48) ( t (cid:48) ) = 1 √ πN (cid:48)(cid:48) N (cid:48) N (cid:48)(cid:48) (cid:88) j =1 e i q j e i v (cid:48) j t (cid:48) d t (cid:48) . (18)The particles are taken to be, initially, distributed in-dependently, uniformly in position. In velocity space,it will be useful to set particles to be distributed, ini-tially, in equally spaced beams containing one particleeach. This guarantees that, in the N → ∞ limit, theparticle distribution converges to a “waterbag” [21] overa band S π × [ − p , p ] (see figure 1). In the limit N → ∞ ,this distribution is a stationary solution to the Vlasovequation, and a standard initial data for numerical sim-ulations [15, 16].To implement our “beam” model, we write v (cid:48) j = j − / N (cid:48)(cid:48) − N (cid:48) N -dimensional vectors q j and v (cid:48) j into N (cid:48) × N (cid:48)(cid:48) matrices q m,n and v (cid:48) m,n with j = n + ( m − N (cid:48)(cid:48) to get the proper limit in the next section. In this spirit,the new representation for initial velocity is v (cid:48) m,n = ( m −
1) + n − / N (cid:48)(cid:48) − N (cid:48) , (20)where m = 1 , ..., N (cid:48) and n = 1 , ..., N (cid:48)(cid:48) , while the ini-tial positions are simply q mn and remain uniformly dis-tributed. With this new formulation, (18) readsd W N (cid:48) N (cid:48)(cid:48) ( t (cid:48) ) = 1 √ πN (cid:48)(cid:48) N (cid:48)(cid:48) (cid:88) n =1 N (cid:48) (cid:88) m =1 e i q mn e i[( m − n − / (2 N (cid:48)(cid:48) ) − N (cid:48) / t (cid:48) d t (cid:48) (21)where one recognizes in N (cid:48)(cid:48) a number of samples for tak-ing a central limit theorem and in N (cid:48) a bandwidth scalewhich contributes higher frequencies and allows for a non-smooth limit process W . THE N → ∞ LIMIT
According to [8], in the limit N (cid:48) , N (cid:48)(cid:48) → ∞ , the pro-cess W N (cid:48) N (cid:48)(cid:48) ( t (cid:48) ) approaches a Wiener process ( W t (cid:48) ) in C .Thus, calculus with differentials d (cid:60) ( W Nt (cid:48) ) and d (cid:61) ( W Nt (cid:48) )becomes Stratonovich’s [18], denoted with ◦ d,d q i = p (cid:48) i d t (cid:48) , (22)d p (cid:48) i = √ π N (cid:48)(cid:48) / N (cid:48) − α [sin( q i ) ◦ d (cid:60) ( W t (cid:48) ) − cos( q i ) ◦ d (cid:61) ( W t (cid:48) )] , (23)and we select the scaling N (cid:48)(cid:48) / = N (cid:48) − α . Along with N (cid:48)(cid:48) = N (cid:48) α = N/N (cid:48) , this yields α = 2 / N (cid:48) = N / and N (cid:48)(cid:48) = N / . By Proposition 4.2 of [8], these equationsyield q i ( t (cid:48) ) = q i + p (cid:48) i t (cid:48) + √ π (cid:90) t (cid:48) B i ( s ) d s, (24) p (cid:48) i ( t (cid:48) ) = p (cid:48) i + √ π B i ( t (cid:48) ) , (25) -3 -2 -1 0 1 2 3 p q (a) -3 -2 -1 0 1 2 3 p q (b) -3 -2 -1 0 1 2 3 p q (c) -3 -2 -1 0 1 2 3 p q (d) FIG. 1:
Initial distribution of particles in monokinetic beams inthe ( q, p ) space. Distribution of: (a) 10 particles, (b) 20 particles,(c) 100 particles and (d) 1000 particles. We can see how thedistribution approaches a “waterbag” on increasing N . where B i is one realization of the standard one-dimensional brownian motion.Computing the variations of (25), δq i ( t (cid:48) ) ∼ t (cid:48) δp (cid:48) i ∼ O (cid:16) t (cid:48) / (cid:17) , (26) δp (cid:48) i ( t (cid:48) ) ∼ √ π B ( t (cid:48) ) ∼ O (cid:16) t (cid:48) / (cid:17) . (27)So, as a correction to ballistic motion, we get q j ( t ) = q j + v j t + δq i ( t (cid:48) ) ∼ q j + v j t + O (cid:18) tN (cid:48)(cid:48) (cid:19) / . (28)The ballistic approximation is valid, then, at least for tN (cid:48)(cid:48) (cid:28) , (29)which is guaranteed for any finite time in the N (cid:48) , N (cid:48)(cid:48) →∞ limit. For finite N , the approach breaks down when t ∼ N (cid:48)(cid:48) ∼ N / .Moreover, any fixed number of particles are indepen-dent in the limit, viz. “molecular chaos propagates”[8, 10, 11]. NUMERICAL RESULTS
To test our analytical findings, we simulate the evolutionof the system ruled by the hamiltonian (1) and observethe fluctuations in particle velocity. We use a moleculardynamics code with a fourth order symplectic numericalintegrator [19].We note that condition (19) ensures that the (con-served) total momentum is exactly zero. The ( q, p )-spaceportrait for this initial configuration (at t = 0), with auniform distribution of positions in the interval [ − π, π ], ispresented, along with the ( q, p )-space portrait for a latertime ( t = 200), in figure 2.As a first illustration of the motion of particles in veloc-ity space, we simulate the system with N = 10 particles,randomly choose 6 particles and plot each of the respec-tive P j ( t ) = p j ( t ) − p j with a timestep of the numericalintegrator δt = 0 .
1. Results are shown in figure 3.It is known that a Wiener process W ( t ) is a Gaussianprocess with independent increments characterized by W (0) = 0 , E ( W ( t )) = 0 , Var( W ( t )) = t, (30)where E is the expectation operator and Var the variance.Therefore, to test whether our P j is indeed a brownianmotion, we define the operator (cid:104)•(cid:105) by its action on aprocess R j as (cid:104) R j (cid:105) = 1 N s (cid:88) j R j , (31)where N s is the sample size. With this, introduce thequantity S ( t ) = (cid:113)(cid:10) P j ( t ) (cid:11) , (32) -4 -2 0 2 4 q -40000040000 p (a) -4 -2 0 2 4 q -40000040000 p (b) FIG. 2: ( q, p )-space portraits: (a) initial state, (b) state at t = 200. Note that the system remains in a homogeneous state.System size N = 10 . and verify whether S ( t ) grows linearly with time t , inagreement with (30). Results are presented in figure 4.We get a linear behaviour for S ( t ) in various systemsizes. Note that the value of S ( t ) scales as N − / , as theevolutions of N · S ( t ) coincide for the simulated systems.To further check whether P j is a brownian motion,we can analyze the moments of its distribution to seewhether they satisfy [12, 14] (cid:10) P kj ( t ) (cid:11) S k ( t ) = 0 , for odd k (33) (cid:10) P kj ( t ) (cid:11) S k ( t ) · · · · · ( k −
1) = 1 , for even k . (34)We run this test with the same configuration used infigure 4 and plot the results in figure 5. Both figuresshow that numeric simulation is in good agreement withthe analytical results of the previous section. The strongfluctuations around the expected values in figure 5 areassociated with the finite number of particles used in thesimulation. t P ( t ) FIG. 3: Evolution of velocity for 6 different particles.Initial conditions given by (19). t N . S ( t ) N = 10 N = 5.10 N = 10 FIG. 4: Evolution of momentum deviation averagesquare S ( t ). System sizes N = 10 , N = 5 · and N = 10 , sample size N s = 0 . N , total simulation time t = 300, time step δt = 0 . t -0,500,511,52 nd -moment4 th -moment6 th -moment t -0,500,511,52 st -moment3 rd -moment5 th -moment FIG. 5: Moments of P j ( t ) rescaled as (33)-(34). Systemsize N = 10 , sample size N s = 10 , total simulationtime t = 200, time step δt = 0 . CONCLUSIONS
Starting from a “particles in monokinetic beams” initialcondition (illustrated by figure 1) approximating a wa-terbag, we show analytically that the velocities of par-ticles in the repulsive XY HMF N -body system displayBrownian corrections to the ballistic motion implied bythe Vlasov limit for N → ∞ , and that these correctionspropagate initial independence (molecular chaos).As we show in equation (23), the motion of particlesin velocity space can be written as a stochastic differen-tial equation driven by Wiener processes which are dueto particle interaction, i.e. the coupling of the parti-cles with the mean field quantity of the system. In suchsystems, the mean field forces are usually postulated aswhite noises based on numerical evidence (see [9] and ref-erences therein) ; here, we show rigorously that process(10) properly rescaled as (21) converges to a Wiener pro-cess, and moreover that particles in the same mean fielddo behave independently of each other (thanks to theirdifferent initial positions and velocities). We present alower time estimate for the validity of our approximationsalong with numerical results confirming our findings. ACKNOWLEDGEMENTS
This work benefited from fruitful discussions withT. M. da Rocha Filho and participants to the XIV LatinAmerican Workshop on Nonlinear Phenomena. Com-ments by D. D. A. Santos are gratefully acknowledged,as well as constructive comments by the anonymous ref-erees. Author B. V. Ribeiro acknowledges CAPES forfinancial support. ∗ Electronic address: [email protected][1] Antoni M., Ruffo S.: Clustering and relaxation in hamil-tonian long-range dynamics.
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