Slow dynamics and ergodicity in the one-dimensional self-gravitating system
SSlow dynamics and ergodicity in the one-dimensionalself-gravitating system
LF Souza a , TM Rocha Filho b a Centro de Ciências Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia and Instituto de Física, Universidade de Brasília, UnB -BrasÃŋlia, DF, 70297-400 b Instituto de Física and International Center for Condensed Matter PhysicsUniversidade de Brasília, CP: 04455, 70919-970 - Brasília, Brazil
A B S T R A C T
We revisit the dynamics of the one-dimensional self-gravitating sheets models. We show thathomogeneous and non-homogeneous states have different ergodic properties. The former isnon-ergodic and the one-particle distribution function has a zero collision term if a proper limitis taken for the periodic boundary conditions. Non-homogeneous states are ergodic in a timewindow of the order of the relaxation time to equilibrium, as similarly observe in other systemswith a long range interaction. For the sheets model this relaxation time is much larger than othersystems with long range interactions if compared to the initial violent relaxation time.
1. Introduction
Lower dimensional models retaining the main characteristics of realistic systems has always been an important toolto grasp the phenomenology in Statistical Physics. They have been particularly important in understanding the non-equilibrium dynamics and equilibrium properties of systems with long range interactions, which often present unusualproperties not observed if the interaction is short-ranged, as non-ergodicity, anomalous diffusion, non-Gaussian quasi-stationary states, negative microcanonical heat capacity, ensemble inequivalence, and a very long relaxation time tothermodynamic equilibrium, diverging with the particle number 𝑁 [1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 10, 11, 12, 13, 14, 16, 17].Some one-dimensional models have been extensively studied in the literature, such as one-dimensional plasmas [18],one-dimensional self-gravitating systems: the sheets and shell models [19], and derived models, e. g. the Ring [20] andthe Hamiltonian Mean Field (HMF) models [21]. The dynamics of systems with long range interactions can typicallybe divided in three stages: a violent collisionless relaxation from the initial condition into a quasi-stationary state (oran oscillating state close to it), occurring in a very short time [22], followed by a very slow evolution to thermody-namic equilibrium, caused by the small cumulative effects of collisions (graininess). The final and third stage is thethermodynamic equilibrium, that may never be attained in the 𝑁 → ∞ limit, when the mean-field description becomesexact and the collisional contributions to the Kinetic equation vanish. In this limit, and under suitable conditions, thedynamics is exactly described by the Vlasov equation [23, 7].Let us consider a system of identical particles described by the Hamiltonian: 𝐻 = 𝑁 ∑ 𝑖 =1 𝐩 𝑖 𝑚 + 1 𝑁 𝑁 ∑ 𝑖<𝑗 =1 𝑉 𝑖𝑗 , (1)with the interparticle potential 𝑉 𝑖𝑗 ≡ 𝑉 ( | 𝐫 𝑖 − 𝐫 𝑗 | ) , 𝐩 𝑖 , 𝐫 𝑖 the momentum and positions for particle 𝑖 , respectively, and 𝑚 the mass of the particles. The factor 𝑁 in the potential energy term in Eq. (1) is introduced such that the totalenergy is extensive [24] (the so-called Kac factor). The one-particle distribution function 𝑓 ( 𝐩 , 𝐫 ; 𝑡 ) then satisfies theVlasov equation: ̇𝑓 ≡ 𝑑𝑓𝑑𝑡 = 𝜕𝑓𝜕𝑡 + 𝐩 𝑚 ⋅ 𝜕𝑓𝜕 𝐫 + 𝐅 ⋅ 𝜕𝑓𝜕 𝐩 = 0 , (2)where the mean-field force is given by: 𝐅 ( 𝐫 ; 𝑡 ) = − 𝜕𝜕 𝐫 ∫ 𝑉 ( 𝐫 − 𝐫 ′ ) 𝑓 ( 𝐫 ′ , 𝐩 ′ ; 𝑡 ) d 𝐩 ′ d 𝐫 ′ . (3) ORCID (s): (T.R. Filho)
LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 1 of 10 a r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l low dynamics and ergodicity Collisional effects modify the Vlasov equation such that ̇𝑓 = 𝐼 [ 𝑓 ] where the collisional integral 𝐼 [ 𝑓 ] is a functionalof 𝑓 , usually obtained using some approximation such as the weak coupling limit, the interparticle force is taken to beof order 𝛼 ≪ and 𝐼 [ 𝑓 ] is computed up to order 𝛼 , or retaining terms of order 𝑁 . The resulting kinetic equationsare called respectively the Landau and Balescu-Lenard equations [25]. For one-dimensional systems the collisionalintegral in the Balescu-Lenard, Landau and Boltzmann equations vanish identically in a homogeneous state and onemust go to the next term in the approximation, i. e. by computing 𝐼 [ 𝑓 ] up to order 𝛼 or 𝑁 [26, 27, 8, 9].Let us consider the example of a system with a vanishing collisional integral for both homogeneous and non-homogeneous states is given by 𝑁 identical particles in one dimension interacting only through zero-distance hard-core potential. In this case the interaction causes a swap of particle velocities, and by simply relabeling the particlesat the moment of the collision one obtains a statistically equivalent system of free particles such that the one-particledistribution function only evolves due to the free flux, and the corresponding kinetic equation if then given by theone-dimensional Liouville equation with zero force: 𝜕𝑓𝜕𝑡 + 𝑝𝑚 𝜕𝑓𝜕𝑥 = 0 , (4)where 𝑚 is the mass, 𝑥 is the position, and 𝑝 the momentum. For a homogeneous state the one-particle distributionfunction is strictly constant, i. e. the collisional integral vanishes identically.Another yet simple model, but with long range interacting particles and real collisions (due to the discontinuityin the force at zero distance) is the one-dimensional self-gravitating system of identical particle with unit mass andHamiltonian [28] 𝐻 = 𝑁 ∑ 𝑖 =1 𝑝 𝑖 𝑁 𝑁 ∑ 𝑖<𝑗 =1 ||| 𝑥 𝑖 − 𝑥 𝑗 ||| . (5)The force on particle 𝑖 is given by 𝐹 𝑖 = ( 𝑁 ( 𝑖 )− − 𝑁 ( 𝑖 )+ )∕ 𝑁 , where 𝑁 ( 𝑖 )+ and 𝑁 ( 𝑖 )− are the number of particles to theright and left of the particle 𝑖 , respectively, and particles can cross each other freely. The potential in this Hamiltonianis obtained from the solution of the Poisson equation in one spatial dimension and corresponds to a system of 𝑁 infinite sheets with total finite mass. The dynamics of this model has been studied in the literature in the last fewdecades, with the recurrent question if the system does relax to thermodynamic equilibrium, due to the extreme slowslow dynamics of its macroscopic parameters [29, 30, 28, 31, 32]. Joyce and Worrakitpoonpon introduced an orderparameter to measure the distance to equilibrium and showed that this system in a non-homogeneous state evolves tothermodynamic equilibrium [33]. They showed this for a number of particles up to 𝑁 = 800 , and yet requiring a verylarge simulation time to observe the complete relaxation. This implies that the contribution of the collisional integralof the corresponding kinetic equation is very small.The very slow relaxation towards equilibrium also manifests in the ergodic properties of the system. A systemwith long range interactions is ergodic if averages of observables over the history of a single particle are equal to theensemble average, i. e. to an average computed at a fixed time for the 𝑁 particles in the system. This approach wasused for the HMF model [15, 34] and for a two-dimensional self-gravitating system [10]. In the limit 𝑁 → ∞ thesesystems are non-ergodic, and never reach the true thermodynamic equilibrium, while for finite 𝑁 they are ergodiconly after a time window of the order of the relaxation time to equilibrium. Here we show that this results are alsovalid for the one-dimensional self-gravitating system with Hamiltonian in Eq. (5) in a non-homogeneous state, but notthe homogeneous case. Indeed in the former, we show that by properly considering periodic boundary conditions andthen taking the limit of the size of the unit cell going to infinity, while keeping the density constant, the one-particledistribution function does not evolve in time, i. e. the collisional effects vanish.The paper is structured as follows: in Section 2 we discuss separately the ergodic properties of homogeneous andnon-homogeneous states of the sheets model. The kinetic equation for the homogeneous state is obtained in Sec. 3with identically vanishing collisional contributions. We close the paper with some concluding remarks in Sec. 4.
2. Slow dynamics and ergodicity
We investigate the ergodic properties of the sheets model system using the approach in Ref. [10]. The system isergodic if time averages taken over a given time window of length 𝑡 𝑒 equals the ensemble average over the 𝑁 -particles LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 2 of 10low dynamics and ergodicity at this same fixed time 𝑡 𝑒 , which we call ergodicity time. We define the time average of the momentum of the 𝑘 -thparticle: 𝑝 𝑘 ( 𝑡 ) = 1 𝑀 𝑀 ∑ 𝑗 =1 𝑝 𝑘 ( 𝑗 Δ 𝑡 ) , (6)and similarly the time average of its position: 𝑥 𝑘 ( 𝑡 ) = 1 𝑀 𝑀 ∑ 𝑗 =1 𝑥 𝑘 ( 𝑗 Δ 𝑡 ) , (7)with a fixed time step Δ 𝑡 , 𝑀 = 𝑡 ∕Δ 𝑡 . We also consider the time dependent standard deviations (supposing the averagesover all particles vanish ⟨ 𝑥 ⟩ = 0 and ⟨ 𝑝 ⟩ = 0 ): 𝜎 𝑝 ≡ √√√√ 𝑁 𝑁 ∑ 𝑘 =1 𝑝 𝑘 ( 𝑡 ) , (8)and 𝜎 𝑥 ≡ √√√√ 𝑁 𝑁 ∑ 𝑘 =1 𝑥 𝑘 ( 𝑡 ) . (9)Ergodicity for a system with long range interaction is then equivalent to [10] 𝜎 𝑝 ( 𝑡 ) → and 𝜎 𝑥 ( 𝑡 ) → for 𝑡 → 𝑡 𝑒 . (10)It was shown for the HMF model and for a two-dimensional self-gravitating system that 𝑡 𝑒 ≈ 𝑡 𝑟 , with 𝑡 𝑟 the relaxationtime to thermodynamic equilibrium [10, 15, 34].We now consider separately the ergodic properties of non-homogeneous and homogeneous states of the sheetsmodel. In order to put in evidence the very large value of the ergodic time 𝑡 𝑒 we implemented a molecular dynamicssimulation of an open 𝑁 -particle system (no spatial boundary conditions) with Hamiltonian in Eq. (5) using and event-driven algorithm [35]. The dynamics between two successive particle crossings is integrable, and can be computedup to machine precision. Collisions are then implemented straightforwardly by updating the force on the particlesafter each crossing. Due to very high local densities at the core of the spatial distribution, a high numeric precision isrequired and we used quadruple precision in order to avoid missing any collision due to round-off errors (which indeedoccur for double precision). The initial state is a waterbag state defined by 𝑓 ( 𝑥, 𝑝 ; 0) = { 𝑝 𝑟 , if − 𝑥 < 𝑥 < 𝑥 and − 𝑝 < 𝑝 < 𝑝 , , otherwise , (11)with 𝑥 and 𝑝 given constants. To measure the distance to the Gaussian distribution we use the reduced moments: 𝜇 𝑘 ≡ ⟨ 𝑝 𝑘 ⟩⟨ 𝑝 ⟩ 𝑘 ∕2 . (12)The reduced moment of order 4 is called the Kurtosis of the distribution, and for any Gaussian distribution we havethat 𝜇 = 3 and 𝜇 = 15 . The left panel of Fig. 1 shows the time evolution of 𝜇 and 𝜇 for the system, with 𝑥 = 10 . and 𝑝 = 0 . for the initial condition. In this case the relaxation time to equilibrium is of the order of 𝑡 𝑟 ≈ 10 . Theright panel of Fig. 1 shows that the condition for ergodicity stated in Eq. (10) is satisfied for a value of time of the orderof magnitude of the relaxation time for equilibrium 𝑡 𝑒 ≈ 𝑡 𝑟 . LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 3 of 10low dynamics and ergodicity
In order to discuss the physical meaning of ergodicity for a long range interacting system, we define the one-particlemomentum and position probability densities 𝜙 ( 𝑝 ; 𝑡 ) and 𝜌 ( 𝑥 ; 𝑡 ) at a given time 𝑡 as the probability density for the givenvalue of 𝑝 and 𝑥 , respectively. We also define the density distribution for the values of 𝑝 and 𝑥 for a fixed particle, saythe 𝑘 -th particle, along its history, up to time 𝑡 , denoted by 𝑔 ( 𝑝 ; 𝑡 ) and ℎ ( 𝑥 ; 𝑡 ) respectively. Then, in the present case,ergodicity is equivalent to the relations 𝜙 ( 𝑝 ; 𝑡 ) = 𝑔 ( 𝑝 ; 𝑡 ) , (13)and 𝜌 ( 𝑥 ; 𝑡 ) = ℎ ( 𝑥 ; 𝑡 ) , (14)for 𝑡 ⪆ 𝑡 𝑟 ≈ 𝑡 𝑒 . Figures 2 and 3 show these distributions for a few values of time, and also the spatial distributionfunction at equilibrium given by 𝜌 ( 𝑥 ) = 𝐶 sech( 𝑥 ∕Λ) , with Λ = 4 𝑒 ∕3 and 𝑒 the mean-field energy per-particle [36],and the momentum Gaussian distribution at equilibrium. It is evident that the time and ensemble distributions becomevery close as 𝑡 approaches 𝑡 𝑟 . So the momentum 𝜙 and 𝑔 , and spatial 𝜌 and ℎ , distribution functions satisfy Eqs. (13)and (14) and are also equal to the equilibrium distribution for a time of the order of magnitude of the relaxation timeto equilibrium, as it was also observed for other long range interacting systems [15, 34, 10]. t σ x σ p × × × t µ µ Figure 1:
Left: Reduced moments 𝜇 and 𝜇 as a function of time for 𝑁 = 100 and a waterbag initial state with 𝑥 = 10 . and 𝑝 = 0 . for the system with Hamiltonian in Eq. (5) and open boundary conditions. A running average was performedover a time window of 𝛿𝑡 = 10000 . The straight lines correspond to the equilibrium values of 𝜇 = 3 and 𝜇 = 15 introducedfor comparison purposes. Right: Standard deviations for 𝑝 𝑘 and 𝑥 𝑘 in Eqs. (8) and (9). We now turn to the case of a homogeneous state. Periodic boundary conditions can be implemented using an Ewaldsum with a unit cell 𝑥 ∈ [− 𝐿, 𝐿 ] such that the force on each particle, due to the particles in the unit cell and the infinitenumber of images, is determined by a direct sum over replicas [37]. For the one-dimensional self-gravitating system aclosed analytical form was obtained by Miller and Rouet [38] as an additional potential representing all replicas, andgiven by: 𝑉 Ewald = − 1 𝑁 𝑁 ∑ 𝑖 =1 ( 𝑥 − 𝑥 𝑖 ) 𝐿 . (15)The full effective Hamiltonian with periodic boundary conditions is then 𝐻 = 𝑁 ∑ 𝑖 =1 𝐩 𝑖 𝑚 + 𝑉 ( 𝐱 ) , (16) LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 4 of 10low dynamics and ergodicity -6 -4 -2 0 2 4 6 p t=10 -6 -4 -2 0 2 4 6 p t=10 -6 -4 -2 0 2 4 6 p t= -6 -4 -2 0 2 4 6 p t= Figure 2:
Distributions 𝜙 ( 𝑝 ; 𝑡 ) (dotted line) and 𝑔 ( 𝑝 ; 𝑡 ) (histogram) for the same simulation as in Fig. 4 and for few valuesof 𝑡 . The dashed line is the equilibrium Gaussian with 𝛽 = 0 . . This value of 𝛽 was obtained by averaging the kineticenergy for a time window of size 𝛿𝑡 = 10 000 at the end of the simulation. The precision for the histogram for 𝑝 𝑘 ( 𝑡 ) wasincreased by collecting the values of the momenta of all particle from time 𝑡 −100 up to 𝑡 , justified by an expected negligiblechange in the statistical distribution for a relatively short period of time. with 𝐱 ≡ ( 𝑥 , … , 𝑥 𝑁 ) and 𝑉 ( 𝐱 ) = 1 𝑁 𝑁 ∑ 𝑖<𝑗 =1 ||| 𝑥 𝑖 − 𝑥 𝑗 ||| − 1 𝑁 𝑁 ∑ 𝑖 =1 ( 𝑥 − 𝑥 𝑖 ) 𝐿 . (17)The resulting equations of motion are then integrated using a fourth order symplectic integrator [39, 40]. The reducedmoments 𝜇 and 𝜇 as a function of time, up to 𝑡 = 10 , are shown in Fig. 4, for an initial waterbag state with 𝑥 = 𝐿 = 1 and 𝑝 = 3 . The system remains in a homogeneous state for the whole simulation time. We observe that the timeevolution is extremely slow if compared to the non-homogeneous case, with only a very small variation in 𝜇 visiblein the graphic. Figure 5 shows the distribution functions 𝑔 ( 𝑝 ; 𝑡 ) and 𝑔 ( 𝑥 ; 𝑡 ) at the final time, also clearly at varianceto what is observed for the non-homogeneous cases. Although the spatial distribution 𝑔 ( 𝑥 ; 𝑡 ) is roughly uniform, asexpected since particles can cross each other and the mean-field force is very small, the momentum distribution 𝑔 ( 𝑝 ; 𝑡 ) is not even symmetrical, as it is the case for the non-homogeneous systems at all time values, except for a very shortinitial time. We conclude that the time for ergodicity, if finite, is certainly many orders of magnitude greater that for anon-homogeneous state. We will shed some light and explain the physical origin of this difference, and of the peculiardynamics of the homogeneous state, in the next section by discussing the kinetic theory for a homogeneous state.
3. Kinetic equation for a homogeneous state
The statistical dynamics of a system of many particle systems can be studied by determining a kinetic equationdescribing the time evolution of the one-particle distribution function. We first define the 𝑁 -particle distribution LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 5 of 10low dynamics and ergodicity -40 -20 0 20 40 x t=10 -40 -20 0 20 40 x t=10 -40 -20 0 20 40 x t=10 -40 -20 0 20 40 x t=5x10 Figure 3:
Same as Fig. 2 but for ℎ 𝑡 ( 𝑥 ) and 𝜌 ( 𝑥 ; 𝑡 ) . The dashed line is the spatial distribution function at equilibrium 𝜌 ( 𝑥 ) = 𝐶 sech( 𝑥 ∕Λ) , Λ = 4 𝑒 ∕3 and 𝑒 the mean-field energy per-particle [36]. t µ µ Figure 4:
Reduced moments of 𝑝 for a homogeneous state with a waterbag initial condition with 𝑥 = 1 . and 𝑝 = 3 . . function 𝑓 𝑁 ( 𝑥 , 𝑝 , … , 𝑥 𝑛 , 𝑝 𝑛 ; 𝑡 ) as the probability density in the 𝑁 -particle phase space, which satisfies the Liouvilleequation. An usual starting point to determine a kinetic equation is the BBGKY hierarchy for the reduced distributionfunctions [25, 41]: 𝜕𝜕𝑡 𝑓 𝑠 (1 , … , 𝑠 ; 𝑡 ) = − 𝑠 ∑ 𝑘 =1 𝑝 𝑘 𝜕𝜕𝑥 𝑘 𝑓 𝑠 (1 , … , 𝑠 ; 𝑡 ) + 12 𝑠 ∑ 𝑘,𝑙 =1( 𝑘 ≠ 𝑙 ) 𝑉 ′ 𝑘𝑙 𝜕 𝑘𝑙 𝑓 𝑠 (1 , … , 𝑠 ; 𝑡 ) LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 6 of 10low dynamics and ergodicity -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 p x Figure 5:
Left: Distribution function 𝑔 𝑡 ( 𝑝 ) for 𝑡 = 10 for the same simulation as in Fig. 5. Right: distribution function 𝜌 𝑡 ( 𝑥 ) for the same simulation. +( 𝑁 − 𝑠 ) 𝑠 ∑ 𝑘 =1 ∫ d( 𝑠 + 1) 𝑉 ′ 𝑘, ( 𝑠 +1) 𝜕𝜕𝑝 𝑘 𝑓 𝑠 +1 (1 , … , 𝑠 + 1; 𝑡 ) , (18)where 𝑉 𝑗𝑘 ≡ 𝑉 ( 𝑥 𝑗 − 𝑥 𝑘 ) , 𝜕 𝑘𝑙 ≡ 𝑝 𝑘 𝜕 ∕ 𝜕𝑥 𝑘 − 𝑝 𝑙 𝜕 ∕ 𝜕𝑥 𝑙 , ≡ 𝑥 , 𝑝 , ≡ 𝑥 , 𝑝 , . . . , d1 ≡ d 𝑥 d 𝑝 , d2 ≡ d 𝑥 d 𝑝 , . . . , andso on. The 𝑠 -particle distribution function is defined by: 𝑓 𝑠 (1 , … , 𝑠 ; 𝑡 ) ≡ ∫ d1 ⋯ d( 𝑠 + 1) 𝑓 𝑁 (1 , … , 𝑁 ; 𝑡 ) . (19)The case with 𝑠 = 1 leads to the prototypical kinetic equation: [ 𝜕𝜕𝑡 + 𝑝 𝜕𝜕𝑥 ] 𝑓 (1; 𝑡 ) = ( 𝑁 − 1) 𝜕𝜕𝑝 ∫ d2 𝑉 ′12 𝑓 (1 , 𝑡 ) . (20)In order to obtain a close-form expression for the kinetic equation one must determine an expression for the two-particledistribution 𝑓 in terms of 𝑓 . For uncorrelated particles we have 𝑓 (1 , 𝑡 ) = 𝑓 (1; 𝑡 ) 𝑓 (2; 𝑡 ) and Eq. (20) then resultsin the Vlasov equation (2).For non-correlated particles we perform the cluster expansion [25]: 𝑓 (1 , 𝑡 ) = 𝑓 (1; 𝑡 ) 𝑓 (2; 𝑡 ) + 𝑔 (1 , 𝑡 ) ,𝑓 (1 , , 𝑡 ) = 𝑓 (1; 𝑡 ) 𝑓 (2; 𝑡 ) 𝑓 (3; 𝑡 ) + 𝑓 (1; 𝑡 ) 𝑔 (2 , 𝑡 ) + 𝑓 (2; 𝑡 ) 𝑔 (1 , 𝑡 )+ 𝑓 (3; 𝑡 ) 𝑔 (1 , 𝑡 ) + 𝑔 (1 , , 𝑡 ) , (21)and so on, where 𝑔 𝑠 is the 𝑠 -particle correlation function. By plugging Eq. (21) into Eq. (18) for 𝑠 = 1 we have 𝜕𝜕𝑡 𝑓 (1; 𝑡 ) = 𝑁 ∫ d2 𝑉 ′12 𝜕 [ 𝑓 (1; 𝑡 ) 𝑓 (2; 𝑡 ) + 𝑔 (1 , 𝑡 ) ] . (22)From Eqs. (18), (21) and (22) we obtain the following equation for the two-particle correlation function [25]: ( 𝜕𝜕𝑡 + 𝑝 𝜕𝜕𝑥 + 𝑝 𝜕𝜕𝑥 ) 𝑔 (1 , , 𝑡 ) = 𝑉 ′12 𝜕 𝑓 (1; 𝑡 ) 𝑓 (2; 𝑡 ) + 𝑉 ′12 𝜕 𝑔 (1 , 𝑡 )+ 𝑁 ∫ d3 [ 𝑉 ′13 𝜕 𝑓 (1; 𝑡 ) 𝑔 (2 , 𝑡 ) + 𝑉 ′23 𝜕 𝑓 (2; 𝑡 ) 𝑔 (1 , 𝑡 )+ ( 𝑉 ′13 𝜕 + 𝑉 ′23 𝜕 ) { 𝑓 (3; 𝑡 ) 𝑔 (1 , 𝑡 ) + 𝑔 (1 , , 𝑡 ) }] . (23)Then one determines a solution for 𝑔 in terms of 𝑓 , and also a solution for 𝑔 if it is not negligible (see [8] and [27]for systems where three-particle correlations are important), and use the result in Eq. (20) LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 7 of 10low dynamics and ergodicity -1 -0.5 0 0.5 1 x -0.03-0.02-0.0100.010.020.03 F g r a v N=2048N=20480N=204800N=2048000
Figure 6:
Force as a function of position for the Hamiltonian in Eq. (16) for the one-dimensional self-gravitating systemwith an additional potential from the Ewald sum, for different number of particles but same density 𝑛 = 𝑁 ∕ 𝐿 . The positionwas rescaled to the interval [−1 , for comparison purposes. We unit cell for 𝑁 = 2048 is given by 𝐿 = 20 , and is obtainedaccordingly for other values of 𝑁 in order to keep the spatial density constant. Now we turn to the one-dimensional self-gravitating system with Hamiltonian in Eq. (5) in a homogeneous state.The derivative of the total potential 𝑉 in Eq. (17) appears in Eq. (23) and we must consequently account for thesingularity of its derivative at zero inter-particle distance. We consider the following relabeling of particle indices: atthe moment two particles (sheets) cross each other, we interchange their labels. In this way, at each collision (at zerodistance) particles simple exchange their momenta and the force is constant in time. If the particles are initially labeledsuch that 𝑥 𝑖 < 𝑥 𝑗 if 𝑖 < 𝑗 , the ordering in position is preserved. Then the force on particle 𝑖 due to particle 𝑗 can bewritten as 𝐹 = 𝐹 grav + 𝐹 HC where 𝐹 grav = − 𝑉 ′ ( 𝑥 𝑖 − 𝑥 𝑗 ) , with 𝑉 given in Eq. (17), and 𝐹 HC stands for the hard-coreforce that swaps particle momenta when they collide at zero distance. The contribution of 𝐹 grav to Eq. (23) vanishesin the limit 𝐿 → ∞ as the gravitational force in a homogeneous state vanishes. To illustrate this fact, Figure 6 showsthe force 𝐹 grav due to both the self-gravitating potential and the Ewald sum, for a few values of the number of particles 𝑁 but for keeping the density 𝑛 = 𝑁 ∕ 𝐿 constant. We note that increasing 𝑁 in this way is not equivalent to considerthe thermodynamic limit that would correspond to take 𝑁 → ∞ but keeping 𝐿 constant. We observe that as thesize 𝐿 = 𝑁 ∕ 𝑛 of the unit cell increases 𝐹 grav approaches zero. As a consequence, only contributions from hard-corecollisions are retained in Eq. (23). This result in fact proves the validity of the Jeans Swindle for the model consideredhere, i. e. that the contribution of the background interaction to the infinite homogeneous contribution vanish, and onemust consider only the effects of local fluctuations in density [42, 43]. These fluctuations vanish as the size of the unitcell goes to infinity.The same reasoning can be used in an analogous way for the BBGKY hierarchy, which then take exactly the sameform as the hierarchy obtained for a system of particles with a hard-core potential at zero distance as only interaction.For such a system in an homogeneous state, the one-particle distribution function 𝑓 ( 𝑝 ; 𝑡 ) is strictly constant in time asthe interaction only swaps the momenta of two particles at each collision, and three-particles processes are nonexistent(the probability that three particles collide at the same time at the same point is zero). For the same initial condition, theBBGKY hierarchy being identical for both systems, the time evolution for the reduced distribution functions must bethe same, and therefore the distribution 𝑓 ( 𝑝 ; 𝑡 ) for a homogeneous one-dimensional self-gravitating system is constantin time. Small deviations from this are expected to occur in numerical simulations due to spurious non-physical effectsresulting from a finite value of 𝐿 , that result in small fluctuations of the value of the force around zero.
4. Concluding Remarks
We showed that the sheets model describing a one-dimensional self-gravitating system has profoundly differentdynamic properties weather it is in a homogeneous or a non-homogeneous state. In the former case we showed thatby considering a proper limit in the periodic boundary conditions the one-particle evolution function does not evolvein time, as its kinetic equation is essentially a Boltzmann-like equation. For the non-homogeneous state, the systemhas a slow dynamics to equilibrium, with a relaxation time much greater than other long range interacting systems if
LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 8 of 10low dynamics and ergodicity one uses the violent relaxation time for comparison. The non-homogeneous system is ergodic but only after a timeof the order of the relaxation time to equilibrium, as also observed for other long range interacting systems, but it isnon-ergodic in a homogeneous state, as illustrated by simulations presented here.A possible way to shed some light on the slow dynamics of this system in a non-homogeneous states is to obtaina kinetic equation, which for the present model is a challenging task as it requires the determination of action-anglevariables for the mean-field description of the system [44, 45], and has been possible only for very special cases(see [46] and references therein). This is the subject of ongoing research.
Acknowledgments
LFS was financed by CNPq (Brazil). TMRF was partially financed by CNPq under grant no. 305842/2017-0.
References [1] A. Campa, T. Dauxois, D. Fanelli and S. Ruffo,
Physics of Long-Range Interacting Systems , (Oxford Univ. Press, Oxford, 2014).[2]
Dynamics and Thermodynamics of Systems with Long-Range Interactions , T. Dauxois, S. Ruffo, E. Arimondo and M. Wilkens Eds. (Springer,Berlin, 2002).[3]
Dynamics and Thermodynamics of Systems with Long-Range Interactions: Theory and Experiments , A. Campa, A. Giansanti, G. Morigi andF. S. Labini (Eds.), AIP Conf. Proceedings Vol. 970 (2008).[4]
Long-Range Interacting Systems, Les Houches 2008, Session XC , T. Dauxois, S. Ruffo and L. F. Cugliandolo Eds. (Oxford Univ. Press, Oxford,2010).[5] A. Campa, T. Dauxois and S. Ruffo, Phys. Rep. , 57 (2009).[6] T. M. Rocha Filho, A. Figueiredo and M. A. Amato, Phys. Rev. Lett. , 190601 (2005).[7] T. M. Rocha Filho, M. A. Amato, A. E. Santana, A. Figueiredo and J. R. Steiner, Phys. Rev. E , 032116 (2014).[8] T. M. Rocha Filho, A. E. Santana, M. A. Amato and A. Figueiredo, Phys. Rev. E , 032133 (2014).[9] C. R. Lourenço and T. M. Rocha Filho, Phys. Rev. E , 012117 (2015).[10] C. H. Silvestre and T. M. RochaFilho, Phys. Lett. A , 337 (2016).[11] Y. Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois and S. Ruffo, Physica A , 36 (2004).[12] A. Campa, A. Giansanti and G. Morelli, Phys. Rev. E , 041117 (2007).[13] A. Campa, P. H. Chavanis, A. Giansanti and G. Morelli, Phys. Rev. E , 040102(R) (2008).[14] K. Jain, F. Bouchet and D. Mukamel, J. Stat. Mech. P11008 (2007).[15] A. Figueiredo, T. M. Rocha Filho and M. A. Amato, Europhys. Lett. , 30011 (2008).[16] T. M. Rocha Filho, M. A. Amato, and A. Figueiredo, Phys. Rev. E , 062103 (2012).[17] T. M. Rocha Filho, M. A. Amato, B. A. Mello, and A. Figueiredo, Phys. Rev. E , 041121 (2011).[18] J. Dawson, Phys. Fluids , 445 (1962).[19] B. N. Miller, K. Yawn and P. Youngkins, Ann. N. Y. Acad. Sci. , 268 (2008).[20] Y. Sota, O. Iguchi, M. Morikawa, T. Tatekawa and K. I. Maeda, Phys. Rev. E , 056133 (2001).[21] M. Antoni and S. Ruffo, Phys. Rev. E , 2361 (1995).[22] D. Lynden-Bell, Mon. Not. R. Astr. Soc. , 101 (1967).[23] W. Braun and K. Hepp, Commun. Math. Phys. , 125 (1977).[24] M. Kac, G. Uhlenbeck and P. Hemmer, J. Mathy. Phys. , 216 (1963).[25] R. Balescu, Statistical Mechanics - Matter out of Equilibrium , Imperial College Press (London, 1997).[26] M. M. Sano, J. Phys. Soc. Japan , 024008 (2012).[27] J.-B. Fouvry, B. Bar-Or and P. H. Chavanis, Phys. Rev.E , 052142 (2019).[28] K. R. Yawn and B. N. Miller, Phys. Rev. E , 056120 (2003).[29] C. J. Reidl Jr. and B. N. Miller, Astrophys. J. , 248 (1987).[30] T. Tsuchiya, N. Gouda and T. Konishi,Astrophys. Space Sci. , 319 (1997)[31] P. Valageas. Phys. Rev. E , 016606 (2006).[32] A. Gabrielli, M. Joyce and F.Sicard3, Phys. Rev. E , 041108 (2009).[33] M. Joyce and T. Worrakitpoonpon, J. Stat. Mech. P10012 (2010).[34] A. Figueiredo, T. M. Rocha Filho, M. A. Amato, Z. T. Oliveira Jr. and R. Matsushita, Phys. Rev. E , 022106 (2014).[35] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids , Clarendon Press (Oxford, 1987).[36] G. Rybicki G, Astrophys. Sp. Sci. , 56 (1971).[37] L. Hernquist, F. R. Bouchet, and Y. Suto, Astrophys. J. Suppl. Ser. , 231 (1991).[38] B. N. Miller and J-L. Rouet, Phys. Rev. E , 066203 (2010).[39] H. Yoshida, Phys. Lett. A , 262 (1990).[40] T. M. Rocha Filho, Comp. Phys. Comm. , 1364 (2014).[41] R. L. Liboff, Kinetic Theory - Classical, Quantum, and Relativistic Descriptions , 3rd ed, Springer-Verlag (New York, 2003).[42] M. K.-H. Kiessling, Adv. Appl. Math. , 132 (2003).[43] M. Falco, S. H. Hansen, R. Wojtak1 and G. A. Mamon, MNRASL , L6 (2013).[44] P.- H. Chavanis, Physica A , 3680 (2012). LF Souza and TM Rocha Filho:
Preprint submitted to Elsevier
Page 9 of 10low dynamics and ergodicity [45] P.- H. Chavanis, Physica A , 469 (2007).[46] F. P. C. Benetti and B. Marcos, Phys. Rev. E , 022111 (2017). LF Souza and TM Rocha Filho: