Velocity and Energy Distributions in Microcanonical Ensembles of Hard

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By the study of Boltzmann I have been un- a consequence of the CLT. Unfortunately this argument able to understand him. He could not under- is only approximately true in the case of large systems stand me on account of my shortness, and his and false for smaller systems. length was and is an equal stumbling block to In Sec. II we obtain the theoretical probability den- me. sity functions of the individual energies, velocity moduli and velocity components, starting from the fundamental More details on the relationship between Maxwell and uniform distribution law in phase space. In Sec. III we Boltzmann and on the influence of Maxwell on Boltz- present the molecular dynamics method used to simulate mann’s thought have been collected by Uffink [19]. hard spheres. Interestingly, the same distributions can be Tolman’s analysis of classical binary collisions for hard reproduced by a simple Monte Carlo stochastic model in- spheres led to rate equations which can be interpreted as troduced in Sec. IV. The numerical results are presented transition probabilities for a Markov chain after proper in Sec. V together with some statistical goodness-of-fit normalization. The interested reader can consult chap- tests. Indeed, it turns out that an equilibrium distri- ter V of Tolman’s classic book [13], in particular the bution of the velocity components seems to be reached discussion around Eq. (45.3) on page 129. The connec- already for N = 2 particles and without using any coarse tion with Markov chains was made explicit by Costantini graining. When N grows the equilibrium distribution ap- and Garibaldi [20, 21], who used a model due to Bril- proaches the normal distribution, Eq. (2). A discussion louin [22]. Before Costantini and Garibaldi, Penrose sug- and a summary follow in Sec. VI. gested that a Markovian hypothesis could justify the use of standard statistical mechanical tools [23]. According to our interpretation of Penrose, due to the limits in hu- man knowledge naturally leading to coarse graining, sys- II. THEORY tems of many interacting particles effectively behave as Markov chains. Moreover, the possible number of states We consider a fluid of N hard spheres in d dimensions of such a chain is finite even if very large, therefore only with the same diameter σ and mass m in a cuboidal the theory of finite Markov chains is useful. Statistical box with sides L , α = 1,...,d. The positions r , equilibrium is reached when the system states obey the α i i = 1,...,N, are confined to a d-dimensional box with equilibrium distribution of the finite Markov chain; this d equilibrium distribution exists, is unique and coincides volume V = α=1 Lα, i.e., each position component riα can vary in the interval [ Lα/2,Lα/2]. Elastic collisions with the stationary distribution if the chain is irreducible Q − and aperiodic. This point of view is also known as Marko- transfer kinetic energy between the particles, while the vianism. Indeed, in a recent paper on the Ehrenfest urn, total energy of the system or dogs and fleas model, we showed that, after appropri- N ate coarse graining, a Markov chain well approximates 1 1 E = m v2 = mv2 (4) the behavior of a realistic model for a fluid [24]. 2 i i 2 i=1 Here, we study the velocity distribution in a system of X N smooth elastic hard spheres in d dimensions. Even if the evolution of the system is deterministic, we can con- does not change in time, i.e., it is a constant of the mo- v sider the velocity components of each particle as random tion. Therefore, the velocities i are confined to the sur- variables. We do not consider a finitary [25] version of face of a hypersphere with radius R = 2E/m given by the model by discretizing velocities, but keep them as real the constraint that the total energy is E, i.e., each velocity component v can vary in thep interval [ R, R] variables. Then a heuristic justification of Eq. (2) can be iα − based on the central limit theorem (CLT). Here is the ar- with the restriction on the sum of the squares given q gument. Following Maxwell’s idea, one can consider the by Eq. (4). In other words, the rescaled positions velocity components of each particle independent from with qiα = riα/Lα are confined to the unit hypercube each other. Further assuming that velocity jumps after in dN dimensions, while the rescaled velocity compo- u v collisions are independent and identically distributed ran- nents = /R are confined to the surface of the unit dom variables, one obtains for the velocity component α hypersphere in dN dimensions defined by the constraint u = √u u = 1. of a particle i at time t · The state of the system is specified by the phase space n(t) vector of all velocities and positions Γ = (v, r), i.e., by viα(t)= viα(0) + ∆viα,j , (3) 2dN variables: the velocity components viα and the po- j=1 sition components riα. However, these variables are not X independent because of constraints. For spheres with where n(t) is the number of collisions for that particle up random velocities and positions confined in a container to time t and ∆viα,j is the change in velocity at collision with hard reflecting walls, the total energy E is conserved j. If the hypotheses stated above are valid, Eq. (3) de- and thus the degrees of freedom are g = 2dN 1; this fines a continuous-time random walk and the distribution is the microcanonical ensemble (constant NV E−). Peri- fviα (x, t) approaches a normal distribution for large t as odic boundary conditions conserve also the total linear 3 momentum is convenient to study the relationship between Eq. (10)

N N and the symmetric Dirichlet distribution with parameter a. P = m v = m v (5) i i i The PDF of the n-dimensional Dirichlet distribution i=1 i=1 X X with parameter vector a is and the generator of Galilean transformations to other n inertial frames of reference def 1 f D(x; a) = xai−11 (x); (11) X B(a) i S N N i=1 G = Pt m r = m (v t r ), (6) Y − i i i − i i=1 i=1 it is zero outside the unit simplex X X n where the coordinates ri are not reboxed upon a crossing n N S = x R : i xi 0 xi =1 . (12) of the unit cell boundaries (if P = 0, G/ i=1 mi = ∈ ∀ ≥ ∧ − ( i=1 ) G/(Nm) is the position of the center of mass), and thus X the− number of independent variables drops to Pg =2d(N − The normalization factor is the multinomial beta func- 1) 1=2dN 2d 1; this is the molecular dynamics tion, which can be deﬁned through the gamma function, ensemble− (constant− −NV EPG) [26–28]. Symmetries in the positions and velocities may reduce g further; e.g. if n Γ(a ) B(a)= i=1 i . (13) all components i of Γ are pairwise symmetric with respect Γ( n a ) to the origin, with both kinds of boundary conditions Q i=1 i this point symmetry will stay on forever and g = dN 1 The multinomial beta functionP is a generalization of the or g = 2d(N/2 1) 1 = dN 2d 1 respectively.− − − − − beta function or Euler integral of the ﬁrst kind, B(x, y)= For the sake of simplicity, in presenting the theory we 1 x−1 y−1 0 t (1 t) dt, which can be expressed through the will treat explicitly only the microcanonical case without gamma function− or Euler integral of the second kind, symmetries. R ∞ −t z−1 Γ(z) = 0 e t dt, as B(x, y) = Γ(x)Γ(y)/Γ(x + y). Following Khinchin [29], one can assume as the starting In the symmetric Dirichlet distribution all elements of point of statistical mechanics that the distribution in the R the parameter vector a have the same value ai = a, accessible portion of phase space is uniform, although so far this has not been rigorously proved in general. In n D def Γ(na) a−1 our case, the measure of the accessible region of phase f (x; a) = x 1S(x). (14) X [Γ(a)]n i space is the product of the volume of the hypercube V N i=1 Y times the surface of the hypersphere with radius R in dN dimensions, Notice that a = 1 gives the uniform distribution on S. It is convenient to work with dimensionless variables. 2πdN/2 The rescaling uiα = viα/R introduced above gives the Ω= V N RdN−1. (7) Γ(dN/2) PDF Γ(dN/2) Then Khinchin’s Ansatz that the probability density x 1 x fu( )= dN/2 {x:x=1}( ). (15) function (PDF) for points (v, r) in the permitted region 2π of phase space is uniform leads to the joint PDF for ve- A second transformation locities and positions w = u2 (16) 1 iα iα fv,r(x, y)= 1{x:x=R}(x)1{y:−Lα/2≤yiα≤Lα/2}(y), (8) Ω leads to a set of dN random variables each one with support in [0,1] and such that where 1A(x) is the indicator function of the set A,

N d def 1 if x A 1A(x) = ∈ . (9) wiα =1. (17) (0 if x / A i=1 α=1 ∈ X X As the energy does not depend on positions, one can inte- The Jacobian for this transformation is grate over the latter, yielding a uniform PDF for particle N d velocities on the surface of a hypersphere, ∂u 1 = w−1/2. (18) ∂w 2dN iα i=1 α=1 Γ(dN/2) 1−dN fv(x)= R 1 (x). (10) Y Y 2πdN/2 {x:x=R} Multiplying it by a factor 2dN because each u results ± iα The marginalization of this joint PDF leads to the distri- in the same wiα and by another factor 2 because of the butions of individual particle energies as well as of veloc- constraint given by Eq. (17) (for details see Song and ity moduli and velocity components. To this purpose, it Gupta [30]), and replacing √π = Γ(1/2), the joint PDF 4 of the variables wiα can be expressed through the sym- invoking the Dirichlet and beta distributions, by Shirts metric Dirichlet PDF with parameter a =1/2, et al. [28, Eq. (9)]. In the thermodynamic limit (N,V,E with D → ∞ fw(x)= fw (x;1/2) N/V = ρ = constant and E/N = E¯ = constant), N d Eq. (24) converges to a gamma distribution, as discussed Γ(dN/2) = x−1/21 (x). (19) by Garibaldi and Scalas [25, pages 121–122]. The gamma [Γ(1/2)]dN iα S i=1 α=1 PDF is Y Y xa−1 x Now the normalized energy per particle f γ (x; a,b) def= exp 1 (x). (25) X baΓ(a) − b [0,∞) 2 2 d Ei mv v ε = = i = i = w (20) A scale parameter b is usually included in the deﬁnition i E 2E R2 iα α=1 of the gamma distribution, but it can always be set to 1 X absorbing it into the argument, is the sum of d variables distributed according to Eq. (19). As a consequence of the aggregation law for Dirichlet x 1 x 1 f γ (x; a,b)= f γ ; a, 1 f γ ; a . (26) distributions, one ﬁnds that the joint PDF of all εi is X X b b ≡ X b b D fε(x)= fε (x; d/2) Coming back to the thermodynamic limit of Eq. (24) an- N ticipated above, this is a gamma distribution with shape Γ(dN/2) ¯ = xd/2−11 (x). (21) parameter a = d/2 and scale parameter b =2E/(dm)= [Γ(d/2)]N i S 2 i=1 R /(dN), Y ¯ It is interesting to notice that this is a uniform distribu- γ d 2E f i (x)= f x; , tion for d = 2; because of this, Boltzmann’s 1868 method E Ei 2 dm [10] works in d = 2 dimensions, but fails for d = 3. dm d/2 xd/2−1 dmx The PDF of the normalized energies of single parti- = exp 1 (x), (27) cles can be obtained by a further marginalization of the 2E¯ Γ(d/2) − 2E¯ [0,∞) symmetric Dirichlet distribution given by Eq. (21), using again the aggregation law. The result is a beta distribu- which is the familiar Boltzmann or Boltzmann-Gibbs dis- tion, whose PDF is tribution for d = 2. The PDF of the velocity moduli, or speeds, of in-

β def 1 a−1 b−1 dividual particles can be obtained from fEi (x) replac- fX (x; a,b) = x (1 x) 1[0,1](x), (22) 2 B(a,b) − ing Ei = mvi /2. The result is a transformed beta- Stacy distribution with the same exponents a = d/2 i.e. a Dirichlet distribution, Eq. (11), with n = 2. In and b = d(N 1)/2 as for the energies, but argument other words, the Dirichlet distribution is a multivariate mx2/(2E) = (x/R− )2, generalization of the beta distribution. Our case has the 2 2 exponents a = d/2 and b = d(N 1)/2, β x d d(N 1) d x − fvi (x)=f i ; , − , v R2 2 2 dx R2 β d d(N 1) d−1 f i (x)=f x; , − Γ(dN/2) 2x ε εi 2 2 = Γ(d/2)Γ(d(N 1)/2) Rd d Γ(dN/2) −1 d−(N−1) 2 −1 = x x2 2 Γ(d/2)Γ(d(N 1)/2) 1 1 (x) (28) d N−− 2 [0,R] ( 1) −1 × − R (1 x) 2 1 (x). (23) × − [0,1] for N > 1, and fvi (x) = δ(x R) for N = 1. Also The transformation of variables Ei = Eεi immediately this result has been obtained with− a different method leads to the beta-Stacy PDF of particle energies, [28, 31, 32]. In the thermodynamic limit, Eq. (28) converges to a β x d d(N 1) d x 2 f i (x)=f ; , − transformed gamma distribution with argument x /2, E Ei E 2 2 dx E shape parameter a = d/2 and scale parameter b = d −1 2 Γ(dN/2) x 2 R /(dN)=2E/¯ (dm), = Γ(d/2)Γ(d(N 1)/2) E − d(N−1) x2 d 2E¯ d x2 x 2 −1 1 γ fvi (x)= f ; , 1 1[0,E](x) (24) vi 2 2 dm dx 2 × − E E d dm 2 (x/2)d−1 dmx2 for N > 1, and fEi (x) = δ(x E) for N = 1. This = exp 1 (x), (29) − E¯ Γ(d/2) − 4E¯ [0,∞) result has been obtained with a different method, without 5 which is the familiar Maxwell distribution for d = 3. normal distribution for velocity components is recovered, The transformation from hyperspherical to as well as their independence. Finally, in the NV EPG 2 d 2 ensemble the constraint given by Eq. (5) leads to simi- cartesian coordinates vi = α=1 viα and d/2 d−1 d lar distributions with different parameter values for the (2π /Γ(d/2))v dvi = dviα leads from Eq. (28) i α=1 P relevant quantities introduced above. This will become to the PDF fvi (x) of the single-particle velocity vectors Q clearer in the following. d(N−1) 2 2 −1 Γ(dN/2) x 1{x:x=R}(x) fvi (x)= 1 , Γ(d(N 1)/2) − R2 (√πR)d − (30) III. MOLECULAR DYNAMICS SIMULATIONS an equation which has been obtained before too [28, 31]. The direct marginalization [30] of the joint PDF of all In molecular dynamics (MD) with continuous poten- velocities, Eq. (10), leads to the PDF fviα (x) of velocity tials, the equations of motion are integrated numerically components, a result obtained integrating over all i ex- using a constant time step; this approach is called time- cept one and over all α except one. This is the quantity driven. The larger the forces, the smaller the time step discussed by Maxwell [1], and its derivation for any N is necessary to ensure energy conservation. With step po- one of the main results in this paper. It turns out that the tentials there are no forces acting on a distance, only distribution of the velocity components is a transformed impulsive ones at the exact time of impact. Therefore an beta with argument 1/2+ x/(2R) and equal exponents event-driven approach is more appropriate: rather than a = b = (dN 1)/2, until a fixed time step, the system is propagated until − either the next collision or the next boundary crossing 1 [33–36]. fviα (x)= B (dN 1)/2, (dN 1)/2 The collision time t between two particles i, j can be − − ij dN−3 calculated from the mutual distance rij = ri rj and the 1 x 1 x 2 1[−R,R](x) − + relative velocity vij = vi vj . If bij = vij rij > 0 the × 2 2R 2 − 2R 2R − · particles are moving away from each other and will not dN−3 2 2 collide. Otherwise impact may happen at time tij when Γ(dN 1) x 1[−R,R](x) = − 1 their distance becomes equal to the sum of their radii, Γ2 (dN 1)/2 − R2 2dN−2R − i.e., rij + tij vij = σ. This is a second order problem 1 x dN 1 dN 1 withk solutions k =f β + ; − , − viα 2 2R 2 2 2 2 2 2 d 1 x bij bij vij (rij σ ) + . (31) t± = − ± − − . (34) × dx 2 2R ij q v2 ij In the thermodynamic limit Eq. (31) converges to a Gaus- If the solutions are complex, no collision occurs. If the 2 2 sian with average µ = 0 and variance σ = R /(dN) = solutions are real, the smaller one, t−, corresponds to dE/¯ (2m), i.e. the familiar Maxwell-Boltzmann distribu- ij when the particles first meet, while the larger one, t+, to tion ij when they leave each other assuming they are allowed to m mx2 interpenetrate. A negative collision time means that the f iα (x)= exp . (32) v dπE¯ − dE¯ event took place in the past. Because of the condition r + − bij < 0, at least tij > 0. If tij < 0 the particles overlap, This is again related to a gamma distribution, since the which indicates an error. So the collision time is given − positive half of a Gaussian can be expressed as by tij , provided it is a positive real number. For a system of N hard spheres, at impact, assuming 2 x2 x2 1 d x2 exp 1 (x)= f γ ; , σ2 . an elastic collision, the total kinetic energy E and the √ −2σ2 [0,∞) X 2 2 dx 2 total linear momentum P are conserved (usually one sets 2πσ (33) P = 0 at the beginning of the simulation by subtracting In summary, all the known results for the relevant dis- 1 N N i=1 vi from each vi). Assuming smooth surfaces, tributions of the NV E ensemble can be obtained observ- the impulse acts along the line of centers of the collision P ing that the normalized individual particle energies εi = partners i and j given by the unit vector ˆrij = rij /rij ; Ei/E follow a symmetric multivariate Dirichlet distribu- with equal masses, vi changes to vi +∆vi and vj changes tion with parameter a = 1/2 given by Eq. (19). This is to vj ∆vi with a direct consequence of the uniform-distribution assump- − tion in Eq. (8) via a simple change of variables. Only for b r ∆v = ij ij = (v ˆr ) ˆr = vk , (35) the velocity components it is necessary to marginalize the i − σ2 − ij · ij ij − ij uniform distribution directly on the surface of the hypersphere and not on the simplex. Maxwell’s Ansatz is vin- where rij and vij are evaluated at the instant of collision, dicated by the fact that, in the thermodynamic limit, a and thus rij = σ. 6

A few computational details for our event-driven and hard spheres (d = 3) from theory (Sec. II) as well molecular dynamics simulation of hard spheres are given as from MD (Sec. III) and MC simulations (Sec. IV) are in the appendix. shown in Fig. 1 for a microcanonical ensemble (constant NV E, hard reflecting walls) and in Fig. 2 for a molecular dynamics ensemble (constant NV EPG, periodic bound- IV. MONTE CARLO SIMULATIONS ary conditions). Simulations with N = 10000 were done too, but are not shown because the distributions over- Except for especially ordered initial conditions, in- lap perfectly with those where N = 1000. In reduced terparticle collisions computed by MD as explained in units [35] the particle mass is m = 1, the particle diameter is σ = 1, the energy per particle is E¯ = 1, the Sec. III have mutual distance versors at collision ˆrij uniformly distributed on a unit half sphere in d dimensions Boltzmann constant is kB = 1, and the number density is ρ =2/3d. The density appears only in MD, and the re- such that, given relative velocities vij , the scalar prod- sults are largely independent of this parameter, as long as uct vij ˆrij is negative. Therefore the same distributions of velocities,· and thus of derived quantities like energies, it is not too large (in this case the particles cannot move as in MD with periodic boundaries can be obtained by freely) or too small (in this case the particles hardly ever Monte Carlo (MC): after initializing the velocities of all collide). The initial total momentum is P = 0, except hard spheres, the MC cycles consist in selecting a pair ij for N = 2 with hard reflecting walls, and the initial posi- and a random versor ˆr such that v ˆr < 0, and then tion of the center of mass is in the origin. The numerical ij ij ij 5 in updating the velocities according to· Eq. (35). Hard simulations were equilibrated over 5 10 collisions and 6 × reflecting walls can be included in the MC scheme by se- sampled over 10 collisions. lecting with a certain frequency a sphere i and inverting We did not do the case with d = 1 because then with equal masses Eq. (35) becomes ∆v = vk = v ; thus one of its velocity components viα. This scheme gives a i − ij − ij useful insight into the mechanism of energy and momen- after a collision vi becomes vj and vice versa, and with tum transfer. It is much easier to code and faster to run particles just exchanging their velocities, the velocity dis- than MD, especially for large numbers of particles N, be- tribution does not change with time, making equilibra- cause no event list management is necessary: on the same tion impossible. For this reason, one-dimensional models computer used for the MD benchmarks shown in Fig. 3, need systems of molecules with different masses [17]. the CPU time for 105 MC collisions with N = 10000 The agreement between theory, MD and MC is excel- spheres is 0.3 s. lent, with little systematic deviations only for MD at the For a given initial state, i.e. a set of particle veloci- smallest values of N. To check that these deviations are ties, the MC dynamics defined above provides the real- genuine, we tried with different starting configurations ization of a Markov chain with a symmetric transition and densities, and with both MD programs discussed in kernel, meaning that P (v′ v) = P (v v′), where v is the Sec. III, obtaining identical results. For further details old velocity vector before| the transition| and v′ is the on our MD simulations see Gabriel [39, Chapter 3]. new velocity vector after the transition. This Markov For d = 2 and N = 2 in the NV EPG ensemble fviα (x) chain is homogeneous, as the transition probability does is the arcsine PDF not depend on the time step. Invoking detailed balance, 1 ′ ′ ′ fviα (x)= , (36) P (v v)P (v)= P (v v )P (v ), the symmetry of the tran- π√R2 x2 sition| kernel implies| that the stationary distribution of − this chain is uniform over the set of accessible states. If which is bimodal. The name is due to its cumulative this set coincides with the surface of the velocity hyper- distribution function sphere, then the Markov chain is ergodic and one can 1 x 1 hope to prove that the uniform distribution over the Fviα (x)= arcsin + . (37) π R 2 hypershere is also the equilibrium distribution for the Markov chain; see Sigurgeirsson [37, Chapter 5] for the For d = 2 and N = 2 in the NV E ensemble and for discussion of a related problem, and Meyn and Tweedie d = 2 and N = 3 in the NV EPG ensemble, fviα (x) is [38] for general methods. The results of MC simulations the Wigner semicircle PDF [40] described below corroborate this conjecture and the al- 2 gorithm outlined above is indeed an effective way of sam- 2 2 fviα (x)= 2 R x . (38) pling the uniform distribution on the surface of a hyper- πR − p sphere. Its cumulative distribution function again contains an arcsine:

x 2 2 1 x 1 V. NUMERICAL RESULTS F iα (x)= R x + arcsin + . (39) v πR2 − π R 2 p Probability density functions fviα of the velocity com- For the same systems, fEi (x) is the uniform distribution ponents, fvi of the velocity modulus, and fEi of the en- on [0, R]. For d = 3 and N = 2 in the NV EPG ensem- ergy for N = 2, 3, 4, 10, 100, 1000 hard disks (d = 2) ble, f iα (x) is the uniform distribution on [ √R, √R]. v − 7

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( (

− − − − − − x x

1.0 1.0 d=2, walls d=3, walls MD, N=2 MD, N=2 MD, N=3 MD, N=3 0.8 MD, N=4 0.8 MD, N=4 MD, N=10 MD, N=10 MD, N=100 MD, N=100 MD, N=1000 MD, N=1000 0.6 MC, N=2 0.6 MC, N=2

) MC, N=3 ) MC, N=3 x x

( MC, N=4 ( MC, N=4 i i

v MC, N=10 v MC, N=10 f f 0.4 MC, N=100 0.4 MC, N=100 MC, N=1000 MC, N=1000

0.2 0.2

0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 x x

1.0 1.0 d=2, walls d=3, walls MD, N=2 MD, N=2 MD, N=3 MD, N=3 0.8 MD, N=4 0.8 MD, N=4 MD, N=10 MD, N=10 MD, N=100 MD, N=100 MD, N=1000 MD, N=1000 0.6 MC, N=2 0.6 MC, N=2

) MC, N=3 ) MC, N=3 x x

( MC, N=4 ( MC, N=4 i i

E MC, N=10 E MC, N=10 f f 0.4 MC, N=100 0.4 MC, N=100 MC, N=1000 MC, N=1000

0.2 0.2

0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 x x

FIG. 1. (Color online) Probability density functions fviα of the velocity components (top), fvi of the velocity modulus (middle), and fEi of the energy (bottom) for d = 2 (left) and d = 3 (right) with E¯ = 1 in the microcanonical ensemble (constant NVE, hard reﬂecting walls): theory (lines), MD (full symbols), and MC (empty symbols).

All distributions are given by Eqs. (24), (28) and (31) 10 000. In all cases the null hypothesis of data distributed inserting the appropriate values of d, N and E = NE¯ according to the model equation cannot be rejected at or R = 2E/m for the NV E ensemble, while for the the 5% signiﬁcance level. The empirical density fviα (x) NV EPG ensemble N must be substituted by N 1 be- is well approximated by a normal law for N 1000 hard cause ofp the additional constraint on the linear momen-− disks, as shown in Tab. II, where the results≥ of two non- tum and thus on the center of mass. parametric tests for normality, Lilliefors [44] and Jar- que and Bera [45, 46], are presented for the MC veloc- To quantify the visual impression, in Tab. I we show ity components of the systems with periodic boundary Kolmogorov-Smirnov goodness-of-ﬁt tests [41–43] com- conditions, d = 2 and N = 10, 100, 1000, 10000; these paring Eq. (31) with the empirical cumulative distri- tests were made with the Matlab functions lillietest and bution function of the MC velocity components in the jbtest. NV EPG ensemble for d = 2 and N = 2, 3, 10, 100, 1000, 8

) ) x x

( (

− − − − − − x x

1.0 1.0 d=2, periodic d=3, periodic MD, N=2 MD, N=2 MD, N=3 MD, N=3 0.8 MD, N=4 0.8 MD, N=4 MD, N=10 MD, N=10 MD, N=100 MD, N=100 MD, N=1000 MD, N=1000 0.6 MC, N=2 0.6 MC, N=2

) MC, N=3 ) MC, N=3 x x

( MC, N=4 ( MC, N=4 i i

v MC, N=10 v MC, N=10 f f 0.4 MC, N=100 0.4 MC, N=100 MC, N=1000 MC, N=1000

0.2 0.2

0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 x x

1.0 1.0 d=2, periodic d=3, periodic MD, N=2 MD, N=2 MD, N=3 MD, N=3 0.8 MD, N=4 0.8 MD, N=4 MD, N=10 MD, N=10 MD, N=100 MD, N=100 MD, N=1000 MD, N=1000 0.6 MC, N=2 0.6 MC, N=2

) MC, N=3 ) MC, N=3 x x

( MC, N=4 ( MC, N=4 i i

E MC, N=10 E MC, N=10 f f 0.4 MC, N=100 0.4 MC, N=100 MC, N=1000 MC, N=1000

0.2 0.2

0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 x x

FIG. 2. (Color online) Probability density functions fviα of the velocity components (top), fvi of the velocity modulus (middle), and fEi of the energy (bottom) for d = 2 (left) and d = 3 (right) with E¯ = 1 in the molecular dynamics ensemble (constant NVEPG, periodic boundary conditions): theory (lines), MD (full symbols), and MC (empty symbols). Delta functions are made visible by a vertical line for the theory and by rescaling down to 1 the data point that would otherwise be out of scale.

VI. DISCUSSION AND CONCLUSIONS tions with different arguments and shape or scale parameters, corresponding respectively to the Gaussian, i.e. Maxwell-Boltzmann, the Maxwell, and the Boltzmann To summarize what we have done, in a system of N or Boltzmann-Gibbs distribution. We showed this the- hard balls in a d-dimensional volume V the velocity com- oretically using Khinchin’s Ansatz, and performed sta- ponents, the velocity modulus and the energies of the tistical goodness-of-fit tests on systematic MD and MC spheres or disks are well reproduced by transformed beta computer simulations of an increasing number N of hard distributions with different arguments and shape param- disks or spheres starting from 2 in the microcanonical eters depending on N, d, the total energy E, and the ensemble (constant NV E, hard reflecting walls) and in boundary conditions; in the thermodynamic limit these the molecular dynamics ensemble (constant NV EPG, distributions converge to transformed gamma distribu- 9

N Kolmogorov-Smirnov pKS latter both by MD and MC with a systematic investi- 2 4.1 × 10−4 0.99 gation of parameter values and boundary conditions; we 3 1.0 × 10−3 0.24 obtained also the PDF of velocity components, Eq. (31), 10 1.3 × 10−3 0.07 which was not possible with previous approaches; we re- 100 7.1 × 10−4 0.69 alized that all these distributions are variants of the beta 1000 6.2 × 10−4 0.84 or the gamma distribution, and can be derived from the 3 10 000 1.1 × 10− 0.15 Dirichlet distribution; we pointed out that for values of N as low as 2 or 3 the shapes of these distributions can be quite different from those in the thermodynamic limit: TABLE I. Comparison between the empirical probability density functions of the velocity components from MC with in particular, they can become uniform or even bimodal; periodic boundary conditions and d = 2 by means of the last, we discussed the slight deviations of the MD results Kolmogorov-Smirnov (KS) test. In each case the sample size from theory. is 2 × 106. At the 5% significance level the critical value is The significance of our investigations goes beyond the 4 9.6 × 10− . The null hypothesis of equally distributed data foundations of statistical mechanics: few-body systems can never be rejected. and microclusters are of current practical interest in applied fields such as nanotechnology and biophysics, and there is an increasing effort to understand the statistical N Lilliefors pL Jarque-Bera pJB 10 0.008* < 10−3 7.4 × 103* < 10−3 mechanics of such systems, which departs from the tra- 100 7.36 × 10−4* 0.01 29.6* < 10−3 ditional approach in the thermodynamic limit [47]. An- 1000 4.79 × 10−4 0.35 5.70 0.06 other application of our results may be in the theory of 10 000 4.37 × 10−4 0.50 2.36 0.31 thermostats or heat baths with a finite thermal capacity [48]. The MD simulations presented above corroborate TABLE II. Results of two non-parametric normality tests for Boltzmann’s ergodic hypothesis [49] for both the NV E the empirical probability density function of the velocity com- and the NV EPG ensembles. Sinai [50] updated this hy- ponents from MC with periodic boundary conditions when d = 2: Lilliefors (L) and Jarque and Bera (JB). In each case pothesis translating it in modern mathematical terms. the sample size is 2 × 106. At the 5% significance level the One should prove that every hard-ball system on a flat critical value is 6.43 × 10−4 for the L test and 5.99 for the JB torus, after fixing its total energy, momentum, and cen- test. The star indicates that the null hypothesis of normally ter of mass, is fully hyperbolic and ergodic; hyperbolic distributed data can be rejected. means that its Lyapunov exponent is non-zero almost everywhere with respect to the Liouville measure. This rephrasing of Boltzmann’s hypothesis is known as the periodic boundary conditions). The MC simulations are Boltzmann-Sinai ergodic hypothesis. The proofs of er- a simple stochastic model based on a generalization of godicity for similar systems use the so-called Chernov- Eq. (3) able to reproduce the same empirical equilibrium Sinai Ansatz, namely the almost sure hyperbolicity of distribution for the random variables viα, vi and Ei as singular orbits [51]. More recently, after proving the er- obtained deterministically with canonic dynamics by MD godiciy of hard disks [52], Simányi published a proof of simulations. While the MC results overlap perfectly with the Boltzmann-Sinai ergodic hypothesis in full generality theory, there is a slight disagreement of the MD results for hard-ball systems [53]. for the smallest values of N. A definitive explanation of MD simulations show that Khinchin’s Ansatz is justi- this will require further investigation. However, one can fied for systems of hard balls. We would like to stress recall that our MC scheme acts only on velocities inde- that this is a consequence of the microscopic dynamics pendently of positions and the volume V , with the inter- and not of any a priori maximum-entropy principle. The particle versor at collision sampled uniformly on a unit uniform distribution on the accessible phase-space region half sphere in d dimensions. On the contrary, due to the is indeed the maximum-entropy distribution. Therefore, canonical dynamics, in MD velocities and positions are maximum-entropy methods do work well and all the dis- mutually dependent; therefore geometrical constraints in tributions in Sec. II could be obtained by maximum- the smallest systems may lead to a non perfectly uni- entropy methods: the beta and gamma distributions are form sampling of the state space. Moreover, in the MC actually the maximum-entropy distributions with given systems with walls the correlation among velocities at first moment, and possibly some other constraint, on a impact is below 0.001 for every value of N (random vari- finite and a semi-infinite interval respectively. However, ables uniformly distributed on the hypersphere are un- this is so only because the dynamics uniformly samples correlated even if dependent), so that the assumptions of the accessible phase-space region and not the other way molecular chaos are always fulfilled, whereas a computer round. In different frameworks, e.g. in biology or eco- simulation of a one-dimensional system has shown that nomics, maximum-entropy assumptions might lead to in MD this happens only with growing N [17]. wrong results for the equilibrium distribution of a sys- We presented comprehensively both analytical deriva- tem, if its dynamics is not specified or carefully studied. tions with a new approach and numerical checks, the The distributions derived in Sec. II are a benchmark for 10 random partition models popular in econophysics. Pure 107 search over all pairs exchange models often lead to the same distributions [25, 2 6 f(x) = 0.044x 54–59]. 10 105

4 ACKNOWLEDGMENTS 10 103

We are thankful to Ubaldo Garibaldi for pointing us to 2 the Dirichlet distribution as the multivariate generaliza- CPU time (s) 10 tion of the beta distribution, and to Mauro Politi for ob- 101 serving that the beta distribution maximizes the entropy 0 on a finite interval given two constraints that include the 10 first moment, even if we did not use this property in the 10-1 0 1 2 3 4 theory exposed in Sec. II. This paper was partially sup- 10 10 10 10 10 ported by Italian MIUR grant PRIN 2009 H8WPX5 “The N growth of firms and countries: distributional properties 200 search over cells and economic determinants — Finitary and non-finitary f(x) = 0.017x + 2.269 probabilistic models in economics”. The support of the UK Economic and Social Research Council (ESRC) in 150 funding the Systemic Risk Centre is gratefully acknowl- edged (grant number ES/K002309/1). 100 4 pairs 3 cells

APPENDIX: COMPUTATIONAL DETAILS OF CPU time (s) 2 THE MOLECULAR DYNAMICS SIMULATIONS 50 1 CPU time (s) 0 2 3 4 5 6 7 8 When a particle reaches a side of the unit box, peri- N odic boundary conditions may require to “rebox” it by 0 reintroducing it on the other side, while hard reflecting 0 2000 4000 6000 8000 10000 12000 walls require to invert the velocity component perpen- N dicular to the wall. After an event, be it a collision with 5 another particle, a boundary crossing or a reflection at FIG. 3. (Color online) CPU time for 10 collisions on a 2 GHz a boundary, the event calendar must be re-evaluated for Intel Core2 Duo as a function of the number of hard spheres pairs involving one of the event participants or a particle N for our two event-based MD programs, one with a simple search over all pairs (top), the other with an optimized scheduled to collide with one of the event participants. search over cells (bottom). The data are fitted respectively by All other particles are not influenced. Thus not every a quadratic and a linear function, which cross-over between scheduled event actually takes place, because it can be N = 3 and N = 4 (inset). The CPU times for the search over invalidated by another earlier event, in which case it is all pairs with N ≥ 1000 were extrapolated from the CPU erased from the priority queue. The latter is most com- times for 100 collisions. monly handled by means of a binary tree [60], which we realized with a multimap of the C++ Standard Template Library [61]. The efficiency of this and alternative data structures for event scheduling has been analyzed exten- sively [62, 63]. The computational effort to search for mini,j tij grows as the square of the number of particles; see Fig. 3 (top). For large systems it is advisable to divide the simulation box into cells [26], which makes the dependence of the CPU time on the number of particles linear; see Fig. 3 (bottom). Provided cell boundary crossings are consid- ered too in the event list, two particles can collide only if they are located in the same cell or in adjacent cells. We chose cells with a side larger than a particle diameter. For more details on this and other algorithmic aspects in event-driven MD see Refs. [64–66]. For a parallel imple- mentation see Miller and Luding [67]. 11

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