Hyperbolic Anosov C-systems. Exponential Decay of Correlation Functions
NNRCPS-HE-05-2017
Hyperbolic Anosov C-systemsExponential Decay of Correlation Functions
George Savvidy and Konstantin SavvidyInstitute of Nuclear and Particle PhysicsDemokritos National Research Center, Ag. Paraskevi, Athens, Greece
Abstract
The uniformly hyperbolic Anosov C-systems defined on a torus have exponential instability oftheir trajectories, and as such C-systems have mixing of all orders and nonzero Kolmogorov entropy.The mixing property of all orders means that all its correlation functions tend to zero and thequestion of a fundamental interest is a speed at which they tend to zero. It was proven that thespeed of decay in the C-systems is exponential, that is, the observables on the phase space becomeindependent and uncorrelated exponentially fast. It is important to specify the properties of theC-system which quantify the exponential decay of correlations. We have found that the upper boundon the exponential decay of the correlation functions universally depends on the value of a systementropy. A quintessence of the analyses is that local and homogeneous instability of the C-systemphase space trajectories translated into the exponential decay of the correlation functions at therate which is proportional to the Kolmogorov entropy, one of the fundamental characteristics ofthe Anosov automorphisms. This result allows to define the decorrrelation and relaxation times ofa C-system in terms of its entropy and characterise the statistical properties of a broad class ofdynamical systems, including pseudorandom number generators and gravitational systems. a r X i v : . [ m a t h - ph ] F e b Introduction
A uniformly hyperbolic Anosov C-systems defined on a torus have exponential instability of alltrajectories [1] and as such have mixing of all orders and nonzero Kolmogorov entropy [1, 2, 3, 4, 5, 6,7, 8]. The statistical properties of a deterministic dynamical system essentially depend on behaviourof the correlation functions defined on a corresponding phase space. The question of a fundamentalinterest is a speed at which correlation functions of C-systems tend to zero [9, 10, 12, 13, 17, 18].It was proven that for the hyperbolic Anosov C-systems the speed of decay is exponential, thatis, the observables on the phase space become independent and uncorrelated exponentially fast[21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].
It is important to specify the properties of theC-system which quantify the exponential decay of the correlation functions .In this paper we shall study statistical properties of observables { f ( x ) } defined on an N -dimensional torus phase space M ( x ∈ M ) of the Anosov C-system diffeomorthisms T and specifythe rate at which the exponential decay takes place. The statistical properties of the deterministicdynamical system defined by the map {∀ x ∈ M : x → x n = T n x } are characterised by the be-haviour of the corresponding correlation functions D n ( f, g ). The very fact that the C-systems havemixing of all orders means that the correlation function of any two observables f ( x ) and g ( x ) tendsto zero ∗ when iteration/interaction time t = n tends to infinity n → ∞ : D n ( f, g ) = h f ( x ) g ( T n x ) i − h f ( x ) ih g ( x ) i → . This function measures the dependence between the values of f ( x ) at zero time and values of g ( x )at the time n and tells that the overlapping integral between observables f ( x ) and g ( T n x ) tends tozero, so that they become independent and uncorrelated. A fundamental question which was raisedin this respect is a question of a speed at which a correlation function D n ( f, g ) tends to zero in (1.1).It was proven in the mathematical literature [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] thatfor the hyperbolic Anosov diffeomorphisms and continuous flows the speed of decay is exponential: | D n ( f, g ) | = |h f ( x ) g ( T n x ) i − h f ( x ) ih g ( x ) i| ≤ C ( f, g ) Λ n = C ( f, g ) exp ( − n log 1 / Λ) , where the exponential upper bound on the correlation functions Λ ( T,f,g ) < T and of the observables f ( x ) and g ( x ). A constant C ( f, g ) depends only onthe observables f ( x ) and g ( x ). This outstanding result tells that the observables on the phase spaceof a C-system become uncorrelated exponentially fast and represent independent random variables.In statistical physics the autocorrelation functions D n ( f, f ) define the important physical propertiesof a dynamical system T , such as its relaxation time τ , as well as temperature, diffusion, viscosityand other macroscopic characteristics [11, 13, 18, 19]. ∗ In fact, correlation functions of any number of observables h f ( T n x ) f ( T n x ) ....f r ( T n r x ) i of a C-system tend tozero as n i → ∞ [1, 17]. We denote by h f i the average value of a function f with respect to invariant measure on M .
1t is important to express Λ and the relaxation time τ in terms of C-system quantitative char-acteristics. For that one should specify the properties of a C-system which quantify the exponentialdecay of the correlation functions in (1.1). We have found that the upper bound on the exponentialdecay of the correlation functions is universal and is defined by the value of the system entropy h ( T ): | D n ( f, g ) | ≤ C e − nh ( T ) ν , (1.1)where C ( f, g ) and ν ( f, g ) depend only on observables and are positive numbers. This result allowsto define the decorrelation time τ for a physical observable f ( x ) as τ = 1 h ( T ) ν f . (1.2) A local and homogeneous instability of the C-system phase trajectories is translated into the exponen-tial decay of the correlation functions at rate which is expressed in terms of the system entropy h ( T ) . The expression of the decorrelation time in terms of system entropy allows to characterise statisticalproperties of a broad class of dynamical systems, including gravitational systems and pseudorandomnumber generators [13, 18, 14, 15, 16].When the dimension N of the C-system (2.10) on a torus is increasing, its index ν f is increasinglinearly with dimension ν f = 2 pN , where p is the order of smoothness of the observable/function f ( x ). The entropy h ( T ) of the C-system (2 .
11) increases linearly as well h ( T ) = π N , therefore τ = π pN . (1.3)Considering a set of initial trajectories occupying a small volume δv in the phase space of a C-system, one can ask how fast this small phase volume will be uniformly distributed over the wholephase space. This characteristic time interval τ defines the relaxation time at which the systemreaches a stationary distribution. Because the entropy defines the expansion rate of the phase spacevolume one can derive that [13] τ = 1 h ( T ) ln 1 δv . (1.4)Thus there are three characteristic time scales associated with the C-system [13]: Decorrelation timeτ = π pN < Interaction timet int = n = 1 < Stationary distribution timeτ = h ( T ) ln δv . (1.5)This result defines important physical characteristics of the C-systems and measures the ”level ofchaos” developed in the system and justifies the statistical/probabilistic description of the system[11]. Indeed, the appearance of well developed statistical properties has important consequencesin the form of the central limit theorem for Anosov diffeomorphysms. The time average of theobservable f ( x ) on M ¯ f n ( x ) = 1 n n − X i =0 f ( T i x )2ehaves as a superposition of quantities which are statistically independent [36]. It has been proventhat the fluctuations of the time averages (1.6) from the phase space average h f i = Z M f ( x ) dµ ( x )multiplied by √ n have at large n → ∞ the Gaussian distribution [35, 36, 37, 38, 39]:lim n →∞ µ (cid:26) x : √ n ¯ f n ( x ) − h f i ! < z (cid:27) = 1 q πσ f Z z −∞ e − y σ f dy , where the value of the standard deviation σ f is a sum σ f = h f ( x ) i − h f ( x ) i + 2 + ∞ X n =1 [ h f ( T n x ) f ( x ) i − h f ( x ) i ] . Using our result (1.1) one can explicitly estimate the standard deviation in terms of entropy: σ f ≤ C e − h ( T ) ν − e − h ( T ) ν . The earlier publications concerning the application of the modern results of the ergodic theoryto concrete physical systems can be found in [11, 12, 13, 17, 18, 19]. These articles contain reviewmaterial as well. The present paper is organised as follows. In section two we shall overview thebasic properties of a C-system defined on a high dimensional torus, its spectral properties and itsentropy. In section three we shall calculate the correlation functions and shall express the upperbound on the correlation functions in terms of entropy for the system on two-dimensional torus.In the fourth section we shall extend these results to the high dimensional C-systems and shalldefine three characteristic time scales associated with the C-systems. In section five the time scalesassociated with the MIXMAX pseudorandom number generators [13, 14, 15, 16] will be estimated.In conclusion we summaries the results. C-sytems on a Torus
A particular system chosen for investigation is the one realising linear automorphisms of the unithypercube M N in Euclidean space E N with coordinates ( x , ..., x N )[1, 13, 14, 15]: x ( k +1) i = N X j =1 T ij x ( k ) j mod 1 , k = 0 , , , ... (2.6)where the components of the vector x ( k ) are defined as x ( k ) = ( x ( k )1 , ..., x ( k ) N ) . The phase space M N of the systems (2.6 ) can also be considered a N -dimensional torus [1, 13, 14, 15], appearing atfactorisation of the Euclidean space E N with coordinates x = ( x , ..., x N ) over an integer lattice Z N .The operator T acts on the initial vector x (0) and produces a phase space trajectory x ( n ) = T n x (0) on a torus. 3he dynamical system defined by the integer matrix T has a determinant equal to one Det T = 1and has no eigenvalues on the unit circle [1]. The spectrum { λ , ..., λ N } of the matrix T fulfilstherefore the following two conditions:1) Det T = λ λ ...λ N = 1 , | λ i | 6 = 1 , ∀ i. (2.7)The Liouville’s measure dµ = dx ...dx N is invariant under the action of T , and T is an automorphismof the unit hypercube onto itself. The conditions (2.7) on the eigenvalues of the matrix T are sufficientto prove that the system represents an Anosov C-system [1] and therefore as such it also represents aKolmogorov K-system [2, 3, 4, 5, 6] with mixing of all orders and of nonzero entropy. The eigenvaluesof the matrix T can be divided into two sets { λ α } and { λ β } with modulus smaller and larger thanone: 0 < | λ α | < α = 1 ...d < | λ β | < ∞ for β = d +1 ...N. (2.8)There exist two hyperplanes X = { X α } and Y = { Y β } which are spanned by the correspondingeigenvectors { e α } and { e β } . These invariant planes define invariant spaces on which the phasetrajectories are expanding and contracting under the transformation T at an exponential rate. TheC-system (2.6) has a nonzero Kolmogorov entropy h ( T ) [1, 4, 6, 7, 8, 15]: h ( A ) = X β ln | λ β | (2.9)which is expressed in terms of the eigenvalues λ β of the operator T . The entropy quantitativelycharacterises the instability of a C-system trajectories and its value depends on the spectral propertiesof the evolution operator T . We shall consider a family of operators of dimension N introduced in[14]: T = ... ... ... ... ... N N − N − ... (2.10)The operator T fulfils the C-condition (2.7) and represents a C-system [13]. The spectrum of theoperator T and of its inverse T − are presented in Fig.1 [14, 16]. The entropy of the C-system T can be calculated for large values of N [14, 16]: h ( A ) = X β ln | λ β | ≈ π N (2.11)4 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 1: On the left figure is the distribution of the eigenvalues of the operator T and on the rightis the distribution of the eigenvalue of the inverse operator T − for the N = 256.and increases linearly with the dimension N . Our aim is to study the behaviour of the observables { f ( x ) } defined on the torus phase space M N of the dynamical system T (2.10) and, in particular,a speed at which the correlation functions decay. Correlation Functions
The general form of the correlations we are intending to consider are: D n ( f, g ) = h f ( x ) g ( T n x ) i − h f ( x ) ih g ( x ) i , (3.12)where the f ( x ) and g ( x ) are the observables/functions defined on the torus phase space M N . Weshall consider the observables belonging to a general class of functions which are p-times differentiable f, g ∈ C p , where p is an integer, or functions which are in the α -H¨older class.In this section the operator T is a two-dimensional matrix , N = 2 in (2.10): T x = x x mod , x x → { x + x }{ x + 2 x } , (3.13)where { x } ≡ x mod
1. To have an idea about how the correlation D n behaves as a function of thetime step n we shall calculate a ”one-step” correlation D ( f, g ) when the observables are separatedby one unit of time n = 1 and f and g are some polynomial functions. A simple example will be ofthe form D ( f, g ) = < f ( x ) T x > − < f ( x ) >< x >, where f = x x r or f = ( x x ) r , r = 0 , , ... and g = x . The result of calculations is presentedon Fig.2 and demonstrates that a one-step correlation decreases, D ( r ) →
0, as the order of thepolynomials f increases, r → ∞ . In order to confirm this behaviour analytically let us consider the5
50 100 1500.0000.0010.0020.0030.0040.005 r D - - - - - r K Figure 2: The correlation function D ( r ) = < x x r { x + x } > − < x x r >< { x + x } > and thecorrelator K ( r ) = < x x { x + 2 x } r > − < x x >< { x + 2 x } r > , where { x } ≡ x mod f ( x ) = ∞ X i ,i =1 a i i sin(2 πi x ) cos(2 πi x ) , < f ( x ) > = 0 , g ( x ) = x, < g ( x ) > = 12 , where the numbers ( i , i ) define the oscillation frequencies of the observable f . The correlationfunction (3.14) will take the form D ( f, g ) = ∞ X i ,i =1 a i i Z dx dx sin(2 πi x ) cos(2 πi x ) { x + x } = − ∞ X r =1 a rr πr , (3.14)where we used a trigonometric representation of the mod x + x ) mod ≡ { x + x } = 12 − π ∞ X r =1 r sin(2 πr ( x + x )) . In order to estimate the behaviour of the Fourier coefficients in (3.14) we shall consider the functions f ( x ) which have (2 p , p ) continuous partial derivatives f ( x , x ) ∈ C p , p ( M ). The behaviour ofthe Fourier coefficients can be found performing a partial integrations and using the periodicity ofthe functions f ( x + 1) = f ( x ). Thus the Fourier coefficients can be represented in the form [40] † : a i i = 4( − p + p (2 πi ) p (2 πi ) p Z dx dx sin(2 πi x ) cos(2 πi x ) ∂ (2 p ) x ∂ (2 p ) x f ( x , x ) . (3.15)This representation allows to estimate the one-step correlation function D ( f, g ): | D ( f, g ) | = | < f ( x ) T x > − < f ( x ) >< x > | = | ∞ X r =1 a rr πr | ≤ ∞ X r =1 M p (2 πr ) p +2 p +1 , (3.16)where | ∂ (2 p ) x ∂ (2 p ) x f ( x , x ) | ≤ M p . † If a function is finite and continuous together with its p − p − M p such that | a i | < M p /i p [40] . D ( f, g ) decreases when the oscillation frequency r of the observable f increases: D ( r ) ∼ r p +2 p +1 → . (3.17)The oscillations frequencies of the observable f ( x ) are defined by ( i , i ) in (3.14) and we are con-sidering the limit when ( i , i ) ∼ r → ∞ . Performing a similar calculation one can get convincedthat the result holds for more general functions f ( x ) in (3.14): f ( x ) = ∞ X i ,i =0 a i i cos(2 πi x ) cos(2 πi x ) + ∞ X i ,i =1 b i i cos(2 πi x ) sin(2 πi x ) + ... The other independent correlation function of
T x is of the form < f ( x ) T x > = ∞ X i ,i =1 a i i Z dx dx sin(2 πi x ) cos(2 πi x ) { x + 2 x } = − ∞ X r =1 a r,r +2 πr . As one can see, the second index of the Fourier coefficient was shifted by two units a r,r +2 . Examiningits behaviour by using (3.15) we conclude that the coefficients decay faster: | D ( f, g ) | = | ∞ X r =1 a r,r +2 πr | ≤ ∞ X r =1 M p (2 πr ) p +1 (2 π ( r + 2)) p . (3.18)Of our primary interest is to find out the behaviour of the correlations D n ( f, g ) for observables f ( x ) and g ( T n x ) separated by n time steps (3.12). For that one should generalise our previouscalculation in two directions, considering n ≥ g ( x ) in (3.12). Thus weshall consider a Fourier representation of the function g ( x ) on a torus of the form g ( x , x ) = ∞ X j ,j =1 b j j cos(2 πj x ) cos(2 πj x ) . (3.19)In order to calculate the observable g ( x ) after n -steps g ( T n x ) = g ( { x + x } , { x + 2 x } ) we haveto define a mod g ( x ). The mod πjx ) = sin(2 πj ( { x } + integer )) = sin(2 πj { x } ) , cos(2 πjx ) = cos(2 πj ( { x } + integer )) = cos(2 πj { x } ), therefore g ( { x } , { x } ) = ∞ X j ,j =1 b j j cos(2 πj x ) cos(2 πj x ) . (3.20)This is an important observation for a successful calculation of D n ( f, g ), because if one considers apolynomial expansion of f and g , then the mod f ( x ) = ∞ X i ,i =1 a i i cos(2 πi x ) cos(2 πi x ) , { T n x } = { a n x + b n x }{ c n x + d n x } , (3.21) g ( T n x ) = ∞ X j ,j =1 b j j cos(2 πj ( a n x + b n x )) cos(2 πj ( c n x + d n x )) , a n , b n , c n , d n ) define the n’th power of the operator T and can be expressedin terms of Fibonacci numbers F n : T n = a n b n c n d n = F n − F n − F n F n F n − F n − , F n = λ n − λ − n √ . (3.22)The λ = √ > T and detT n = a n d n − b n c n = 1 , d n > b n = c n > a n . (3.23)Using the above formulas we can express the general correlation functions in the form h f ( x ) g ( T n x ) i = (3.24)= ∞ X i ,i ,j ,j =1 a i i b j j (cid:18) δ i ,j a n + j c n δ i ,j b n + j d n + δ j a n ,i + j c n δ j b n ,i + j d n ++ δ j a n ,i + j c n δ j d n ,i + j b n + δ j c n ,i + j a n δ j d n ,i + j b n + δ j c n ,i + j a n δ j b n ,i + j d n (cid:19) . In order to execute the last four delta functions in (3.24) one should solve the linear equations a n j − c n j = i ± b n j ∓ d n j = i , − a n j + c n j = i ∓ b n j ± d n j = i (3.25)with respect to the j , j . All these equations have unique solutions because the correspondingdeterminants are equal to ± j , j . The lastdelta function in (3.24) has no positive solutions and therefore does not contribute to the correlation.Thus we have h f ( x ) g ( T n x ) i = 18 ∞ X j ,j =1 a j a n + j c n ,j b n + j d n b j j + 18 ∞ X i ,i =1 a i ,i b i d n + i c n ,i b n + i a n + (3.26)+ 18 ∞ X i d n >i c n ; i b n >i a n a i ,i b i d n − i c n ,i b n − i a n + 18 X i c n >i d n ; i a n >i b n a i ,i b − i d n + i c n , − i b n + i a n . The subtraction terms in (3.12) are: < f ( x ) > = X i ,i =1 a i i Z dx dx cos(2 πi x ) cos(2 πi x ) = 0 ,< g ( x ) > = X j ,j =1 b j j Z dx dx cos(2 πj x ) cos(2 πj x ) = 0 , and the total expression for the correlation will take the following form: D n ( f, g ) = < f ( x ) g ( T n x ) > − < f ( x ) >< g ( x ) > = (3.27)= 18 ∞ X j ,j =1 a j a n + j c n ,j b n + j d n b j j + 18 ∞ X i ,i =1 a i ,i b i d n + i c n ,i b n + i a n ++ 18 ∞ X i d n >i c n ; i b n >i a n a i ,i b i d n − i c n , i b n − i a n + 18 ∞ X i c n >i d n ; i a n >i b n a i ,i b − i d n + i c n , − i b n + i a n .
8e have to estimate each term in the above expression. Using the representation (3.15) we shallevaluate the first term in the correlator D n ( f, g ):18 | ∞ X r ,r =1 a r a n + r c n ,r b n + r d n b r r | ≤ ∞ X r ,r =1 M p (2 π ( r a n + r c n ) p (2 π ( r b n + r d n )) p M q (2 πr ) q (2 πr ) q =2( b n ) p ( c n ) p ∞ X r ,r =1 M p (2 π ( r + r a n c n ) p (2 π ( r + r d n b n )) p M q (2 πr ) q (2 πr ) q ≤ e − p ln b n − p ln c n ∞ X r ,r =1 M p (2 πr ) p (2 πr ) p M q (2 πr ) q (2 πr ) q , where | ∂ (2 p ) x ∂ (2 p ) x f ( x , x ) | ≤ M p , | ∂ (2 q ) x ∂ (2 q ) x g ( x , x ) | ≤ M q . (3.28)The logarithm in the exponent can be expressed asln b n = ln c n = ln F n = ln λ n − λ − n √ n ln λ + ln(1 − λ − n ) −
12 ln 5 > n ln λ. Thus we have18 | ∞ X r ,r =1 a r a n + r c n ,r b n + r d n b r r | ≤ e − n (2 p +2 p ) ln λ ∞ X r ,r =1 M p M q (2 πr ) q +2 p (2 πr ) p +2 q and remembering that the entropy of the system is h ( T ) = ln λ (2.9), we have for the first term18 | ∞ X r ,r =1 a r a n + r c n ,r b n + r d n b r r | ≤ C ( f, g ) e − nh ( T ) ν ( f,g ) , (3.29)where the numerical factors C ( f, g ) = ∞ X r ,r =1 M p M q (2 πr ) q +2 p (2 πr ) p +2 q , ν ( f, g ) = 2 p + 2 p depend only on the observables and are independent of the system dynamics T . For the second termwe have a similar estimate:18 | ∞ X r ,r =1 a r ,r b r d n + r c n ,r b n + r a n | ≤ ∞ X r ,r =1 M p (2 πr ) p (2 πr ) p M q (2 π ( r d n + r c n )) q (2 π ( r b n + r a n )) q =2( b n ) q ( c n ) q ∞ X r ,r =1 M p (2 πr ) p (2 πr ) p M q (2 π ( r + r d n c n ) q (2 π ( r + r a n b n )) q ≤ e − q ln b n − q ln c n ∞ X r ,r =1 M p (2 πr ) p (2 πr ) p M q (2 πr ) q (2 πr ) q ≤ e − n (2 q +2 q ) ln λ ∞ X r ,r =1 M p M q (2 πr ) p +2 q (2 πr ) q +2 p , | ∞ X r ,r =1 a r ,r b r d n + r c n ,r b n + r a n | ≤ C ( f, g ) e − nh ( T ) ν ( f,g ) , (3.30)where the numerical factors are: C ( f, g ) = ∞ X r ,r =1 M p M q (2 πr ) p +2 q (2 πr ) q +2 p , ν ( f, g ) = 2 q + 2 q , and they are independent of the system dynamics T . The third and the fourth terms of D n ( f, g ) in(3.27) are+ 18 ∞ X i d n >i c n ; i b n >i a n a i ,i b i d n − i c n ,i b n − i a n + 18 ∞ X i c n >i d n ; i a n >i b n a i ,i b − i d n + i c n , − i b n + i a n and using the representation (3.15) for the Fourier coefficients one can find that18 | ∞ X r d n >r c n ; r b n >r a n a r ,r b r d n − r c n ,r b n − r a n | ≤≤ ∞ X r d n >r c n ; r b n >r a n M p (2 πr ) p (2 πr ) p M q (2 π ( r d n − r c n )) q (2 π ( r b n − r a n )) q == 2( b n ) q ( d n ) q ∞ X r d n >r c n ; r b n >r a n M p (2 πr ) p (2 πr ) p M q (2 π ( r − r c n d n ) q (2 π ( r − r a n b n )) q ≤≤ b n ) q ( d n ) q ∞ X r ,r =1 M p (2 πr ) p (2 πr ) p M q (2 π ) q (2 π ) q ≤≤ e − n (2 q +2 q ) ln λ ∞ X r ,r =1 M p (2 πr ) p (2 πr ) p M q (2 π ) q (2 π ) q . Thus for the third term we have18 | ∞ X r d n >r c n ; r b n >r a n a r ,r b r d n − r c n ,r b n − r a n | ≤ C ( f, g ) e − nh ( T ) ν ( f,g ) , (3.31)where the numerical factors are C ( f, g ) = ∞ X r ,r =1 M p (2 πr ) p (2 πr ) p M q (2 π ) q (2 π ) q , ν ( f, g ) = 2 q + 2 q , and they are T independent. The fourth term has an identical upper bound:18 | ∞ X i c n >i d n ; i a n >i b n a i ,i b − i d n + i c n , − i b n + i a n | ≤ C ( f, g ) e − nh ( T ) ν ( f,g ) . We arrive to the following upper bound on the correlation function: | D n ( f, g ) | ≤ C e − nh ( T ) ν + C e − nh ( T ) ν + 2 Ce − nh ( T ) ν . (3.32)10t follows that the dependence on the system dynamics T appears in the exponential factor e − nh ( T ) ν through its fundamental characteristic, the entropy h ( T ). The coefficients C i ( f, g ) , C ( f, g ) and ν i ( f, g ) , ν ( f, g ) depend only on observables through their smoothness indices p i and q i and the upperbounds on derivatives M p and M q . The above calculation provides a qualitative understanding of how the exponential decay of thecorrelation functions appears and its rate. Under repeated action of the dynamic system T on theobservable g ( T n x ) its oscillating frequencies are stretching apart toward the high frequency modesand the overlapping integral with the fixed observable f ( x ) falls exponentially .In order to estimate the upper bound (3.32) let us consider the observables of the same order ofsmoothness p = p = q = q = p . In that case the formula simplifies: | D n ( f, g ) | ≤ C ( f, g ) e − nh ( T ) ν , (3.33)where ν = 4 p, C ( f, g ) = 72 M p (16 π ) p ∞ X r =1 r p ! . This result allows to define the decorrelation time for the physical observable f ( x ) as in [11, 13, 18]: τ = 1 h ( T ) ν f . (3.34)The value of the standard deviation σ f is a sum σ f = + ∞ X n = −∞ [ h f ( T n x ) f ( x ) i − h f ( x ) i ]and using the (4.40) one can explicitly estimate the standard deviation σ f σ f ≤ C f e − h ( T ) ν f − e − h ( T ) ν f . (3.35) High Dimensional C-systems
Let us now consider the operators T of high dimension, N > T n similar to the formula (3.22). We shall use a computer tocalculate the coefficients of the matrix T n . What we need is the rate at which the coefficients growas a function of n . The numerical experiments demonstrate that the fastest growing coefficients ineach row are the ones which are the next to the last column and for the columns they are on thelast row. Let us consider N = 3: T = , T n = a n a n a n a n a n a n a n a n a n . (4.36)11here is only one eigenvalue which is outside of the unit circle λ = 4 . .. . The largest coefficients ineach row are a n , a n , a n correspondingly and the largest coefficients in each column are a n , a n , a n correspondingly, and they all grow at the rate ∝ λ n . The observables are f ( x ) = ∞ X i ,i ,i =1 a i i i cos(2 πi x ) cos(2 πi x ) cos(2 πi x ) , (4.37) g ( T n x ) = ∞ X j ,j ,j =1 b j j j cos(2 πj ( a n x + a n x + a n x )) cos(2 πj ( a n x + a n x + a n x ))cos(2 πj ( a n x + a n x + a n x )) . A typical term in the correlation function D n ( f, g ) will have a form similar to the (3.26): ∞ X j ,j ,j =1 a j a n + j a n + j a n , j a n + j a n + j a n , j a n + j a n + j a n b j j j , (4.38)and it can be bound from above: | ∞ X j ,j ,j =1 a j a n + j a n + j a n , j a n + j a n + j a n , j a n + j a n + j a n b j j j | ≤ a n ) p ( a n ) p ( a n ) p ∞ X r ,r ,r =1 M p (2 πr ) p (2 πr ) p (2 πr ) p M q (2 πr ) q (2 πr ) q (2 πr ) q ≤ e − (2 p +2 p +2 p ) n ln λ ∞ X r ,r ,r =1 M p (2 πr ) p +2 p +2 p M q (2 πr ) q (2 πr ) q (2 πr ) q , where ln λ = h ( T ) and | ∂ (2 p ) x ∂ (2 p ) x ∂ (2 p ) x f ( x , x , x ) | ≤ M p , | ∂ (2 q ) x ∂ (2 q ) x ∂ (2 p ) x g ( x , x , x ) | ≤ M q . (4.39)One can estimate the upper bound on the correlation function considering for simplicity the observ-ables of the same order of smoothness p i = q i = p . In that case the formula takes the followingform: | D n ( f, g ) | ≤ C ( f, g ) e − nh ( T ) ν , (4.40)where ν = 6 p, C ( f, g ) = M p (2 π ) p ∞ X r =1 r p ! . Generalising this calculation to the operators T on dimension N we shall get ν f = 2 pN, C ( f, g ) = M p (2 π ) pN ∞ X r =1 r p ! N , and the formula for the decorrelation time takes the form τ = 1 ν f h ( T ) = 12 pN h ( T ) . (4.41)12 he exponential decay of the correlation functions is getting faster as the dimension N of the op-erators T is increasing. Taking into consideration that the entropy h ( T ) of our system is linearlyincreasing with N , h ( T ) ≈ π N in (2.11), we shall get the following expression for the decorrelationtime : τ = π pN . (4.42)Considering a set of initial trajectories occupying a small volume δv in the phase space of the C-system it is important to know how fast the volume δv will be spread/distributed over the wholephase space during the evolution of the system. This characteristic time interval τ defines the time atwhich the system reaches a stationary distribution. The entropy of the system defines the expansionrate of the phase space volume elements and one can derive therefore that [13] τ = 1 h ( T ) ln 1 δv . (4.43)Thus there are three characteristic time scales associated with the C-system [13]: Decorrelation timeτ = π pN < Interaction timet int = 1 < Stationary distribution timeτ = h ( T ) ln δv . (4.44)This result defines important physical characteristics of the C-systems and measures the ”level ofchaos” developed in the system and justifies the physical conditions at which the statistical/probabilisticdescription of the system is available. MIXMAX Random Number Generator
One of the interesting applications of the Anosov C-systems (2.10) is associated with the so calledMIXMAX generator of pseudorandom numbers [13, 14, 15, 16]. It was demonstrated in [14] thatthe MIXMAX pseudorandom number generators are passing strong statistical U01-tests [41] whenthe entropy of the generators is larger than fifty, h ( T ) >
50. Using the formulas (4.42), (4.43) and(4.44) of the last section one can estimate characteristic time scales associated with the MIXMAXgenerators. The generator N = 256 in Table 1 of the article [16] has the entropy h ( T ) = 194 andthe smallest phase volume is of order δv = 2 − · , therefore the characteristic time scales for thisgenerator are Decorrelation timeτ = 0 . < Interaction timet int = 1 < Stationary distribution timeτ = 95 . (5.45)The MIXMAX generator which has much higher entropy was presented in Table 3 of the article [16].It has the entropy h ( T ) = 8679 and the smallest volume δv = 2 − · , therefore the characteristic13ime scales for this generator are Decorrelation timeτ = 0 . < Interaction timet int = 1 < Stationary distribution timeτ = 1 . . (5.46)Both generators have very short decorrelation time. The second generator N = 240 has muchbigger entropy and therefore its relaxation time τ is much smaller, of order 1 .
17, and is close tothe interaction time. In that sense it has very strong stochastic/chaotic properties, it much fasterspreads trajectories over the whole phase space and reaches the equilibrium. Therefore it should notbe surprising that these generators are passing all the tests in the BigCrush U01-suite [41]. Thesegenerators have the best combination of speed, reasonable size of the state and are currently availablegenerators in the ROOT and CLHEP software packages at CERN for Monte-Carlo simulations andscientific calculation [42, 43, 44]. Conclusion
Our analyses of the N dimensional hyperbolic Anosov C-system (2.10) indicates that its basic statis-tical characteristics are expressible in terms of entropy. The decorrelation time τ and the relaxationtime τ are inversely proportional to the entropy of the system and indicate that these time scalesbecome shorter as entropy increases. This is an intuitively appealing result because the entropymeasures the uncertainty in the description of the physical systems and here it is translated intothe important time scales characteristics. As a result a perfectly deterministic dynamical systemshows up a fast thermalisation and well developed statistical properties. When measuring differentobservables of the hyperbolic Anosov C-system it will be difficult to recognise that in reality thedata are coming out from a perfectly deterministic dynamical system.The exponential decay of the correlation functions has been found earlier in classical dynamicsof the N-body gravitating systems and can be used to justify a statistical description of globularclusters and elliptic galaxies [18, 19, 20].It was suggested in the literature that the outgoing Hawking radiation [45] may be not exactlythermal, but had subtle correlations [46, 47, 48, 49]. In that respect one can suppose that theeffective description of the black hole radiation can be understood in analogy with the behaviour ofthe hyperbolic systems of the type considered above [50]. Acknowledgement
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