aa r X i v : . [ m a t h . D S ] J a n Deterministic Diffusion
L. P. Nizhnik, I. L. NizhnikInstitute of Mathematics NAS of Ukraine, Kiev, [email protected], [email protected]
Abstract
In the present paper, we give a series of definitions and properties of Lifting Dy-namical Systems (LDS) corresponding to the notion of deterministic diffusion. Wepresent heuristic explanations of the mechanism of formation of deterministic diffusionin LDS and the anomalous deterministic diffusion in the case of transportation in longbilliard channels with spatially periodic structures. The expressions for the coefficientof deterministic diffusion are obtained.
Key words: dynamical system, one-dimensional lifting dynamical system, deterministicdiffusion, anomalous diffusion, diffusion coefficient, billiard channel, nonideal reflection law.
One-dimensional dynamical systems on the entire axis with discrete time are defined by therecurrence relation x n +1 = f ( x n ) , (1)where f ( x ) is a real function given on the entire axis and x is a given initial value [8, 14].Equation (1) determines a trajectory ( x , x , ..., x n , ... ) = x in the dynamical system (1)according to the initial value x and the form of the function f. For the dynamical systemsadmitting the chaotic behavior of trajectories, the problem of construction of the entiretrajectory or even of determination of the values of x n for large n is quite complicatedbecause, as a rule, the numerical calculations are performed with a certain accuracy andthe dependence of the subsequent values of x n on the variations of the previous values isunstable. Moreover, from the physical point of view, the initial value x is specified witha certain accuracy. Therefore for the investigation of the behavior of trajectories for largevalues of time, we can analyze not the evolution of system (1) but the evolution of measureson the axis generated by this evolution.If a probability measure µ (a normalized measure for which the measure of the entireaxis is equal to 1) with density ρ : µ ( A ) = R A ρ ( x ) dx is given at the initial time, then, fora unit of time, system (1) maps this measure into µ : µ ( A ) = µ ( f − ( A )) , where f − ( A ) isa complete preimage of the set A under the map f. The operator mapping the measure µ into the measure µ is called a Perron–Frobenius operator. The Perron–Frobenius operator1 is linear even for nonlinear dynamical systems (1) and maps the density ρ of the initialmeasure into the density ρ of the measure µ as the integral operator with singular kernelcontaining a Dirac δ function: ρ ( x ) = F ρ ( x ) = Z δ ( x − f ( y )) ρ ( y ) dy = X y k ∈ f − ( { x } ) | f ′ ( y k ) | ρ ( y k ) . (2)The investigation of the asymptotic behavior of the density ρ n = F n ρ as n → ∞ isreduced to the investigation of the behavior of the semigroup F n . There are examples of dy-namical systems (1) with locally stretching maps f for which the densities ρ n are asymptoti-cally Gaussian as n → ∞ independently of the choice of the density of the initial probabilitymeasure. In this case, it is said that deterministic diffusion occurs in the dynamical system(1).The aim of the present paper is to consider examples of dynamical systems with deter-ministic diffusion. We restrict ourselves to the so-called lifting dynamical systems (LDS)with piecewise linear functions f ( x ) in DS (1).We briefly consider the mechanisms of appearance of the anomalous deterministic diffu-sion in the process of transportation in long billiard channels with spatially periodic struc-tures. Consider the dynamical system (1) on the entire axis, where the function f ( x ) is given onthe main interval I = [ − , ) . The function s ( x ) = f ( x ) − x with x ∈ I has the sense of ashift of the point x under the map f. We split the entire axis ( −∞ , ∞ ) into disjoint intervals I k = [ k − , k + ) , where k ∈ Z are integers. Assume that the function f is extended fromthe interval I onto all intervals I k so that the shift of a point under the map f on eachinterval k is the same as on I , that is s ( k + x ) = f ( k + x ) − ( k + x ) = s ( x ) = f ( x ) − x. Thisgives a periodic shift function s ( x ) and implies the periodicity of the function f with lift 1 f ( k + x ) = k + f ( x ) , | x | < , k ∈ Z. (3)The dynamical system (1) with a function f satisfying property (3) is called a LDS. Thisdynamical system (DS) is well known and thoroughly investigated [4, 8, 9]. Lemma.
Let a function f defined on the interval I = [ − , ) be a piecewise monotonestretching function on a finite partition { I ,j } mj =1 of the interval I = m S j =1 I ,j , i.e., there exists λ > such that | f ( x ) − f ( y ) | ≥ λ | x − y | (4) for x, y ∈ I ,k , and, in addition, the function f ( x ) has finite nonzero discontinuities at thepoints of joint of the intervals I ,j . Then the periodic extension (3) with lift 1 specifies alocally stretching function f on the entire axis.Proof. If the points x and y belong to the same interval I ,k , then, by virtue of (4), | f ( x ) − f ( y ) | = | f ( x − k ) − f ( y − k ) | ≥ λ | x − y | .
2f the points x and y belong to neighboring intervals, then the function has a discontinuityat the point of joint of these intervals and the value of this discontinuity is not smaller thana certain a > . In view of the monotonicity of the function f, there exists ε > | x − y | < ε , the points x and y belong either to the same interval I ,k or to neighboringintervals. Hence, | f ( x ) − f ( y ) | ≥ a ≥ λε, where ε = min( ε , a λ ) . Thus, for | x − y | < ε, we have | f ( x ) − f ( y ) | ≥ λ | x − y | . Let x = ( x , x , ..., x n , ... ) be a trajectory for the LDS (1)–(3). By k, we denote the numberof the interval I k = [ k − , k + ) containing a number a ∈ I k as follows: k = [ a ) is the nearestinteger for the number a. Then the trajectory x is associated with an integer-valued sequence: M = ([ x ) , [ x ) , ..., [ x n ) , ... ) = ( m , m , ..., m n , ... ) of the numbers of intervals containing thevalues x n . The sequence M is called the route of the trajectory x and contains the numbersof intervals successively “visited” by the phase point in the LDS [8]. Proposition 1.
Let the function f determining the LDS (1)–(3) be stretching. Then thetrajectory of the LDS is uniquely determined by its route. Proof.
Assume that two initial conditions x (1)0 and x (2)0 generate trajectories of the LDS withthe same route. This means that the points x (1) n and x (2) n lie in the same interval I m n . In viewof the stretching property of the map f, the inequality | x (1) n +1 − x (2) n + | ≥ λ | x (1) n − x (2) n | , λ > , is true. Since any two points from the same interval I m n differ by at most 1, the successiveapplication of this inequality yields the inequality | x (1)0 − x (2)0 | ≤ λ n for any n. Since λ > , this implies that x (1)0 = x (2)0 . Definition 1.
If any integer-valued sequence in the LDS is a route of a certain trajectoryand the trajectory is uniquely determined by its route, then we say that the LDS possessesthe Bernoulli property [8].
Proposition 2.
If a function f defined in the interval I = [ − , ) is odd, continuous,monotonically increasing, stretching, and unbounded, then the LDS (1)–(3) correspondingto this function f possesses the Bernoulli property. Proof.
The uniqueness of reconstruction of a trajectory according to its route is proved inAppendix 1. Let M = ( m , m , ..., m n , ... ) be an arbitrary integer-valued sequence. In theinterval I m , we consider a sequence of embedded contracting closed intervals U n constructedas follows:Let U (1) be the closure of the preimage of the interval I m n under the map f lying on I m n − : U (1) = f − ( I m n ) . Since the map f is stretching, the length mes U (1) ≤ λ − . Let U (2) be thepreimage of U (1) in the interval I m n − and let, by induction, U ( k ) be the preimage of U ( k − inthe interval I m n − k . The sets U n = U ( n ) ⊂ I m and mes U n ≤ λ n . As n → ∞ , we get a systemof embedded closed sets U n +1 ⊂ U n in the interval I m k and mes U n → n → ∞ . Thecommon limit point x for all U n lies in I m and its trajectory x = ( x , f ( x ) , ..., f n ( x ) , ... )has the route M. Markov Partition of the Phase Space of LDS. Piece-wise Linear LDS
The system of intervals I k = [ k − , k + ) , k ∈ Z, for the LDS (1)–(3) can be regarded asa Markov partition of the phase space [8]. Consider a finer Markov subpartition. Assumethat the main interval I is split into finitely many m subintervals I ,j = [ x j , x j +1 ) , where x = − , x j < x j +1 , x m = , j = 0 , , , ..., m, and I ,j = [ x j − , x j ) . An integer-valuedshift of this partition leads to the decomposition of the intervals I k into subintervals I k,j =[ k + x j − , k + x j ) . Hence, the entire axis is split into intervals { I k,j } , k ∈ Z, j = 1 , , ..., m. Definition 2.
We say that the Markov partition { I k,j } k ∈ Z, ≤ j ≤ m of the entire axis is consis-tent with the LDS (1)–(3) if the LDS maps, for a unit of time, any probability measure withconstant densities in each set I k,j into a measure with constant densities in each interval I k,j . Definition 3.
We say that a function f specifying the LDS (1)–(3) is consistent with theMarkov partition { I k,j } k ∈ Z, ≤ j ≤ m defined with the help of the numbers − = x , x , ..., x m = if it is linear and nonconstant on each interval I k,j and, at each end of the interval I k,j , takes values equal to an integer plus one of the numbers x j , j = 0 , , ..., m. Example 1.
Let f ( x ) = Λ x be a linear function in the interval I = [ − , ) , where Λ = 2 l +1is an odd number. Then this function is consistent with the Markov partition { I k } k ∈ Z , I k =[ k − , k + ) . Example 2.
Let f ( x ) = Λ x, where Λ = 2 l is an even number. Then this function isconsistent with the Markov partition { I k, ± } k ∈ Z , where I k, + = [ k, k + ) and I k, − = [ k − , k ) . An important example of the function f satisfying Definition 3 is given by the followingassertion: Proposition 3.
Assume that a function f ( x ) determining the LDS in the interval I ispiecewise linear, takes half-integer values at the ends of each its linear pieces, and moreover, f ′ ( x ) = 0 almost everywhere. Then, by Definition 3, the function f is consistent with theMarkov partition { I k }| k ∈ Z . Proof.
The proof follows from the verification of the conditions of Definition 3 for points { x j } , that is for all points at which the function f takes half-integer values.A piecewise linear function f from Proposition 3 will be called a piecewise linear functiontaking half-integer values at the ends of the linear pieces.The equivalence of Definitions 2 and 3 for LDS yields the following statement: Proposition 4.
For the Markov partition { I n,j } n ∈ Z, | j |≤ m to be consistent with the action ofthe LDS (1)–(3) by Definition 2, it is necessary and sufficient that the function f determiningthe LDS (1)–(3) be consistent with this Markov partition by Definition 3. Proof.
We now show that if f satisfies the conditions of Definition 3, then the LDS transformsthe probability measure with constant densities on I n,j into a measure with constant densitieson I n,k , i.e., the LDS satisfies the conditions of Definition 2. Since operator (2) is linear, itsuffices to consider the case where the density of the initial measure is constant on the fixed4nterval I n ,k . On this interval, the function f ( x ) is linear and nonconstant and maps I n ,k into the union S I n,k of several neighboring intervals. Therefore, the inverse map of theseintervals is linear and, hence, the measure has a constant density in each of these intervals.Let the conditions of Definition 2 be satisfied. If the density of the initial measure isconstant in the interval I n ,k , then the map f gives a new measure µ on the entire axis. Itis clear that supp µ coincides with the closure of a certain union f − ( I n,k ) = S I ,j because,otherwise, there exist an interval I ¯ n, ¯ k and its part A such that µ ( I ¯ n, ¯ k ) = µ ( A ) = 0 and µ ( A ) = 0 . This contradicts the condition of Definition 2 according to which the measure µ has a constant density. Let I ¯ n, ¯ k be one of the intervals f ( I n ,k ) and let B = f − ( I ¯ n, ¯ k ) . Since B ⊂ I n ,k , the initial measure on B has a constant density, f bijectively maps B onto I ¯ n, ¯ k , and, according to the Perron–Frobenius operator, the density ρ ( x ) of the measure µ on I ¯ n, ¯ k is expressed via the density ρ of the initial measure on I n ,k by the equality ρ ( x ) = ρ ( x ) | f ′ ( x ) | . Thus, we arrive at the conclusion that f ′ ( x ) ≡ const on the interval B andthat the function maps the ends of the interval B into the ends of the interval I ¯ n, ¯ k . Thus,the function f satisfies the conditions of Definition 3.The following question arises: Is it possible to construct Markov partitions of the en-tire axis consistent with the linear function f ( x ) = Λ x for the values of Λ other than inExamples 1 and 2?Since the linear function f ( x ) = Λ x is odd, the numbers x j specifying a consistent Markovsubpartition of the interval I are symmetric about the middle of the interval I . Hence, itis sufficient to define solely the positive values of x j and indicate that the number m ofsubintervals { I ,j } ≤ j ≤ m is even or odd, because s = 0 belongs to the set { s j } for even m and does not belong to this set for odd m. We enumerate the ends of the intervals { I k,j } located on the positive half axis in theorder of increase starting from the first positive number. This yields a sequence 0 < s
The parameters of the Markov subpartition { I k,j } , i.e., the parity of thenumber m and the integer-valued vector ~n = ( n , n , ..., n ˆ m ) uniquely define the quantity Λ , the slope of the linear function f ( x ) = Λ x, consistent with the Markov subpartition { I k,j } and a collection of numbers { s j } ˆ mj =1 specifying the ends of the intervals I k,j = [ k + s j , k + s j +1 )of this subpartition. Proof.
Let the number of components of the vector ~n be equal to ˆ m. Then the number m of subintervals { I ,j } in the Markov subpartition is given by the equality m = 2 ˆ m for even m and the equality m = 2 ˆ m − m (the parity of m is specified by the quantity s = ( − m ). We always have s ˆ m = . For odd m, the sequence s , s , ..., s k , ... has the5orm s , s , ..., s ˆ m +1 , , − s ˆ m − , − s ˆ m − , ..., − s , s , s , ..., s ˆ m − , , − s ˆ m − , ... and is explicitly expressed via s , s , ..., s ˆ m = . Hence, we can explicitly express each termof the sequence s , s , ..., s k , ... in terms of an integer and one of the numbers s , s , ..., s ˆ m . Thus, equalities (5) turn into a linear system for s , s , ..., s ˆ m whose coefficients are eitherintegers or the quantity Λ . The consistency condition for this system can be formulated asthe equality of the determinant of this system to zero. This gives the following algebraicequation for Λ: R ~n (Λ) = 0 . If we find Λ , then all s , s , ..., s ˆ m can be uniquely determined from (5). Similarly, weconsider the case of even m. In this case, the sequence s , s , ..., s k , ... contains integer valuesand has the form s , s , ..., s ˆ m = , − s ˆ m − , − s ˆ m − , ..., − s , , s , s , .... An implementation of this scheme by a constructive example is given in the next Propo-sition.
Proposition 6.
A linear function f ( x ) = Λ x determining the LDS (1)–(3) will be consistentwith the Markov partition { [ k − , ξ ) , [ k − ξ, k + ξ ) , [ k + ξ, k + ) } k ∈ Z , if for integers 0 < m < n and the values ε = ± ε = ± n + ε + p (2 n − ε ) + 8 mε ,ξ = 2 m n − ε + p (2 n − ε ) + 8 mε . Proof.
This is a consequence of the equalities (5), in this case having the formΛ ξ = m + ε ξ, Λ2 = n + ε ξ. It is worth noting that the set of values of the slope Λ of the linear function f ( x ) =Λ x, x ∈ I = [ − , ) , for which it is possible to construct Markov partitions consistent withthe LDS (1)–(3) is everywhere dense on the half axis (2 , ∞ ) [4, 9]. We consider the LDS (1)–(3) for which the collection of intervals { I k } k ∈ Z , I k = [ k − , k + ) , forms a Markov partition of the phase space consistent with the action of the LDS. Thismeans that, for a time unit, the LDS maps the measure µ with unit density in the interval I into a probability measure with constant densities p k ≥ I k and P k ∈ Z p k = 1 . As a result of multiple application of the LDS, the initial measure µ is transformed into a6easure with constant densities in the intervals I k . Let P k ( n ) be the density of the measurein the interval I k after the n -fold action of the LDS upon the initial measure µ . Then P k ( n + 1) = X l ∈ Z p k − l P l ( n ) (6)and P k (0) = δ k, due to the choice of the initial measure µ . The asymptotic behavior ofthe quantities P k ( n ) for large n, as solutions of Eq. (6), is described by the well-knowncentral limit theorem of the probability theory [5, 6]. Indeed, if we consider the sum ξ = ξ + ξ + ... + ξ n of n independent random variables each of which takes only integer values k with probability p k , then we get Eq. (6), where P k ( n ) is the probability of the event that ξ takes the value k. Theorem 1.
Let the numbers { p k } k ∈ Z take nonnegative values such that P k ∈ Z p k = 1 , let thegreatest common divisor of the numbers k with p k > be equal to 1, and let there exist thefirst and second moments σ = X k ∈ Z kp k , σ = X k ∈ Z k p k . (7) Then the solution of Eq. (6) as n → ∞ has an asymptotics P k ( n ) − σ √ πn e − ( x − ξn )22 σ n → uniformly in k. The character of convergence in (8) depends on additional conditions, e.g.,on the existence of the third moment (the Laplace theorem).Proof.
We now briefly present the well-known scheme of the proof of Theorem 1 based onthe fact that Eq. (6) is a difference analog of the convolution equation. As a result of theFourier transformation, the convolution turns into the product. Let P ( λ, n ) = P k ∈ Z P k ( n ) e ikλ be the characteristic function of the solution of Eq. (6), P ( λ ) = P k ∈ Z p k e ikλ . Thus, we get thefollowing formula from Eq. (6): P ( λ, n ) = [ P ( λ )] n P ( λ, . (9)Since the initial measure is concentrated on I and its density is constant, we concludethat P ( λ,
0) = 1 and P k ( n ) = 12 π π Z − π [ P ( λ )] n e − ikλ dλ (10)As λ → , P ( λ ) = P k ∈ Z p k e ikλ = 1 + iσ λ − σ λ + o (1) = e iσ λ − σ λ + o (1) . Hence,[ P ( λ )] n = e iσ nλ − σ nλ + o (1) . Substituting this result in (10) and integrating along the entire axis, we obtain (8).7he degree of closeness of two probability measures µ and ν with densities p ( x ) and q ( x )on the axis is often estimated by the value d ( µ, ν ) of the deviation (uniform in x ) of thedistribution functions P ( x ) = x Z −∞ p ( s ) ds = µ (( −∞ , x )) ,Q ( x ) = x Z −∞ q ( s ) ds = ν (( −∞ , x )) , i.e., d ( µ, ν ) = sup x | P ( x ) − Q ( x ) | (11) Definition 4.
We say that two sequences of measures µ n and ν n are asymptotically equiv-alent as n → ∞ if lim n →∞ d ( µ n , ν n ) = 0 . For normal measures ν n with variances σ n and means ξ n , i.e., in the case where the density of measures has the form of a Gauss curve q n ( x ) = 1 p πσ n e − ( x − ξn )22 σ n , (12)we say that the sequence of measures µ n is asymptotically normal as n → ∞ . In addition, if σ n = Dn, then we say that the sequence of measures µ n determines a normal diffusion withdiffusion coefficient D. If the variance σ n depends nonlinearly on n, then the deterministicdiffusion is anomalous.The result of Theorem 1 for the LDS can be interpreted as follows: The initial measurewith unit density in the interval I can be regarded as randomly specified initial data x forthe LDS uniformly distributed over the interval I . Then, for large times, as n → ∞ , theposition of x n is randomly distributed according to the normal law with mean value ξ n andvariance σ n . In other words, in this case, we have the deterministic diffusion with the diffusion coeffi-cient D = σ n n . Our first aim is to study this phenomenon.First, we reformulate the result of Theorem 1 for the LDS (1)–(3) with a piecewise-linearfunction f. Theorem 2.
Assume that a function f determining the LDS (1)–(3) is a piecewise linearfunction taking different half-integer values at the ends of all linear parts.Suppose that there exists D = 12 Z − | f ( x ) | dx − . (13) Then, after n iterations in the LDS, the initial measure µ with unit density in theinterval I is asymptotically mapped into a measure with normal distribution and the diffusioncoefficient D. roof. In view of Theorem 1 and Propositions 3 and 4, it is necessary to show that σ = X k ∈ Z k p k = Z − | f ( x ) | dx − . (14)This can readily be proved because the integral of the piecewise linear function f caneasily be taken. If the function f ( x ) is linear in the segment [ a, b ] and takes values k − and k + at the ends of this segment, then b R a | f ( x ) | dx = ( k + )( b − a ) . We get p k = µ ( I k ) = µ ( f − ( I k )) . The set f − ( I k ) T I is formed by several subintervals I ,j of the interval I in which the function f ( x ) takes values from the interval [ j − , j + ) . Hence, R f − ( I k ) T I | f ( x ) | dx = ( k + ) P µ ( I ,j ) = ( k + ) p k . As a result of summation over k, we obtain Z − | f ( x ) | dx = X k ∈ Z ( k + 112 ) p k , which is equivalent to (14). Example 3.
Let f ( x ) = Λ x , where Λ is a positive number. Then function f satisfies thecondition of Theorem 2 if and only if Λ is an odd number and Λ >
1. Formula (13) gives D = 1 / − Example 4. (zig-zag map) On the interval [ − / , / , let us consider an odd piecewiselinear function f that takes the half-integer value f ( ξ ) = p + 1 / ξ (0 < ξ < / f (0) = 0 and f (1 /
2) = 1 / . Then the diffusion coefficient, according to (13), has thefollowing form: D = p + 112 (2 p + 1 − ξ ) . It argees with known results (see [12] and the references therein.)By using the results of Theorems 1 and 2, we can give the following definition of deter-ministic diffusion for the dynamical system (1):
Definition 5.
We say that the one-dimensional DS (1) has a deterministic diffusion if, forany initial probability measure µ with bounded density, there exist a sequence of numbers σ n > ξ n such that the sequence of measures µ n = F n µ obtained from the initialmeasure by the n -fold action of the DS is asymptotically equivalent, as n → ∞ , to a sequenceof normal measures with variances σ n and mean values ξ n The main problems connected with the deterministic diffusion for DS is to establish thefact of existence of this diffusion in terms of the functions f specifying the DS (1). Theconstruction of an efficient algorithm for the determination the coefficient of deterministicdiffusion D and drift ξ n seems to be an important problem, especially from the viewpoint of9pplications. At present, a series of expressions is deduced and various numerical methodsfor the analysis of the dependence of the diffusion coefficient on the form of the functions f are developed. Note that the dependence of the coefficient of deterministic diffusion for theLDS (1)–(3) on Λ is quite complicated (nowhere differentiable fractal dependence) even forthe linear function f ( x ) = Λ x in the interval I = [ − , ) , [4, 9].We now present heuristic arguments for the existence of deterministic diffusion for theLDS with the linear function f ( x ) = Λ x, Λ > , in the interval I = [ − , ) . In this case,the LDS (1)–(3) can be represented in the form x n +1 = x n + (Λ − { x n ) , (15)where { x n ) = x − [ x ) is the fractional part of the number x and [ x ) is the nearest integer forthe number x. Equation (15) can be represented in the equivalent form x n +1 = x + n X k =1 (Λ − { x k ) . (16)If the map f is stretching, i.e., Λ > , and the initial value x takes values in the interval I with a certain (e.g., constant) probability density, then we can assume that the quantities { x k ) , the fractional parts of x k are uniformly distributed over the interval I and independentfor different k. The rigorous substantiation of the uniformity of distributions of the fractionalparts for different stretching maps can be found in [7]. According to (16), the quantities x n can be regarded as the sum of x and n independent identically distributed random variables.Thus, by the central limit theorem, the quantities x n , as n → ∞ , are distributed accordingto the normal law with zero mean value and the variance equal to the sum of variances ofthe terms. Since the variance of { x ) , regarded as a variable uniformly distributed in theinterval [ − , ) , is equal to σ = R − x dx = , we get σ ( x n +1 ) = (Λ − . This yields theapproximate relation for the diffusions coefficients D = (Λ − for any linear map f ( x ) = Λ x in the LDS (1)–(3).Let us turn to the case where the Markov subpartition is consistent with the LDS ac-cording to Definition 2. In this case, by P k,j ( n ) , we denote the densities in each subinterval I k,j of the interval I k , k ∈ Z, j = 1 , , ..., m, for n iterations. If we consider the collectionof P k,j ( n ) j = 1 , , ..., m, as the components of the vector P k ( n ) = col ( P ( n ) k, , ..., P ( n ) k,m ) , thenwe get an analog of Eq. (6), where the quantities P k ( n ) are vectors, p k = { p k,i,j }| mi,j =1 is thetransition matrix for the Perron–Frobenius operator, and p k,i,j is the density of measure in I k,i in the case of single application of the LDS to the measure with unit density on I θ,j . This vector analog of Eq. (6) is also well studied and the solution P k ( n ) leads to the normaldistribution in the m -dimensional space [5, 3]. Theorem 3.
Assume that the Markov subpartition { I k,j } k ∈ Z, ≤ j ≤ n of the axis is consistentwith the action of the LDS (1)–(3) and that the function f ( x ) is stretching and maps theinterval I = [ − , ) into a finite interval [ a, b ) of length greater than 2. Then, after the n -fold action of the LDS, the initial measure µ in I with bounded density is transformedinto a measure µ n asymptotically equivalent, as n → ∞ , to a normal measure with densities k,j ( n ) in the intervals I k,j P k,j ( n ) = α j √ πDn e − ( k − ξn )24 Dn , j = 1 , , ..., m, (17) where ξ n is the drift, D is the coefficient of deterministic diffusion, and the parameters α j > , j = 1 , , ..., m, specify the distributions of densities in the subintervals I k,j of theintervals I k , k ∈ Z. Proof.
As already indicated, the vectors P k ( n ) = col ( P k ( n, , ..., P k,m ( n )) satisfy the equa-tion P k ( n + 1) = X P j P k − j ( n ) , (18)where the matrix P j is expressed via the translation matrices in the considered LDS. Notethat the matrix E = P j P j is equivalent to the stochastic irreducible matrix DED − , whereis a D -diagonal matrix with the lengths of subintervals I , , ..., I ,m on the diagonal. As aresult of the Fourier transformation, Eq. (18) is transformed into the difference equation P ( λ, n + 1) = P ( λ ) P ( λ, n ) , (19)where the matrix P ( λ ) = P j P j e ijλ and the vector P ( λ, n ) = P k P k ( n ) e ikλ . The solution of Eq. (19) is explicitly expressed via the eigenvalues and eigenvectors(including the adjoined vectors in the case of multiple eigenvalues) of the matrix P ( λ ) . If z ( λ ) is the maximum eigenvalue of the matrix P ( λ ) (this eigenvalue is simple), then thesolution of Eq. (19) as n → ∞ can be represented in the form P ( λ, n ) = z n ( λ ) α ( λ ) + o (1) , n → ∞ , (20)where α ( λ ) is the eigenvector of the matrix P ( λ ) corresponding to the eigenvalue z ( λ ) . Notethat z (0) = 1 and all components of the vector α (0) are positive. If we perform the inverseFourier transformation, then we get relation (17) with D = − ∂ ∂λ z ( λ ) | λ =0 , ξ = i ∂z ( λ ) ∂λ | λ =0 from (20). Remark . By using Theorem 3, one can deduce the explicit “parametric” dependence of D on Λ for the linear function f ( x ) = Λ x specifying the LDS. In this case, the role ofparameters is played by the characteristics ( s, ~n ) of the Markov subpartition { I k,j } consistentwith the LDS, that is by the parity of m and the integer-valued vector ~n. By using theseparameters, we can explicitly construct three polynomials R ~n ( x ) , P ~n ( x ) , and Q ~n ( x ) withinteger-valued coefficients such that the slope Λ is the maximum root of the polynomial R ~n ( x ) , i.e., R ~n ( x ) = 0 (see Proposition 5) and D = P ~n ( x ) Q ~n ( x ) . Example 5.
As an example, we consider the case of LDS with a linear function f ( x ) = Λ x, whereΛ = 2 s is an even number. In this case, the Markov subpartition I k is formed by twosubintervals I k, + = [ k, k + ) and I k, − = [ k − , k ) . If the first components of the vectorsare referred to the intervals I k, + and the second components are referred to the intervals11 k, − , then p = 1Λ (cid:18) (cid:19) , p j = 1Λ (cid:18) (cid:19) , p s = (cid:18) (cid:19) , p − j = 1Λ (cid:18) (cid:19) , and p − s = 1Λ (cid:18) (cid:19) , j = 1 , , ..., s − . In this case, the characteristic function P ( λ, n ) = P k P k ( n ) e iλk is vector-valued and P ( λ ) = X j p j e ijλ = sin λ s s sin λ e i λ ( s − e − i λ ( s +1) e i λ ( s +1) e − i λ ( s − ! . The determinant of the matrix P ( λ ) is equal to zero and the trace sp p ( λ ) = sin λ ss sin λ cos λ s − . Hence, the nontrivial eigenvalue z ( λ ) of the matrix P ( λ ) coincides with the trace of thematrix P ( λ ) . This yields the following well-known explicit expression [4] for the diffusioncoefficients for even Λ = 2 s : D = − ∂ ∂λ sp p ( λ ) | λ =0 = (Λ − − . (21)As it follows from the relations of Example 3 and (21) for the coefficient of deterministicdiffusion in the case of a linear function f ( x ) = Λ x with integer Λ , D (Λ) is a monotonicfunction of Λ . As Λ increases, i.e., the degree of stretching of the map f ( x ) increases, thecoefficient of deterministic diffusion D (Λ) increases. However, if we compare the relationsfor even and odd Λ , then, e.g., we get D = for Λ = 3 and D = for Λ = 4 . Thus, atfirst sight, these results seem to be intuitively strange.We consider this problem in more detail. Assume that the initial probability measureis uniformly distributed over the interval [ , , that is its density is constant and equal to2. As a result of the one-time action of the LDS, we get a measure with constant densityin the interval [ ,
2) for the mapping with Λ = 3 and, in the interval [0 ,
2) for the mappingwith Λ = 4 . Clearly, the variance of a measure with constant density in the interval [0 , , , whichagrees with our assumption that the higher the degree of stretching, the greater the variance.A similar picture is observed in the case where the initial measure has a constant density inthe interval [ − , − ) . For Λ = 3 , the mapping realized by the LDS leads to a measure withconstant density in the interval [ − , − ) and the mapping with Λ = 4 yields a measure withconstant density in the interval [ − , . However, if the initial probability measure has a constant density in the union of intervals[ − , − ) S [ , , then, in view of the linearity of the Perron–Frobenius operator, for Λ = 3 , we get a measure in the union of intervals A = [ − , − ) S [ , , whereas for the mappingwith Λ = 4 , the measure has a constant density in the interval A = [ − , . Clearly, thevariance of the probability measure on A is greater than the variance on A . Thus, thehigher degree of stretching not always corresponds to a greater variance.
If stretching isdirected toward the mean value, then the variance decreases.
The relations for the values of D for even and odd Λ can be replaced by a single relationvalid for all Λ by introducing a function ω (Λ) , namely, D (Λ) = 124 (Λ − − ω (Λ)) , (22)12here the function ω (Λ) is equal to 2 for even Λ and to –1 for odd Λ . For any Λ , the function ω (Λ) is characterized by a complex fractal dependence on Λ caused by the fractal dependenceof D on Λ (see [4, 9]). The function ω (Λ) can be approximately regarded as a 2-periodicfunction such that ω (Λ) = 2 − | Λ − | for Λ ∈ [3 , . The plot of this function is depicted inFig. 1 and gives the first approximation of the behavior of D as a function of Λ . Figure 1: Graph of the function D (Λ).Let us consider some examples of a linear function f ( x ) = Λ x, that illustrated the resultsof theorem 3. Example 6.
1. Markov subpartition of m = 3 subintervals: I k, = [ k − , k − ξ ) , I k, =[ k − ξ, k + ξ ) , I k, = [ k + ξ, k + ) , k ∈ Z.
2. A system of two equations for ξ and the value of slope Λ :Λ ξ = 12 , Λ 12 = 2 − ξ.
3. An equation for Λ : Λ −
4Λ + 1 = 0 .
4. A value of Λ : Λ = 2 + √ ≈ . .
5. A value of ξ : ξ = −√ ≈ . .
6. A matrix P ( λ ) : P ( λ ) = 1Λ e − iλ e iλ + e iλ e − iλ e iλ e − iλ + e − iλ e iλ .
7. A value of greatest eigenvalue z ( λ ) of matrix P ( λ ) : z ( λ ) = λ (1 + cos λ + √ cos λ + 2 cos λ ) .
13. A value of deterministic diffusion coefficient: D = √ ≈ . .
9. Values of densities α j on subintervals I k,j : α = α = √ ≈ . , α = √ ≈ . . Example 7.
1. Markov subpartition of m = 3 subintervals: I k, = [ k − , k − ξ ) , I k, = [ k − ξ, k + ξ ) , I k, = [ k + ξ, k + ) , k ∈ Z.
2. A system of two equations for ξ and the value of slope Λ :Λ ξ = 32 , Λ 12 = 3 − ξ.
3. An equation for Λ : Λ −
6Λ + 3 = 0 .
4. A value of Λ : Λ = 3 + √ ≈ . .
5. A value of ξ : ξ = −√ ≈ . .
6. A matrix P ( λ ) : P ( λ ) = 1Λ e − iλ e iλ + e − iλ e iλ + e iλ e − iλ e iλ + e − iλ e iλ e − iλ + e − iλ e iλ + e − iλ e iλ .
7. An equation for greatest eigenvalue z ( λ ) of matrix P ( λ ) :Λ z − Λ z (1 + 2 cos λ + 2 cos 2 λ ) − Λ z (1 + 2 cos λ ) + 1 + 2 cos λ = 0 .
8. A value of deterministic diffusion coefficient: D = 31Λ − − ≈ . .
9. Values of densities α j on subintervals I k,j : α = α = √ ≈ . , α = √ ≈ . . Example 8.
1. Markov subpartition of m = 3 subintervals: I k, = [ k − , k − ξ ) , I k, = [ k − ξ, k + ξ ) , I k, = [ k + ξ, k + ) , k ∈ Z.
2. A system of two equations for ξ and the value of slope Λ :Λ ξ = 32 , Λ 12 = 2 + ξ.
3. An equation for Λ : Λ − − .
4. A value of Λ : Λ = 2 + √ ≈ . .
5. A value of ξ : ξ = √ − ≈ . .
6. A matrix P ( λ ) : P ( λ ) = 1Λ e iλ + e − iλ e iλ e − iλ e iλ + e − iλ e iλ e − iλ e iλ + e − iλ .
7. An equation for greatest eigenvalue z ( λ ) of matrix P ( λ ) : − Λ z + Λ z (1 + 2 cos λ ) + Λ z (1 + 2 cos 2 λ (1 + 2 cos λ )) + 1 + 2 cos λ = 0 .
8. A value of deterministic diffusion coefficient: D = 9Λ + 213Λ − √ ≈ . .
9. Values of densities α j on subintervals I k,j : α = α = + √ , α = + √ . Example 9.
1. Markov subpartition of m = 4 subintervals: I k, = [ k − , k − ξ ) , I k, = [ k − ξ, k ) , I k, = [ k, k + ξ ) , I k, = [ k + ξ, k + ) , k ∈ Z.
14. A system of two equations for ξ and the value of slope Λ :Λ ξ = 1 , Λ 12 = 1 + ξ.
3. An equation for Λ : Λ − − .
4. A value of Λ : Λ = 1 + √ ≈ . .
5. A value of ξ : ξ = √ − ≈ . .
6. A matrix P ( λ ) : P ( λ ) = 1Λ e iλ e − iλ e iλ e − iλ e iλ e − iλ .
7. A value of greatest eigenvalue z ( λ ) of matrix P ( λ ) : z ( λ ) = (1 + √ λ ) .
8. A value of deterministic diffusion coefficient: D = −√ ≈ . .
9. Values of densities α j on subintervals I k,j : α = α = √ ≈ . , α = α = √ ≈ . . Example 10.
1. Markov subpartition of m = 4 subintervals: I k, = [ k − , k − ξ ) , I k, = [ k − ξ, k ) , I k, = [ k, k + ξ ) , I k, = [ k + ξ, k + ) , k ∈ Z.
2. A system of two equations for ξ and the value of slope Λ :Λ ξ = 1 , Λ 12 = 2 − ξ.
3. An equation for Λ : Λ −
4Λ + 2 = 0 .
4. A value of Λ : Λ = 2 + √ ≈ . .
5. A value of ξ : ξ = −√ ≈ . .
6. A matrix P ( λ ) : P ( λ ) = 1Λ e − iλ e iλ e iλ e − iλ e iλ e − iλ e iλ e − iλ e − iλ e iλ .
7. A value of greatest eigenvalue z ( λ ) of matrix P ( λ ) : z ( λ ) = (1 + cos λ + p − sin λ ) .
8. A value of deterministic diffusion coefficient: D = .
9. Values of densities α j on subintervals I k,j : α = α = √ − ≈ . , α = α = −√ ≈ . . Example 11.
1. Markov subpartition of m = 3 subintervals: I k, = [ k − , k − ξ ) , I k, = [ k − ξ , k − ξ ) , I k, = [ k − ξ , k + ξ ) , I k, = [ k + ξ , k + ξ ) , I k, =[ k + ξ , k + ) , k ∈ Z.
2. A system of equations for ξ and the value of slope Λ :Λ ξ = 32 , Λ ξ = 2 − ξ , Λ 12 = 2 + ξ .
15. An equation for Λ : Λ − −
4Λ + 3 = 0 .
4. A value of Λ : Λ ≈ . .
5. A value of ξ : ξ = , ξ = − .
6. A matrix P ( λ ) : P ( λ ) = 1Λ e iλ + e − iλ e iλ e − iλ e iλ + e − iλ e iλ e − iλ e iλ + e − iλ e iλ e − iλ e iλ + e − iλ e iλ e − iλ e iλ + e − iλ .
7. An equation for greatest eigenvalue z ( λ ) of matrix P ( λ ) :Λ z − Λ z (1 + 2 cos λ ) − Λ z (1 + 2 cos λ (1 + 2 cos λ )) − Λ z (1 + 2 cos λ + 2 cos 2 λ ) − z +1 + 2 cos λ = 0 .
8. A value of deterministic diffusion coefficient: D = 81Λ + 69Λ − + 99Λ − ≈ . .
9. Values of densities α j on subintervals I k,j : α = α = Λ +4Λ − − Λ) , α = α = Λ +4Λ − − Λ) ,α = − − Λ) . Example 12.
1. Markov subpartition of m = 6 subintervals: I k, = [ k − , k − ξ ) , I k, = [ k − ξ , k − ξ ) , I k, = [ k − ξ , k ) , I k, = [ k, k + ξ ) , I k, =[ k + ξ , k + ξ ) , I k, = [ k + ξ , k + ) , k ∈ Z.
2. A system of equations for ξ and the value of slope Λ :Λ ξ = ξ , Λ ξ = 2 − ξ , Λ 12 = 2 + ξ .
3. An equation for Λ : Λ − + Λ − .
4. A value Λ : Λ ≈ . .
5. A value ξ : ξ = +1 , ξ = +1 .
6. A matrix P ( λ ) : P ( λ ) = 1Λ e − iλ e iλ + e iλ e − iλ e iλ + e iλ e − iλ e − iλ e iλ e iλ e − iλ e − iλ e iλ e iλ e − iλ + e − iλ e iλ e − iλ + e − iλ e iλ .
7. An equation for greatest eigenvalue z ( λ ) of matrix P ( λ ) : − Λ z + Λ z (1 + 2 cos λ ) + Λ z (1 + 2 cos λ ) + z (1 + 2 cos λ + 2 cos 2 λ + 2 cos 3 λ ) + 2 +2 cos 2 λ + 4 cos λ cos 2 λ = 0 .
8. A value of deterministic diffusion coefficient: D = 5Λ + 26Λ + 2218Λ + 18Λ + 56 ≈ . .
9. Values of densities α j on subintervals I k,j : α = α = Λ +13Λ − , α = α = Λ +23Λ − ,α = α = Λ +Λ+53Λ − . Example 13.
1. Markov subpartition of m = 7 subintervals: I k, = [ k − , k − ξ ) , I k, = [ k − ξ , k − ξ ) , I k, = [ k − ξ , k − ξ ) , I k, = [ k − ξ , k + ξ ) , I k, =16 k + ξ , k + ξ ) , I k, = [ k + ξ , k + ξ ), I k, = [ k + ξ , k + ) k ∈ Z.
2. A system of equations for ξ and the value of slope Λ :Λ ξ = ξ , Λ ξ = ξ , Λ ξ = 12 , Λ 12 = 2 − ξ .
3. An equation for Λ : Λ − + 1 = 0 .
4. A value of Λ : Λ ≈ . .
5. A value of ξ : ξ = , ξ = , ξ = .
6. A matrix P ( λ ) : P ( λ ) = 1Λ e − iλ e iλ + e iλ e − iλ e iλ + e iλ e − iλ e iλ + e iλ e − iλ e iλ e − iλ + e − iλ e iλ e − iλ + e − iλ e iλ e − iλ + e − iλ e iλ .
7. An equation for greatest eigenvalue z ( λ ) of matrix P ( λ ) :Λ z − Λ z (2 + 2 cos λ ) + 1 = 0 .
8. A value of deterministic diffusion coefficient: D = − ≈ . .
9. Values of densities α j on subintervals I k,j : α = α = Λ4(Λ − , α = α = Λ4 , α = α = (Λ − − − , α = Λ − − − . The results of theorem 3 and above-considered examples allow us to give the followingdefinition of deterministic diffusion of LDS (1)-(3).
Definition 6.
We say that the LDS (1)–(3) has a deterministic diffusion if, for any initialprobability measure µ with bounded density, there exist a sequence of numbers σ n > ξ n and 1–periodic function α ( x ) ≥ , / R − / α ( x ) dx = 1 , such that the sequence of mea-sures µ n = F n µ , obtained from the initial measure by the n -fold action of the LDS, isasymptotically equivalent, as n → ∞ to a sequence of measures with densities ρ n ( x ) = α ( x ) σ n √ π e − ( x − ξn )22 σ n . Remark . The LDS (1)–(3) on the entire axis generates the associated DS x n +1 = ˆ f ( x n ) onthe finite segment [ − , ) , where the value of the function ˆ f ( x n ) is equal to the fractionalpart of f ( x ) , i.e., ˆ f ( x n ) = { f ( x )) , and the function ˆ f maps the segment [ − , ) into itself.We denote this DS, which is a compactification of the LDS, by CLDS. The function α ( x ) ≥ The results of numerical experiments carried out in [2] demonstrate that the deterministicdiffusion in long billiard channels is anomalous. There are different theoretical models of17his type of diffusion transport in long channels. One of these models can be found in [1].The essence of this model can be described as follows:Consider a long billiard channel with boundaries in the form of periodically repeated arcsslightly distorting the straight lines of boundaries. Assume that the upper boundary of thechannel is symmetric to the lower boundary. Then the trajectory of motion of a billiard ballregarded as a material point obeying the ideal law of reflections from both boundaries can bestudied in a channel of half width with symmetric reflections of the trajectory in the upperpart of the channel into its lower part relative to the straight middle line of the channel.Hence, the reflections from this middle line can be regarded as ideal.We approximately assume that the reflection from the lower part of the distorted bound-ary is realized in its linear approximation. However, the normal ν to the surface of actualreflection does not coincide with normal to the rectilinear approximation of the channeland is described by a known function periodic along the axis of the channel. Assume that x n − , x n , and x n +1 are the abscissas of points of three consecutive reflections of the billiardball and that the vector ν ( x n ) makes an angle α ( x n ) with the normal to the surface (seeFig. 2). Then the analysis of the ideal reflection at the point x n leads to the equality of theangle of incidence ϕ n and the angle of reflection ψ n relative to the vector ν. It is clear that x n − x n − h = tan( ϕ − α ) and x n +1 − x n h = tan( ψ + α ) . Hence, x n +1 − x n h = x n +1 − x n h + tan(2 α )1 − tan(2 α ) · x n +1 − x n h . (23)For large h, we get the following approximate relation from (23): x n +1 − x n = x n − x n − + h tan(2 α ( x n )) . h ψϕ αx n − x n x n +1 ν Figure 2: Reflections of the billiard ball in a channel of half width.Let the function f ( x ) = tan(2 αx ) be periodic in x with period 1. Then we arrive at thefollowing model of a trajectory in the billiard channel: x n +1 − x n = x n − x n − + f ( x n ) . (24)The dynamical system (24) is a two-dimensional dynamical system discrete in time whichgeneralizes the dynamical system (1). The initial conditions for x and x are assumed tobe given. Let x = 0 and let x ∈ I k = [ k − , k + ) . Equation (24) can be represented inthe equivalent form as follows: x n +1 = x · n + n X k =1 ( n + 1 − k ) f ( x k ) . (25)18ssume that the function f ( x ) = Λ · { x ) in Eq. (25) is linear with respect to the fractionalpart of the argument. Then, for Λ > x uniformly distributed over the interval I k , theterms in (25) can be regarded uniformly distributed independent random variables. Thisleads to the normal distribution of x n +1 with variance equal to the sum of variances of allterms in (25).Hence, the variance of the distribution of x n +1 is equal to σ = 112 n +Λ n ( n + 1)(2 n + 1)6 and nonlinearly depends on n → ∞ . The deterministic diffusion inthis billiard strip is anomalous. Thus, there are two factors leading to the anomaly of thedeterministic diffusion, namely, the quadratic dependence of the variance on distribution ofthe initial values even for the ideal billiard and the growth of coefficients of the terms in sum(25) leading to the cubic dependence of the variance of x n +1 on n. We now make an important remark. We have studied the distribution of positions of thebilliard ball after n → ∞ reflections. From the physical point of view, it is more importantto get the distribution of the abscissas of billiard balls for large values of time t because,for the the same period of time, the billiard ball makes different numbers of reflections fordifferent trajectories.In the ordinary billiards, the time between two successive collisions is proportional to thecovered distance. However, in the model of bouncing ball [13] over an irregular surface, thetime between two consecutive collisions is maximal if the points of consecutive reflectionscoincide. The problem of investigation of the anomalous deterministic diffusion in billiardchannels as t → ∞ is of significant independent interest.Note that the Gaussian density in the case of ordinary diffusion is the Green function ofthe Cauchy problem for the partial differential equation ∂u∂t − D ∆ u = 0 . In the investigation of anomalous diffusion, we consider a differential equation with frac-tional derivatives. Equations of this kind are studied in numerous works (see [11] and thereferences therein).
Acknowledgments
The authors express their gratitude to Prof. A. Katok for a remarkable course of lectureson the contemporary theory of dynamical systems delivered in May, 2014 in Kiev and forthe informal discussions which stimulated the authors to write this paper.
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