The Relationship between Tsallis Statistics, the Fourier Transform, and Nonlinear Coupling
Kenric P. Nelson and Sabir Umarov
Abstract — Tsallis statistics (or q-statistics) in other q functions disappears when q = 1 . This definition nonextensive statistical mechanics is a one-parameter originates from the raising of probabilities to the power q [8], description of correlated states. In this paper we use a translated entropic index: 1 q→ q . The essence of but it does not utilize the symmetry of the real numbers about this translation is to improve the mathematical zero. symmetry of the q-algebra and make q directly Moreover, the physical interpretation of what the q proportional to the nonlinear coupling. A conjugate parameter represents has remained obscure. These conceptual transformation is defined qˆ ≡ −2q which provides a dual difficulties are eliminated by a simple translation. In the 2+q expressions for q Gaussians and q algebra, the term mapping between the heavy-tail q-Gaussian distributions, whose translated q parameter is between (1− q ) appears frequently. By an explicit translation of the −2 entropy applications can be shown to have translation, has non trivial consequences. First, the physical parameters proportional to the nonlinear generalization is now centered about the natural symmetry of statistical coupling. the real numbers, so for q = 0 , the q Gaussian is the familiar
bell shaped Gaussian. This symmetry about zero makes it Index Terms — nonextensive entropy, nonlinear coupling, q- possible to express the fundamental equations of q statistics Gaussian, stochastic processes, Tsallis statistics and q algebra without burdensome numerical constants. The structure of the simplified q algebra is introduced in Section II and detailed in Appendix A. In section III the mathematical I. INTRODUCTION symmetry is utilized to define a conjugate dual between heavy HE developments in the field of nonextensive entropy tail and compact support distributions. This important T[1] have had a significant impact on analytical methods for relationship is used to extend and improve the definition of the modeling statistical behavior of nonlinear systems. The q Fourier transform in Sections IV and V. Most significantly, physics of nonextensive systems have been shown to provide the proposed variable q emerges as a direct measure of the important analytical techniques for analysis of turbulence [2], nonlinear coupling in the statistical system, which is discussed communication signals [3, 4], dynamics of the solar wind [5], in Section VI. Thus the simplification of the q statistic neural networks [6] and other complex phenomena. equations leads to symmetric mathematical relationships and a In this paper the expressions for q statistics [7] are stronger connection to physical principles. simplified and connected more directly to nonlinear analysis, by a simple translation of the q parameter. As currently expressed, the nonlinearity inherent in the q Gaussian and
K. P. Nelson is with Raytheon Company, Woburn, MA 01801 USA (corresponding author 603 508 9827; [email protected]). S. Umarov., is with Tufts University, Medford, MA 02155 USA ([email protected]). Tsallis statistics and the Fourier transform 2
The nonextensive entropy function provides a starting point II. PRELIMINARIES for interpreting the proposed translation of the q parameter. Entropy is the average of the ‘surprisal’, ln 1 : Tsallis [8] showed that raising probabilities by a power ( pi ) leads to a nonextensive generalization of the Boltzmann n 1 Gibbs Shannon entropy function relevant to the S=ln()p = − pi ln p i . Following from the q i ∑ thermodynamic properties of nonlinear systems. Further i=1 developments in this field have led to a q algebra [10], which exponential, the logarithm can be generalized to a function of encapsulates the nonlinear relationships in q statistics in a xq −1 fractional power by ln (x ) ≡ , where q is the translated form which provides analogs to basic mathematical q q relationships. The building block for the q algebra is the q parameter of q statistics, and the natural logarithm is recovered exponential function and the q addition which defines how the as q approaches zero. Using the generalized logarithm, the exponents of q exponentials are combined. However, the nonextensive entropy is defined as current expression for these two functions suffers from the n need to include 1 q in the definition. As currently expressed, −1 + p1−q n p−q −1 ∑ i the q exponential is S≡ln 1 = p i = i =1 . (4) q qp ∑ i 1 i q q (1−q ') i=1 ex ≡(1 + (1 − q ') x ) q ' + With the translated nonextensive entropy function, the power 1 (1−q ') (1) of the probability is 1− q . The one represents the probability = (1+− (1qx ') ) 1 +−≥ (1 qx ') 0 required for the average, and the negative sign is due to the 0 1+ (1 −q ') x < 0 ‘surprisal’ being the inverse of the probability, and the and two exponents combined by q addition is nonlinear coupling term q follows from the generalization of x⊕q ' y =++ x y xy − qxy' . Scaling the q addition and the the logarithm. The defining expression for nonextensive other functions of q algebra to many variables is complicated entropy, which relates the q entropy for two independent by the accumulation of numerical constants. However, by systems A and B, now has the nonlinear term scaled by q translating 1− q ' to q, the q addition not only is easier to SABq(+= ) SA qq () + SB () + qSASB qq ()() (5) express, but makes explicit that q is the nonlinear coupling Using the escort probability with a power of 1 q as a between the variables x and y. Now q addition constraint leads to the q Gaussian distribution as the maximum is x⊕q y = x + y + qxy and the nonlinear coupling is zero at q entropy solution [11]. The escort probability is defined as 1−q q = 0 . The q exponential function simplifies to pi Pi = . (6) 1 n 1 q 1−q x q (1+qx ) 1 + qx ≥ 0 ∑ pi eq ≡(1 + qx ) + = (2) i=1 0 1+qx < 0 . Further insight into this form of the escort probability is x x For lim eq = e and there is no nonlinear coupling, but as q possible by multiplying the numerator and denominator by all q→0 of the non zero probabilities raised to the q power increases so does the coupling between the variables. m Negative values of q are associated with a ‘decoupling’ or p p q pi i∏ j anti correlation between the variables. A full description of q j=1 pi j≠ i the translated q algebra is provided in appendix A. Pq, i =n = n m , (7) pi q A simple illustration of the utility of the q exponential ∑ q ∑ pi∏ p j i=1 pi i=1 j=1 function is that it is the solution for the nonlinear differential j≠ i equation [8] where n is the number of states and m is the number of states dy dy y1−q=− or y =− y q . (3) with non zero probability. Here the escort probability is dt dt expressed to show that the statistical properties of the i th state 1 of the nonlinear system include the nonlinear coupling via q to The solution to this equation is ye=−t =(1 − qt ) q . For q = 0 , q all the other states of the system with non zero probability. the differential equation is linear and its solution is the The numerator is expanded to further illustrate this expression exponential decay function. For q > 0 the nonlinear coupling q ppppi[]12⋅... iim− 1 ⋅ p + 1 ... p . (8) increases the rate of decay. Instead of decaying to zero In this form, the numerator of the escort probability as t → ∞ , the solution “decays” to zero at the finite value of explicitly includes the nonlinear coupling between all the t = 1 . This faster than exponential decay is referred to as q states of a system. The denominator normalizes this ‘compact support’ if the negative values beyond the zero root expression. Because of the significance of nonlinear coupling are truncated to the value zero. For q < 0 the nonlinear and its explicit expression in this probability distribution, the coupling decreases the rate of decay. The solution now decays raising by the power 1 q and renormalization of a probability at a slower than exponential rate, referred to as a ‘heavy tail’. distribution will be referred to as the coupled probability Tsallis statistics and the Fourier transform 3
distribution. d d 1 1 −1 eax =+()1 qaxq =+ a() 1 qax q q0 ≠ . (14) In continuous variables, the coupled probability density dxq dx function and the generalized entropy are The power 1 −1 = 1−q is equivalent to the q parameter [ f( x ) ]1−q q q q fq ( x ) = +∞ (9) transitioning from q → . Expressing the derivative with the 1−q 1−q f( x ) dx th ∫ [] q exponential notation and extending to the n derivative −∞ ∞ results in 1−q 1−q −1 + f ( xdx ) d q ∫−∞ eax =+− a1q (1 qax ) = a exp (1 − qaxq ) , ≠ 1 Sq = . (10) q ()1−q q [] q dx + 1−q (15) d n n 1 The q mean and q variance of a random variable X are eax = a n 1 −− iq 1 exp (1 − nqaxq ) , ≠ . n q ∏()() q [] defined using the coupled probability density function dx i=1 1−nq n ∞ The integral of the q exponential has the following form q= xf q ( x ) dx ∫−∞ 1+q (11) ax q q ∞ edx=1 1 + (1 + qax ) σ2=−(X ) 2 = ( x − )(). 2 f x dx ∫ q a(1+ q )() 1 + q q qq ∫−∞ qq 1 =a(1+ q ) expq [] (1 +qax ) + c,1 q ≠− 1 Reference [12] includes a development of the all the q 1+q moments for arbitrary densities. The maximum of the q n (16) ...edxdxax ...=1 1 exp (1 + nqax ) entropy with finite q variance is the q Gaussian ∫() ∫ q an ∏ 1+nq q [] i=1 1+nq β1 β 2 n q 2 q q −β(x − ) q q i−1 1 Gxq()= 1 − qxβ qq ( − ) ≡ e q (12) +c x , q ≠ − . C + C ∑ i q q i=1 n x where eq is the q exponential as defined in (2), The expressions for the derivative and integral of the q 2 β=[(2 + q ) σ 2 ] − 1 and the normalization term C is Gaussian are complicated by the x term in the exponent. q q q Rather than develop the full form, it is sufficient for the current th discussion to note that the n integral (excluding the integer 1+q th Γ( q ) polynomial) and the n derivative of the q Gaussian have the π ⇐q > 0 q 2+ 3 q following relationship Γ()2q th 2 2 power of n q Gaussian derivative: q→ q − n Cq = π ⇐ q = 0 (13) (17) th 2 2 2+q power of n q Gaussian integral: q→ q + n Γ()−2q π ⇐ −<2q < 0 These q values are the same as those for the nth q Fourier −q 1 Γ th ()−q Transform. Summarizing, the n q parameter is defined as 2q −1 Thus, the q Gaussian has its origin in the nonlinear coupling zqq()≡≡ =+1 n n =±± 0, 1, 2, ... (18) of statistical states, as defined in the coupled probability n n 2 + nq q 2 density and the generalized entropy. For positive values of q Positive values of n are related to the n th integral of the q the nonlinear coupling between the states strengthens the Gaussian. Negative values of n are related to the n th decay, resulting in a distribution with compact support. In this derivative. The numeral 2 is related to the power of 2 for the case, the probability is zero for x − > 1 . At q=0 there q qβq Gaussian distribution and generalizes to 0<α ≤ 2 for the is no nonlinear statistical coupling and the traditional Gaussian (q ,α ) distributions [6, 8 10] 1 distribution, common throughout linear systems analysis, −β x α α q Gxae( )≡ = a 1 − qxβ (19) holds. For negative values of q the decoupled states result in q,α q ( ) weakening of the decay, resulting in a heavy tail distribution. αq −1 z≡ q ≡ =+1 n 2 (,)an (,) an q α Beyond q < − 3 the classic variance is divergent, but the q α + nq (20) variance is finite. Beyond q < − 2 the distribution can not be 0<≤α 2; n = 0, ±± 1, 2, ... normalized; i.e. Cq is divergent. The (q ,α ) distribution is part of the generalization of the The arguments of the gamma function within the alpha stable Lévy distributions. Using the translated q normalization term are related to a sequence of q values which parameter the q sequence can be expressed without numerical were defined as a property of repeated application of the q constants. The q(α ,n ) term will be abbreviated by the Fourier transform [13]. Here we show that the q sequence is 2q equivalent expression q2 n , since q= q 2 n = 2 n . For α (α ,n ) α also related to the derivative and integral of q Gaussians. The 2+()α q derivative of the q exponential is double sided q exponential functions, α = 1; for Gaussian functions α = 2 . The ability to express this important relationship in a simple intuitive form is in sharp contrast to Tsallis statistics and the Fourier transform 4
the original expression −1 −2qˆ− 2( − q1 ) qˆ =2+qˆ = 2() + − q = q . The q conjugate of a function is n+(α + n ) q ' 1 q '(α ,n ) = (21) determined by applying the q conjugate to each of the q α +n(1 − q ') parameters of the function. However, care must be taken if the The clarity of the translated expression for the q sequence conjugate is applied to the value ±qk . A broader hat will be simplifies the establishment of many important relationships used to clarify that the conjugate includes the sign and the within the q algebra. For example the normalizing coefficient sequence value. for the q Gaussian Cq can be expressed as Definition 2 The q conjugate for ±qk and the inverse are 1 Γ −2( ± qk ) π ( q2 ) ±qk ≡ = ∓ q k ±1; q ⇐q > 0 Γ 1 2+ ( ± qk ) ()q3 (27) ∓ −2(qk ±1 ) C=π ⇐ q = 0 (22) ∓qk±1 = = ± q k . q 2+ (∓q ) k ±1 Γ 1 th π ()−q1 This notation is required to contrast with the n sequence of −q ⇐ −<2q < 0 th Γ 1 a q conjugate. The n sequence value of a q conjugate is ()−q 2(−q1 ) ±=±qzqˆk k( ˆ ) =± = ∓ q 1− k (28) The normalization can be simplified in other ways by applying 2+k ( − q 1 ) When applying the conjugate to a q function care is needed to the relationship Γ(x + 1) =Γ x ( x ) : Γ()1 = ( 1 +=Γ 1) 11 () and q2 q qq apply the conjugate to the same q sequence value throughout Γ()1 =Γ ( 1 + 1) = 11 Γ () . the function. q3 q 1 qq 11 Definition 3: Let X be the set of functions which depend qk III. CONJUGATE PAIRS OF HEAVY TAIL AND COMPACT on qk . Then the q conjugate operator and its inverse are SUPPORT Q GAUSSIANS defined as The q sequence can be used to define a mapping between T(qq ,ˆ ) : X q→ X q ˆ the heavy tail q Gaussians, with 2 < q < 0 and the compact k k . (29) T= T−1 = T support q Gaussians, with 0 < q < ∞. For guidance in this (,)qqˆ (,) qq ˆ (,) qq ˆ mapping, first consider the Fourier transform for a power law ∈ ɶ ≡ = ɶ For fq(k ; x ) X q ; fqx(;)k T( qqk ,ɶ ) ((;)) fqx fqx (;) k . function, which the heavy tail q Gaussians approach k 2 2 asymptotically as x goes to infinity. For 2 < q < 0, the tail of A set of conjugate q Gaussian pairs with σq= σ q ˆ = 1 is the q Gaussian approaches a power law shown in Figure 2. On the left of Figure 2 are the compact 2 1 2 e−β x ∼(− qxβ 2 )q ∼ Ο (), xx q →∞ (23) support distributions which proceed from the Gaussian q distribution for values close to zero and converge to the The Fourier transform of a power law is also a power law uniform distribution as q goes to infinity. On the right are the with the following form [14, 15] conjugate heavy tail distributions, which converge to a 2 −(2 + 1) q 2 2 π q uniform distribution of infinite extent and infinitesimal density x ⇔Γ+π (q 1)sin( − q )ω for −<< 2q 0 (24) at q = − 2 . If β is invariant the compact support distribution Where ω is the dual variable (or frequency). In the frequency converges to a delta function as q goes to infinity. domain the power is −(2 + 1) which rearranged in the form of q The integral of the unnormalized q Gaussian and qˆ a q Gaussian, shows the following relationship with the q Gaussian functions have the following relationship. Restating sequence 2+q 1 the definitions − − 22q 2 q ∞ w= w 1 (25) 2 C edx−βq x =q =2 + qCσ ∫ q q q So the Fourier transform of a power law is suggestive of an −∞ βq (30) important mapping between a q Gaussian and a −q Gaussian. ∞ 1 −β x2 C edxqˆ =qˆ =2 + qCˆσ In fact the mapping between G↔ G constitutes a ∫ qˆ qˆ q ˆ q− q 1 −∞ βqˆ conjugate dual relating the heavy tail q Gaussians to the If −2 0 is mapped to qˆ 1 1 1 Γ−q −Γ− q q the heavy tail range −2 <−q1 < 0 and vice versa. This π( −1) π ( 1) ( 1 ) Cqˆ = = (31) −q11 − q 1 1 1 relationship leads to the definition for the q conjugate dual . Γ() − q ()−q Γ() − q −2 Definition 1 The q conjugate dual is defined as The ratio between C and C is −2q qˆ q qˆ ≡ − z1 ( q ) = 2+q . (26) The inverse of the conjugate is the same function: Tsallis statistics and the Fourier transform 5
Γ − 1 2 2 q ( q1 ) q Gaussian is transformed from → − , but the form of the π 3 3 q q 1 − q() q 2 2 1 1 Γ − 1 Cqˆ ()q q 2 + q = = = (32) function is no longer a q Gaussian, the q Fourier transform Γ − 1 ()q Cq π 1 q 1 2 − q [13, 20, 21] preserves the q Gaussian form with the power Γ − 1 ()q shifting from 2→ 2 . Previously, the q Fourier transform was This ratio also holds if 0 normal distribution and V Ff[]()ω = eixω ⊗ fxdx () = fxe () ixw[ f ( x )] dx . (34) V q∫supp f qq ∫ q ν −∞ has a chi squared distribution with ν degrees of freedom. The Definition 5: Let q > 0 . The q Fourier transform for the ν +1 − ν +1 Γ( ) 2 ()2 compact support domain is defined as distribution is f( t )=2 1 + t ; which can be ν ()ν vπ Γ(2 )
2 β −βt 2 Ff[]()ω = T Te [ix ω ⊗ fxdx ()] equated to a heavy tail q Gaussian eq with q = − ( ) q(,) qqˆ (,) qqqq ˆ Cq ν +1 ∫supp f ν +1 1 2 ∞ (35) and β =( 2ν ) = 2 +q , which is equivalent to σ q = 1 . The ixw[ f ( x )] − q = T(,)qqˆ T (,) qq ˆ [() fxe q ] dx . 2 ∫ −∞ complement of q is qˆ = ν , which demonstrates an inverse relationship between the q conjugate and the degree of Lemma 6: The functional form of Fq [ f ](ω ) is equal for the freedom. In Section VI the interpretation of physical compact support domain ( q > 0 ) and the heavy tail domain applications with heavy tail distributions will also be −2 ≤q <∞ . simplified by utilizing the q conjugate parameter. Proof: Since Leubner and Voros [18, 19] have discussed the relationship ixω ix ω between Tsallis statistics and the κ distribution, used by the Te(qqqq ,ˆ ) [⊗ fxe ( )] = qq ˆ ⊗ ˆ fx ( ) and space physics community to model power law distributions in ix ω eˆ⊗ ˆ fxdx() = Ffw ˆ []() , ∫supp f q q q plasma velocities. Both approaches have the same form, and as defined by Leubner and Voros have the simple translation then Ffq[]()ω= T( qqqˆ , ) [ Ff ˆ []()] ω and Definition 4 and κ = 1 q using the proposed q parameter or κ =1 (1 − q ') using Definition 5 have the same functional form. □ the original parameter. The q sequence transitions defined by Corollary 7: Let q > − 2 . For the q Gaussian 2 (18) are easier to express, κ= κ + n , since κ represents the G( x ) = ae −β x the q Fourier Transform is n 2 q q 2 power of the generalized exponential directly rather than the −Bx aC q (2+q ) FGxw()() = Ae , A = , B = 2 q . (36) q q q 1 β 8β a inverse. However, the exponential and Gaussian functions are recovered as κ → ∞ , which complicates the relationship This relationship is derived in Example 9. between these fundamental functions and there generalization Example 8: As an illustration of the q Fourier transform to ‘compact support’ and ‘heavy tail’ distributions. applied to a general function, consider the transformation of In the next two sections the conjugate pairs of q parameters the uniform distribution as a function of q. The uniform are used to extend and improve the nonlinear generalization of distribution is defined as the Fourier transform. 1 2 x ≤ 1 U() x = . (37) IV. THE q FOURIER TRANSFORM FOR COMPACT SUPPORT 0 elsewhere DOMAIN In contrast to the Fourier transform where the power of the Tsallis statistics and the Fourier transform 6
For −2 0 , can be shown to q2 q = =sincq [(1 + q )2 w ]. have the same form. (1 + q)2 q w 2 Starting with a compact support q Gaussian function, The solution makes use of the q sin [22], which is defined as fx()= ae−β x , q > 0 , the conjugate heavy tail function is eix− e − ix q sin (x ) = q q ; and the q sinc, which is defined as determined using conjugate mapping q→ q ˆ . For purposes of q 2i defining the q Fourier transform for compact support q sinq (w ) q w . The properties of sinc [(1+ q )2 w ] are examined in q2 Gaussians, the constants a and β are treated as independent of Figure 4. For q = 0 , the solution is the sinc function which is q. The definition still holds if either of these constants is a consistent with the Fourier transform. For negative values of q function of q and is thus transformed. The conjugate q the oscillations of the q sin and q sinc functions are dampened. ˆ −βˆx function is then fxT()=(q , qˆ ) fxae () =ˆ q ˆ , −<<2 q ˆ 0 , Between −1
0 using the Ffxˆ()()( w= aqˆ e 4 βa ) 2 qˆ β q ˆ conjugate transformation. In this case in accordance with (35) ˆ 2 ˆ −Bw 2(−q1 ) ∞ =Aeqˆ ; qˆ1 = =− q ; (41) − q 1 2+ ( − q1 ) FUx[()]()ω = T UxT () [ eixw[ U ( x )] ] dx q(,) qqˆ∫ (,) qqq ˆ C ˆ qˆ (2+ q ) −∞ Aˆ =a ; B ˆ = . β 2qˆ 1 8β a −qˆ 1 1 1 qˆ =Tˆ [1 + qixwˆ ()2 ] dx (q , q ) ∫ 2 The function −q Gaussian is transformed to q1 Gaussian −1 qˆ by repeating the transformation T this time with the parameters 1+1 1 + 1 =T 1+qˆ (1 + qiwˆ 2qˆ )qˆ −− (1 qiw ˆ 2 q ˆ ) q ˆ (39) −q = qˆ → q . Applying T to (41) results in (qˆ , q ) (2)i wq ˆ 2 qˆ 1 1 −Bwˆ 2 iqw(1+ )2q − iqw (1 + )2 q ˆ 1 Ffxq[ ( )]( w )= TFTfx qˆ [ ( )]( w ) = TAe q ˆ =q e − e 1 (2i )(1+ q )2 w q2 q 2 , (42) 2 C (2+q ) q −Bw q =Ae, Aa = , B = 2q sin [(1+ q )2 w ] q1 β 8βa =q2 = sinc [(1 + q )2q w ]. (1+ q )2 q w q2 which is identical to (40), the result for heavy tail q Gaussians. The solution makes use of the transformation from qˆ to q , Thus, the definition for the q Fourier transform is shown to be 2 2 consistent for the full range of q Gaussian distributions, defined in (28). Figure 4c shows the q Fourier transform for q > − 2. the uniform distribution for several values between 0.01≤q ≤ 1 . In this region, the oscillations of the q sinc Definition 10 : Let q > − 2 . The q Fourier transform is function are amplified. A logarithmic plot is used to show the defined for fx()∈ LR1 () as changing scale of the oscillations. Eventually the eixw ⊗ fxdx( ) −<≤ 2 q 0, ∫supp f q q amplification overwhelms the oscillations. Near q = 0.5 the F[ fx ( )]( w ) = q ixw ˆ Tˆ e ˆ⊗ ˆ fxdx( ) q > 0 (43) function no longer resembles a sinc function and is negative (qq , ) ∫supp f q q except for values of w near zero. At q = 1 the function is one fxˆ( )= T [ fx ( )] for all frequencies. Although not shown, for values of q > 1 (q , q ˆ ) For simplicity of expression, the conjugate transformation the amplification is dominant and the function approaches of the integrand in the domain q > 0 , is implied by specify qˆ infinity without any oscillations. Tsallis statistics and the Fourier transform 7 as the nonlinear parameter. Following the integration, the as conjugate transformation is specified directly. In the original Ffxw= eixw ⊗ fxdxɶ −<≤ qɶ . (47) qɶ[()]()∫ q ɶ q ɶ () 2 0 q algebra notation the extended definition of the q Fourier supp f transform has the following form. Definition 15 (The compact support qɶ − FT ): Let qɶ be in ɶ > Definition 11 : Let q '< 3 . The q ' Fourier transform is the compact support q Gaussian domain q 0 . Then the conjugate q Fourier transform is defined as defined for fx()∈ LR1 () as Ffxw[ ( )]( )= T eixw ⊗ fxdxqɶˆ ( )ɶ > 0, qɶ( qqɶɶˆ , ) ∫ qqɶɶ ˆ ˆ eixw ⊗ fxdx() 1 ≤ q '3, < supp f ∫supp f q' q ' − qɶ −2( − q ) (48) ˆ 2 2 Fq '[ fx ( )]( w ) = qɶ = = = q . ixw ˆ +ɶ + − 1 T e ⊗ fxdx() q '1 < (44) 2q 2() q 2 (',')qq∫supp f qq ' ' Lemma 16: The conjugate q FT of f(x) is equal to the q fxˆ()= T [ fx ()]; q ' = 5− 3q ' (q ', q ') 3−q ' exponential conjugate of the q FT An important consequence of the extension of the q FT to ɶ = qFTf- []() w T(q , qɶ ) [- qFTf []()] w . include the domain of compact is its implications for the q Proof: For −2
Gaussian is for the q parameter to be transformed to −q2 prior function with damped oscillations; and for q < 0 the –q sinc to applying the q FT. The transformation between function has amplified oscillations. q and −q is also a conjugate dual, but is related to the heavy 2 Example 18: Likewise, the qɶ FT for a q Gaussian is tail and compact support 2 sided exponential distributions. w2 Definition 12: Let α = 1 for the (α ,q ) exponential family. − 2 aC ɶ β 2qɶ 2+qɶ −βx qɶ 4 qɶa Fae()( w= e ) 2 The q exponential conjugate dual is defined using (20) as qɶ q β q ɶ ɶ 2 −q =Aeɶ −Bw qɶ ==−=2 qɶ qq (50) qɶ ; 12+qɶ 1 ˆ ; ɶ ≡− =− = 1 q zq( ) q ɶ (1,1) 2 aC +ɶ (2−q ) 1 + q ɶ q ɶ (2q ) 2 (45) A= ; B =2qɶ = − 2 q . −qɶ β 8βa 8βa 2 qɶ−1 = qɶ = = q . 1 + qɶ Again, the coefficients a and β are treated as independent of q. The solution is a qˆ Gaussian with coefficients A and B Definition 13 : The q exponential conjugate and its inverse are functions of qɶ = − q . Thus for compact support q for the set of q functions ( X ) is 2 qk Gaussian with q > 0 , the qɶ FT is a heavy tail q Gaussian with → T(qq ,ɶ ) : X q X q ɶ −2 VI. THE RELATIONSHIP BETWEEN q AND NONLINEAR example of q exponential model in which case q is equal to the 1 STATISTICAL COUPLING nonlinear coupling. In this example q = β H , where β is the The conjugate pair of q parameters provides a key insight inverse temperature and H is the total energy. into the relationship between Tsallis Statistics and the coupling The applications examined here lead to a conjecture strength of nonlinear systems. This section will provide an regarding nonlinear statistical coupling . Whereas, the interpretation of the applications discussed in [1] and other nonlinear coupling defined by q from the q algebra references, which shows that for applications with ‘compact expressions can be any real number, the nonlinear statistical support’ q Gaussian distributions q is proportional to the coupling is constrained by the requirement of a finite mean nonlinear coupling, and that applications with ‘heavy tail’ q distribution for q > − α . Nevertheless, it is conjectured that Gaussian distributions the conjugate qˆ is proportional to the the nonlinear statistical coupling strength is the positive real nonlinear coupling. Table 1 provides a synopsis of this numbers for both the heavy tail and compact support regions. interpretation. As a demonstration of the relationship between For the heavy tail region the strength of the nonlinear Tsallis statistics and nonlinear coupling, we will review an statistical coupling is determined using the conjugate application of particular relevance to information systems, relationship. multiplicative noise. Conjecture 19: Let φ be the nonlinear statistical coupling Drawing upon the analysis by Anteneodo and Tsallis [23], strength which results in the (q ,α ) distribution being consider a stochastic process influenced by multiplicative and characteristic of a statistical properties of the system. Then φ additive Gaussian white noise ()m () a is related to the entropic index q and the power of the q dX= fX() + gX ()(2 MdW )(2 + AdW ) (51) ttt t t exponential variable α by the following relationship where X( t ) is the stochastic variable, f is the deterministic q q ≥ 0 (m , a ) α drift and g is the noise induced drift, and dW t are φ = −q (56) independent Wiener processes which define the multiplicative −α 1 ixw x+ y q e⊗ fxdx( ) −<≤ 2 q 0, eq =[1 + qxy ( + )] ∫supp f q q Fq [ fx ( )]( w ) = 1 ixw ˆ =[(1 +qx ) ++ (1 qy ) − 1] q (58) Tˆ e ˆ⊗ ˆ fxdx( ) q > 0 (qq , ) ∫supp f q q 1 =[eeqx + qy − 1] q =⊗ ee x y In turn the expanded definition of the q Fourier transform 1 1 q q q should enable the extension of the q Central Limit Theorem to xq −1 Combinations of q logarithms, ln q x = q , follow from these the compact support domain. The conjugate relationship is definitions. Thus, the q logarithm can be combined using the also used to define an alternative definition for the q Fourier Transform which provides a transformation between conjugate following relationships lnqq(xy⊗) = ln q x + ln q y and pairs of q Gaussians. While this conjugate q Fourier lnq (xy )= ln qqq x ⊕ ln y . Transform requires an additional transformation related to To emphasize the benefits of the proposed translation of the conjugate q exponentials, the resulting pair of functions is q parameter, compare the expressions for three terms and then closer in form to the original Fourier transform, i.e. wide examine the simplicity of adding n terms with the new functions transform to narrow functions and vice versa. expression From a variety of different perspectives, the translated q original expression: parameter is shown to be directly proportional to nonlinear x⊕ y ⊕ z =+++ x y z xy + xz + yz coupling. First, the translated definitions for the q entropy q' q ' sum of independent systems and the q sum have nonlinear −q '( xy +++− xz yz ) xyz 2' q xyz + q ' 2 xyz terms which are scaled by q. Second, the escort probability or translated expression: (59) coupled probability distribution is expressed in (8) to x⊕⊕=+++ y z x y z qxy( ++ xz yz ) + qxyz2 demonstrate a nonlinear coupling between the statistical states. q q N N NN−1 1 i+ N − 1 N Finally, the physical parameters for many nonextensive N−1 N ∑∑qiix=+ xq∑ ∑ xxq ij +... ∑ ∏ xqx j + ∏ i applications are shown to be proportional to q in the compact i=0 i = 0 iji ==+ 01 i = 0 j= i i = 0 ˆ support domain or q in the heavy tail domain. By simplifying the relation to nonlinear analysis it is anticipated that the analytical tools of q statistics will find The q product expression is also simplified: wider utility in nonlinear information theory. Nonlinear Original expression: dynamics affect information systems at the device level, where 1 1'−q 1' − q 1' − q 1−q ' the details of quantum mechanics [25] are increasingly xyzx⊗⊗=q' q ' + y + z − 2 important, at the circuit board level, where multiplicative noise Translated expression: (60) [23] impacts signal transmission and detection, and at the 1 system level, where disperse correlations between system q q q q x⊗⊗=q y q z xyz ++− 2 components [26] create nonlinear behavior. The analytical N 1 tools discussed here provide methods for the modeling and q q q q ∏ qx i≡ xx1 ++ 2 ... x N −− (1) N simulation of nonlinear statistical systems in a simple, intuitive i=1 form. The q algebra provides a methodology for concise expression of the relationships between groups of q exponents and q logarithms. For this reason, the connection between q APPENDIX I addition and q product is not direct: xxx⊕q ⊕ q ≠⊗3 q x . THE TRANSLATED q ALGEBRA EXPRESSIONS Rather the q addition of like q exponents connects with raising Combinations of the q exponential and q logarithm function 3 of q exponentials to a power: exp x⊕ x ⊕ x = e x . And form the basis of the q algebra. The q exponential functions, q( qq) ( q ) 1 the q product of like q exponentials connects with the ex =[1 + qx ] q , can be combined using q addition ⊕ in the q + q multiplication of q exponents: ex⊗ e x ⊗ ee x = 3 x . These following manner q qq qq q 1 1 two expressions are not equal, because raising a q exponential eex y =[1 + qx ]q [1 + qy ] q q q to a power rescales both the exponent and the q parameter 1 q γ =+[1q ( x ++ y qxy )] (57) γ 1 ⋅γ q q ex =+1 qxq =+ 1 γ xe = γ x (61) q [] ()γ q x⊕q y γ = eq x3 3 x 3x Thus (eq ) = e q which does not equal eq unless q=0, As this example shows, q addition typically combines 3 exponents of a q exponent. The q product typically combines which recovers the standard exponential function. The q 1 q subtraction and q division operations follow in a similar the q exponential functions f⊗ g =[ fq + g q − 1] . For q + manner and are defined as follows example, Tsallis statistics and the Fourier transform 10 x− y [20] S. Umarov, and C. Tsallis, “On a representation of the inverse Fq x y = . transform,” Physics Letters A, vol. 372, no. 29, pp. 4874 4876, 7 July, q 1+ qy (62) 2008. 1 [21] C. Tsallis, and S. M. D. Queiros, “Nonextensive statistical mechanics ⊘ q q q and central limit theorems II Convolution of independent random xq y= x − y + 1 variables and q product,” AIP Conference Proceedings, vol. 965, no. 21, 2007. Table A.1 summarizes the parameter translation from 1 q to [22] S. Umarov, C. Tsallis, M. Gell Mann et al. , “Symmetric (q, α) Stable Distributions. Part I: First Representation,” cond mat/0606038v2, 2008. q of the basic expressions of the q algebra. [23] C. Anteneodo, and C. Tsallis, “Multiplicative noise: A mechanism leading to nonextensive statistical mechanics,” J. Math. Phy., vol. 44, ACKNOWLEDGMENT 2003. [24] J. J. S. de Andrade, M. P. Almeida, A. A. Moreira et al. , "A The authors are grateful for conversations with Constantino Hamiltonian Approach for Tsallis Thermostatistics," Nonextensive Tsallis. P. K. Rastogi provided valuable comments during Entropy: Interdisciplinary Applications , Santa Fe Instiitute Studies in preparation of the manuscript. the Sciences of Complexity M. Gell Mann and C. Tsallis, eds., pp. 123 138, New York: Oxford University Press, 2004. [25] S. Abe, "Generalized Nonadditive Information Theory and Quantum REFERENCES Entanglement," Nonextensive Entropy: Interdisciplinary Applications , Santa Fe Instiitute Studies in the Sciences of Complexity M. Gell Mann [1] M. Gell Mann, and C. Tsallis, Nonextensive Entropy : Interdisciplinary and C. Tsallis, eds., pp. 55 62, New York: Oxford University Press, Applications , New York: Oxford University Press, 2004. 2004. [2] F. M. Ramos, C. R. Neto, and R. R. Rosa, “Generalized [26] S. Thurner, “Nonextensive statistical mechanics and complex thermostatistical description of intermittency and non extensivity in networks,” Europhysics News, vol. 36, no. Special Issue on turbulence and financial markets,” cond mat/0010435, 2000. Nonextensive Statistical Mechanics, pp. 218 220, Nov/Dec, 2005. [3] Karmeshu, and S. Sharma, “q exponential product form solution of [27] C. Tsallis, "Nonextensive Statistical Mechanics: Construction and packet distribution in queueing networks: Maximization of Tsallis Physical Interpretation," Nonextensive Entropy: Interdisciplinary entropy,” IEEE Comm. Letters, vol. 10, pp. 585 587, 2006. Applications , Santa Fe Instiitute Studies in the Sciences of Complexity [4] S. Sharma, and Karmeshu, “Bimodal packet distributions in loss M. Gell Mann and C. Tsallis, eds., pp. 1 54, New York: Oxford systems using maximum Tsallis entropy principle,” IEEE Trans. University Press, 2004. Comm., vol. 56, pp. 1530 1535, 2008. [28] G. A. Tsekouras, A. Provata, and C. Tsallis, “Nonextensivity of the [5] L. F. Burlaga, and A. F. Viñas, “Triangle for the entropic index q of cyclic Lattice Lotka Volterra model,” Phy. Rev. E, vol. 69, 2004. non extensive statistical mechanics observed by Voyager 1 in the distant [29] N. O. B. Paujuelo, "unpublished [see details in C. Tsallis, Introduction heliosphere,” Physica A, vol. 361, pp. 173 179, 2006. to Nonextensive Statistical Mechanics Approaching a Complex [6] A. D. Anastasiadis, “Neural network training and applications using World]," Springer, 2008, in press. biological data,” Doctoral Thesis, School of Computer Science and [30] C. Tsallis, M. Gell Mann, and Y. Sato, “Asymptotically scale invariant Information Systems, University of London, London, 2005. occupancy of phase space makes the entropy Sq extensive ” Proc. [7] H. Suyari, “Law of Error in Tsallis Statistics,” IEEE Trans. Inf. Theory, National Acad. Sci., vol. 102, no. 43, pp. 15377 15382, 2005. vol. 51, no. 2, pp. 753 757, February, 2005. [8] C. Tsallis, “Possible generalization of Boltzmann Gibbs statistics,” J. Stat. Phys., vol. 52, no. 1 2, pp. 479 487, July, 1988. [9] F. Topsøe, “Factorization and escorting in the game theoretical approach to non extensive entropy measures,” Physica A, vol. 365, pp. 91 95, 2006. [10] E. P. Borges, “A possible deformed algebra and calculus inspired in nonextensive thermostatistics ” Physica A, vol. 340, no. 1 3, pp. 95 101, 1 September, 2004. [11] H. Suyari, “Generalization of Shannon Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy,” IEEE Trans. Inf. Theory, vol. 50, no. 7, pp. 1783 87, August, 2004. [12] C. Tsallis, A. R. Plastino, and R. F. Alvarez Estrada, “Escort mean values and the characterization of power law decaying probability densities,” cond mat/0802.1698v1, 2008. [13] S. Umarov, C. Tsallis, and S. Steinberg, “On a q Central Limit Theorem Consistent with Nonextensive Statistical Mechanics,” Milan J. Math, vol. Online First, 2008. [14] S. Smith. "An Interesting Fourier Transform 1/f Noise," http://www.dsprelated.com/showarticle/40.php . [15] W. H. Beyer, “CRC Standard Mathematical Tables,” Boca Raton, Florida: CRC Press, 1984. [16] R. V. Hogg, and A. T. Craig, "Introduction to Mathematical Statistics," p. 182, New York: Macmillan, 1978. [17] "Student's t distribution," http://en.wikipedia.org/wiki/Student's_t distribution . [18] M. P. Leubner, and Z. Vörös, “A nonextensive entropy path to probability distributions in solar wind turbulence,” Nonlinear Processes in Geophysics, vol. 12, pp. 171 180, 1 Feb, 2005. [19] M. P. Leubner, and Z. Vörös, “A nonextensive entropy approach to solar wind intermittency,” The Astrophysical Journal, vol. 618, pp. 547 555, 2005. Tsallis statistics and the Fourier transform 11 Table 1: Examples of the relationship between the entropic index and the nonlinear coupling as defined by (56). q and α parameters Nonlinear Statistical Application Physical Parameters range of q Coupling, φφφ −2 ν degree of freedom Degree of q =, α = 2 1 Freedom for v +1 ν Student T −2 −2 γ L index of Lévy distribution Lévy like q = , α = 2 1 anomalous γ L + 1 γ L diffusion [27] −2 q =1 , α = 1 d – dimension of growth Lotka Volterra d 1 d Model [27, 28] q > 0 β inverse temperature Thermostatics q =1 , α = 1 β H 1 with finite bath β H H – total energy of system and [24] † q > 0 bath * The parameter ν is defined for all real numbers and therefore all real values of q are appropriate. This application is similar to (3) which is a general solution to a nonlinear equation. † The reference defines q as the negative of the value defined by Tsallis. Tsallis statistics and the Fourier transform 12 Table VII.1 Comparison of the original and translated q algebra expressions. Function Name Original Parameter, q ' Proposed Translation is Gaussian q=−1 q ' and 1 −= qq ' q '= 1 q = 0 is Gaussian Exponential 1 1 x 1−q ' x q eq ' ≡[1 + (1 − q ') x ] + eq ≡[1 + qx ] + Logarithm x1−q ' −1 xq −1 ln x ≡ ln x ≡ q ' 1− q ' q q Escort or Coupled Probability q ' 1−q pi pi Distribution W W pq ' p1−q ∑ j=1 j ∑ j=1 j W W Entropy 1− pq ' −1 + p1−q S = ∑ i=1 i S = ∑ i=1 i q ' q '− 1 q q q Entropy expressed as average of q 1 1 surprise Sq'= ln q ' Sq= ln q pi pi q addition x⊕q ' yxy =++−(1 qxy ') x⊕q y = x + y + qxy q subtraction x− y x− y x⊕ y = x⊕ y = q ' 1+ (1 − q ') y q 1+ qy q product 1 1 1'−q 1' − q 1−q ' q q q x⊗=q ' yx[ + y − 1] + x⊗q y =[ x + y − 1] + q division 1 1 1'−q 1' − q 1−q ' q q q x⊗=q ' yx[ − y + 1] + x⊗q y =[ x − y + 1] + q Gaussian sequence 2q '+ n (1 − q ') 2q n =0, ± 1, ± 2,... q'= z ( q ') = qn= z n ( q ) = n n 2+n (1 − q ') 2 + nq −1 1 n = + q 2 q alpha sequence for αq'+ n (1 − q ') αq q alpha distribution q'n= z n ( q ') = qn= z n ( q ) = n =0, ± 1, ± 2,... α +n(1 − q ') α − nq −1 1 n = + q α Conjugate 5− 3q ' −2q q Gaussian Dual −q' = −q = 1 3− q ' 1 2 + q Conjugate 3− 2q ' −q q exponential Dual −q' = −q = 2 2− q ' 2 1+ q Additive Duality q'a ( q ')= 2 − q ' qa ( q ) = − q Multiplicative Inversion q' ( q ') = 1 −q 1 m q ' q( q ) = = m 1− q 1− q−1 Multiplication & Difference 1 qqnn−+11⋅ = q n − 1 − q n + 1 q 'n−1 + = 2 q 'n+1 * Harmonic Mean 2 1 1 2 1 1 = + = + 1−q '1 − q '1n − q ' − n q qn q − n *Pajuelo [29] has shown an interesting relationship between the harmonic, arithmetic, and geometric mean which is consistent with the triplet of q values measured for the solar wind [5, 30]. Tsallis statistics and the Fourier transform 13 Parameters of Conjugate q-Gaussians 121212 101010 888 −2q −2( − q1 ) −q1 = and q = 2+q 2() + − q 1 666 Heavy-Tail to Compact-Support 444 q-Conjugateq-Conjugate ParameterParameter q-Conjugateq-Conjugate ParameterParameter 222 Compact-Support to Heavy-Tail 000 Gauss. (0) -2-2-2 -2-2-2 000 222 444 666 888 101010 121212 141414 Var. Diverges (-2/3) q - Nonlinear Parameter Uniform (Inf) Cauchy (-1) Figure 1: The complementary q parameters for heavy tail ( −2 Tsallis statistics and the Fourier transform 14 0.01 0.50.50.5 -0.010 000 0.1 0.50.50.5 0.10.1 -0.095 000 -0.667 0.50.50.5 1.01.01.0 000 101010 ProbabilityProbability DensityDensity -1.667 ProbabilityProbability0.50.50.5 DensityDensity -1.667 000 100 0.50.50.5 100 -1.961 000 -2-2-2 000 222 -2-2-2 000 222 xxx xxx 2 Figure 2: The conjugate pairs of q Gaussian distributions with σ q = 1 . The value of q is inset in each figure. On the left are the compact support qGaussians with q > 0 . On the right are the conjugate heavy tail distributions with 2 < q < −2q 0. The conjugate distribution has qˆ = − q 1 = 2+q . As q approaches 2 the density is spread infinitesimally to the tails of the distribution. Tsallis statistics and the Fourier transform 15 20 18 Ratio 16 14 12 10 Normalization 8 6 -- NormalizationNormalization ConstantConstant -- NormalizationNormalization ConstantConstant q q q q Conjugate C C C C 4 Normalization 2 0 -2 0 2 4 6 8 10 12 14 q - Nonlinear Parameter Figure 3: The qGaussian pdf normalization constant, Cq and the conjugate normalization constant, C . The ratio (line) of the conjugate normalization −q1 C 1.5 1.5 − q1 2+q q constants is equal to C=( 2 ) = ( q ) (circles). q 1 Tsallis statistics and the Fourier transform 16 111 q =-0.010 q =-0.005 0.80.80.8 q =0.000 q =0.005 0.60.60.6 q =0.010 w] w] w] w] 0.40.40.4 q q q q 0.20.20.2 [(2+q)2 [(2+q)2 [(2+q)2 [(2+q)2 2 2 2 2 q q q q 000 sinc sinc sinc sinc -0.2 -0.4 -0.6 -0.8 -20-20-20 -10-10-10 000 101010 202020 www 151515 101010 111 q =-0.010 q =0.010 q =-0.050 q =0.050 q =-0.100 q =0.100 0.8 0.80.8 q =-0.333 q =0.500 101010 q =-0.500 101010 q =1.000 w] w] w] w] w] w] 0.60.60.6 w] w] q q q q q q q q 0.40.40.4 555 101010 [(2+q)2 [(2+q)2 [(2+q)2 [(2+q)2 [(2+q)2 [(2+q)2 [(2+q)2 [(2+q)2 2 2 2 2 2 2 2 2 q q q q q q q q 0.20.20.2 sinc sinc sinc sinc sinc sinc sinc sinc 000 101010 000 -0.2 -5-5-5 101010 000 555 101010 151515 000 202020 404040 606060 808080 100 www www q Figure 4: The q Fourier Transform of the uniform distribution is sincq [(q+ 1)2 w ] . a) Near q = 0 the oscillations of a sinc 2 function are evident. Negative values of q dampen the oscillations and positive values of q amplify the oscillations. b) As q 1 varies from 0.01 to 0.04 the oscillations are further dampened. q = − 3 is the critical damping value. c) For q between 0 and 0.1 the oscillations are amplified. A semilog plot is used to show the exponential scale of the oscillations. The gaps are negative values of the functions. For q near 0.5 the function is negative except near w = 0 . For q = 1 the function is one for all values of w. For q > 1 the function increases monotonically. − α b) physical examples with α ≠ 1,2 and a 2 (53) more detailed description of the q generalized α stable Dx()≡ A + Mgx[] () distributions c) when a nonlinear stochastic process has If the deterministic and noise induced drift are related by a solutions with divergent mean q < − α what is the relationship τ 2 potential Vx( )= 2 [ gx ( )] , then fx()=− Vx '() =− τ gxgx ()'() . between q and nonlinearity for the three regions (compact The drift term simplifies to support, heavy tail, and divergent mean). M Jx( )= fx ( )(1 − τ ) (54) which clarifies that the deviation from deterministic drift VII. CONCLUSION M induced by the multiplicative noise has a strength of τ . In conclusion, q statistics and its associated q algebra has been simplified, aligned with the symmetry of the natural In this case, the stationary solution pX ( x ) for (52) is a q numbers, and associated more directly with nonlinear Gaussian coupling. The translation q'=− 1 q or 1 −= qq ' which enables −β[g ( x )] 2 τ +M − 2 M pxX()lim= pxte X (,) ∝ q ; β = ; q = (55) the simplification has provided new insights into the t→0 2Aτ + M relationship between compact support and heavy tail q 2M The q conjugate transformation (29) is qˆ = , which Gaussians. The conjugate transformation τ −2q 53' − q ˆ qˆ = or q ' = demonstrates that the q parameter is directly proportional to 2+q 3 − q ' the nonlinearity of the system M . As Table 1 shows this τ between these two domains provides a method for extending simple, intuitive relationship is consistent for many of the definitions and theorems in one domain to the other. Using nonextensive entropy applications. Some analytical models, this transformation, the definition for the q Fourier transform such as the nonlinear Fokker Planck equation, in which the has been extended to the compact support domain: parameter (in this case ν ) is valid for all real values do not follow the pattern just described. The thermodynamics of system in a finite bath described by Anrade, et. al. [24] is an Tsallis statistics and the Fourier transform 9
− 2 Not applicable * Fokker Planck like equation. diffusion [27]