Tsallis Statistics of the Magnetic Field in the Heliosheath Lf Burlaga,1 Af Vin

TSALLIS STATISTICS OF THE MAGNETIC FIELD IN THE HELIOSHEATH L. F. Burlaga,1 A. F. Vin˜ as,1 N. F. Ness,2 and M. H. Acun˜a3 Received 2006 March 27; accepted 2006 May 3; published 2006 May 25

ABSTRACT The spacecraft Voyager 1 crossed the termination shock on 2004 December 16 at a distance of 94 AU from the Sun, and it has been moving through the heliosheath toward the interstellar medium since then. The distributions of magnetic field strength B observed in the heliosheath are Gaussian over a wide range of scales, yet the measured profile appears to be filamentary with occasional large jumps in the magnetic field strengthB(t) . All of the probability distributions of changes in B,dBn { B(t ϩ t) Ϫ B(t) on scales t from 1 to 128 days, can be fit with the symmetric Tsallis distribution of nonextensive statistical mechanics. At scales ≥32 days, the distributions are Gaussian, but on scales from 1 through 16 days the probability distribution functions have non-Gaussian tails, suggesting that the inner heliosheath is not in statistical equilibrium on scales from 1 to 16 days. Subject headings: magnetic fields — solar wind — turbulence — waves

1. INTRODUCTION is expressed by the form of u in the distribution function. In The solar wind moves supersonically beyond a few solar Tsallis statistical mechanics, the probability that a system is in a state with energy u is proportional toexp (Ϫ u) . In classical radii from the Sun to a “termination shock” at ≈94 AU, carrying q b statistical mechanics, the probability that a system is in a state solar magnetic fields with it. The heliosheath is a subsonic Ϫ region extending from the termination shock to the interstellar with energy u is proportional toexp ( bu) , the Boltzmann-Gibbs distribution. For example, in a gas of molecules, the probability medium (Axford 1972; Parker 1963; Hundhausen 1972). Voy- p ager 1 entered the heliosheath by crossing the termination of finding molecules with energyu E is proportional to exp (ϪE/kT), where T is the temperature and k is Boltzmann’s shock on 2004 December 16 (Stone et al. 2005; Gurnett & p Kurth 2005; Decker et al. 2005; Burlaga et al. 2005), and it constant (b 1/kT ). has been moving through the inner heliosheath since that time. This Letter demonstrates that the PDFs of the fluctuations Magnetic field observations have been made by the MAG ex- of B observed in the heliosheath from DOY 1 through 256, periment on Voyager 1 (Behannon et al. 1977) from the launch 2005 on scales from 1 to 128 days can be described by the in 1977 to the present. Tsallis distribution. We show that the PDF of the heliosheath A generalization of Boltzmann-Gibbs statistical mechanics magnetic field strength is Gaussian (consistent with statistical 116 days, but it is not in Gaussian at was introduced by Tsallis (1988, 2004a) and has been applied equilibrium) at scales scales between 1 and 4 days. The observation of the Tsallis to many complex physical systems that are not in thermal equi- distribution of fluctuations of B in the heliosheath and the mea- librium (Tsallis 2004b; Tsallis & Brigatti 2004; Gell-Mann & surement of the variation of the parameters of this distribution Tsallis 2004). The statistical mechanics of Tsallis introduces a with scale describe the multiscale state of magnetic strength new entropy function, which is appropriate for nonequilibrium, fluctuations in the heliosheath and place strong constraints on nonextensive systems in which phase space has a hierarchical any model of these fluctuations. multiscale structure (rather than being uniformly occupied as it is for an equilibrium Gaussian distribution). By extremizing the entropy function subject to a normalization condition and 2. OBSERVATIONS OF DAILY AVERAGES OF MAGNETIC FIELD a constraint on energy, Tsallis derived a probability distribution STRENGTH IN HELIOSHEATH AND THEIR DISTRIBUTIONS function that describes many types of nonlinear physical sys- Daily averages of the magnetic field strength B as a function tems, e.g., “chaotic systems,” turbulence, cosmic-ray spectra, of time (DOY, day of year) from DOY 1 to 256, 2005 are plasma physics, and solar wind physics. shown in the top panel of Figure 1. The error bars on B are The probability distribution function (PDF) derived by Tsal- ≈0.015 nT, the estimated 1 j error due to both systematic and lis is proportional to the “q-exponential function” random effects. Two components of B are determined by rolls about the z-axis (directed toward the Sun) at the times indicated Ϫ { ϩ Ϫ 1/(1Ϫq) expq ( bu) [1 (q 1)bu] , (1) by the asterisks in the top panel of Figure 1. The third com- ponent (along the z-axis) is estimated at the time of a roll by p where b and q are constants. This function reduces to an ex- assumingABz S 0 in an interval of 52 days centered on a roll. ponential in the variable u when q approaches 1, and it ap- Corrections for varying contributions to B from the spacecraft proaches a power law in u when u is large. When q approaches and instrument are made between the rolls, using the infor- p 22Ϫ Ϫ 1 andu y ,expq ( bu)exp( approachesby ) , which is mation from both the primary and secondary instruments (Be- proportional to a Gaussian distribution. hannon et al. 1977). The physics for a particular application of the theory of Tsallis The fluctuations in daily averages of B from DOY 1 to 256 have a distribution that is approximately Gaussian with a mean 1 Laboratory for Solar and Space Physics, Code 612.2, NASA Goddard ABS p 0.11 nT, as shown in the bottom panel of Figure 1. The Space Flight Center, Greenbelt, MD 20771; [email protected]. p ϩ 2 solid curve is a fit to the Gaussian function B B0 Institute for Astrophysics and Computational Sciences, Catholic University 1/2Ϫ Ϫ 2 of America, Washington, DC 20064; [email protected]. {A/[w(p/2) ]} exp { 2[(B ABS)/w]}; the dotted curves are 3 Planetary Magnetospheres Laboratory, Code 695, NASA Goddard Space the 95% confidence bands based on a t-test computed by the Flight Center, Greenbelt, MD 20771; [email protected]. plotting program Origin. L83 L84 BURLAGA ET AL. Vol. 644

Fig. 1.—Top: Daily averages of the magnetic field strength B vs. time measured by Voyager 1 from DOY 1 to 256, 2005 in the heliosheath. Bottom: Distribution of B (filled squares) as well as a fit to a Gaussian distribution function (solid curve) and the 95% confidence bands (dotted curves). Fig. 2.—The symbols are PDFs of relative changes in the magnetic field strength measured by Voyager 1 in the heliosheath from DOY 1 to 256, 2005 The quality of the fit is given by the coefficient of determi- on scales from 1 to 128 days. The solid curves are fits of the data to the Tsallis nationR 2 p 0.89 , and the parameters derived from the fit are distribution function. p p ע p ע p Ϫ B0 1.8 6.0 nT,ABS 0.099 0.004 nT, w 2 j ע p ע 0.10 0.02 nT, andA 5.6 1.6 . bins that were used in constructing the histograms; each his- Gaussian distributions of B were also found at smaller scales togram is plotted a factor of 100 above the one below it for and in different intervals in the heliosheath (Burlaga et al. clarity. The error bars, the short horizontal segments above and 2005). However, the magnetic field strength fluctuations in the below the observed points in Figure 2, were computed as the top panel of Figure 1 have a filamentary structure rather than square root of the countsCi in bin i; thus, the error bars are 1/2 ע simple Gaussian fluctuations. For example, there are relatively plotted atCii(C ) with a shift of 100 for adjacent distri- large jumps at steps and peaks inB(t) , which we shall describe butions introduced for clarity. quantitatively and examine in some detail. These jumps and Using the Levenberg (1944) and Marquardt (1963) method, other changes related to various heliosheath features can be we fit these observed PDFs with the symmetric Tsallis described quantitatively by the probability distributions, as dis- distribution cussed in § 3. They are also associated with the multifractal structure observed in the interval under consideration (L. F. Ϫ Ϫ A exp [Ϫ (dBn)]22{ A [1 ϩ (q Ϫ 1) dBn ]1/(q 1) , (2) Burlaga et al. 2006, in preparation). qqb q qb q

where q is the “entropic index” or “nonextensivity” factor, 3. MULTISCALE PROBABILITY DISTRIBUTION FUNCTIONS Ϫϱ ! q ≤ 3, which depends on scale (Tsallis 1988; Burlaga &

From the daily averages of observations of B in the helio- Vinãs 2004, 2005).ABqq and refer to a Tsallis distribution with sheath (Fig. 1), we computed a set of PDFs (histograms) de- a particular q; they too depend on scale. Forq p 1 , the Tsallis { ϩ Ϫ scribing increments in B,ondBn(tin; t ) B(t it n) B(t i) distribution reduces to the Boltzmann-Gibbs (Gaussian) distri- p n p scalestn 2 days, wheren 0 , 1, 2, 3, 4, 5, 6, and 7. Thus, bution. Burlaga & Vinãs (2004, 2005, 2006) showed that this we consider scales from 1 to 128 days. The plasma is being Tsallis distribution (eq. [2]) provides good fits to PDFs of the convected past the spacecraft, but the speed is not measured magnetic field strength B observed in the supersonic solar wind directly; hence, we cannot determine the length scale associated on scales from 1 to 128 days from 1 to ≈87 AU. with t. The fits of the Tsallis distribution (eq. [2]) to the observed

The distributions ofdBn(tin; t )/ ABS for the daily averages of distributions ofdBn(ti )/ ABS of the heliosheath data in Figure 2 B observed in the heliosheath (Fig. 1, top) on scales from are shown by the solid curves in Figure 2, for all of the distri- 1 day (n p 0 ) to 128 days (n p 7 ) from DOY 1 to 256, 2005 butions of dBn on scales from 1 to 62 days. The ﬁt to the are shown by the points marked as the ﬁlled squares in Fig- distribution of dBn for 128 days is a quadratic, corresponding ure 2. The ordinate is the fraction of days in each of the 13 to a Tsallis distribution withq p 1 . No. 1, 2006 TSALLIS STATISTICS OF MAGNETIC FIELD IN HELIOSHEATH L85

The Tsallis q-exponential distribution shown by the curves in Figure 2 ﬁts all of the observed PDFs, on scales from 1 to 128 days. This result supports the hypothesis that the heliosheath might be in a state described by the nonextensive statistical mechanics of Tsallis. The distributions ofdBn in Fig- ure 2 are approximately Gaussian on scales ≥2 5 p 32 days, and non-Gaussian with long tails with inﬂection points on scales of 1 to at least 8 days.

4. MOMENTS AND FIT PARAMETERS VERSUS SCALE The variation of the distribution functions as a function of scale can be summarized by the variations of the entropic index (q), the kurtosis (K), and the standard deviation (SD) with scale [t(days) p 2n], which are shown as a function of n in Fig- ures 3a,3b, and 3c, respectively. The values of q and its error bars were obtained from the Levenberg/Marquart algorithm. The p standard deviation (SD) and kurtosis (K)ofxiindBn(t ; t ) { Ϫ Ϫ 21/2 { are defined asSD {[1/(N 1)]S(xiiAx S)} and K Ϫ Ϫ 44 {[1/(N 1)]S(xiiAx S) }/SD , respectively; hereAx iS is the mean ofxi , N is the number of points in the sample, and the sum is over xi from 1 to N. The kurtosis as defined above is 3 for a Gaussian distribution. The kurtosis and standard deviation shown by the points in Figures 3b and 3c were computed for each scale t p n 2 from the time series ofdBn(tin; t ) . The “error bars” for SD are measures of the uncertainty of SD, based on the difference between Fig. 3.—The points (filled squares) show variation with scale (days), t p 2n,of(a) the entropic index q,(b) the values of the kurtosis K, and (c)the the SD computed from the time seriesdBn(tin; t ) and the SD computed from the measured distribution function at the scale t. standard deviation SD. The solid curves are fits to the data. The error bar for K was similarly computed. The value of q versus n is plotted as the points in Figure 3a. the slope of SD(n) increases with increasing n below the in- The solid curve is a fit of these points to the function y p flection point, and it decreases with increasing n above the ,inflection point. The fluctuations are larger at the larger scales ע p ע ϩ p A12A exp (x/x 0), whereA 10.81 0.45 , A 20.95 -and there is some indication that the fluctuations approach lim ע p 0.41, and x0 5.0 44. The value of q is significantly greater than 1 forn ≤ 4 (≤16 days), and it approaches values consistent iting values at large and small scales. with 1 for large scales. The correlation coefficient as a function of scale is defined as The kurtosis computed from the observeddBn(tin; t ) is plotted as a function of n in Figure 3b. It is large at the smallest { ϩ Ϫ Ϫ Ϫ 2 scale (n p 0 , 1 day), and it decreases to ≈3 atn p 3 (8 days), C(t) GH[B(tit) GHB(t iii)][B(t ) GHB(t )] /A[(B(t ii) GHB(t )]S. scattering about a value near 3 forn ≥ 4 . The large kurtosis at (4) the smallest scales reflects the large non-Gaussian tails of the corresponding PDFs in Figure 2. The curve is a fit of the data A plot of the pointsC(t) on a log-log scale (not shown) scatter ע p ϩ p to the functiony A12A exp (x/x 0) , where A 12.6 about a straight line for 1 day ≤ t ≤ 16 days with slope s p ע p ע p p Ϫ 2 ע A2 6.2 0.4 , and x0 1.2 0.2, approaching the Ϫ,0.2 valueK p 3 for a Gaussian distribution for large n. 0.55 0.09 (R 0.96 ). This is similar to the observation of Burlaga & Vinãs (2006) showing thatC(t) is a power law Figures 3a and 3b show an evolution from non-Gaussian ≈ ≈ distributions at small scales to Gaussian distributions at scales at 45 and 85 AU in the supersonic solar wind. However, ≥ C(t) for the heliosheath observations is also consistent (within 32 days. We regard the Gaussian distributions as representing p an “equilibrium” state, as in the case of a Boltzmann-Gibbs the uncertainties) with an exponential distribution C(t) y ϩ A exp (Ϫ / ). An exponential fit to the heliosheath ob- distribution. The non-Gaussian distributions represent “non- 0 t te servations for DOY 1–256, 2005 givesR 2 p 0.98 , y p Ϫ0.14 equilibrium” states at small scales. It is significant that the PDFs 0 days for the correlation 2 ע A p 0.911, andp 12 ,0.03 ע in the heliosheath are non-Gaussian on the approximately same 1 te time. Thus, the correlation time is of the order of 12 days, range of scales in which multifractal structure was observed te (L. F. Burlaga et al. 2006, in preparation). The entropy function approximately half a solar rotation. introduced by Tsallis (1988) was motivated by the need to understand systems with multifractal structure (Tsallis 2004b). 5. SUMMARY AND DISCUSSION The standard deviation computed from the observed dBn(tin; t ) When Voyager 1 crossed the termination shock, it was a (Fig. 3b) increases as a function of increasing scale. The variation surprise to discover that the magnetic field strength had a Gaus- of the SD with scale n is described by the curve in Figure 3b, which sian distribution over a wide range of scales, in contrast to the is a fit of the data to the equation nearly lognormal distributions found in the solar wind between 1 and at least 85 AU. This suggests that the solar wind material p ϩ Ϫ ϩ Ϫ y A212(A A )/{1 exp [(x x 0)/w]}, (3) and magnetic field are thermalized when they are heated in their passage through the termination shock, bringing them p ע p p p Ϫ with A1 0.30, A2 0.65, x0 2.8 0.3, and w close to a thermal equilibrium. The SD has an inflection point atn ≈ 3 (8 days); It was also surprising to discover that the magnetic field .0.8 ע 1.17 L86 BURLAGA ET AL. Vol. 644 strength profile in the heliosheath appears to have a filamentary strength fluctuations in the heliosheath. No theory or model of structure, despite the observed Gaussian distribution of the the solar wind and heliosheath is complete if it does not predict magnetic field and the nearly isotropic turbulence relative to the Tsallis distribution and the observed scale dependence of the mean magnetic field (Burlaga et al. 2005). The boundaries its parameters. For example, Burlaga et al. (2003) found that of the filaments are observed as large jumps in B on relatively the form of observed multiscale distribution functions of small scales. The probability distribution of dBn due to the changes in the magnetic field in the distant solar wind at boundaries of filaments and to other types of changes in B is ≈60 AU can be derived from an MHD model with realistic described by the Tsallis distribution of nonequilibrium statis- input functions derived from observations made by spacecraft tical mechanics at all scales from 1 to 128 days. At scales closer to the Sun. (3) As discussed above, the Tsallis distri- t ≥ 32 days, the PDFs are Gaussian, consistent with statistical bution function is derived from a physical principle, an extre- equilibrium. At scales from 1 to 8 days, the PDFs have large mum of an entropy function, rather than being just an arbitrary non-Gaussian tails. “fit” to the observations. The observed applicability of the Tsallis distribution to the heliosheath (and solar wind) is important in three respects. The data in this Letter are from the magnetic field experiment (1) It provides a quantitative description of one aspect of the on Voyager 1. N. F. Ness was partially supported by grant multiscale state of the magnetic fields in the inner heliosheath. NNG04GB71G to the Catholic University of America. T. (2) This multiscale Tsallis distribution provides a very strong McClanahan and S. Kramer carried out the processing of the constraint on theories and models of large-scale magnetic field data. The “0-tables” were computed by Robert Borda.

REFERENCES

Axford, W. I. 1972, in Solar Wind, ed. C. P. Sonett, J. Coleman, P. J., & Gurnett, D. A., & Kurth, W. S. 2005, Science, 309, 2025 J. M. Wilcox (NASA SP-308; Washington, DC: NASA), 598 Hundhausen, A. J. 1972, Coronal Expansion and Solar Wind (New York: Behannon, K. W., Acunã, M. H., Burlaga, L. F., Lepping, R. P., Ness, N. F., Springer) & Neubauer, F. M. 1977, Space Sci. Rev., 21, 235 Levenberg, K. 1944, Q. Appl. Math., 2, 164 Burlaga, L. F., Ness, N. F., Acunã, M. H., Lepping, R. P., Connerney, Marquardt, D. 1963, SIAM J. Appl. Math., 11, 431 J. E. P., Stone, E. C., & McDonald, F. B. 2005, Science, 309, 2027 Parker, E. N. 1963, Interplanetary Dynamical Processes (New York: Burlaga, L. F., & Vinãs, A. F. 2004, J. Geophys. Res., 109, A12107, doi: Interscience) 10.1029/2004JA010763 Stone, E. C., Cummings, A. C., McDonald, F. B., Heikkila, B. C., Lal, N., & ———. 2005, J. Geophys. Res., 110, A07110 Webber, W. R. 2005, Science, 309, 2017 ———. 2006, Physica A, 356, 375 Tsallis, C. 1988, J. Stat. Phys., 52, 479 Burlaga, L. F., Wang, C., Richardson, J. D., & Ness, N. F. 2003, ApJ, 585, Tsallis, C. 2004a, Physica A, 340, 1 1158 ———. 2004b, in Nonextensive Entropy: Interdisciplinary Applications, ed. Decker, R. B., Krimigis, S. M., Roelof, E. C., Hill, M. E., Armstrong, T. P., M. Gell-Mann & C. Tsallis (New York: Oxford Univ. Press), 1 Gloeckler, G., Hamilton, D. C., & Lanzerotti, L. J. 2005, Science, 309, 2020 Tsallis, C., & Brigatti, E. 2004, Continuum Mech. Thermodynamics, 16, 223 Gell-Mann, M., & Tsallis, C., eds. 2004, Nonextensive Entropy: Interdisci- plinary Applications (New York: Oxford Univ. Press)