The Self-organized Criticality Behaviors of Two New Parameters in SGR J1935+2154

The minimum variation timescale (MVT) and spectral lag of hundreds of X-ray bursts (XRBs) from soft gamma-ray repeater (SGR) J1935+2154 were analyzed in detail for the first time in our recent work, which are important probes for studying the physical mechanism and radiation region. In this work, we investigate their differential and cumulative distributions carefully and find that they follow power-law models. Besides, the distributions of fluctuations in both parameters follow the Tsallis $q$-Gaussian distributions and the $q$ values are consistent for different scale intervals. Therefore, these results indicate that both parameters are scale-invariant, which provides new parameters for the study of self-organized criticality systems. Interestingly, we find that the $q$ values for MVT and spectral lag are similar with duration and fluence, respectively.

The Self-organized criticality (SOC) in slowly driven nonlinear dissipative systems was proposed by Bak et al. (1987), Aschwanden (2012) and Aschwanden et al. (2016) to explain the phenomenon of self-similarities (i.e.power-law size distributions), the spatial dimension and the classical diffusion of the fractal-diffusive SOC system can be estimated through the power-law index for the parameters' length scales, durations, peak fluxes and fluences.On the other hand, the distribution for the parameter fluctuations at different times in a SOC system was found to follow a Tsallis -Gaussian function, and ★ E-mail: xiaoshuo@gznu.edu.cn† E-mail: zhangsn@ihep.ac.cn ‡ E-mail: xiongsl@ihep.ac.cn § E-mail: dili@nao.cas.cn the  values are constant for different scale intervals (i.e.scale invariance of the avalanche size differences) (Caruso et al. 2007).The power-law index of the parameter distribution and the  value of the parameter difference distribution can be inter-convertible (Caruso et al. 2007;Celikoglu et al. 2010).
In our recent work, we reported the small spectral lags (the distribution peaks ∼ 1.3 ms) (Xiao et al. 2023b) and the minimum variation timescale (MVT, the distribution peaks at ∼ 2 ms) (Xiao et al. 2023a) by analyzing hundreds of XRBs from SGR J1935+2154 observed by GECAM, HXMT and GBM.The two parameters are important probes to studying the physical mechanism and radiation region of XRBs (Schmidt 1978;Fenimore et al. 1993;Titarchuk et al. 2007;Zhang et al. 2009;Huppenkothen et al. 2013;Ackermann et al. 2014;Golkhou & Butler 2014;Golkhou et al. 2015).However, to our knowledge, whether these parameters are also scale-invariant has not been studied, which is the focus of this work.) for =1, the yellow lines are fitted by a Tsallis -Gaussian distribution ( = 2.36 ± 0.12), and reduced- 2 is 0.8.Right panel: the fitted q values for different the yellow lines are fitted by a constant, which is  = 2.19 ± 0.01 with reduced- 2 is 0.9.In this work, we present our sample selection and scale invariance analysis in Section 2 and a discussion and summary are given in Section 3.

Sample selection and method
The bursts from SGR J1935+2154 and the values of MVT and spectral lag observed by GECAM, HXMT and GBM are collected from our previous work (Xiao et al. 2023a,b).Since all three satellites have large fields of view and high sensitivity (Xiong 2020;Zhang et al. 2020;Meegan et al. 2009), we have a relatively comprehensive sample.For bursts observed jointly by multiple satellites, we adopt the GBM measurements by default.In total, 669 bursts are selected from July 2014 to January 2022 after removing those with "timing glitches" (a known GBM hardware anomaly with dips and peaks in a light curve (Briggs et al. 2013).)or saturation.
To investigate whether the parameters are scale-invariant or not, we adopt two approaches like Refs.(Chang et al. 2017;Wei et al. 2021;Sang & Lin 2022;Li et al. 2023).The first is to investigate whether the differential and cumulative distributions of the parameters follow power-law models (Aschwanden 2015) (see Equations 1 and 2) and the power-law index are consistent within the uncertainties, where  0 ,   and   are normalization constant, the power-law index of differential and cumulative distributions, respectively. is the number of bursts,  0 and  0 are constants due to the threshold effects.
The second is to test whether the distributions of the differences of the parameters   follow the Tsallis -Gaussian distribution (Tsallis 1988;Tsallis et al. 1998) (Equation 4) and the  values for different time scales  are consistent for different scale intervals, where   is the value of the parameter (i.e.MVT and spectral lag) of the th burst ordered according to time and  is the temporal interval scale.We obtain   by dividing the difference   by a scale factor (   ), which is the standard deviation of   .

𝑓 (𝑥
where  is the normalization factor, the  and  (the parameter we are interested in) affect the width and sharpness of the peak, respectively.We adopted the Markov Chain Monte Carlo (MCMC) method to fit the data, and tests the goodness-of-fit by reduced- 2 (i.e.close to 1 implies a good fit).

Minimum variation timescales
The MVT in 8-100 keV of each burst is calculated in Xiao et al. (2023a).The left panel of Figure 1 shows the differential distribution, which can be fitted fairly well (reduced- 2 = 2 /  =37.4/19=1.9)by the power-law model (Equation 1) with  d = 2.56±0.40 and  0d = 0.010 ± 0.003.Note that less than ∼0.001 ms does not fit well due to threshold effects (Aschwanden 2015).The cumulative distribution of MVT is shown in the right panel of Figure 1, which also can be fitted well (reduced- 2 = 2 /  =12.6/23=0.6)by the power-law model (Equation 2) with  c = 2.89 ± 0.35 and  0c = 0.014 ± 0.003.Note that  d and  c are consistent within the uncertainties, which indicates that the MVT parameter of XRBs from SGR J1935+2154 exhibits SOC behavior.
The distributions of fluctuations for different time scales  are investigated, for example, the left panel of Figure 2 shows the distribution for =1, and it can be fitted well (reduced- 2 = 2 /  =58.1/76=0.8)by the Tsallis -Gaussian distribution (Equation 4) with  = 2.36 ± 0.12.The q-values obtained by fitting the distributions for different scales  are shown in the right panel of Figure 2, which are steady and can be fitted well with a constant ( = 2.19 ± 0.01, with reduced- 2 = 2 /  =83.3/98=0.9),which also verifies that the MVT parameter of XRB from SGR J1935+2154 exhibits SOC behavior.

Spectral lags
The spectral lag between light curves in the 10-20 and 60-100 keV energy bands is calculated in Xiao et al. (2023b), and a positive lag is defined as that low-energy photons follow high-energy photons.The differential and cumulative distributions are shown in Figure 3, both follow power-law models well.The result of the fit to the difference distribution is  d = 1.54 ± 0.16 and  0d = 0.003 ± 0.001 with reduced- 2 = 2 /  =43.9/19=2.3.For the cumulative distribution, the result of the fit is  c = 1.80 ± 0.11 and  0c = 0.003 ± 0.001, with reduced- 2 = 2 /  =11.5/13=0.9.Note that the  d and  c are also consistent within the uncertainties, which indicates that the spectral lag parameter of XRB from SGR J1935+2154 exhibits SOC behavior.
The left panel of Figure 4 shows the distribution of fluctuations for time scale  = 1, which can be fitted well (reduced- 2 = 2 /  =52.3/36=1.5)by the Tsallis -Gaussian distribution (Equation 4) with  = 2.94 ± 0.20.The q-values obtained by fitting the distributions for different scales  are also consistent within the uncertainties and fitted with a constant ( = 2.86±0.02,reduced- 2 = 2 /  =103.7/98=1.1),which also verifies that the spectral lag parameter of XRB from SGR J1935+2154 exhibit SOC behavior.

DISCUSSION AND CONCLUSION
In this work, we perform a detailed study of the statistical properties of the two parameters, i.e., MVT and spectral lag, of hundreds of XRBs from SGR J1935+2154 observed by GECAM, HXMT and GBM.With two methods, we report that both parameters exhibit SOC behaviors, that is, the differential and cumulative distributions of the parameters well follow power-law models and the powerlaw index ( d and  c ) are consistent within the uncertainties, as well as the distributions of fluctuations follow the Tsallis -Gaussian distribution and the  values for different time scales are consistent for different scale intervals.
Interestingly, the  values (i.e. = 2.19 ± 0.01) for MVT are similar to the result (i.e.2.28 ± 0.15) for the duration reported by Wei et al. (2021).The duration of a burst only describes the total emission properties, but MVT captures the information concerning individual pulses, that is, the MVT is approximately equal to the rise time of the shortest pulse in a burst.In a previous work, we did not find a significant correlation between duration and MVT for XRBs from SGR J1935+2154 (Xiao et al. 2023a).The MVT of XRBs can be used to estimate the emission region in pulsar-like models, as well as the radius and Lorentz factor of the relativistic jet in GRB-like models (see Zhang 2020; Zhang 2022 for review).Therefore, the SOC behavior of MVT implies that the radius of the radiation region may have a similar origin in pulsar-like models.
On the other hand, although the spectral lag is a time-domain parameter, the  values (i.e. = 2.86 ± 0.02) for spectral lag are similar to the result (i.e.2.78 ± 0.12) for fluence reported by Wei et al. (2021).The relationship between fluence and spectral lag has been found in some transient sources such as GRBs (Ukwatta et al. 2012), but there is a lack of studies for XRBs.Further work is required to investigate this.
According to the study of Caruso et al. (2007) and Celikoglu et al. (2010), there is an exact relation (i.e. = ( + 2)/2) between the power-law index ( d or  c ) and the  values based on the assumption that the sizes of two events have no correlation.We find that this relationship is also satisfied for MVT and spectral lag within the uncertainties.However, MVT and spectral lag are different from other parameters, such as duration, peak flux, and fluence or energy, which have detailed theoretical explanations, e.g. the spatial dimension can be estimated based on the Fractal-diffusive Avalanche Model (Aschwanden 2012).Therefore, we believe that the SOC behaviours of MVT and spectral lag will provide new insights in future SOC studies of these phenomen.

Figure 1 .Figure 2 .
Figure 1.The differential (left panel) and cumulative (right panel) distributions of MVTs observed by GECAM, HXMT and Fermi/GBM.The fitted yellow lines are obtained by MCMC with reduced- 2 1.9 and 0.6, respectively.

Figure 3 .Figure 4 .
Figure 3.The differential (left panel) and cumulative (right panel) distributions of spectral lags observed by GECAM, HXMT and Fermi/GBM.The fitted yellow lines are obtained by MCMC, and reduced- 2 are 2.3 and 0.9, respectively.