Fractional derivatives and negative probabilities

Commun Nonlinear Sci Numer Simulat 79 (2019) 104913

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Fractional derivatives and negative probabilities J. Tenreiro Machado Institute of Engineering of Polytechnic of Porto, Dept. of Electrical Engineering, Porto, Portugal

a r t i c l e

i n f o

Article history: Received 7 May 2019 Revised 22 June 2019 Accepted 9 July 2019 Available online 10 July 2019

a b s t r a c t This paper addresses the theories of Fractional Calculus and Negative Probabilities. The fundamental concepts are discussed and a number of analogies are established between the two mathematical areas. The proposed ideas are tested by means of several numerical experiments that shed new light into possible synergies and new advances in their application.

Keywords: Fractional calculus Negative probabilities Control Entropy

© 2019 Elsevier B.V. All rights reserved.

1. Introduction The progress of science expands through two different and independent mechanisms. The most usual process in the advance of science is based on cumulative steps that improve current knowledge. The second course of action, that is more rare and often poses difficulties to be immediately recognized, is established by abrupt innovations towards completely new directions. These quantum leaps in the human knowledge often have the mark of being ‘exotic’ and, therefore, of not having an immediate application [1]. Nonetheless, history reveals that these preconceptions lead to subsequent perplexity when the scientific society grasps the new horizons raised by such disrupting ideas. The concept of Fractional Calculus (FC) emerged in 1695, with the letters exchanged between Gottfried Leibniz (1646– 1

1716) and other mathematicians about the meaning of D 2 f . The classical differential involves the standard, integer-order, operations of differentiation and the notion of a derivative of 12 order seems ‘exotic’, but important mathematicians contributed to the development of FC [2]. Two decades ago, the scientific community recognized that FC is a relevant tool for describing phenomena with non-locality and today FC is an important research area. The concept of probability came forth in 1654, in the follow-up of the letters between Pierre de Fermat (1607–1665) and Blaise Pascal (1623–1662). Nonetheless, the present day theory of probability is mostly credited to Andrey Kolmogorov (1903–1987) [3]. Standard probabilities lie in the range [0, 1] and, consequently, the concept of Negative probability (NP), that is, of probabilities with values outside [0, 1], seems ‘exotic’. During the twentieth century, the Nobel laureates Paul Dirac (1902–1984) and Richard Feynman (1918–1988) proposed NP in the sphere of quantum physics, but the application of NP remained limited to a few studies until recently. The concepts of FC and NP extend our horizons, but, in spite of their historical resemblances, both areas are usually considered apart. This paper highlights some analogies between the two areas and is structured as follows. Section 2 introduces the fundamental mathematical concepts and the main aspects relevant in FC and NP. Section 3 proposes an analogy between the differential calculus and the probability theories. Based on the new perspective, several prototype experiments

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in two distinct fields are presented. Two application, first in control theory and second in entropy, are discussed. Finally, Section 4 summarizes the main conclusions. 2. Fundamental concepts 2.1. Fractional calculus In 1695, in the follow-up of the letters between Leibniz and Bernoulli the questions ‘Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?’ and ‘What if the order will be 21 ?’ were raised. Leibniz wrote ‘It will lead to a paradox, from which one day useful consequences will be drawn’. Nonetheless, FC remained a closed topic for centuries, in spite of the work developed by well-known mathematicians, such as, Euler, Liouville and Riemann, just to name a few. Several historical surveys can be found in [2,4–9]. At the turn of the 20th century several scientists [10] started using FC. We can cite: Heaviside [11], in relation with his research in electromagnetism and operational calculus, Cole and Cole [12–14], for modeling electrical processes in living tissues, and Gemant [15,16], when tackling experimental data for the dynamic viscosity. The topic remained ‘exotic’, but in the last years a plethora of interesting applications emerged [17–24]. We find in the literature several classical definitions of FD, being the most well known the so-called Riemann–Liouville, Caputo and Grünwald–Letnikov (GL) formulations [25]. If we consider the GL definition of FD we have a Taylor series given by: GL α a Dt

∞ 1  f (t ) = lim α (−1 )k γ (α , k ) f (t − kh ), h→0 h

(1a)

k=0

γ (α , k ) =

(α + 1 ) , (α − k + 1 )

(1b)

where t > a,  (·) denotes the Euler gamma function and h is the step time increment. It is worth mentioning the Laplace transform of a FD of order α ∈ R that, for null initial conditions, can be written as:

L{Dα0 x(t )} = sα X (s ),

(2)

where L(· ) and s denote the Laplace operator and variable, respectively. The GL definition (1) is commonly adopted in signal processing and control theory [26], because it produces straightforwardly discrete-time algorithms by using the sampling period T for approximating the time increment h:

Z {Dtα f (t )} =

1 T

1 − z−1

α

Z { f (t )},

(3)

where Z (· ) and z denote the Z-transform operator and variable, respectively. This expression can be written in the form of a Taylor series:

Z {Dtα f (t )} =

∞ 1  (−1 )k γ (α , k )z−k . Tα

(4)

k=0

In practical implementations the series is truncated with r terms

Z {Dtα f (t )} ≈

r 1  (−1 )k γ (α , k )z−k , Tα

(5)

k=0

that can be also written in terms of a Padé approximation. We must note also that expressions (1)–(5) are also valid for complex orders ν = α + jβ ∈ C. The geometrical interpretation of FD has been the subject of debate and several perspectives were proposed [27,28]. The probabilistic interpretation [28,29] is particularly suited to the present study. For α ∈ R the FD calculates the slope of a triangle where the left point represents an average of all the past values of the function weighted by the coefficients of the Taylor series expansion as represented in Fig. 1. 2.2. On fractional transfer functions Many phenomena in physics and engineering, that evolve in continuous time or space, can be described alternatively in the Fourier or the Laplace domains. This description leads to the concepts of transfer function (TF) and the simplest if the so-called Debye TF given by [30]:

G (s ) =

K , 1 + ωs0

(6)

J.T. Machado / Commun Nonlinear Sci Numer Simulat 79 (2019) 104913

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Fig. 1. Probabilistic interpretation of the FD for α ∈ R.

where K, ω0 ∈ R+ and s denotes the Laplace variable. However, this expression reveals considerable limitations when modeling real-world phenomena. Two fractional heuristic models of particular interest [31], are the Cole-Cole and Davidson-Cole [12–14,32–34] TF given respectively by:

G (s ) = G (s ) =

K

 s β ,

(7)

1 + ω0



K 1 + ωs0

α ,

(8)

where α , β ∈ R. Obviously, expressions (7)–(8) generalize (6), since for α = 1 and β = 1 we obtain the Debye model. The two fractional TF can be combined producing to the Havriliak–Negami TF given by [35,36]:

G (s ) =



K

 s β α .

(9)

1 + ω0

The heuristic expressions (7)–(9) are ubiquitous in many scientific areas [10,37–44] and are often considered ‘building blocks’ for more complex fractional TF. 2.3. Negative probabilities In 1942, Paul Dirac introduced the concepts of negative energies and NP [45]. Dirac noted that ‘Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money’. In 1950, Richard Feynman [46,47] examined the notion in the scope of quantum physics. Feynman reminded that we do not object about having arithmetic calculations with negative numbers, but that the concept of ‘minus three apples’ is not valid in the real world. M. Bartlett [48] developed the first formal treatment of NP, that were discussed later by Allen [49]. Székely [50] proposed the idea of ‘half-coins’ as abstract objects leading to NP. Let us consider a fair coin having two sides, ‘0’ and ‘1’, with identical probabilities. variable X with values {0, 1}, the probability generating function (PGF) is defined as GX (z ) =  For  adiscrete random k E zX = ∞ k=0 P (X = k )z , z ∈ C, where P (X = k ) is the probability mass function of X. The PGF is equivalent to, and also called as, the Z-transform of the probability mass function. Furthermore, for the addition of two independent random variables, that is, for X = X1 + X2 , we have for PGF the multiplication of the corresponding PGF, namely, GX (z ) = GX1 (z ) · GX2 (z ). For a discrete random variable X with values {0, 1} the PGF of a fair coin is GX (z ) = 12 (1 + z ). Therefore, the sum of n ∈ N fair coins has PGF GX (z ) =

GX ( z ) =

1 2

1

2 (1

n

+ z ) . Székely proposed the ‘half-coin’ (i.e., n =

 12 (1 + z ) .

1 2)

with generalized PGF:

(10)

The abstraction is compatible with the real-world complete coin, because if we flip two half coins then the sum of the outcomes is either 0 or 1, with probability 21 each, as if we flipped a fair coin. Furthermore, we have the outcome of an infinite number of sides and the emergence of NP. Indeed, expanding (10) in Taylor series we obtain

1 2

 12 1 1 1 1 (1 + z ) = √ 1 + z − z2 + z3 − 2

2

8

16



5 √ z4 + · · · , 128 2

(11)

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that includes NP, such as, for example, P (X = 2 ) = − For other fractional values α ∈ R we can write:

GX ( z ) =

1 2

1 √ 8 2

and P (X = 4 ) = −

5√ . 128 2

α ∞ 1  P ( X = k )z k (1 + z ) = α 2

k=0

∞ 1  = α γ (α , k )zk . 2

(12)

k=0

Additionally, Székely considered also biased coins and dice, with probabilities different from the values 21 in (10). Nonetheless, the term ‘half fair coin’ proposed by Székely is not totally clear and we should ‘half flipping of one fair coin’, instead. NP follow the so-called quasiprobability distributions, relaxing the Kolmogorov’s axioms of probability theory. Quasiprobability distributions share several features of the standard probabilities, but they violate the first axiom P(A) ≥ 0, where A belongs to the event space, and the third axiom stating that for any countable sequence of disjoint sets Ak , k = 1, . . . , we  have P (∪∞ A )= ∞ k=1 P (Ak ). k=1 k NP have been applied mainly in physics [51–56], but remained almost untouched in other scientific areas. Nevertheless, we can mention the works of Haug [57], that applied the notion of NP in mathematical finance, Tijms and Staats [58], that used NP in the scope of waiting-time probabilities, and Burgin and Meissner [59], that considered NP in the description of financial option pricing. Other recent contributions can be found in [60–69]. 3. A relationship between FC and NP The generalization of the classical notions in the domains of differential calculus and probability, becomes evident when considering transforms and Taylor series. Indeed, we have FD when using the Z {·} or L{·} operators, for the discrete time and continuous time in calculus, and NP when adopting the PGF in probability. Therefore, we can ask if some relationship between the two topics can be established [70–72]. If we compare (4) and (12), then we can say that the FD, in differential calculus, mimics the fractional flipping of one fair coin, in probability theory. Therefore, we can draw the analogy sketched in Fig. 2, where α ∈ R, K, T ∈ R+ and z ∈ C. We now sketch some possible application of the analogy in two distinct areas. 3.1. Control systems We start by checking the series expansions of the binomials [ T1 (1 − z−1 )]α and [ T1 (1 + z−1 )]α for α = {− 12 , + 21 }:

G1 ( z ) =

G2 ( z ) =

G3 ( z ) =

G4 ( z ) =

1 T

1 T

1 T

1 T

1 − z−1

1 − z−1

1 + z−1

1 + z−1

− 12  12

=

− 12  12

=

1

T−2 1

T

=



1

=



1 2

1−



1 T 1 1

T2

− 12



1+

(13a)

1 −1 1 −2 1 −3 5 −4 z − z − z − z − ... 2 8 16 128

1−

1+

1 −1 3 −2 5 −3 35 −4 z + z + z + z + ... 2 8 16 128

(13b)

1 −1 3 −2 5 −3 35 −4 z + z − z + z − ... 2 8 16 128

(13c)

5 −4 1 −1 1 −2 1 −3 z − z + z − z + ... 2 8 16 128

(13d)

The expressions G1 (z) and G2 (z) describe the 12 -order fractional integral and derivative, respectively, in the Z-transform domain. Their counterpart correspond to the 21 -anti-flipping and 12 -flipping of one fair coin, respectively, in the probability domain. On the other hand, G3 (z) and G4 (z) are the Z-transform versions of the 12 -anti-flipping and 21 -flipping of 1 fair coin, respectively. However, they do not have an usual interpretation in the scope of Z-transform TF. If we adopt the probabilistic interpretation of FD mentioned in sub-Section 2.3, then it means that we assign NP to some samples in the past of the input signal. Let us now analyze the time response of the expressions (13). Figs. 3 and 4 depict the time response for a unit step input 1

1

1

of the algorithms (13) with truncation of the series with r = 5 terms. The responses of the ideal D 2 and D− 2 = I 2 are also included. We analyze the time response of a unit-feedback closed-loop control system with TF G p (s ) = s(s1+1 ) in the direct loop, under the action of a discrete-time controller C (z ) = K · Gi (z ), i = 1, . . . , 4, r = 5, a sampling frequency T = 1 s and a unit step reference input R(s ) = 1s . Moreover, no tuning technique was applied for the gain K = 0.2 that remains identical in all experiments just to simplify comparisons.

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Fig. 2. Analogy between the fractional derivative and Davidson-Cole TF, in the discrete and continuous time/space formulations, versus the fractional flipping of 1 fair coin, in probability.

Fig. 4 depicts the time response and the control effort of the closed-loop system with the algorithms (13) for a truncation order of r = 5. The closed-loop output time response shows a clear pattern, where G3 (z) and G4 (z) somehow interpolate the extreme cases G2 (z) and G1 (z). This observation, follows the ideas drawn in the conceptual diagram represented in Fig. 2.

3.2. Entropy Information theory was developed by Shannon in [73] and has been applied in many scientific areas. The fundamental cornerstone is the information content of some event having probability of occurrence P (X = k ):

I (P (X = k ) ) = − ln (P (X = k ) ).

(14)

6

J.T. Machado / Commun Nonlinear Sci Numer Simulat 79 (2019) 104913

1

1

1

Fig. 3. Time response of the expressions Gi (z), i = 1, 2, 3, 4, and D 2 and D− 2 = I 2 , for T = 1, r = 5 terms and a unit step input.

Fig. 4. Time response and control effort of the unit-feedback closed-loop system with transfer function G p (s ) = s(s1+1 ) , K = 0.2, T = 1 and the control algorithms C (z ) = K · Gi (z ), i = 1, . . . , 4, with series truncation r = 5.

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Fig. 5. Locus (Re(S), Im(S), α ) for the probability distributions of (1) and (12) with 0 ≤ α ≤ 1 and series truncation r = 600.

The expected value, called Shannon entropy [74,75], becomes:

S = E [− ln (P (X = k ) )] =



[− ln (P (X = k ) )] · P (X = k ),

(15)

i

where E(·) denotes the expected value operator. We now evaluate the probability distributions produced by γ (α , k) in the range 0 ≤ α ≤ 1, under the light of the Shannon entropy S. Therefore, we compare (1) and the probability distribution P (X = k ) = (−1 )k+1 γ (α , k ), k = 1, . . . , ∞, for the FD, versus (12) and the probability distribution P (X = k ) = 21α γ (α , k ), k = 0, . . . , ∞, for the NP. We must also note that in the case of NP we obtain complex values for S. Fig. 5 shows the locus (Re(S), Im(S), α ) for the series truncation r = 600, where Re(S) and Im(S) denote the real and imaginary components of S. We verify that the two cases reveal an almost similar evolution with α , but exhibiting a orthogonal behavior, that is, showing values mainly in the real and imaginary components. This pattern seem to support again the conceptual diagram of Fig. 2 and motivates further experiments for exploring other analogies between FC and NP. 4. Conclusions This paper reviewed the progress of two ‘exotic’ scientific areas, namely FC and NP. The holistic observation of several gedankenexperiment by means of their generalized mathematical formulation revealed new horizons, hidden when looking separately to each domain. Two simple exercises, namely with control systems and entropy, explored the new avenues of thought by taking advantage of the emerging affinities. References [1] Flexner A. The usefulness of useless knowledge. In: Harper’s Magazine, (179); 1939. p. 544–52. [2] Kochubei A, Luchko Y, editors. Handbook of fractional calculus with applications: basic theory, Vol. 1 of De Gruyter Reference. Berlin: De Gruyter; 2019. ISBN 978-3-11-057162-2. [3] Kolmogorov AN. Foundations of the theory of probability. De Gruyter Reference. New York: Chelsea Publishing Company; 1950. [4] Oldham K, Spanier J. The fractional calculus: theory and application of differentiation and integration to arbitrary order. New York: Academic Press; 1974. [5] Miller K, Ross B. An introduction to the fractional calculus and fractional differential equations. New York: John Wiley and Sons; 1993. [6] Samko S, Kilbas A, Marichev O. Fractional integrals and derivatives: theory and applications. Amsterdam: Gordon and Breach Science Publishers; 1993. [7] Dugowson S. Les différentielles métaphysiques (histoire et philosophie de la généralisation de l’ordre de dérivation). Paris: Thèse, Université Paris Nord; 1994.

8

J.T. Machado / Commun Nonlinear Sci Numer Simulat 79 (2019) 104913

[8] Machado JT, Kiryakova V, Mainardi F. Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul 2011;16(3):1140–53. doi:10.1016/j. cnsns.2010.05.027. [9] Machado JT, Lopes AM. Fractional Jensen-Shannon analysis of the scientific output of researchers in fractional calculus. Entropy 2017;19(3):127. doi:10. 3390/e19030127. [10] Valério D, Machado JT, Kiryakova V. Some pioneers of the applications of fractional calculus. Fract Calculus Appl Anal 2014;17(2):552–78. doi:10.2478/ s13540- 014- 0185- 1. [11] Heaviside O. On operators in physical mathematics. Proc R Soc 1893;52:504–29. [12] Cole KS, Cole RH. Electric impedance of Arbacia eggs. J Gen Physiol 1936;19(4):625–32. [13] Cole KS, Cole RH. Dispersion and absorption in dielectrics - I Alternating current characteristics. J Chem Phys 1941;9:341–52. [14] Cole KS, Cole RH. Dispersion and absorption in dielectrics - II Direct current characteristics. J Chem Phys 1942;10:98–105. [15] Gemant A. A method of analyzing experimental results obtained from elasto-viscous bodies. Physics 1936;7:311–17. [16] Gemant A. On fractional differentials. Philos Mag 1938;25:540–9. [17] Oustaloup A. Systèmes asservis linéaires d’ordre fractionnaire: théorie et pratique. Paris: Masson; 1983. [18] Hilfer R. Application of fractional calculus in physics. Singapore: World Scientific; 20 0 0. [19] Tarasov V. Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. New York: Springer; 2010. [20] Mainardi F. Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. London: Imperial College Press; 2010. [21] Caponetto R, Bohannan G, Biswas K, Machado JAT, Lopes AM. Fractional-order devices. SpringerBriefs in Nonlinear Circuits. Cham: Springer; 2017. ISBN 978-3-319-54460-1. [22] Milici C, Dra˘ ga˘ nescu G, Machado JT. Introduction to fractional differential equations. Nonlinear Systems and Complexity. Cham: Springer; 2019. ISBN 978-3-030-00895-6. [23] Tarasov VE, editor. Handbook of fractional calculus with applications: applications in physics, part A Vol. 4 of De Gruyter Reference. Berlin: De Gruyter; 2019. ISBN 978-3-11-057170-7. [24] Petráš I, editor. Handbook of fractional calculus with applications: applications in control, Vol. 6 of De Gruyter Reference. Berlin: De Gruyter; 2019. ISBN 978-3-11-057174-5. [25] Kilbas A, Srivastava H, Trujillo J. Theory and applications of fractional differential equations, 204. Amsterdam: North-Holland Mathematics Studies, Elsevier; 2006. [26] Machado JT. Analysis and design of fractional-order digital control systems. Syst Anal Model Simul 1997;27(2–3):107–22. [27] Podlubny I. Geometric and physical interpretation of fractional integration and fractional differentiation. Fract Calculus Appl Anal 2002;5(4):367–86. [28] Machado JAT. A probabilistic interpretation of the fractional-order differentiation. Fract Calculus Appl Anal 2003;6(1):73–80. [29] Machado JAT. Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun Nonlinear Sci Numer Simul 2009;14(9–10):3492–7. doi:10.1016/j.cnsns.20 09.02.0 04. [30] Debye P. Polar molecules. New York: Chemical Catalog; 1929. [31] Machado JAT. Matrix fractional systems. Commun Nonlinear Sci Numer Simul 2015;25(1–3):10–18. doi:10.1016/j.cnsns.2015.01.006. [32] Cole KS, Cole RH. Electric impedance of Asterias eggs. J Gen Physiol 1936;19(4):609–23. [33] Davidson DW, Cole RH. Dielectric relaxation in glycerine. J Chem Phys 1950;18:1417. [34] Davidson DW, Cole RH. Dielectric relaxation in glycerol, propylene glycol, and n-propanol. J Chem Phys 1951;19(12):1484–90. [35] Havriliak S, Negami S. A complex plane analysis of α -dispersions in some polymer systems. J Polym Sci Part C 1966;14(1):99–117. [36] Havriliak S, Negami S. A complex plane representation of dielectric and mechanical relaxation processes in some polymers. Polymer 1967;8(4):161–210. [37] Nigmatullin R, Ryabov YE. Cole-Davidson dielectric relaxation as a self-similar relaxation process. Phys Solid State 1997;39(1):87–90. doi:10.1134/1. 1129804. [38] Jeon T-I, Grischkowsky D. Observation of a Cole-Davidson type complex conductivity in the limit of very low carrier densities in doped silicon. Appl Phys Lett 1998;72(18):2259–61. doi:10.1063/1.121271. [39] Calderwood J. A physical hypothesis for Cole-Davidson behavior. IEEE Trans Dielectr Electr Insul 2004;10(6):1006–11. doi:10.1109/TDEI.2003.1255778. [40] Tarasov VE. Fractional hydrodynamic equations for fractal media. Ann Phys 2005;318(2):286–307. doi:10.1016/j.aop.2005.01.004. [41] Sabatier J, Aoun M, Oustaloup A, Grégoire G, Ragot F, Roy P. Fractional system identification for lead acid battery state of charge estimation. Signal Process 2006;86(10):2645–57. doi:10.1016/j.sigpro.2006.02.030. [42] Sommacal L, Melchior P, Oustaloup A, Cabelguen J-M, Ijspeert AJ. Fractional multi-models of the frog gastrocnemius muscle. J Vib Control 2008;14(9– 10):1415–30. doi:10.1177/1077546307087440. [43] Khamzin A, Nigmatullin R, Popov I. Description of the anomalous dielectric relaxation in disordered systems in the frame of the Mori-Zwanzig formalism. J Phys 2012;394(1):012013. doi:10.1088/1742-6596/394/1/012013. [44] Caponetto R, Graziani S, Pappalardo FL, Sapuppo F. Experimental characterization of ionic polymer metal composite as a novel fractional order element. Adv Math Phys 2013;43:10. doi:10.1155/2013/953695. Article ID 953695. [45] Dirac PAM. The physical interpretation of quantum mechanics. Proc R Soc Lond 1942;A(180):1–39. [46] Feynman RP. The concept of probability theory in quantum mechanics. In: Second Berkeley symposium on mathematical statistics and probability theory. Berkeley, California: University of California Press; 1951. p. 533–41. [47] Feynman RP. Negative probability. In: Basil J Hiley FDP, editor. Quantum implications: essays in honour of David Bohm. London, New York: Routledge & Kegan Paul Ltd; 1987. p. 235–48. [48] Bartlett MS. Negative probability. Math Proc Camb Philos Soc 1945;41(1):71–3. [49] Allen EH. Negative probabilities and the uses of signed probability theory. Philos Sci 1976;43(1):53–70. doi:10.1086/288669. [50] Székely GJ. Half of a coin: negative probabilities. In: Wilmott Magazine, (July); 2005. p. 66–8. [51] Bell JS. On the Einstein Podolsky Rosen paradox. Physics 1964;1(3):195–200. [52] Mückenheim W, Ludwig G, Dewdney C, Holland PR, Kyprianidis A, Vigier JP, et al. A review of extended probabilities. Phys Rep 1986;133(6):337–401. doi:10.1016/0370-1573(86)90110-9. [53] Leibfried D, Pfau T, Monroe C. Shadows and mirrors: reconstructing quantum states of atom motion. Phys Today 1998;51(4):22–8. doi:10.1063/1. 882256. [54] Khrennikov A. Interpretations of probability. The Netherlands: VSP; 1999. [55] Hofmann HF. How to simulate a universal quantum computer using negative probabilities. J Phys A 2009;42(275304):1–9. doi:10.1088/1751-8113/42/ 27/275304. [56] Penchev V. Negative and complex probability in quantum information. Philos Altern 2012(1):63–77. [57] Haug EG. Why so negative to negative probabilities?, what is the probability of the expected being neither expected nor unexpected? Wilmott Magazine, (Mar/Apr) 2007:34–8. [58] Tijms H, Staats K. Negative probabilities at work in the M/D/1 queue. Probab Eng Inf Sci 2007;21(1):67–76. doi:10.1017/S0269964807070040. [59] Burgin M, Meissner G. Negative probabilities in financial modeling. Wilmott Mag 2012;58(March):60–5. [60] Halliwell JJ, Yearsley JM. Negative probabilities, Fine’s theorem, and linear positivity. Phys Rev A 2013;87:022114. doi:10.1103/PhysRevA.87.022114. [61] Bracken A, Melloy G. Waiting for the quantum bus. Stud Hist Philos Sci Part B 2014;48:13–19. doi:10.1016/j.shpsb.2014.09.001. [62] de Barros JA, Oas G, Suppes P. Negative probabilities and counterfactual reasoning on the double-slit experiment. 2014. arXiv:1412.4888. [63] de Barros JA, Oas G. Negative probabilities and counter-factual reasoning in quantum cognition. Physica Scripta 2014;2014(T163):014008. doi:10.1088/ 0031-8949/2014/T163/014008.

J.T. Machado / Commun Nonlinear Sci Numer Simulat 79 (2019) 104913

9

[64] Da B, Mao S, Ding Z. Negative probability sampling in study of reflection surface electron spectroscopy spectrum. 2015. arXiv:1505.06806v1. [65] de Barros JA, Kujala J, Oas G. Negative probabilities and contextuality. J Math Psychol 2016;74:34–45. doi:10.1016/j.jmp.2016.04.014. [66] Pownuk A, Kreinovich V. (hypothetical) negative probabilities can speed up uncertainty propagation algorithms. Tech. Rep. 1110. University of Texas at El Paso, Departmental Technical Reports; 2017. [67] Blass A, Gurevich Y. Negative probabilities, ii: what they are and what they are for. Bull Eur Assoc Theor Comput Sci 2018;125. arXiv: 1807.10382. [68] Hudson R. A short walk in quantum probability. Philos Trans A 2018;376:1–13. doi:10.1098/rsta.2017.0226. [69] Sokolovskia D, Gurvitz S. Paths, negative “probabilities”, and the Leggett-Garg inequalities. 2019. arXiv:1902.08060v2. [70] Machado JAT. Fractional coins and fractional derivatives. Abstr Appl Anal 2013;377(ID 205097):5. doi:10.1155/2013/205097. [71] Machado JT, Lopes AM, Duarte FB, Ortigueira MD, Rato RT. Rhapsody in fractional. Fract Calculus Appl Anal 2014;17(4):1188–214. doi:10.2478/ s13540- 014- 0206- 0. [72] Machado JAT. Entropy analysis of systems exhibiting negative probabilities. Commun Nonlinear Sci Numer Simul 2016;36:58–64. doi:10.1016/j.cnsns. 2015.11.022. [73] Shannon CE. A mathematical theory of communication. Bell Syst Tech J 1948;27(3):623–56. 379–423. [74] Gray RM. Entropy and information theory. New York: Springer-Verlag; 1990. [75] Beck C. Generalised information and entropy measures in physics. Contemp Phys 2009;50(4):495–510. doi:10.1080/00107510902823517.