In this manuscript, we have used the recently developed Fα-calculus to calculate the energy straggling function through the fractal distributed structures. We have shown that such a fractal structure of space causes the fractal pattern of the energy loss. Also, we have offered Fα-differential Fokker–Planck equation for thick fractal absorbers.

References

1.
Hilfer
,
R.
, ed.,
2000
,
Applications of Fractional Calculus in Physics
,
World Scientific
, Singapore.
2.
Carpinteri
,
A.
, and
Mainardi
,
F.
, eds.,
2014
,
Fractals and Fractional Calculus in Continuum Mechanics
, Vol.
378
,
Springer
, Berlin.
3.
Sabatier
,
J.
,
Agrawal
,
O. P.
, and
Tenreiro Machado
,
J. A.
,
2007
,
Advances in Fractional Calculus
, Vol.
4
,
Springer
,
Berlin
.
4.
Ma
,
L.
, and
Li
,
C.
,
2016
, “
Center Manifold of Fractional Dynamical System
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
2
), p.
021010
.
5.
Velmurugan
,
G.
, and
Rakkiyappan
,
R.
,
2016
, “
Hybrid Projective Synchronization of Fractional-Order Chaotic Complex Nonlinear Systems With Time Delays
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
3
), p.
031016
.
6.
Baleanu
,
D.
,
Muslih
,
S. I.
,
Rabei
,
E. M.
,
Golmankhaneh
,
A. K.
, and
Golmankhaneh
,
A. K.
,
2010
, “
On Fractional Dynamics on the Extended Phase Space
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
4
), p.
041011
.
7.
Agrawal
,
O. P.
,
2008
, “
Fractional Optimal Control of a Distributed System Using Eigenfunctions
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
2
), p.
021204
.
8.
Özdemir
,
N.
, and
Iskender
,
B. B.
,
2010
, “
Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
2
), p.
021002
.
9.
Murillo
,
J. Q.
, and
Yuste
,
S. B.
,
2011
, “
An Explicit Difference Method for Solving Fractional Diffusion and Diffusion-Wave Equations in the Caputo Form
,”
ASME J. Comput. Nonlinear Dyn.
,
6
(
2
), p.
021014
.
10.
Zaslavsky
,
G. M.
,
1994
, “
Fractional Kinetic Equation for Hamiltonian Chaos
,”
Phys. D
,
76
(
1
), pp.
110
122
.
11.
Riewe
,
F.
,
1997
, “
Mechanics With Fractional Derivatives
,”
Phys. Rev. E
,
55
(
3
), p.
3581
.
12.
Bouchaud
,
J.-P.
, and
Georges
,
A.
,
1990
, “
Anomalous Diffusion in Disordered Media: Statistical Mechanisms, Models and Physical Applications
,”
Phys. Rep.
,
195
(
4–5
), pp.
127
293
.
13.
Metzler
,
R.
,
Barkai
,
E.
, and
Klafter
,
J.
,
1999
, “
Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker–Planck Equation Approach
,”
Phys. Rev. Lett.
,
82
(
18
), p.
3563
.
14.
Klages
,
R.
,
Radons
,
G.
, and
Sokolov
,
I. M.
, eds.,
2008
,
Anomalous Transport: Foundations and Applications
,
Wiley
, Hoboken, NJ.
15.
Metzler
,
R.
,
Barkai
,
E.
, and
Klafter
,
J.
,
1999
, “
Anomalous Transport in Disordered Systems Under the Influence of External Fields
,”
Phys. A
,
266
(
1
), pp.
343
350
.
16.
Metzler
,
R.
,
Glöckle
,
W. G.
, and
Nonnenmacher
,
T. F.
,
1994
, “
Fractional Model Equation for Anomalous Diffusion
,”
Phys. A
,
211
(
1
), pp.
13
24
.
17.
Zaslavsky
,
G. M.
,
2002
, “
Chaos, Fractional Kinetics, and Anomalous Transport
,”
Phys. Rep.
,
371
(
6
), pp.
461
580
.
18.
Zaslavsky
,
G. M.
,
2005
, “
Hamiltonian Chaos and Fractional Dynamics
,” Oxford University Press, Oxford, UK.
19.
Zaslavsky
,
G. M.
,
2007
,
The Physics of Chaos in Hamiltonian Systems
,
Imperial College Press
,
London, UK
.
20.
Ben-Avraham
,
D.
, and
Havlin
,
S.
,
2000
,
Diffusion and Reactions in Fractals and Disordered Systems
,
Cambridge University Press
, New York.
21.
O'Shaughnessy
,
B.
, and
Procaccia
,
I.
,
1985
, “
Analytical Solutions for Diffusion on Fractal Objects
,”
Phys. Rev. Lett.
,
54
(
5
), p.
455
.
22.
Mainardi
,
F.
,
Pagnini
,
G.
, and
Gorenflo
,
R.
,
2007
, “
Some Aspects of Fractional Diffusion Equation of Single and Distributed Order
,”
Appl. Math. Comp.
,
187
(
1
), pp.
295
305
.
23.
Gorenflo
,
R.
,
Mainardi
,
F.
,
Moretti
,
D.
,
Pagnini
,
G.
, and
Paradisi
,
P.
,
2002
, “
Discrete Random Walk Models for Space-Time Fractional Diffusion
,”
Chem. Phys.
,
284
(1–2), pp.
521
544
.
24.
Gafiychuk
,
V.
, and
Datsko
,
B.
,
2012
, “
Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives
,”
ASME J. Comput. Nonlinear Dyn.
,
7
(
3
), p.
031001
.
25.
Parvate
,
A.
,
Satin
,
S.
, and
Gangal
,
A. D.
,
2011
, “
Calculus on Fractal Curves in Rn
,”
Fractals
,
19
(
1
), pp.
15
27
.
26.
Golmankhaneh
,
A. K.
,
Golmankhaneh
,
A. K.
, and
Baleanu
,
D.
,
2015
, “
About Schrödinger Equation on Fractals Curves Imbedding in R3
,”
Int. J. Theor. Phys.
,
54
(
4
), pp.
1275
1282
.
27.
Golmankhaneh
,
A. K.
,
Golmankhaneh
,
A. K.
, and
Baleanu
,
D.
,
2013
, “
Lagrangian and Hamiltonian Mechanics on Fractals Subset of Real-Line
,”
Int. J. Theor. Phys.
,
52
(
11
), pp.
4210
4217
.
28.
Kolwankar
,
K. M.
, and
Gangal
,
A. D.
,
1998
, “
Local Fractional Fokker–Planck Equation
,”
Phys. Rev. Lett.
,
80
(
2
), p.
214
.
29.
Kolwankar
,
K. M.
, and
Gangal
,
A. D.
,
1996
, “
Fractional Differentiability of Nowhere Differentiable Functions and Dimensions
,”
Chaos: Interdiscip. J. Nonlinear Sci.
,
6
(
4
), pp.
505
513
.
30.
Kolwankar
,
K. M.
, and
Gangal
,
A. D.
,
1997
, “
Hölder Exponents of Irregular Signals and Local Fractional Derivatives
,”
Pramana
,
48
(
1
), pp.
49
68
.
31.
Kolwankar
,
K. M.
, and
Gangal
,
A. D.
,
1999
, “
Local Fractional Calculus: A Calculus for Fractal Space-Time
,”
Fractals
,
Springer
,
Berlin
, pp.
171
181
.
32.
Leo
,
W. R.
,
1994
,
Techniques for Nuclear and Particle Physics Experiments
,
Springer-Verlag
,
Berlin
.
33.
Fano
,
U.
,
1963
, “Penetration of Protons, Alpha Particles, and Mesons,”
Ann. Rev. Nucl. Sci.
,
13
(
1
), pp. 1–66.
34.
Vavilov
,
P. V.
,
1957
, “Ionization Losses of High-Energy heavy Particles,”
Sov. Phys. JETP.
,
5
(4), pp. 749–751.
35.
Symon
,
K. R.
,
1948
, “Fluctuations in Energy Lost by High Energy Charged Particles in Passing through Matter,” Ph.D. thesis, Harvard University, Cambridge, MA.
36.
Tsoulfanidis
,
N.
, and
Sheldon
,
L.
,
2011
,
Measurement and Detection of Radiation
,
CRC Press
, Boca Raton, FL.
37.
Knoll
,
G. F.
,
2010
,
Radiation Detection and Measurement
,
Wiley
, Hoboken, NJ.
38.
Payne
,
M. G.
,
1969
, “
Energy Straggling of Heavy Charged Particles in Thick Absorbers
,”
Phys. Rev.
,
185
(
2
), p.
611
.
39.
Landau
,
L.
,
1944
, “
On the Energy Loss of Fast Particles by Ionization
,”
J. Phys.
,
8
(4), pp.
201
205
.
40.
Hille
,
E.
, and
Tamarkin
,
J. D.
,
1929
, “
Remarks on a Known Example of a Monotone Continuous Function
,”
Am. Math. Mon.
,
36
(
5
), pp.
255
264
.
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