Lévy noise is a paradigmatic noise used to describe out-of-equilibrium systems. Typically, properties of Lévy noise driven systems are very different from their Gaussian white noise driven counterparts. In particular, under action of Lévy noise, stationary states in single-well, super-harmonic, potentials are no longer unimodal. Typically, they are bimodal; however, for fine-tuned potentials, the number of modes can be further increased. The multimodality arises as a consequence of the competition between long displacements induced by the non-equilibrium stochastic driving and action of the deterministic force. Here, we explore robustness of bimodality in the quartic potential under action of the Lévy noise. We explore various scenarios of bounding long jumps and assess their ability to weaken and destroy multimodality. In general, we demonstrate that despite its robustness it is possible to destroy the bimodality, however it requires drastic reduction in the length of noise-induced jumps.

1.
Stochastic Dynamics, edited by L. Schimansky-Geier and T. Pöshel (Springer Verlag, Berlin, 1997).
2.
J.
Klafter
,
S. T.
Lim
, and
R.
Metzler
,
Fractional Dynamics: Recent Advances
(
World Scientific Publishing
,
Singapore
,
2012
).
3.
T.
Tomé
and
M. J.
De Oliveira
,
Stochastic Dynamics and Irreversibility
(
Springer Verlag
,
Berlin
,
2016
).
4.
C. W.
Gardiner
,
Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences
(
Springer Verlag
,
Berlin
,
2009
).
5.
H.
Risken
,
The Fokker–Planck Equation. Methods of Solution and Application
(
Springer Verlag
,
Berlin
,
19996
).
6.
L. E.
Reichl
,
A Modern Course in Statistical Physics
(
John Wiley
,
New York
,
1998
).
7.
A. V.
Chechkin
,
J.
Klafter
,
V. Y.
Gonchar
,
R.
Metzler
, and
L. V.
Tanatarov
,
Chem. Phys.
284
,
233
(
2002
).
8.
A. V.
Chechkin
,
J.
Klafter
,
V. Y.
Gonchar
,
R.
Metzler
, and
L. V.
Tanatarov
,
Phys. Rev. E
67
,
010102(R)
(
2003
).
9.
K.
Capała
and
B.
Dybiec
,
J. Stat. Mech.
(
2019
)
033206
.
10.
Q.
Jacquet
,
E.-J.
Kim
, and
R.
Hollerbach
,
Entropy
20
,
613
(
2018
).
11.
T.
Guggenberger
,
A.
Chechkin
, and
R.
Metzler
,
J. Phys. A: Math. Theor.
54
,
29LT01
(
2021
).
12.
A. V.
Chechkin
,
V. Y.
Gonchar
,
J.
Klafter
, and
R.
Metzler
, in
Fractals, Diffusion, and Relaxation in Disordered Complex Systems: Advances in Chemical Physics, Part B
, edited by W. T. Coffey and Y. P. Kalmykov (
John Wiley & Sons
,
New York
,
2006
), Vol. 133, pp.
439
496
.
13.
A. V.
Chechkin
,
R.
Metzler
,
J.
Klafter
, and
V. Y.
Gonchar
, in
Anomalous Transport: Foundations and Applications
, edited by R. Klages, G. Radons, and I. M. Sokolov (
Wiley-VCH
,
Weinheim
,
2008
), pp.
129
162
.
14.
A. A.
Dubkov
,
B.
Spagnolo
, and
V. V.
Uchaikin
,
Int. J. Bifurcation Chaos. Appl. Sci. Eng.
18
,
2649
(
2008
).
15.
B.
Dybiec
and
E.
Gudowska-Nowak
,
J. Stat. Mech.
2009
,
P05004
(
2009
).
16.
T. H.
Solomon
,
E. R.
Weeks
, and
H. L.
Swinney
,
Phys. Rev. Lett.
71
,
3975
(
1993
).
17.
T. H.
Solomon
,
E. R.
Weeks
, and
H. L.
Swinney
,
Phys. D
76
,
70
(
1994
).
18.
D.
del Castillo-Negrete
,
Phys. Fluids
10
,
576
(
1998
).
19.
M. F.
Shlesinger
,
G. M.
Zaslavski
, and
J.
Klafter
,
Nature
363
,
31
(
1993
).
20.
J.
Klafter
,
M. F.
Shlesinger
, and
G.
Zumofen
,
Phys. Today
49
(
2
),
33
(
1996
).
21.
A. V.
Chechkin
,
V. Y.
Gonchar
, and
M.
Szydłowski
,
Phys. Plasmas
9
,
78
(
2002
).
22.
D.
del Castillo-Negrete
,
B. A.
Carreras
, and
V. E.
Lynch
,
Phys. Rev. Lett.
94
,
065003
(
2005
).
23.
H.
Katori
,
S.
Schlipf
, and
H.
Walther
,
Phys. Rev. Lett.
79
,
2221
(
1997
).
24.
C.-K.
Peng
,
J.
Mietus
,
J. M.
Hausdorff
,
S.
Havlin
,
H. E.
Stanley
, and
A. L.
Goldberger
,
Phys. Rev. Lett.
70
,
1343
(
1993
).
25.
R.
Segev
,
M.
Benveniste
,
E.
Hulata
,
N.
Cohen
,
A.
Palevski
,
E.
Kapon
,
Y.
Shapira
, and
E.
Ben-Jacob
,
Phys. Rev. Lett.
88
,
118102
(
2002
).
26.
M. A.
Lomholt
,
T.
Ambjörnsson
, and
R.
Metzler
,
Phys. Rev. Lett.
95
,
260603
(
2005
).
27.
G. M.
Viswanathan
,
V.
Afanasyev
,
S. V.
Buldyrev
,
E. J.
Murphy
,
P. A.
Prince
, and
H. E.
Stanley
,
Nature
381
,
413
(
1996
).
28.
D.
Brockmann
,
L.
Hufnagel
, and
T.
Geisel
,
Nature
439
,
462
(
2006
).
29.
M. R.
Evans
and
S. N.
Majumdar
,
Phys Rev. Lett.
106
,
160601
(
2011
).
30.
M. R.
Evans
,
S. N.
Majumdar
, and
G.
Schehr
,
J. Phys. A: Math. Theor.
53
,
193001
(
2020
).
31.
S.
Gupta
and
A. M.
Jayannavar
,
Front. Phys.
10
,
789097
(
2022
).
32.
A. V.
Chechkin
,
V. Y.
Gonchar
,
J.
Klafter
,
R.
Metzler
, and
L. V.
Tanatarov
,
J. Stat. Phys.
115
,
1505
(
2004
).
33.
A. A.
Dubkov
and
B.
Spagnolo
,
Acta Phys. Pol. B
38
,
1745
(
2007
); available at https://www.actaphys.uj.edu.pl/R/38/5/1745/pdf.
34.
M.
Cieśla
,
K.
Capała
, and
B.
Dybiec
,
Phys. Rev. E
99
,
052118
(
2019
).
35.
K.
Szczepaniec
and
B.
Dybiec
,
Phys. Rev. E
90
,
032128
(
2014
).
36.
K.
Capała
and
B.
Dybiec
,
Chaos
29
,
093113
(
2019
).
37.
K.
Capała
,
B.
Dybiec
, and
E.
Gudowska-Nowak
,
Chaos
30
,
073140
(
2020
).
38.
G.
Samorodnitsky
and
M. S.
Taqqu
,
Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance
(
Chapman and Hall
,
New York
,
1994
).
39.
A. A.
Kilbas
,
H. M.
Srivastava
, and
J. J.
Trujillo
, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies Vol. 204 (Elsevier Science Inc., New York, 2006).
40.
I.
Podlubny
,
Fractional Differential Equations
(
Academic Press
,
San Diego
,
1999
).
41.
A. V.
Chechkin
,
R.
Metzler
,
V. Y.
Gonchar
,
J.
Klafter
, and
L. V.
Tanatarov
,
J. Phys. A: Math. Gen.
36
,
L537
(
2003
).
42.
T.
Koren
,
M. A.
Lomholt
,
A. V.
Chechkin
,
J.
Klafter
, and
R.
Metzler
,
Phys. Rev. Lett.
99
,
160602
(
2007
).
43.
T.
Koren
,
A. V.
Chechkin
, and
J.
Klafter
,
Phys. A
379
,
10
(
2007
).
44.
A.
Wardak
,
J. Phys. A: Math. Theor.
53
,
375001
(
2020
).
45.
S.
Jespersen
,
R.
Metzler
, and
H. C.
Fogedby
,
Phys. Rev. E
59
,
2736
(
1999
).
46.
V. V.
Yanovsky
,
A. V.
Chechkin
,
D.
Schertzer
, and
A. V.
Tur
,
Phys. A
282
,
13
(
2000
).
47.
D.
Schertzer
,
M.
Larchevêque
,
J.
Duan
,
V. V.
Yanovsky
, and
S.
Lovejoy
,
J. Math. Phys.
42
,
200
(
2001
).
48.
M.
Kwaśnicki
,
Fract. Calc. Appl. Anal.
20
,
7
(
2017
).
49.
P.
Garbaczewski
and
V.
Stephanovich
,
Phys. Rev. E
99
,
042126
(
2019
).
50.
M.
Kwaśnicki
,
J. Funct. Anal.
262
,
2379
(
2012
).
51.
E.
Katzav
and
M.
Adda-Bedia
,
Europhys. Lett.
83
,
30006
(
2008
).
52.
A.
Zoia
,
A.
Rosso
, and
M.
Kardar
,
Phys. Rev. E
76
,
021116
(
2007
).
53.
P.
Garbaczewski
and
M.
Żaba
,
J. Phys. A: Math. Theor.
55
,
305005
(
2022
).
54.
A.
Janicki
and
A.
Weron
,
Simulation and Chaotic Behavior of α-Stable Stochastic Processes
(
Marcel Dekker
,
New York
,
1994
).
55.
P. D.
Ditlevsen
,
Geophys. Res. Lett.
26
,
1441
, https://doi.org/10.1029/1999GL900252 (
1999
).
56.
M.
Mercadier
,
W.
Guerin
,
M. M.
Chevrollier
, and
R.
Kaiser
,
Nat. Phys.
5
,
602
(
2009
).
57.
E.
Barkai
,
E.
Aghion
, and
D. A.
Kessler
,
Phys. Rev. X
4
,
021036
(
2014
).
58.
T. A.
Amor
,
S. D. S.
Reis
,
D.
Campos
,
H. J.
Herrmann
, and
J. S.
Andrade
,
Sci. Rep.
6
,
20815
(
2016
).
59.
P.
Barthelemy
,
J.
Bertolotti
, and
D.
Wiersma
,
Nature
453
,
495
(
2008
).
60.
V.
Fioriti
,
F.
Fratichini
,
S.
Chiesa
, and
C.
Moriconi
,
Int. J. Adv. Rob. Syst.
12
,
98
(
2015
).
61.
S. C.
Lera
and
D.
Sornette
,
Phys. Rev. E
97
,
012150
(
2018
).
62.
S. G.
Samko
,
A. A.
Kilbas
, and
O. I.
Marichev
,
Fractional Integrals and Derivatives: Theory and Applications
(
Gordon and Breach Science Publishers
,
Yverdon
,
1993
).
63.
A.
Padash
,
A. V.
Chechkin
,
B.
Dybiec
,
I.
Pavlyukevich
,
B.
Shokri
, and
R.
Metzler
,
J. Phys. A: Math. Theor.
52
,
454004
(
2019
).
64.
F.
Song
,
C.
Xu
, and
G. E.
Karniadakis
,
SIAM J. Sci. Comput.
39
,
A1320
(
2017
).
65.
N.
Cusimano
,
F.
del Teso
,
L.
Gerardo-Giorda
, and
G.
Pagnini
,
SIAM J. Numer. Anal.
56
,
1243
(
2018
).
66.
B.
Dybiec
,
I. M.
Sokolov
, and
A. V.
Chechkin
,
J. Stat. Mech.
2010
,
P07008
.
67.
B.
Dybiec
,
E.
Gudowska-Nowak
, and
I. M.
Sokolov
,
Phys. Rev. E
76
,
041122
(
2007
).
68.
S. I.
Denisov
,
W.
Horsthemke
, and
P.
Hänggi
,
Phys. Rev. E
77
,
061112
(
2008
).
69.
B.
Dybiec
,
E.
Gudowska-Nowak
,
E.
Barkai
, and
A. A.
Dubkov
,
Phys. Rev. E
95
,
052102
(
2017
).
70.
R. N.
Mantegna
and
H. E.
Stanley
,
Phys. Rev. Lett.
73
,
2946
(
1994
).
72.
R.
Mannella
,
Int. J. Mod. Phys. C
13
,
1177
(
2002
).
73.
J. M.
Chambers
,
C. L.
Mallows
, and
B. W.
Stuck
,
J. Am. Stat. Assoc.
71
,
340
(
1976
).
75.
L.
Devroye
,
Non-Uniform Random Variate Generation
(
Springer Verlag
,
New York
,
1986
).
76.
M. E. J.
Newman
and
G. T.
Barkema
,
Monte Carlo Methods in Statistical Physics
(
Oxford University Press
,
Oxford
,
1999
).
77.
A. A.
Kharcheva
,
A. A.
Dubkov
,
B.
Dybiec
,
B.
Spagnolo
, and
D.
Valenti
,
J. Stat. Mech.
2016
,
P054039
(
2016
).
78.
W.
Feller
,
An Introduction to Probability Theory and Its Applications
(
John Wiley
,
New York
,
1968
).
79.
B. V.
Gnedenko
and
A. N.
Kolmogorov
,
Limit Distributions for Sums of Independent Random Variables
(
Addison–Wesley
,
Reading, MA
,
1968
).
81.
R.
Metzler
and
J.
Klafter
,
Phys. Rep.
339
,
1
(
2000
).
82.
A.
Pal
,
Ł.
Kuśmierz
, and
S.
Reuveni
,
Phys. Rev. E
100
,
040101
(
2019
).
83.
R.
Metzler
and
J.
Klafter
,
J. Phys. A: Math. Gen.
37
,
R161
(
2004
).
84.
A.
Weron
and
R.
Weron
,
Lect. Notes Phys.
457
,
379
(
1995
).
85.
K. A.
Penson
and
K.
Górska
,
Phys. Rev. Lett.
105
,
210604
(
2010
).
86.
K.
Górska
and
K. A.
Penson
,
Phys. Rev. E
83
,
061125
(
2011
).
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